3.5 Test file Number [79] 4-Trig-functions/4.1-Sine/4.1.7-d-trig-^m-a+b-c-sin-^n-^p

3.5.1 Mathematica

Integral number [399] \[ \int \frac {\cos ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx \]

[C]   time = 0.357346 (sec), size = 394 ,normalized size = 15.15 \[ \frac {\frac {24 \cos (c+d x) (a+b \sin (c+d x))}{4 a+3 b \sin (c+d x)-b \sin (3 (c+d x))}-i \text {RootSum}\left [i \text {$\#$1}^6 b-3 i \text {$\#$1}^4 b+8 \text {$\#$1}^3 a+3 i \text {$\#$1}^2 b-i b\& ,\frac {2 \text {$\#$1}^4 b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-4 i \text {$\#$1}^3 a \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+2 \text {$\#$1} a \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )-6 i \text {$\#$1}^2 b \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )-i b \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )+12 \text {$\#$1}^2 b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-i \text {$\#$1}^4 b \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )-2 \text {$\#$1}^3 a \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )+4 i \text {$\#$1} a \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+2 b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )}{\text {$\#$1}^5 b-2 \text {$\#$1}^3 b-4 i \text {$\#$1}^2 a+\text {$\#$1} b}\& \right ]}{18 a b d} \]

[In]

Integrate[Cos[c + d*x]^4/(a + b*Sin[c + d*x]^3)^2,x]

[Out]

((-I)*RootSum[(-I)*b + (3*I)*b*#1^2 + 8*a*#1^3 - (3*I)*b*#1^4 + I*b*#1^6 & , (2*b*ArcTan[Sin[c + d*x]/(Cos[c +
d*x] - #1)] - I*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + (4*I)*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1 + 2
*a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1 + 12*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - (6*I)*b*Log[1 -
2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (4*I)*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^3 - 2*a*Log[1 - 2*Cos[c
+ d*x]*#1 + #1^2]*#1^3 + 2*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - I*b*Log[1 - 2*Cos[c + d*x]*#1 + #
1^2]*#1^4)/(b*#1 - (4*I)*a*#1^2 - 2*b*#1^3 + b*#1^5) & ] + (24*Cos[c + d*x]*(a + b*Sin[c + d*x]))/(4*a + 3*b*S
in[c + d*x] - b*Sin[3*(c + d*x)]))/(18*a*b*d)

Integral number [400] \[ \int \frac {\cos ^2(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx \]

[C]   time = 0.236978 (sec), size = 273 ,normalized size = 10.5 \[ \frac {\frac {12 \sin (2 (c+d x))}{4 a+3 b \sin (c+d x)-b \sin (3 (c+d x))}-i \text {RootSum}\left [i \text {$\#$1}^6 b-3 i \text {$\#$1}^4 b+8 \text {$\#$1}^3 a+3 i \text {$\#$1}^2 b-i b\& ,\frac {2 \text {$\#$1}^4 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-6 i \text {$\#$1}^2 \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )-i \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )+12 \text {$\#$1}^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-i \text {$\#$1}^4 \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )+2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )}{\text {$\#$1}^5 b-2 \text {$\#$1}^3 b-4 i \text {$\#$1}^2 a+\text {$\#$1} b}\& \right ]}{18 a d} \]

[In]

Integrate[Cos[c + d*x]^2/(a + b*Sin[c + d*x]^3)^2,x]

[Out]

((-I)*RootSum[(-I)*b + (3*I)*b*#1^2 + 8*a*#1^3 - (3*I)*b*#1^4 + I*b*#1^6 & , (2*ArcTan[Sin[c + d*x]/(Cos[c + d
*x] - #1)] - I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + 12*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - (6*I)*Lo
g[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 + 2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - I*Log[1 - 2*Cos[c + d
*x]*#1 + #1^2]*#1^4)/(b*#1 - (4*I)*a*#1^2 - 2*b*#1^3 + b*#1^5) & ] + (12*Sin[2*(c + d*x)])/(4*a + 3*b*Sin[c +
d*x] - b*Sin[3*(c + d*x)]))/(18*a*d)

Integral number [401] \[ \int \frac {1}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx \]

