3.5 Test file Number [79]

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 = 0.280054 (sec), size = 394 ,normalized size = 15.15 \[ \frac {-i \text {RootSum}\left [-i b+3 i b \text {$\#$1}^2+8 a \text {$\#$1}^3-3 i b \text {$\#$1}^4+i b \text {$\#$1}^6\& ,\frac {2 b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-i b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )+4 i a \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}+2 a \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}+12 b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2-6 i b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2-4 i a \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^3-2 a \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^3+2 b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4-i b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4}{b \text {$\#$1}-4 i a \text {$\#$1}^2-2 b \text {$\#$1}^3+b \text {$\#$1}^5}\& \right ]+\frac {24 \cos (c+d x) (a+b \sin (c+d x))}{4 a+3 b \sin (c+d x)-b \sin (3 (c+d x))}}{18 a b d} \]

[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 = 0.184512 (sec), size = 273 ,normalized size = 10.5 \[ \frac {-i \text {RootSum}\left [-i b+3 i b \text {$\#$1}^2+8 a \text {$\#$1}^3-3 i b \text {$\#$1}^4+i b \text {$\#$1}^6\& ,\frac {2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )+12 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2-6 i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2+2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4-i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4}{b \text {$\#$1}-4 i a \text {$\#$1}^2-2 b \text {$\#$1}^3+b \text {$\#$1}^5}\& \right ]+\frac {12 \sin (2 (c+d x))}{4 a+3 b \sin (c+d x)-b \sin (3 (c+d x))}}{18 a d} \]

[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 = 0.371561 (sec), size = 502 ,normalized size = 29.53 \[ \frac {\frac {i \text {RootSum}\left [-i b+3 i b \text {$\#$1}^2+8 a \text {$\#$1}^3-3 i b \text {$\#$1}^4+i b \text {$\#$1}^6\& ,\frac {2 b^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-i b^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )+4 i a b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}+2 a b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}-24 a^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2+12 b^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2+12 i a^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2-6 i b^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2-4 i a b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^3-2 a b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^3+2 b^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4-i b^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4}{b \text {$\#$1}-4 i a \text {$\#$1}^2-2 b \text {$\#$1}^3+b \text {$\#$1}^5}\& \right ]}{a^2-b^2}-\frac {12 b \cos (c+d x) (-3 a+a \cos (2 (c+d x))+2 b \sin (c+d x))}{(a-b) (a+b) (4 a+3 b \sin (c+d x)-b \sin (3 (c+d x)))}}{18 a d} \]

[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 = 1.16388 (sec), size = 845 ,normalized size = 32.5 \[ \frac {-\frac {i b \text {RootSum}\left [-i b+3 i b \text {$\#$1}^2+8 a \text {$\#$1}^3-3 i b \text {$\#$1}^4+i b \text {$\#$1}^6\& ,\frac {16 a^2 b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+2 b^3 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-8 i a^2 b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )-i b^3 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )+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 (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}+8 a b^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}-120 a^2 b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2+12 b^3 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2+60 i a^2 b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2-6 i b^3 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\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}^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 (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^3-8 a b^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^3+16 a^2 b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4+2 b^3 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4-8 i a^2 b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4-i b^3 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4}{b \text {$\#$1}-4 i a \text {$\#$1}^2-2 b \text {$\#$1}^3+b \text {$\#$1}^5}\& \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 a b^2+3 a b^2 \cos (2 (c+d x))+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 = 1.26339 (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 = 1.635 (sec), size = 241 ,normalized size = 9.27 \[\text {result too large to display}\]

[In]

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

[Out]

1/d*(2*(-1/3/a*tan(1/2*d*x+1/2*c)^5+1/3/b*tan(1/2*d*x+1/2*c)^4+4/3/a*tan(1/2*d*x+1/2*c)^3+2/3/b*tan(1/2*d*x+1/
2*c)^2+1/3/a*tan(1/2*d*x+1/2*c)+1/3/b)/(a*tan(1/2*d*x+1/2*c)^6+3*a*tan(1/2*d*x+1/2*c)^4+8*b*tan(1/2*d*x+1/2*c)
^3+3*a*tan(1/2*d*x+1/2*c)^2+a)+2/9/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 = 2.011 (sec), size = 175 ,normalized size = 6.73 \[\text {result too large to display}\]

[In]

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

[Out]

1/d*(2*(-1/3/a*tan(1/2*d*x+1/2*c)^5+1/3/a*tan(1/2*d*x+1/2*c))/(a*tan(1/2*d*x+1/2*c)^6+3*a*tan(1/2*d*x+1/2*c)^4
+8*b*tan(1/2*d*x+1/2*c)^3+3*a*tan(1/2*d*x+1/2*c)^2+a)+2/9/a*sum((_R^4+1)/(_R^5*a+2*_R^3*a+4*_R^2*b+_R*a)*ln(ta
n(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 = 3.128 (sec), size = 349 ,normalized size = 20.53

method	result	size
derivativedivides	\(\frac {\frac {\frac {2 b^{2} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a \left (a^{2}-b^{2}\right )}-\frac {2 b \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 \left (a^{2}-b^{2}\right )}+\frac {8 b^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a \left (a^{2}-b^{2}\right )}+\frac {8 b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 \left (a^{2}-b^{2}\right )}-\frac {2 b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{3 a \left (a^{2}-b^{2}\right )}+\frac {2 b}{3 a^{2}-3 b^{2}}}{a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8 b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a}+\frac {\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\left (3 a^{2}-2 b^{2}\right ) \textit {\_R}^{4}-2 a b \,\textit {\_R}^{3}+6 a^{2} \textit {\_R}^{2}-2 a \textit {\_R} b +3 a^{2}-2 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}}{9 a \left (a^{2}-b^{2}\right )}}{d}\)	\(349\)
default	\(\frac {\frac {\frac {2 b^{2} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a \left (a^{2}-b^{2}\right )}-\frac {2 b \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 \left (a^{2}-b^{2}\right )}+\frac {8 b^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a \left (a^{2}-b^{2}\right )}+\frac {8 b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 \left (a^{2}-b^{2}\right )}-\frac {2 b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{3 a \left (a^{2}-b^{2}\right )}+\frac {2 b}{3 a^{2}-3 b^{2}}}{a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8 b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a}+\frac {\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\left (3 a^{2}-2 b^{2}\right ) \textit {\_R}^{4}-2 a b \,\textit {\_R}^{3}+6 a^{2} \textit {\_R}^{2}-2 a \textit {\_R} b +3 a^{2}-2 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}}{9 a \left (a^{2}-b^{2}\right )}}{d}\)	\(349\)
risch	\(\text {Expression too large to display}\)	\(2453\)

[In]

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

[Out]

1/d*(2*(1/3*b^2/a/(a^2-b^2)*tan(1/2*d*x+1/2*c)^5-1/3/(a^2-b^2)*b*tan(1/2*d*x+1/2*c)^4+4/3*b^2/a/(a^2-b^2)*tan(
1/2*d*x+1/2*c)^3+4/3/(a^2-b^2)*b*tan(1/2*d*x+1/2*c)^2-1/3*b^2/a/(a^2-b^2)*tan(1/2*d*x+1/2*c)+1/3/(a^2-b^2)*b)/
(a*tan(1/2*d*x+1/2*c)^6+3*a*tan(1/2*d*x+1/2*c)^4+8*b*tan(1/2*d*x+1/2*c)^3+3*a*tan(1/2*d*x+1/2*c)^2+a)+1/9/a/(a
^2-b^2)*sum(((3*a^2-2*b^2)*_R^4-2*a*b*_R^3+6*a^2*_R^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 = 6.409 (sec), size = 398 ,normalized size = 15.31 \[\text {result too large to display}\]