[C]   time = 0.476477 (sec), size = 502 ,normalized size = 29.53 \[ \frac {-\frac {12 b \cos (c+d x) (a \cos (2 (c+d x))-3 a+2 b \sin (c+d x))}{(a-b) (a+b) (4 a+3 b \sin (c+d x)-b \sin (3 (c+d x)))}+\frac {i \text {RootSum}\left [i \text {$\#$1}^6 b-3 i \text {$\#$1}^4 b+8 \text {$\#$1}^3 a+3 i \text {$\#$1}^2 b-i b\& ,\frac {2 \text {$\#$1}^4 b^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-4 i \text {$\#$1}^3 a b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+12 i \text {$\#$1}^2 a^2 \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )-24 \text {$\#$1}^2 a^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+2 \text {$\#$1} a b \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )-6 i \text {$\#$1}^2 b^2 \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )-i b^2 \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )+12 \text {$\#$1}^2 b^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-i \text {$\#$1}^4 b^2 \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )-2 \text {$\#$1}^3 a b \log \left (\text {$\#$1}^2-2 \text {$\#$1} \cos (c+d x)+1\right )+4 i \text {$\#$1} a b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+2 b^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )}{\text {$\#$1}^5 b-2 \text {$\#$1}^3 b-4 i \text {$\#$1}^2 a+\text {$\#$1} b}\& \right ]}{a^2-b^2}}{18 a d} \]

[In]

Integrate[(a + b*Sin[c + d*x]^3)^(-2),x]

[Out]

((I*RootSum[(-I)*b + (3*I)*b*#1^2 + 8*a*#1^3 - (3*I)*b*#1^4 + I*b*#1^6 & , (2*b^2*ArcTan[Sin[c + d*x]/(Cos[c +
d*x] - #1)] - I*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + (4*I)*a*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1
+ 2*a*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1 - 24*a^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + 12*b^2*
ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + (12*I)*a^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (6*I)*b^2*
Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (4*I)*a*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^3 - 2*a*b*Log[1
- 2*Cos[c + d*x]*#1 + #1^2]*#1^3 + 2*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - I*b^2*Log[1 - 2*Cos[
c + d*x]*#1 + #1^2]*#1^4)/(b*#1 - (4*I)*a*#1^2 - 2*b*#1^3 + b*#1^5) & ])/(a^2 - b^2) - (12*b*Cos[c + d*x]*(-3*
a + a*Cos[2*(c + d*x)] + 2*b*Sin[c + d*x]))/((a - b)*(a + b)*(4*a + 3*b*Sin[c + d*x] - b*Sin[3*(c + d*x)])))/(
18*a*d)

Integral number [402] \[ \int \frac {\sec ^2(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx \]

[C]   time = 1.60412 (sec), size = 845 ,normalized size = 32.5 \[ \frac {-\frac {i b \text {RootSum}\left [i b \text {$\#$1}^6-3 i b \text {$\#$1}^4+8 a \text {$\#$1}^3+3 i b \text {$\#$1}^2-i b\& ,\frac {2 b^3 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4+16 a^2 b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4-i b^3 \log \left (\text {$\#$1}^2-2 \cos (c+d x) \text {$\#$1}+1\right ) \text {$\#$1}^4-8 i a^2 b \log \left (\text {$\#$1}^2-2 \cos (c+d x) \text {$\#$1}+1\right ) \text {$\#$1}^4-20 i a^3 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^3-16 i a b^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^3-10 a^3 \log \left (\text {$\#$1}^2-2 \cos (c+d x) \text {$\#$1}+1\right ) \text {$\#$1}^3-8 a b^2 \log \left (\text {$\#$1}^2-2 \cos (c+d x) \text {$\#$1}+1\right ) \text {$\#$1}^3+12 b^3 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2-120 a^2 b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2-6 i b^3 \log \left (\text {$\#$1}^2-2 \cos (c+d x) \text {$\#$1}+1\right ) \text {$\#$1}^2+60 i a^2 b \log \left (\text {$\#$1}^2-2 \cos (c+d x) \text {$\#$1}+1\right ) \text {$\#$1}^2+20 i a^3 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}+16 i a b^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}+10 a^3 \log \left (\text {$\#$1}^2-2 \cos (c+d x) \text {$\#$1}+1\right ) \text {$\#$1}+8 a b^2 \log \left (\text {$\#$1}^2-2 \cos (c+d x) \text {$\#$1}+1\right ) \text {$\#$1}+2 b^3 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+16 a^2 b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-i b^3 \log \left (\text {$\#$1}^2-2 \cos (c+d x) \text {$\#$1}+1\right )-8 i a^2 b \log \left (\text {$\#$1}^2-2 \cos (c+d x) \text {$\#$1}+1\right )}{b \text {$\#$1}^5-2 b \text {$\#$1}^3-4 i a \text {$\#$1}^2+b \text {$\#$1}}\& \right ]}{a \left (a^2-b^2\right )^2}+\frac {18 \sin \left (\frac {1}{2} (c+d x)\right )}{(a+b)^2 \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {18 \sin \left (\frac {1}{2} (c+d x)\right )}{(a-b)^2 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {12 b \cos (c+d x) \left (-2 a^3-7 b^2 a+3 b^2 \cos (2 (c+d x)) a+2 b \left (2 a^2+b^2\right ) \sin (c+d x)\right )}{a (a-b)^2 (a+b)^2 (4 a+3 b \sin (c+d x)-b \sin (3 (c+d x)))}}{18 d} \]