[In]

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

[Out]

1/d*(-1/(a+b)^2/(tan(1/2*d*x+1/2*c)-1)-1/(a-b)^2/(tan(1/2*d*x+1/2*c)+1)+2*b/(a-b)^2/(a+b)^2*((-1/3*(2*a^2+b^2)
*b/a*tan(1/2*d*x+1/2*c)^5+(-1/3*a^2+4/3*b^2)*tan(1/2*d*x+1/2*c)^4-4/3*b*(a^2+2*b^2)/a*tan(1/2*d*x+1/2*c)^3+(-2
/3*a^2-10/3*b^2)*tan(1/2*d*x+1/2*c)^2+1/3*(2*a^2+b^2)*b/a*tan(1/2*d*x+1/2*c)-1/3*a^2-2/3*b^2)/(a*tan(1/2*d*x+1
/2*c)^6+3*a*tan(1/2*d*x+1/2*c)^4+8*b*tan(1/2*d*x+1/2*c)^3+3*a*tan(1/2*d*x+1/2*c)^2+a)+1/18/a*sum((b*(-11*a^2+2
*b^2)*_R^4+2*a*(5*a^2+4*b^2)*_R^3-54*a^2*b*_R^2+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 = 8.928 (sec), size = 525 ,normalized size = 20.19 \[\text {result too large to display}\]

[In]

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

[Out]

1/d*(-1/3/(a-b)^2/(tan(1/2*d*x+1/2*c)+1)^3+1/2/(a-b)^2/(tan(1/2*d*x+1/2*c)+1)^2-(a-4*b)/(a-b)^3/(tan(1/2*d*x+1
/2*c)+1)-1/3/(a+b)^2/(tan(1/2*d*x+1/2*c)-1)^3-1/2/(a+b)^2/(tan(1/2*d*x+1/2*c)-1)^2-(4*b+a)/(a+b)^3/(tan(1/2*d*
x+1/2*c)-1)+2*b^2/(a-b)^3/(a+b)^3*((1/3*(a^4+7*a^2*b^2+b^4)/a*tan(1/2*d*x+1/2*c)^5-3*b^3*tan(1/2*d*x+1/2*c)^4+
4*b^2*(2*a^2+b^2)/a*tan(1/2*d*x+1/2*c)^3+(6*a^2*b+6*b^3)*tan(1/2*d*x+1/2*c)^2-1/3*(a^4+7*a^2*b^2+b^4)/a*tan(1/
2*d*x+1/2*c)+2*a^2*b+b^3)/(a*tan(1/2*d*x+1/2*c)^6+3*a*tan(1/2*d*x+1/2*c)^4+8*b*tan(1/2*d*x+1/2*c)^3+3*a*tan(1/
2*d*x+1/2*c)^2+a)+1/18/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*(11*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))))

3.5.3 Fricas

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

[C]   time = 3.2791 (sec), size = 36403 ,normalized size = 1400.12 \[ \text {too large to display} \]

[In]

integrate(cos(d*x+c)^2/(a+b*sin(d*x+c)^3)^2,x, algorithm=""fricas"")

[Out]

1/324*(3*sqrt(2/3)*sqrt(1/6)*(a^2*d - (a*b*d*cos(d*x + c)^2 - a*b*d)*sin(d*x + c))*sqrt(-((a^4 - a^2*b^2)*((-I
*sqrt(3) + 1)*(3/(a^6*b^2*d^4 - a^4*b^4*d^4) - 1/(a^4*d^2 - a^2*b^2*d^2)^2)/(-1/1062882*(a^4 - 16*a^2*b^2 + 64
*b^4)/(a^12*b^4*d^6 - a^10*b^6*d^6) + 1/118098/((a^6*b^2*d^4 - a^4*b^4*d^4)*(a^4*d^2 - a^2*b^2*d^2)) - 1/53144
1/(a^4*d^2 - a^2*b^2*d^2)^3 + 1/1062882*(a^6 + 28*a^4*b^2 - 80*a^2*b^4 + 64*b^6)/((a^2 - b^2)^2*a^10*b^4*d^6))
^(1/3) - 6561*(I*sqrt(3) + 1)*(-1/1062882*(a^4 - 16*a^2*b^2 + 64*b^4)/(a^12*b^4*d^6 - a^10*b^6*d^6) + 1/118098
/((a^6*b^2*d^4 - a^4*b^4*d^4)*(a^4*d^2 - a^2*b^2*d^2)) - 1/531441/(a^4*d^2 - a^2*b^2*d^2)^3 + 1/1062882*(a^6 +
28*a^4*b^2 - 80*a^2*b^4 + 64*b^6)/((a^2 - b^2)^2*a^10*b^4*d^6))^(1/3) - 162/(a^4*d^2 - a^2*b^2*d^2))*d^2 + 3*
sqrt(1/3)*(a^4 - a^2*b^2)*d^2*sqrt(-((a^8*b^2 - 2*a^6*b^4 + a^4*b^6)*((-I*sqrt(3) + 1)*(3/(a^6*b^2*d^4 - a^4*b
^4*d^4) - 1/(a^4*d^2 - a^2*b^2*d^2)^2)/(-1/1062882*(a^4 - 16*a^2*b^2 + 64*b^4)/(a^12*b^4*d^6 - a^10*b^6*d^6) +
1/118098/ ...

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

[C]   time = 8.35384 (sec), size = 70185 ,normalized size = 4128.53 \[ \text {too large to display} \]

[In]

integrate(1/(a+b*sin(d*x+c)^3)^2,x, algorithm=""fricas"")

[Out]

-1/108*(36*a*b*cos(d*x + c)^3 + 36*b^2*cos(d*x + c)*sin(d*x + c) - sqrt(2/3)*sqrt(1/2)*((a^4 - a^2*b^2)*d - ((
a^3*b - a*b^3)*d*cos(d*x + c)^2 - (a^3*b - a*b^3)*d)*sin(d*x + c))*sqrt(-(1458*a^4 + 486*a^2*b^2 - 486*b^4 - (
a^8 - 3*a^6*b^2 + 3*a^4*b^4 - a^2*b^6)*((-I*sqrt(3) + 1)*(3*(3*a^4 + a^2*b^2 - b^4)^2/(a^8*d^2 - 3*a^6*b^2*d^2
+ 3*a^4*b^4*d^2 - a^2*b^6*d^2)^2 - (27*a^2 - 11*b^2)/(a^10*d^4 - 3*a^8*b^2*d^4 + 3*a^6*b^4*d^4 - a^4*b^6*d^4)
)/(-1/1062882*(729*a^4 - 432*a^2*b^2 + 64*b^4)/(a^16*d^6 - 3*a^14*b^2*d^6 + 3*a^12*b^4*d^6 - a^10*b^6*d^6) - 1
/19683*(3*a^4 + a^2*b^2 - b^4)^3/(a^8*d^2 - 3*a^6*b^2*d^2 + 3*a^4*b^4*d^2 - a^2*b^6*d^2)^3 + 1/39366*(3*a^4 +
a^2*b^2 - b^4)*(27*a^2 - 11*b^2)/((a^10*d^4 - 3*a^8*b^2*d^4 + 3*a^6*b^4*d^4 - a^4*b^6*d^4)*(a^8*d^2 - 3*a^6*b^
2*d^2 + 3*a^4*b^4*d^2 - a^2*b^6*d^2)) + 1/1062882*(3375*a^8 - 4573*a^6*b^2 + 2460*a^4*b^4 - 624*a^2*b^6 + 64*b
^8)*b^2/((a^2 - b^2)^6*a^10*d^6))^(1/3) + 2187*(I*sqrt(3) + 1)*(-1/1062882*(729*a^4 - 432*a^2*b^2 + 64*b^4)/(a
^16*d^6 -  ...