[In]

Integrate[Sec[c + d*x]^2/(a + b*Sin[c + d*x]^3)^2,x]

[Out]

(((-I)*b*RootSum[(-I)*b + (3*I)*b*#1^2 + 8*a*#1^3 - (3*I)*b*#1^4 + I*b*#1^6 & , (16*a^2*b*ArcTan[Sin[c + d*x]/
(Cos[c + d*x] - #1)] + 2*b^3*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] - (8*I)*a^2*b*Log[1 - 2*Cos[c + d*x]*#1
+ #1^2] - I*b^3*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + (20*I)*a^3*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1 + (
16*I)*a*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1 + 10*a^3*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1 + 8*a*b^
2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1 - 120*a^2*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + 12*b^3*ArcT
an[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + (60*I)*a^2*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (6*I)*b^3*Lo
g[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (20*I)*a^3*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^3 - (16*I)*a*b^2
*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^3 - 10*a^3*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^3 - 8*a*b^2*Log[1
- 2*Cos[c + d*x]*#1 + #1^2]*#1^3 + 16*a^2*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 + 2*b^3*ArcTan[Sin[
c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - (8*I)*a^2*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 - I*b^3*Log[1 - 2*Cos[
c + d*x]*#1 + #1^2]*#1^4)/(b*#1 - (4*I)*a*#1^2 - 2*b*#1^3 + b*#1^5) & ])/(a*(a^2 - b^2)^2) + (18*Sin[(c + d*x)
/2])/((a + b)^2*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (18*Sin[(c + d*x)/2])/((a - b)^2*(Cos[(c + d*x)/2] +
Sin[(c + d*x)/2])) + (12*b*Cos[c + d*x]*(-2*a^3 - 7*a*b^2 + 3*a*b^2*Cos[2*(c + d*x)] + 2*b*(2*a^2 + b^2)*Sin[c
+ d*x]))/(a*(a - b)^2*(a + b)^2*(4*a + 3*b*Sin[c + d*x] - b*Sin[3*(c + d*x)])))/(18*d)

Integral number [403] \[ \int \frac {\sec ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx \]

[C]   time = 1.74385 (sec), size = 1158 ,normalized size = 44.54 \[ \text {result too large to display} \]

[In]

Integrate[Sec[c + d*x]^4/(a + b*Sin[c + d*x]^3)^2,x]

[Out]

((4*I)*b^2*RootSum[(-I)*b + (3*I)*b*#1^2 + 8*a*#1^3 - (3*I)*b*#1^4 + I*b*#1^6 & , (14*a^4*ArcTan[Sin[c + d*x]/
(Cos[c + d*x] - #1)] + 74*a^2*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] + 2*b^4*ArcTan[Sin[c + d*x]/(Cos[c
+ d*x] - #1)] - (7*I)*a^4*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] - (37*I)*a^2*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]
- I*b^4*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + (144*I)*a^3*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1 + (36*I
)*a*b^3*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1 + 72*a^3*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1 + 18*a*b^3
*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1 - 180*a^4*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - 372*a^2*b^2*Ar
cTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + 12*b^4*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + (90*I)*a^
4*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 + (186*I)*a^2*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (6*I)*b^4*
Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (144*I)*a^3*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^3 - (36*I)*
a*b^3*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^3 - 72*a^3*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^3 - 18*a*b
^3*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^3 + 14*a^4*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 + 74*a^2*b^2*
ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 + 2*b^4*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - (7*I)*a^
4*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 - (37*I)*a^2*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 - I*b^4*Log[1
- 2*Cos[c + d*x]*#1 + #1^2]*#1^4)/(b*#1 - (4*I)*a*#1^2 - 2*b*#1^3 + b*#1^5) & ] + (3*Sec[c + d*x]^3*(48*a^5*b
+ 568*a^3*b^3 + 14*a*b^5 + (78*a^5*b + 606*a^3*b^3 + 81*a*b^5)*Cos[2*(c + d*x)] + 18*a*b^3*(4*a^2 + b^2)*Cos[
4*(c + d*x)] + 2*a^5*b*Cos[6*(c + d*x)] - 30*a^3*b^3*Cos[6*(c + d*x)] - 17*a*b^5*Cos[6*(c + d*x)] + 48*a^6*Sin
[c + d*x] - 244*a^4*b^2*Sin[c + d*x] + 20*a^2*b^4*Sin[c + d*x] - 4*b^6*Sin[c + d*x] + 16*a^6*Sin[3*(c + d*x)]
- 194*a^4*b^2*Sin[3*(c + d*x)] - 86*a^2*b^4*Sin[3*(c + d*x)] - 6*b^6*Sin[3*(c + d*x)] - 14*a^4*b^2*Sin[5*(c +
d*x)] - 74*a^2*b^4*Sin[5*(c + d*x)] - 2*b^6*Sin[5*(c + d*x)]))/(4*a + 3*b*Sin[c + d*x] - b*Sin[3*(c + d*x)]))/
(72*a*(a^2 - b^2)^3*d)

3.5.2 Maple

Integral number [399] \[ \int \frac {\cos ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx \]

[B]   time = 0.965 (sec), size = 550 ,normalized size = 21.15 \[ -\frac {2 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d \left (\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +8 b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +a \right ) a}+\frac {2 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d \left (\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +8 b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +a \right ) b}+\frac {8 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d \left (\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +8 b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +a \right ) a}+\frac {4 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d \left (\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +8 b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +a \right ) b}+\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{3 d \left (\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +8 b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +a \right ) a}+\frac {2}{3 d \left (\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +8 b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +a \right ) b}+\frac {2 \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{4} b +\textit {\_R}^{3} a +\textit {\_R} a +b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{9 d a b} \]

[In]

int(cos(d*x+c)^4/(a+b*sin(d*x+c)^3)^2,x)

[Out]

-2/3/d/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)/a
*tan(1/2*d*x+1/2*c)^5+2/3/d/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/
2*d*x+1/2*c)^2*a+a)/b*tan(1/2*d*x+1/2*c)^4+8/3/d/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*
d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)/a*tan(1/2*d*x+1/2*c)^3+4/3/d/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/
2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)/b*tan(1/2*d*x+1/2*c)^2+2/3/d/(tan(1/2*d*x+1/2*c)
^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)/a*tan(1/2*d*x+1/2*c)+2/3/d/
(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)/b+2/9/d/
a/b*sum((_R^4*b+_R^3*a+_R*a+b)/(_R^5*a+2*_R^3*a+4*_R^2*b+_R*a)*ln(tan(1/2*d*x+1/2*c)-_R),_R=RootOf(_Z^6*a+3*_Z
^4*a+8*_Z^3*b+3*_Z^2*a+a))

Integral number [400] \[ \int \frac {\cos ^2(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx \]

[B]   time = 0.944 (sec), size = 236 ,normalized size = 9.08 \[ -\frac {2 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d \left (\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +8 b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +a \right ) a}+\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{3 d \left (\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +8 b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +a \right ) a}+\frac {2 \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{4}+1\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{9 d a} \]

[In]

int(cos(d*x+c)^2/(a+b*sin(d*x+c)^3)^2,x)

[Out]

-2/3/d/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)/a
*tan(1/2*d*x+1/2*c)^5+2/3/d/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/
2*d*x+1/2*c)^2*a+a)/a*tan(1/2*d*x+1/2*c)+2/9/d/a*sum((_R^4+1)/(_R^5*a+2*_R^3*a+4*_R^2*b+_R*a)*ln(tan(1/2*d*x+1
/2*c)-_R),_R=RootOf(_Z^6*a+3*_Z^4*a+8*_Z^3*b+3*_Z^2*a+a))

Integral number [401] \[ \int \frac {1}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx \]