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

[C]   time = 38.6728 (sec), size = 102913 ,normalized size = 3958.19 \[ \text {too large to display} \]

[In]

integrate(sec(d*x+c)^2/(a+b*sin(d*x+c)^3)^2,x, algorithm=""fricas"")

[Out]

1/108*(108*(a^3*b + 2*a*b^3)*cos(d*x + c)^4 - 108*a^3*b + 108*a*b^3 - sqrt(2)*sqrt(1/2)*((a^6 - 2*a^4*b^2 + a^
2*b^4)*d*cos(d*x + c) - ((a^5*b - 2*a^3*b^3 + a*b^5)*d*cos(d*x + c)^3 - (a^5*b - 2*a^3*b^3 + a*b^5)*d*cos(d*x
+ c))*sin(d*x + c))*sqrt(-(5670*a^6*b^2 + 31590*a^4*b^4 + 2916*a^2*b^6 - 810*b^8 - (a^12 - 5*a^10*b^2 + 10*a^8
*b^4 - 10*a^6*b^6 + 5*a^4*b^8 - a^2*b^10)*((-I*sqrt(3) + 1)*((35*a^6*b^2 + 195*a^4*b^4 + 18*a^2*b^6 - 5*b^8)^2
/(a^12*d^2 - 5*a^10*b^2*d^2 + 10*a^8*b^4*d^2 - 10*a^6*b^6*d^2 + 5*a^4*b^8*d^2 - a^2*b^10*d^2)^2 - 45*(10*a^2*b
^4 - b^6)/(a^14*d^4 - 5*a^12*b^2*d^4 + 10*a^10*b^4*d^4 - 10*a^8*b^6*d^4 + 5*a^6*b^8*d^4 - a^4*b^10*d^4))/(-1/1
9683*(35*a^6*b^2 + 195*a^4*b^4 + 18*a^2*b^6 - 5*b^8)^3/(a^12*d^2 - 5*a^10*b^2*d^2 + 10*a^8*b^4*d^2 - 10*a^6*b^
6*d^2 + 5*a^4*b^8*d^2 - a^2*b^10*d^2)^3 - 1/1062882*(15625*a^4*b^4 - 2000*a^2*b^6 + 64*b^8)/(a^20*d^6 - 5*a^18
*b^2*d^6 + 10*a^16*b^4*d^6 - 10*a^14*b^6*d^6 + 5*a^12*b^8*d^6 - a^10*b^10*d^6) + 5/1458*(35*a^6*b^2 + 195*a^4*
b^4 + 18*a ...

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

[C]   time = 104.276 (sec), size = 133123 ,normalized size = 5120.12 \[ \text {too large to display} \]

[In]

integrate(sec(d*x+c)^4/(a+b*sin(d*x+c)^3)^2,x, algorithm=""fricas"")

[Out]

1/108*(36*(2*a^5*b - 30*a^3*b^3 - 17*a*b^5)*cos(d*x + c)^6 - 36*a^5*b + 72*a^3*b^3 - 36*a*b^5 - 108*(a^5*b - 2
1*a^3*b^3 - 10*a*b^5)*cos(d*x + c)^4 + sqrt(2/3)*sqrt(1/6)*((a^8 - 3*a^6*b^2 + 3*a^4*b^4 - a^2*b^6)*d*cos(d*x
+ c)^3 - ((a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7)*d*cos(d*x + c)^5 - (a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7)*d
*cos(d*x + c)^3)*sin(d*x + c))*sqrt(-(573480*a^8*b^4 + 4293324*a^6*b^6 + 3847662*a^4*b^8 + 159894*a^2*b^10 - 1
7010*b^12 - (a^16 - 7*a^14*b^2 + 21*a^12*b^4 - 35*a^10*b^6 + 35*a^8*b^8 - 21*a^6*b^10 + 7*a^4*b^12 - a^2*b^14)
*((-I*sqrt(3) + 1)*((1180*a^8*b^4 + 8834*a^6*b^6 + 7917*a^4*b^8 + 329*a^2*b^10 - 35*b^12)^2/(a^16*d^2 - 7*a^14
*b^2*d^2 + 21*a^12*b^4*d^2 - 35*a^10*b^6*d^2 + 35*a^8*b^8*d^2 - 21*a^6*b^10*d^2 + 7*a^4*b^12*d^2 - a^2*b^14*d^
2)^2 + 15*(1029*a^4*b^6 - 3173*a^2*b^8 + 119*b^10)/(a^18*d^4 - 7*a^16*b^2*d^4 + 21*a^14*b^4*d^4 - 35*a^12*b^6*
d^4 + 35*a^10*b^8*d^4 - 21*a^8*b^10*d^4 + 7*a^6*b^12*d^4 - a^4*b^14*d^4))/(-1/531441*(1180*a^8*b^4 + 8834*a^6*
b^6 + 7917 ...

3.5.4 Mupad

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

[B]   time = 15.9945 (sec), size = 2431 ,normalized size = 93.5 \[ \text {Too large to display} \]

[In]

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

[Out]

2/(3*d*(a*b + 8*b^2*tan(c/2 + (d*x)/2)^3 + 3*a*b*tan(c/2 + (d*x)/2)^2 + 3*a*b*tan(c/2 + (d*x)/2)^4 + a*b*tan(c
/2 + (d*x)/2)^6)) + symsum(log((638976*a^2*b^4 - 655360*b^6 - 8192*a^6 + 24576*a^4*b^2 - 2949120*root(531441*a
^10*b^8*d^6 + 59049*a^8*b^6*d^4 + 2187*a^6*b^4*d^2 + 48*a^2*b^4 + 15*a^4*b^2 + a^6 - 64*b^6, d, k)*a^3*b^5 + 2
138112*root(531441*a^10*b^8*d^6 + 59049*a^8*b^6*d^4 + 2187*a^6*b^4*d^2 + 48*a^2*b^4 + 15*a^4*b^2 + a^6 - 64*b^
6, d, k)*a^5*b^3 - 9437184*root(531441*a^10*b^8*d^6 + 59049*a^8*b^6*d^4 + 2187*a^6*b^4*d^2 + 48*a^2*b^4 + 15*a
^4*b^2 + a^6 - 64*b^6, d, k)*b^8*tan(c/2 + (d*x)/2) - 786432*a*b^5*tan(c/2 + (d*x)/2) + 98304*a^5*b*tan(c/2 +
(d*x)/2) - 21233664*root(531441*a^10*b^8*d^6 + 59049*a^8*b^6*d^4 + 2187*a^6*b^4*d^2 + 48*a^2*b^4 + 15*a^4*b^2
+ a^6 - 64*b^6, d, k)^2*a^2*b^8 + 18579456*root(531441*a^10*b^8*d^6 + 59049*a^8*b^6*d^4 + 2187*a^6*b^4*d^2 + 4
8*a^2*b^4 + 15*a^4*b^2 + a^6 - 64*b^6, d, k)^2*a^4*b^6 + 2654208*root(531441*a^10*b^8*d^6 + 59049*a^8*b^6*d^4
+ 2187*a^6*b^4*d^2 + 48*a^2*b^4 + 15*a^4*b^2 + a^6 - 64*b^6, d, k)^2*a^6*b^4 - 167215104*root(531441*a^10*b^8*
d^6 + 59049*a^8*b^6*d^4 + 2187*a^6*b^4*d^2 + 48*a^2*b^4 + 15*a^4*b^2 + a^6 - 64*b^6, d, k)^3*a^5*b^7 + 1134673
92*root(531441*a^10*b^8*d^6 + 59049*a^8*b^6*d^4 + 2187*a^6*b^4*d^2 + 48*a^2*b^4 + 15*a^4*b^2 + a^6 - 64*b^6, d
, k)^3*a^7*b^5 - 107495424*root(531441*a^10*b^8*d^6 + 59049*a^8*b^6*d^4 + 2187*a^6*b^4*d^2 + 48*a^2*b^4 + 15*a
^4*b^2 + a^6 - 64*b^6, d, k)^4*a^6*b^8 + 107495424*root(531441*a^10*b^8*d^6 + 59049*a^8*b^6*d^4 + 2187*a^6*b^4
*d^2 + 48*a^2*b^4 + 15*a^4*b^2 + a^6 - 64*b^6, d, k)^4*a^8*b^6 - 1934917632*root(531441*a^10*b^8*d^6 + 59049*a
^8*b^6*d^4 + 2187*a^6*b^4*d^2 + 48*a^2*b^4 + 15*a^4*b^2 + a^6 - 64*b^6, d, k)^5*a^7*b^9 + 1451188224*root(5314
41*a^10*b^8*d^6 + 59049*a^8*b^6*d^4 + 2187*a^6*b^4*d^2 + 48*a^2*b^4 + 15*a^4*b^2 + a^6 - 64*b^6, d, k)^5*a^9*b
^7 + 688128*a^3*b^3*tan(c/2 + (d*x)/2) - 1179648*root(531441*a^10*b^8*d^6 + 59049*a^8*b^6*d^4 + 2187*a^6*b^4*d
^2 + 48*a^2*b^4 + 15*a^4*b^2 + a^6 - 64*b^6, d, k)*a*b^7 + 12976128*root(531441*a^10*b^8*d^6 + 59049*a^8*b^6*d
^4 + 2187*a^6*b^4*d^2 + 48*a^2*b^4 + 15*a^4*b^2 + a^6 - 64*b^6, d, k)*a^2*b^6*tan(c/2 + (d*x)/2) - 6266880*roo
t(531441*a^10*b^8*d^6 + 59049*a^8*b^6*d^4 + 2187*a^6*b^4*d^2 + 48*a^2*b^4 + 15*a^4*b^2 + a^6 - 64*b^6, d, k)*a
^4*b^4*tan(c/2 + (d*x)/2) + 737280*root(531441*a^10*b^8*d^6 + 59049*a^8*b^6*d^4 + 2187*a^6*b^4*d^2 + 48*a^2*b^
4 + 15*a^4*b^2 + a^6 - 64*b^6, d, k)*a^6*b^2*tan(c/2 + (d*x)/2) - 53084160*root(531441*a^10*b^8*d^6 + 59049*a^
8*b^6*d^4 + 2187*a^6*b^4*d^2 + 48*a^2*b^4 + 15*a^4*b^2 + a^6 - 64*b^6, d, k)^2*a^3*b^7*tan(c/2 + (d*x)/2) + 50
429952*root(531441*a^10*b^8*d^6 + 59049*a^8*b^6*d^4 + 2187*a^6*b^4*d^2 + 48*a^2*b^4 + 15*a^4*b^2 + a^6 - 64*b^
6, d, k)^2*a^5*b^5*tan(c/2 + (d*x)/2) + 2654208*root(531441*a^10*b^8*d^6 + 59049*a^8*b^6*d^4 + 2187*a^6*b^4*d^
2 + 48*a^2*b^4 + 15*a^4*b^2 + a^6 - 64*b^6, d, k)^2*a^7*b^3*tan(c/2 + (d*x)/2) - 59719680*root(531441*a^10*b^8
*d^6 + 59049*a^8*b^6*d^4 + 2187*a^6*b^4*d^2 + 48*a^2*b^4 + 15*a^4*b^2 + a^6 - 64*b^6, d, k)^3*a^6*b^6*tan(c/2
+ (d*x)/2) + 5971968*root(531441*a^10*b^8*d^6 + 59049*a^8*b^6*d^4 + 2187*a^6*b^4*d^2 + 48*a^2*b^4 + 15*a^4*b^2
+ a^6 - 64*b^6, d, k)^3*a^8*b^4*tan(c/2 + (d*x)/2) - 859963392*root(531441*a^10*b^8*d^6 + 59049*a^8*b^6*d^4 +
2187*a^6*b^4*d^2 + 48*a^2*b^4 + 15*a^4*b^2 + a^6 - 64*b^6, d, k)^4*a^5*b^9*tan(c/2 + (d*x)/2) + 859963392*roo
t(531441*a^10*b^8*d^6 + 59049*a^8*b^6*d^4 + 2187*a^6*b^4*d^2 + 48*a^2*b^4 + 15*a^4*b^2 + a^6 - 64*b^6, d, k)^4
*a^7*b^7*tan(c/2 + (d*x)/2) - 483729408*root(531441*a^10*b^8*d^6 + 59049*a^8*b^6*d^4 + 2187*a^6*b^4*d^2 + 48*a
^2*b^4 + 15*a^4*b^2 + a^6 - 64*b^6, d, k)^5*a^8*b^8*tan(c/2 + (d*x)/2))/(a^3*b^4))*root(531441*a^10*b^8*d^6 +
59049*a^8*b^6*d^4 + 2187*a^6*b^4*d^2 + 48*a^2*b^4 + 15*a^4*b^2 + a^6 - 64*b^6, d, k), k, 1, 6)/d + (8*tan(c/2
+ (d*x)/2)^3)/(3*d*(3*a^2*tan(c/2 + (d*x)/2)^2 + 3*a^2*tan(c/2 + (d*x)/2)^4 + a^2*tan(c/2 + (d*x)/2)^6 + a^2 +
8*a*b*tan(c/2 + (d*x)/2)^3)) - (2*tan(c/2 + (d*x)/2)^5)/(3*d*(3*a^2*tan(c/2 + (d*x)/2)^2 + 3*a^2*tan(c/2 + (d
*x)/2)^4 + a^2*tan(c/2 + (d*x)/2)^6 + a^2 + 8*a*b*tan(c/2 + (d*x)/2)^3)) + (4*tan(c/2 + (d*x)/2)^2)/(3*d*(a*b
+ 8*b^2*tan(c/2 + (d*x)/2)^3 + 3*a*b*tan(c/2 + (d*x)/2)^2 + 3*a*b*tan(c/2 + (d*x)/2)^4 + a*b*tan(c/2 + (d*x)/2
)^6)) + (2*tan(c/2 + (d*x)/2)^4)/(3*d*(a*b + 8*b^2*tan(c/2 + (d*x)/2)^3 + 3*a*b*tan(c/2 + (d*x)/2)^2 + 3*a*b*t
an(c/2 + (d*x)/2)^4 + a*b*tan(c/2 + (d*x)/2)^6)) + (2*tan(c/2 + (d*x)/2))/(3*d*(3*a^2*tan(c/2 + (d*x)/2)^2 + 3
*a^2*tan(c/2 + (d*x)/2)^4 + a^2*tan(c/2 + (d*x)/2)^6 + a^2 + 8*a*b*tan(c/2 + (d*x)/2)^3))