[B]   time = 0.65 (sec), size = 658 ,normalized size = 38.71 \[ \frac {2 b^{2} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d \left (\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +8 b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +a \right ) a \left (a^{2}-b^{2}\right )}-\frac {2 b \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d \left (\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +8 b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +a \right ) \left (a^{2}-b^{2}\right )}+\frac {8 b^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d \left (\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +8 b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +a \right ) a \left (a^{2}-b^{2}\right )}+\frac {8 b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d \left (\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +8 b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +a \right ) \left (a^{2}-b^{2}\right )}-\frac {2 b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{3 d \left (\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +8 b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +a \right ) a \left (a^{2}-b^{2}\right )}+\frac {2 b}{3 d \left (\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +8 b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +a \right ) \left (a^{2}-b^{2}\right )}+\frac {\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\left (3 a^{2}-2 b^{2}\right ) \textit {\_R}^{4}-2 \textit {\_R}^{3} a b +6 \textit {\_R}^{2} a^{2}-2 a \textit {\_R} b +3 a^{2}-2 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}}{9 d a \left (a^{2}-b^{2}\right )} \]

[In]

int(1/(a+b*sin(d*x+c)^3)^2,x)

[Out]

2/3/d/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)*b^
2/a/(a^2-b^2)*tan(1/2*d*x+1/2*c)^5-2/3/d/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*
c)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)*b/(a^2-b^2)*tan(1/2*d*x+1/2*c)^4+8/3/d/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+
1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)*b^2/a/(a^2-b^2)*tan(1/2*d*x+1/2*c)^3+8/3/d/(ta
n(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)*b/(a^2-b^2)
*tan(1/2*d*x+1/2*c)^2-2/3/d/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/
2*d*x+1/2*c)^2*a+a)*b^2/a/(a^2-b^2)*tan(1/2*d*x+1/2*c)+2/3/d/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+
8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)*b/(a^2-b^2)+1/9/d/a/(a^2-b^2)*sum(((3*a^2-2*b^2)*_R^4-2*_
R^3*a*b+6*_R^2*a^2-2*a*_R*b+3*a^2-2*b^2)/(_R^5*a+2*_R^3*a+4*_R^2*b+_R*a)*ln(tan(1/2*d*x+1/2*c)-_R),_R=RootOf(_
Z^6*a+3*_Z^4*a+8*_Z^3*b+3*_Z^2*a+a))

Integral number [402] \[ \int \frac {\sec ^2(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx \]

[B]   time = 0.956 (sec), size = 1276 ,normalized size = 49.08 \[ \text {result too large to display} \]

[In]

int(sec(d*x+c)^2/(a+b*sin(d*x+c)^3)^2,x)

[Out]

-1/d/(a+b)^2/(tan(1/2*d*x+1/2*c)-1)-1/d/(a-b)^2/(tan(1/2*d*x+1/2*c)+1)-4/3/d*b^2/(a-b)^2/(a+b)^2/(tan(1/2*d*x+
1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)*a*tan(1/2*d*x+1/2*c)^
5-2/3/d*b^4/(a-b)^2/(a+b)^2/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/
2*d*x+1/2*c)^2*a+a)/a*tan(1/2*d*x+1/2*c)^5-2/3/d*b/(a-b)^2/(a+b)^2/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c
)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)*tan(1/2*d*x+1/2*c)^4*a^2+8/3/d*b^3/(a-b)^2/(a+b)^2/
(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)*tan(1/2*
d*x+1/2*c)^4-8/3/d*b^2/(a-b)^2/(a+b)^2/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)
^3+3*tan(1/2*d*x+1/2*c)^2*a+a)*a*tan(1/2*d*x+1/2*c)^3-16/3/d*b^4/(a-b)^2/(a+b)^2/(tan(1/2*d*x+1/2*c)^6*a+3*tan
(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)/a*tan(1/2*d*x+1/2*c)^3-4/3/d*b/(a-b)^
2/(a+b)^2/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2*a+a
)*tan(1/2*d*x+1/2*c)^2*a^2-20/3/d*b^3/(a-b)^2/(a+b)^2/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan
(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)*tan(1/2*d*x+1/2*c)^2+4/3/d*b^2/(a-b)^2/(a+b)^2/(tan(1/2*d*x+1/2*
c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)*a*tan(1/2*d*x+1/2*c)+2/3/
d*b^4/(a-b)^2/(a+b)^2/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+
1/2*c)^2*a+a)/a*tan(1/2*d*x+1/2*c)-2/3/d*b/(a-b)^2/(a+b)^2/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*
b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)*a^2-4/3/d*b^3/(a-b)^2/(a+b)^2/(tan(1/2*d*x+1/2*c)^6*a+3*tan
(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)-1/9/d*b/(a-b)^2/(a+b)^2/a*sum((b*(11*
a^2-2*b^2)*_R^4+2*a*(-5*a^2-4*b^2)*_R^3+54*_R^2*a^2*b+2*a*(-5*a^2-4*b^2)*_R+11*a^2*b-2*b^3)/(_R^5*a+2*_R^3*a+4
*_R^2*b+_R*a)*ln(tan(1/2*d*x+1/2*c)-_R),_R=RootOf(_Z^6*a+3*_Z^4*a+8*_Z^3*b+3*_Z^2*a+a))