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

[B]   time = 15.7458 (sec), size = 1648 ,normalized size = 63.38 \[ \text {result too large to display} \]

[In]

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

[Out]

symsum(log(-((131072*b^2)/243 - (16384*a^2)/243 + (8192*root(531441*a^12*b^4*d^6 - 531441*a^10*b^6*d^6 + 19683
*a^8*b^4*d^4 + 729*a^6*b^2*d^2 - 16*a^2*b^2 + a^4 + 64*b^4, d, k)*a^4*tan(c/2 + (d*x)/2))/27 + (1048576*root(5
31441*a^12*b^4*d^6 - 531441*a^10*b^6*d^6 + 19683*a^8*b^4*d^4 + 729*a^6*b^2*d^2 - 16*a^2*b^2 + a^4 + 64*b^4, d,
k)*b^4*tan(c/2 + (d*x)/2))/27 + (262144*root(531441*a^12*b^4*d^6 - 531441*a^10*b^6*d^6 + 19683*a^8*b^4*d^4 +
729*a^6*b^2*d^2 - 16*a^2*b^2 + a^4 + 64*b^4, d, k)^2*a^2*b^4)/3 - (131072*root(531441*a^12*b^4*d^6 - 531441*a^
10*b^6*d^6 + 19683*a^8*b^4*d^4 + 729*a^6*b^2*d^2 - 16*a^2*b^2 + a^4 + 64*b^4, d, k)^2*a^4*b^2)/3 - 98304*root(
531441*a^12*b^4*d^6 - 531441*a^10*b^6*d^6 + 19683*a^8*b^4*d^4 + 729*a^6*b^2*d^2 - 16*a^2*b^2 + a^4 + 64*b^4, d
, k)^3*a^5*b^3 + 442368*root(531441*a^12*b^4*d^6 - 531441*a^10*b^6*d^6 + 19683*a^8*b^4*d^4 + 729*a^6*b^2*d^2 -
16*a^2*b^2 + a^4 + 64*b^4, d, k)^4*a^6*b^4 + 221184*root(531441*a^12*b^4*d^6 - 531441*a^10*b^6*d^6 + 19683*a^
8*b^4*d^4 + 729*a^6*b^2*d^2 - 16*a^2*b^2 + a^4 + 64*b^4, d, k)^4*a^8*b^2 + 7962624*root(531441*a^12*b^4*d^6 -
531441*a^10*b^6*d^6 + 19683*a^8*b^4*d^4 + 729*a^6*b^2*d^2 - 16*a^2*b^2 + a^4 + 64*b^4, d, k)^5*a^7*b^5 - 59719
68*root(531441*a^12*b^4*d^6 - 531441*a^10*b^6*d^6 + 19683*a^8*b^4*d^4 + 729*a^6*b^2*d^2 - 16*a^2*b^2 + a^4 + 6
4*b^4, d, k)^5*a^9*b^3 + (131072*root(531441*a^12*b^4*d^6 - 531441*a^10*b^6*d^6 + 19683*a^8*b^4*d^4 + 729*a^6*
b^2*d^2 - 16*a^2*b^2 + a^4 + 64*b^4, d, k)*a*b^3)/27 - (65536*root(531441*a^12*b^4*d^6 - 531441*a^10*b^6*d^6 +
19683*a^8*b^4*d^4 + 729*a^6*b^2*d^2 - 16*a^2*b^2 + a^4 + 64*b^4, d, k)*a^3*b)/27 - (131072*root(531441*a^12*b
^4*d^6 - 531441*a^10*b^6*d^6 + 19683*a^8*b^4*d^4 + 729*a^6*b^2*d^2 - 16*a^2*b^2 + a^4 + 64*b^4, d, k)*a^2*b^2*
tan(c/2 + (d*x)/2))/9 - (32768*root(531441*a^12*b^4*d^6 - 531441*a^10*b^6*d^6 + 19683*a^8*b^4*d^4 + 729*a^6*b^
2*d^2 - 16*a^2*b^2 + a^4 + 64*b^4, d, k)^2*a^5*b*tan(c/2 + (d*x)/2))/3 - (131072*root(531441*a^12*b^4*d^6 - 53
1441*a^10*b^6*d^6 + 19683*a^8*b^4*d^4 + 729*a^6*b^2*d^2 - 16*a^2*b^2 + a^4 + 64*b^4, d, k)^2*a^3*b^3*tan(c/2 +
(d*x)/2))/3 + 245760*root(531441*a^12*b^4*d^6 - 531441*a^10*b^6*d^6 + 19683*a^8*b^4*d^4 + 729*a^6*b^2*d^2 - 1
6*a^2*b^2 + a^4 + 64*b^4, d, k)^3*a^6*b^2*tan(c/2 + (d*x)/2) + 3538944*root(531441*a^12*b^4*d^6 - 531441*a^10*
b^6*d^6 + 19683*a^8*b^4*d^4 + 729*a^6*b^2*d^2 - 16*a^2*b^2 + a^4 + 64*b^4, d, k)^4*a^5*b^5*tan(c/2 + (d*x)/2)
- 2654208*root(531441*a^12*b^4*d^6 - 531441*a^10*b^6*d^6 + 19683*a^8*b^4*d^4 + 729*a^6*b^2*d^2 - 16*a^2*b^2 +
a^4 + 64*b^4, d, k)^4*a^7*b^3*tan(c/2 + (d*x)/2) + 1990656*root(531441*a^12*b^4*d^6 - 531441*a^10*b^6*d^6 + 19
683*a^8*b^4*d^4 + 729*a^6*b^2*d^2 - 16*a^2*b^2 + a^4 + 64*b^4, d, k)^5*a^8*b^4*tan(c/2 + (d*x)/2))/a^3)*root(5
31441*a^12*b^4*d^6 - 531441*a^10*b^6*d^6 + 19683*a^8*b^4*d^4 + 729*a^6*b^2*d^2 - 16*a^2*b^2 + a^4 + 64*b^4, d,
k), k, 1, 6)/d - (2*tan(c/2 + (d*x)/2)^5)/(3*d*(3*a^2*tan(c/2 + (d*x)/2)^2 + 3*a^2*tan(c/2 + (d*x)/2)^4 + a^2
*tan(c/2 + (d*x)/2)^6 + a^2 + 8*a*b*tan(c/2 + (d*x)/2)^3)) + (2*tan(c/2 + (d*x)/2))/(3*d*(3*a^2*tan(c/2 + (d*x
)/2)^2 + 3*a^2*tan(c/2 + (d*x)/2)^4 + a^2*tan(c/2 + (d*x)/2)^6 + a^2 + 8*a*b*tan(c/2 + (d*x)/2)^3))

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

[B]   time = 17.893 (sec), size = 1567 ,normalized size = 92.18 \[ \text {result too large to display} \]

[In]

int(1/(a + b*sin(c + d*x)^3)^2,x)

[Out]

symsum(log(- (8192*(80*b^6 - 270*a^2*b^4))/(243*(a^7 + a^3*b^4 - 2*a^5*b^2)) - root(1594323*a^14*b^2*d^6 - 159
4323*a^12*b^4*d^6 + 531441*a^10*b^6*d^6 - 531441*a^16*d^6 - 59049*a^10*b^2*d^4 + 59049*a^8*b^4*d^4 - 177147*a^
12*d^4 + 8019*a^6*b^2*d^2 - 19683*a^8*d^2 + 432*a^2*b^2 - 64*b^4 - 729*a^4, d, k)*((8192*(144*a*b^7 + 648*a^3*
b^5 - 2187*a^5*b^3))/(243*(a^7 + a^3*b^4 - 2*a^5*b^2)) - root(1594323*a^14*b^2*d^6 - 1594323*a^12*b^4*d^6 + 53
1441*a^10*b^6*d^6 - 531441*a^16*d^6 - 59049*a^10*b^2*d^4 + 59049*a^8*b^4*d^4 - 177147*a^12*d^4 + 8019*a^6*b^2*
d^2 - 19683*a^8*d^2 + 432*a^2*b^2 - 64*b^4 - 729*a^4, d, k)*(root(1594323*a^14*b^2*d^6 - 1594323*a^12*b^4*d^6
+ 531441*a^10*b^6*d^6 - 531441*a^16*d^6 - 59049*a^10*b^2*d^4 + 59049*a^8*b^4*d^4 - 177147*a^12*d^4 + 8019*a^6*
b^2*d^2 - 19683*a^8*d^2 + 432*a^2*b^2 - 64*b^4 - 729*a^4, d, k)*((8192*(26973*a^7*b^5 - 20412*a^5*b^7 + 39366*
a^9*b^3))/(243*(a^7 + a^3*b^4 - 2*a^5*b^2)) - root(1594323*a^14*b^2*d^6 - 1594323*a^12*b^4*d^6 + 531441*a^10*b
^6*d^6 - 531441*a^16*d^6 - 59049*a^10*b^2*d^4 + 59049*a^8*b^4*d^4 - 177147*a^12*d^4 + 8019*a^6*b^2*d^2 - 19683
*a^8*d^2 + 432*a^2*b^2 - 64*b^4 - 729*a^4, d, k)*(root(1594323*a^14*b^2*d^6 - 1594323*a^12*b^4*d^6 + 531441*a^
10*b^6*d^6 - 531441*a^16*d^6 - 59049*a^10*b^2*d^4 + 59049*a^8*b^4*d^4 - 177147*a^12*d^4 + 8019*a^6*b^2*d^2 - 1
9683*a^8*d^2 + 432*a^2*b^2 - 64*b^4 - 729*a^4, d, k)*((8192*(236196*a^7*b^9 - 649539*a^9*b^7 + 590490*a^11*b^5
- 177147*a^13*b^3))/(243*(a^7 + a^3*b^4 - 2*a^5*b^2)) + (8192*tan(c/2 + (d*x)/2)*(6561*a^8*b^8 - 13122*a^10*b
^6 + 6561*a^12*b^4))/(27*(a^7 + a^3*b^4 - 2*a^5*b^2))) + (8192*(13122*a^6*b^8 - 85293*a^8*b^6 + 72171*a^10*b^4
))/(243*(a^7 + a^3*b^4 - 2*a^5*b^2)) + (8192*tan(c/2 + (d*x)/2)*(11664*a^5*b^9 - 40824*a^7*b^7 + 37908*a^9*b^5
- 8748*a^11*b^3))/(27*(a^7 + a^3*b^4 - 2*a^5*b^2))) + (8192*tan(c/2 + (d*x)/2)*(3078*a^6*b^6 - 8181*a^8*b^4))
/(27*(a^7 + a^3*b^4 - 2*a^5*b^2))) - (8192*(2592*a^2*b^8 - 11340*a^4*b^6 + 11664*a^6*b^4))/(243*(a^7 + a^3*b^4
- 2*a^5*b^2)) + (8192*tan(c/2 + (d*x)/2)*(1260*a^5*b^5 - 720*a^3*b^7 + 1944*a^7*b^3))/(27*(a^7 + a^3*b^4 - 2*
a^5*b^2))) + (8192*tan(c/2 + (d*x)/2)*(128*b^8 - 688*a^2*b^6 + 1053*a^4*b^4))/(27*(a^7 + a^3*b^4 - 2*a^5*b^2))
) - (8192*tan(c/2 + (d*x)/2)*(32*a*b^5 - 108*a^3*b^3))/(27*(a^7 + a^3*b^4 - 2*a^5*b^2)))*root(1594323*a^14*b^2
*d^6 - 1594323*a^12*b^4*d^6 + 531441*a^10*b^6*d^6 - 531441*a^16*d^6 - 59049*a^10*b^2*d^4 + 59049*a^8*b^4*d^4 -
177147*a^12*d^4 + 8019*a^6*b^2*d^2 - 19683*a^8*d^2 + 432*a^2*b^2 - 64*b^4 - 729*a^4, d, k), k, 1, 6)/d + ((2*
b)/(3*(a^2 - b^2)) + (8*b*tan(c/2 + (d*x)/2)^2)/(3*(a^2 - b^2)) - (2*b*tan(c/2 + (d*x)/2)^4)/(3*(a^2 - b^2)) -
(2*b^2*tan(c/2 + (d*x)/2))/(3*a*(a^2 - b^2)) + (8*b^2*tan(c/2 + (d*x)/2)^3)/(3*a*(a^2 - b^2)) + (2*b^2*tan(c/
2 + (d*x)/2)^5)/(3*a*(a^2 - b^2)))/(d*(a + 3*a*tan(c/2 + (d*x)/2)^2 + 3*a*tan(c/2 + (d*x)/2)^4 + a*tan(c/2 + (
d*x)/2)^6 + 8*b*tan(c/2 + (d*x)/2)^3))

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

[B]   time = 21.2072 (sec), size = 2500 ,normalized size = 96.15 \[ \text {Too large to display} \]

[In]

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

[Out]