Integral number [403] \[ \int \frac {\sec ^4(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx \]

[B]   time = 1.183 (sec), size = 1549 ,normalized size = 59.58 \[ \text {result too large to display} \]

[In]

int(sec(d*x+c)^4/(a+b*sin(d*x+c)^3)^2,x)

[Out]

-1/3/d/(a+b)^2/(tan(1/2*d*x+1/2*c)-1)^3-1/2/d/(a+b)^2/(tan(1/2*d*x+1/2*c)-1)^2-1/d/(a+b)^3/(tan(1/2*d*x+1/2*c)
-1)*a-4/d/(a+b)^3/(tan(1/2*d*x+1/2*c)-1)*b-1/3/d/(a-b)^2/(tan(1/2*d*x+1/2*c)+1)^3+1/2/d/(a-b)^2/(tan(1/2*d*x+1
/2*c)+1)^2-1/d/(a-b)^3/(tan(1/2*d*x+1/2*c)+1)*a+4/d/(a-b)^3/(tan(1/2*d*x+1/2*c)+1)*b+2/3/d*b^2/(a-b)^3/(a+b)^3
/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)*a^3*tan
(1/2*d*x+1/2*c)^5+14/3/d*b^4/(a-b)^3/(a+b)^3/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+
1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)*a*tan(1/2*d*x+1/2*c)^5+2/3/d*b^6/(a-b)^3/(a+b)^3/(tan(1/2*d*x+1/2*c)^6*a+
3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)/a*tan(1/2*d*x+1/2*c)^5-6/d*b^5/(
a-b)^3/(a+b)^3/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^
2*a+a)*tan(1/2*d*x+1/2*c)^4+16/d*b^4/(a-b)^3/(a+b)^3/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(
1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)*a*tan(1/2*d*x+1/2*c)^3+8/d*b^6/(a-b)^3/(a+b)^3/(tan(1/2*d*x+1/2*c
)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)/a*tan(1/2*d*x+1/2*c)^3+12/
d*b^3/(a-b)^3/(a+b)^3/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+
1/2*c)^2*a+a)*tan(1/2*d*x+1/2*c)^2*a^2+12/d*b^5/(a-b)^3/(a+b)^3/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4
*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)*tan(1/2*d*x+1/2*c)^2-2/3/d*b^2/(a-b)^3/(a+b)^3/(tan(1/
2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)*a^3*tan(1/2*d*x
+1/2*c)-14/3/d*b^4/(a-b)^3/(a+b)^3/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3
*tan(1/2*d*x+1/2*c)^2*a+a)*a*tan(1/2*d*x+1/2*c)-2/3/d*b^6/(a-b)^3/(a+b)^3/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*
x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)/a*tan(1/2*d*x+1/2*c)+4/d*b^3/(a-b)^3/(a+b)^3
/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+1/2*c)^2*a+a)*a^2+2/d
*b^5/(a-b)^3/(a+b)^3/(tan(1/2*d*x+1/2*c)^6*a+3*tan(1/2*d*x+1/2*c)^4*a+8*b*tan(1/2*d*x+1/2*c)^3+3*tan(1/2*d*x+1
/2*c)^2*a+a)+1/9/d*b^2/(a-b)^3/(a+b)^3/a*sum(((19*a^4+28*a^2*b^2-2*b^4)*_R^4+18*a*b*(-4*a^2-b^2)*_R^3+6*a^2*(1
1*a^2+34*b^2)*_R^2+18*a*b*(-4*a^2-b^2)*_R+19*a^4+28*a^2*b^2-2*b^4)/(_R^5*a+2*_R^3*a+4*_R^2*b+_R*a)*ln(tan(1/2*
d*x+1/2*c)-_R),_R=RootOf(_Z^6*a+3*_Z^4*a+8*_Z^3*b+3*_Z^2*a+a))