symsum(log(5479612416*a^8*b^36 - 180486144*a^6*b^38 - root(5314410*a^16*b^4*d^6 - 5314410*a^14*b^6*d^6 - 26572
05*a^18*b^2*d^6 + 2657205*a^12*b^8*d^6 - 531441*a^10*b^10*d^6 + 531441*a^20*d^6 + 11514555*a^12*b^4*d^4 + 2066
715*a^14*b^2*d^4 + 1062882*a^10*b^6*d^4 - 295245*a^8*b^8*d^4 + 984150*a^8*b^4*d^2 - 98415*a^6*b^6*d^2 + 15625*
a^4*b^4 - 2000*a^2*b^6 + 64*b^8, d, k)*(tan(c/2 + (d*x)/2)*(764411904*a^6*b^40 - 27805483008*a^8*b^38 + 437297
356800*a^10*b^36 - 3672461721600*a^12*b^34 + 19250011791360*a^14*b^32 - 69150635753472*a^16*b^30 + 18016587200
1024*a^18*b^28 - 352655758540800*a^20*b^26 + 529923028377600*a^22*b^24 - 618699706859520*a^24*b^22 + 563713761
042432*a^26*b^20 - 399760062234624*a^28*b^18 + 218398602240000*a^30*b^16 - 90108039168000*a^32*b^14 + 27130620
764160*a^34*b^12 - 5617221156864*a^36*b^10 + 713536708608*a^38*b^8 - 41803776000*a^40*b^6) - root(5314410*a^16
*b^4*d^6 - 5314410*a^14*b^6*d^6 - 2657205*a^18*b^2*d^6 + 2657205*a^12*b^8*d^6 - 531441*a^10*b^10*d^6 + 531441*
a^20*d^6 + 11514555*a^12*b^4*d^4 + 2066715*a^14*b^2*d^4 + 1062882*a^10*b^6*d^4 - 295245*a^8*b^8*d^4 + 984150*a
^8*b^4*d^2 - 98415*a^6*b^6*d^2 + 15625*a^4*b^4 - 2000*a^2*b^6 + 64*b^8, d, k)*(root(5314410*a^16*b^4*d^6 - 531
4410*a^14*b^6*d^6 - 2657205*a^18*b^2*d^6 + 2657205*a^12*b^8*d^6 - 531441*a^10*b^10*d^6 + 531441*a^20*d^6 + 115
14555*a^12*b^4*d^4 + 2066715*a^14*b^2*d^4 + 1062882*a^10*b^6*d^4 - 295245*a^8*b^8*d^4 + 984150*a^8*b^4*d^2 - 9
8415*a^6*b^6*d^2 + 15625*a^4*b^4 - 2000*a^2*b^6 + 64*b^8, d, k)*(tan(c/2 + (d*x)/2)*(157695787008*a^12*b^38 -
4039140556800*a^14*b^36 + 39183049506816*a^16*b^34 - 212750482120704*a^18*b^32 + 750889290203136*a^20*b^30 - 1
854140141887488*a^22*b^28 + 3327952874029056*a^24*b^26 - 4413464400863232*a^26*b^24 + 4311710468702208*a^28*b^
22 - 3009938035433472*a^30*b^20 + 1359808836452352*a^32*b^18 - 238981192998912*a^34*b^16 - 150898421366784*a^3
6*b^14 + 136937506922496*a^38*b^12 - 52028967665664*a^40*b^10 + 10565134000128*a^42*b^8 - 976165945344*a^44*b^
6 + 12093235200*a^46*b^4) - root(5314410*a^16*b^4*d^6 - 5314410*a^14*b^6*d^6 - 2657205*a^18*b^2*d^6 + 2657205*
a^12*b^8*d^6 - 531441*a^10*b^10*d^6 + 531441*a^20*d^6 + 11514555*a^12*b^4*d^4 + 2066715*a^14*b^2*d^4 + 1062882
*a^10*b^6*d^4 - 295245*a^8*b^8*d^4 + 984150*a^8*b^4*d^2 - 98415*a^6*b^6*d^2 + 15625*a^4*b^4 - 2000*a^2*b^6 + 6
4*b^8, d, k)*(tan(c/2 + (d*x)/2)*(69657034752*a^11*b^41 - 1619526057984*a^13*b^39 + 16404231684096*a^15*b^37 -
99052303417344*a^17*b^35 + 405403942256640*a^19*b^33 - 1203882531618816*a^21*b^31 + 2700324609196032*a^23*b^2
9 - 4688893637296128*a^25*b^27 + 6394933732442112*a^27*b^25 - 6897962008903680*a^29*b^23 + 5886924977995776*a^
31*b^21 - 3949971812646912*a^33*b^19 + 2053768012627968*a^35*b^17 - 806001549115392*a^37*b^15 + 22777850363904
0*a^39*b^13 - 42212163059712*a^41*b^11 + 3970450980864*a^43*b^9 + 52242776064*a^45*b^7 - 34828517376*a^47*b^5)
+ 8707129344*a^12*b^40 - 470184984576*a^14*b^38 + 6308315209728*a^16*b^36 - 44092902998016*a^18*b^34 + 197477
693521920*a^20*b^32 - 623151832891392*a^22*b^30 + 1459506434899968*a^24*b^28 - 2616109254180864*a^26*b^26 + 36
53180601827328*a^28*b^24 - 4009284777738240*a^30*b^22 + 3462677318909952*a^32*b^20 - 2339013569937408*a^34*b^1
8 + 1217047711186944*a^36*b^16 - 473946464452608*a^38*b^14 + 130868154040320*a^40*b^12 - 22777850363904*a^42*b
^10 + 1645647446016*a^44*b^8 + 156728328192*a^46*b^6 - 30474952704*a^48*b^4 + root(5314410*a^16*b^4*d^6 - 5314
410*a^14*b^6*d^6 - 2657205*a^18*b^2*d^6 + 2657205*a^12*b^8*d^6 - 531441*a^10*b^10*d^6 + 531441*a^20*d^6 + 1151
4555*a^12*b^4*d^4 + 2066715*a^14*b^2*d^4 + 1062882*a^10*b^6*d^4 - 295245*a^8*b^8*d^4 + 984150*a^8*b^4*d^2 - 98
415*a^6*b^6*d^2 + 15625*a^4*b^4 - 2000*a^2*b^6 + 64*b^8, d, k)*(tan(c/2 + (d*x)/2)*(39182082048*a^14*b^40 - 70
5277476864*a^16*b^38 + 5994858553344*a^18*b^36 - 31972578951168*a^20*b^34 + 119897171066880*a^22*b^32 - 335712
078987264*a^24*b^30 + 727376171139072*a^26*b^28 - 1246930579095552*a^28*b^26 + 1714529546256384*a^30*b^24 - 19
05032829173760*a^32*b^22 + 1714529546256384*a^34*b^20 - 1246930579095552*a^36*b^18 + 727376171139072*a^38*b^16
- 335712078987264*a^40*b^14 + 119897171066880*a^42*b^12 - 31972578951168*a^44*b^10 + 5994858553344*a^46*b^8 -
705277476864*a^48*b^6 + 39182082048*a^50*b^4) + 156728328192*a^13*b^41 - 2938656153600*a^15*b^39 + 2609526664
3968*a^17*b^37 - 145874891464704*a^19*b^35 + 575506421121024*a^21*b^33 - 1702539829149696*a^23*b^31 + 39166409
21518080*a^25*b^29 - 7169850829799424*a^27*b^27 + 10598909922312192*a^29*b^25 - 12763719955464192*a^31*b^23 +
12573216672546816*a^33*b^21 - 10131310955151360*a^35*b^19 + 6650296421842944*a^37*b^17 - 3524976829366272*a^39
*b^15 + 1486724921229312*a^41*b^13 - 487581829005312*a^43*b^11 + 119897171066880*a^45*b^9 - 20805685567488*a^4
7*b^7 + 2272560758784*a^49*b^5 - 117546246144*a^51*b^3)) - 59982446592*a^11*b^39 + 1080651497472*a^13*b^37 - 6
860250464256*a^15*b^35 + 16482112118784*a^17*b^33 + 27170113388544*a^19*b^31 - 327284061511680*a^21*b^29 + 119
4949984370688*a^23*b^27 - 2698934854606848*a^25...

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

[B]   time = 25.4368 (sec), size = 2500 ,normalized size = 96.15 \[ \text {Too large to display} \]

[In]

int(1/(cos(c + d*x)^4*(a + b*sin(c + d*x)^3)^2),x)

[Out]

symsum(log(26838024192*a^8*b^54 - tan(c/2 + (d*x)/2)*(7962624000*a^7*b^55 - 508612608000*a^9*b^53 + 8841498624
000*a^11*b^51 - 82283765760000*a^13*b^49 + 501714984960000*a^15*b^47 - 2205295497216000*a^17*b^45 + 7379181637
632000*a^19*b^43 - 19451488075776000*a^21*b^41 + 41318016122880000*a^23*b^39 - 71811432161280000*a^25*b^37 + 1
03155513237504000*a^27*b^35 - 123224906907648000*a^29*b^33 + 122756816093184000*a^31*b^31 - 101967282708480000
*a^33*b^29 + 70396872007680000*a^35*b^27 - 40129785593856000*a^37*b^25 + 18687625592832000*a^39*b^23 - 6994754
113536000*a^41*b^21 + 2053854351360000*a^43*b^19 - 455730831360000*a^45*b^17 + 71860690944000*a^47*b^15 - 7177
310208000*a^49*b^13 + 341397504000*a^51*b^11) - 392822784*a^6*b^56 - root(18600435*a^18*b^6*d^6 - 18600435*a^1
6*b^8*d^6 - 11160261*a^20*b^4*d^6 + 11160261*a^14*b^10*d^6 + 3720087*a^22*b^2*d^6 - 3720087*a^12*b^12*d^6 + 53
1441*a^10*b^14*d^6 - 531441*a^24*d^6 - 173879622*a^14*b^6*d^4 - 155830311*a^12*b^8*d^4 - 23225940*a^16*b^4*d^4
- 6475707*a^10*b^10*d^4 + 688905*a^8*b^12*d^4 - 11565585*a^8*b^8*d^2 + 3750705*a^10*b^6*d^2 + 433755*a^6*b^10
*d^2 - 117649*a^4*b^8 + 5488*a^2*b^10 - 64*b^12, d, k)*(tan(c/2 + (d*x)/2)*(764411904*a^6*b^58 - 61439606784*a
^8*b^56 + 2110475575296*a^10*b^54 - 33643637121024*a^12*b^52 + 319697763065856*a^14*b^50 - 2067381036048384*a^
16*b^48 + 9810082122817536*a^18*b^46 - 35797302942326784*a^20*b^44 + 103613766013034496*a^22*b^42 - 2430046994
98881024*a^24*b^40 + 468678655511248896*a^26*b^38 - 750973819695611904*a^28*b^36 + 1006348379003928576*a^30*b^
34 - 1132028278205497344*a^32*b^32 + 1070100496146087936*a^34*b^30 - 848821864657895424*a^36*b^28 + 5626355927
01198336*a^38*b^26 - 309384400894377984*a^40*b^24 + 139566181489975296*a^42*b^22 - 50807786761396224*a^44*b^20
+ 14569217952178176*a^46*b^18 - 3172130021597184*a^48*b^16 + 494158536400896*a^50*b^14 - 49418889191424*a^52*
b^12 + 2463538323456*a^54*b^10 - 14338695168*a^56*b^8) + 95551488*a^7*b^57 + 35879583744*a^9*b^55 - 1812522147
840*a^11*b^53 + 29896430247936*a^13*b^51 - 273690491977728*a^15*b^49 + 1665068560662528*a^17*b^47 - 7358934856
605696*a^19*b^45 + 24887080515133440*a^21*b^43 - 66575487905316864*a^23*b^41 + 144045035942510592*a^25*b^39 -
255939373888192512*a^27*b^37 + 377317716543258624*a^29*b^35 - 464589495171809280*a^31*b^33 + 47947008416012697
6*a^33*b^31 - 415092174607761408*a^35*b^29 + 300910589340991488*a^37*b^27 - 181823043267035136*a^39*b^25 + 908
63416678809600*a^41*b^23 - 37111903240495104*a^43*b^21 + 12175612162301952*a^45*b^19 - 3127996467412992*a^47*b
^17 + 605418993598464*a^49*b^15 - 82897275985920*a^51*b^13 + 7145262637056*a^53*b^11 - 290870673408*a^55*b^9 +
root(18600435*a^18*b^6*d^6 - 18600435*a^16*b^8*d^6 - 11160261*a^20*b^4*d^6 + 11160261*a^14*b^10*d^6 + 3720087
*a^22*b^2*d^6 - 3720087*a^12*b^12*d^6 + 531441*a^10*b^14*d^6 - 531441*a^24*d^6 - 173879622*a^14*b^6*d^4 - 1558
30311*a^12*b^8*d^4 - 23225940*a^16*b^4*d^4 - 6475707*a^10*b^10*d^4 + 688905*a^8*b^12*d^4 - 11565585*a^8*b^8*d^
2 + 3750705*a^10*b^6*d^2 + 433755*a^6*b^10*d^2 - 117649*a^4*b^8 + 5488*a^2*b^10 - 64*b^12, d, k)*(tan(c/2 + (d
*x)/2)*(45578059776*a^9*b^57 - 1988020371456*a^11*b^55 + 21725255172096*a^13*b^53 - 78629462802432*a^15*b^51 -
330769869373440*a^17*b^49 + 5337288405614592*a^19*b^47 - 32144913894998016*a^21*b^45 + 126404118900965376*a^2
3*b^43 - 367050326151462912*a^25*b^41 + 829818883454238720*a^27*b^39 - 1502808604998893568*a^29*b^37 + 2216700
870917750784*a^31*b^35 - 2688523449382600704*a^33*b^33 + 2692902186903011328*a^35*b^31 - 2227622993351147520*a
^37*b^29 + 1515332894269243392*a^39*b^27 - 839694861496221696*a^41*b^25 + 372789943915216896*a^43*b^23 - 12885
4679612424192*a^45*b^21 + 32863270985072640*a^47*b^19 - 5445156193763328*a^49*b^17 + 316457498640384*a^51*b^15
+ 91463986446336*a^53*b^13 - 25165538721792*a^55*b^11 + 2461645209600*a^57*b^9 - 73741860864*a^59*b^7) + root
(18600435*a^18*b^6*d^6 - 18600435*a^16*b^8*d^6 - 11160261*a^20*b^4*d^6 + 11160261*a^14*b^10*d^6 + 3720087*a^22
*b^2*d^6 - 3720087*a^12*b^12*d^6 + 531441*a^10*b^14*d^6 - 531441*a^24*d^6 - 173879622*a^14*b^6*d^4 - 155830311
*a^12*b^8*d^4 - 23225940*a^16*b^4*d^4 - 6475707*a^10*b^10*d^4 + 688905*a^8*b^12*d^4 - 11565585*a^8*b^8*d^2 + 3
750705*a^10*b^6*d^2 + 433755*a^6*b^10*d^2 - 117649*a^4*b^8 + 5488*a^2*b^10 - 64*b^12, d, k)*(root(18600435*a^1
8*b^6*d^6 - 18600435*a^16*b^8*d^6 - 11160261*a^20*b^4*d^6 + 11160261*a^14*b^10*d^6 + 3720087*a^22*b^2*d^6 - 37
20087*a^12*b^12*d^6 + 531441*a^10*b^14*d^6 - 531441*a^24*d^6 - 173879622*a^14*b^6*d^4 - 155830311*a^12*b^8*d^4
- 23225940*a^16*b^4*d^4 - 6475707*a^10*b^10*d^4 + 688905*a^8*b^12*d^4 - 11565585*a^8*b^8*d^2 + 3750705*a^10*b
^6*d^2 + 433755*a^6*b^10*d^2 - 117649*a^4*b^8 + 5488*a^2*b^10 - 64*b^12, d, k)*(tan(c/2 + (d*x)/2)*(6965703475
2*a^11*b^59 - 2855938424832*a^13*b^57 + 46200028299264*a^15*b^55 - 432918470983680*a^17*b^53 + 273299375849472
0*a^19*b^51 - 12560556506480640*a^21*b^49 + 43925900257198080*a^23*b^47 - 119837962587340800*a^25*b^45 + 25765
1619562782720*a^27*b^43 - 433619569038458880*a^...