\(\int \frac {x^5}{(a+b \sin (c+d x^2))^2} \, dx\) [43]

Optimal result

Integrand size = 18, antiderivative size = 663 \[ \int \frac {x^5}{\left (a+b \sin \left (c+d x^2\right )\right )^2} \, dx=\frac {i x^4}{2 \left (a^2-b^2\right ) d}-\frac {x^2 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^2}-\frac {i a x^4 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{2 \left (a^2-b^2\right )^{3/2} d}-\frac {x^2 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^2}+\frac {i a x^4 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a+\sqrt {a^2-b^2}}\right )}{2 \left (a^2-b^2\right )^{3/2} d}+\frac {i \operatorname {PolyLog}\left (2,\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^3}-\frac {a x^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d^2}+\frac {i \operatorname {PolyLog}\left (2,\frac {i b e^{i \left (c+d x^2\right )}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^3}+\frac {a x^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i \left (c+d x^2\right )}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d^2}-\frac {i a \operatorname {PolyLog}\left (3,\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d^3}+\frac {i a \operatorname {PolyLog}\left (3,\frac {i b e^{i \left (c+d x^2\right )}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d^3}+\frac {b x^4 \cos \left (c+d x^2\right )}{2 \left (a^2-b^2\right ) d \left (a+b \sin \left (c+d x^2\right )\right )} \] Output:

1/2*I*x^4/(a^2-b^2)/d-x^2*ln(1-I*b*exp(I*(d*x^2+c))/(a-(a^2-b^2)^(1/2)))/( 
a^2-b^2)/d^2-1/2*I*a*x^4*ln(1-I*b*exp(I*(d*x^2+c))/(a-(a^2-b^2)^(1/2)))/(a 
^2-b^2)^(3/2)/d-x^2*ln(1-I*b*exp(I*(d*x^2+c))/(a+(a^2-b^2)^(1/2)))/(a^2-b^ 
2)/d^2+1/2*I*a*x^4*ln(1-I*b*exp(I*(d*x^2+c))/(a+(a^2-b^2)^(1/2)))/(a^2-b^2 
)^(3/2)/d+I*polylog(2,I*b*exp(I*(d*x^2+c))/(a-(a^2-b^2)^(1/2)))/(a^2-b^2)/ 
d^3-a*x^2*polylog(2,I*b*exp(I*(d*x^2+c))/(a-(a^2-b^2)^(1/2)))/(a^2-b^2)^(3 
/2)/d^2+I*polylog(2,I*b*exp(I*(d*x^2+c))/(a+(a^2-b^2)^(1/2)))/(a^2-b^2)/d^ 
3+a*x^2*polylog(2,I*b*exp(I*(d*x^2+c))/(a+(a^2-b^2)^(1/2)))/(a^2-b^2)^(3/2 
)/d^2-I*a*polylog(3,I*b*exp(I*(d*x^2+c))/(a-(a^2-b^2)^(1/2)))/(a^2-b^2)^(3 
/2)/d^3+I*a*polylog(3,I*b*exp(I*(d*x^2+c))/(a+(a^2-b^2)^(1/2)))/(a^2-b^2)^ 
(3/2)/d^3+1/2*b*x^4*cos(d*x^2+c)/(a^2-b^2)/d/(a+b*sin(d*x^2+c))
 

Mathematica [A] (verified)

Time = 2.85 (sec) , antiderivative size = 513, normalized size of antiderivative = 0.77 \[ \int \frac {x^5}{\left (a+b \sin \left (c+d x^2\right )\right )^2} \, dx=\frac {i d^2 x^4-2 d x^2 \log \left (1+\frac {i b e^{i \left (c+d x^2\right )}}{-a+\sqrt {a^2-b^2}}\right )-\frac {i a d^2 x^4 \log \left (1+\frac {i b e^{i \left (c+d x^2\right )}}{-a+\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}-2 d x^2 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a+\sqrt {a^2-b^2}}\right )+\frac {i a d^2 x^4 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a+\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}+\left (2 i-\frac {2 a d x^2}{\sqrt {a^2-b^2}}\right ) \operatorname {PolyLog}\left (2,-\frac {i b e^{i \left (c+d x^2\right )}}{-a+\sqrt {a^2-b^2}}\right )+\left (2 i+\frac {2 a d x^2}{\sqrt {a^2-b^2}}\right ) \operatorname {PolyLog}\left (2,\frac {i b e^{i \left (c+d x^2\right )}}{a+\sqrt {a^2-b^2}}\right )-\frac {2 i a \operatorname {PolyLog}\left (3,\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}+\frac {2 i a \operatorname {PolyLog}\left (3,\frac {i b e^{i \left (c+d x^2\right )}}{a+\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}+\frac {b d^2 x^4 \cos \left (c+d x^2\right )}{a+b \sin \left (c+d x^2\right )}}{2 \left (a^2-b^2\right ) d^3} \] Input:

Integrate[x^5/(a + b*Sin[c + d*x^2])^2,x]
 

Output:

(I*d^2*x^4 - 2*d*x^2*Log[1 + (I*b*E^(I*(c + d*x^2)))/(-a + Sqrt[a^2 - b^2] 
)] - (I*a*d^2*x^4*Log[1 + (I*b*E^(I*(c + d*x^2)))/(-a + Sqrt[a^2 - b^2])]) 
/Sqrt[a^2 - b^2] - 2*d*x^2*Log[1 - (I*b*E^(I*(c + d*x^2)))/(a + Sqrt[a^2 - 
 b^2])] + (I*a*d^2*x^4*Log[1 - (I*b*E^(I*(c + d*x^2)))/(a + Sqrt[a^2 - b^2 
])])/Sqrt[a^2 - b^2] + (2*I - (2*a*d*x^2)/Sqrt[a^2 - b^2])*PolyLog[2, ((-I 
)*b*E^(I*(c + d*x^2)))/(-a + Sqrt[a^2 - b^2])] + (2*I + (2*a*d*x^2)/Sqrt[a 
^2 - b^2])*PolyLog[2, (I*b*E^(I*(c + d*x^2)))/(a + Sqrt[a^2 - b^2])] - ((2 
*I)*a*PolyLog[3, (I*b*E^(I*(c + d*x^2)))/(a - Sqrt[a^2 - b^2])])/Sqrt[a^2 
- b^2] + ((2*I)*a*PolyLog[3, (I*b*E^(I*(c + d*x^2)))/(a + Sqrt[a^2 - b^2]) 
])/Sqrt[a^2 - b^2] + (b*d^2*x^4*Cos[c + d*x^2])/(a + b*Sin[c + d*x^2]))/(2 
*(a^2 - b^2)*d^3)
 

Rubi [A] (verified)

Time = 2.83 (sec) , antiderivative size = 625, normalized size of antiderivative = 0.94, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {3860, 3042, 3805, 3042, 3804, 2694, 27, 2620, 3011, 2720, 5030, 2620, 2715, 2838, 7143}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {x^5}{\left (a+b \sin \left (c+d x^2\right )\right )^2} \, dx\)

\(\Big \downarrow \) 3860

\(\displaystyle \frac {1}{2} \int \frac {x^4}{\left (a+b \sin \left (d x^2+c\right )\right )^2}dx^2\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} \int \frac {x^4}{\left (a+b \sin \left (d x^2+c\right )\right )^2}dx^2\)

\(\Big \downarrow \) 3805

\(\displaystyle \frac {1}{2} \left (-\frac {2 b \int \frac {x^2 \cos \left (d x^2+c\right )}{a+b \sin \left (d x^2+c\right )}dx^2}{d \left (a^2-b^2\right )}+\frac {a \int \frac {x^4}{a+b \sin \left (d x^2+c\right )}dx^2}{a^2-b^2}+\frac {b x^4 \cos \left (c+d x^2\right )}{d \left (a^2-b^2\right ) \left (a+b \sin \left (c+d x^2\right )\right )}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} \left (-\frac {2 b \int \frac {x^2 \cos \left (d x^2+c\right )}{a+b \sin \left (d x^2+c\right )}dx^2}{d \left (a^2-b^2\right )}+\frac {a \int \frac {x^4}{a+b \sin \left (d x^2+c\right )}dx^2}{a^2-b^2}+\frac {b x^4 \cos \left (c+d x^2\right )}{d \left (a^2-b^2\right ) \left (a+b \sin \left (c+d x^2\right )\right )}\right )\)

\(\Big \downarrow \) 3804

\(\displaystyle \frac {1}{2} \left (-\frac {2 b \int \frac {x^2 \cos \left (d x^2+c\right )}{a+b \sin \left (d x^2+c\right )}dx^2}{d \left (a^2-b^2\right )}+\frac {2 a \int \frac {e^{i \left (d x^2+c\right )} x^4}{2 e^{i \left (d x^2+c\right )} a-i b e^{2 i \left (d x^2+c\right )}+i b}dx^2}{a^2-b^2}+\frac {b x^4 \cos \left (c+d x^2\right )}{d \left (a^2-b^2\right ) \left (a+b \sin \left (c+d x^2\right )\right )}\right )\)

\(\Big \downarrow \) 2694

\(\displaystyle \frac {1}{2} \left (-\frac {2 b \int \frac {x^2 \cos \left (d x^2+c\right )}{a+b \sin \left (d x^2+c\right )}dx^2}{d \left (a^2-b^2\right )}+\frac {2 a \left (\frac {i b \int \frac {e^{i \left (d x^2+c\right )} x^4}{2 \left (a-i b e^{i \left (d x^2+c\right )}+\sqrt {a^2-b^2}\right )}dx^2}{\sqrt {a^2-b^2}}-\frac {i b \int \frac {e^{i \left (d x^2+c\right )} x^4}{2 \left (a-i b e^{i \left (d x^2+c\right )}-\sqrt {a^2-b^2}\right )}dx^2}{\sqrt {a^2-b^2}}\right )}{a^2-b^2}+\frac {b x^4 \cos \left (c+d x^2\right )}{d \left (a^2-b^2\right ) \left (a+b \sin \left (c+d x^2\right )\right )}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (-\frac {2 b \int \frac {x^2 \cos \left (d x^2+c\right )}{a+b \sin \left (d x^2+c\right )}dx^2}{d \left (a^2-b^2\right )}+\frac {2 a \left (\frac {i b \int \frac {e^{i \left (d x^2+c\right )} x^4}{a-i b e^{i \left (d x^2+c\right )}+\sqrt {a^2-b^2}}dx^2}{2 \sqrt {a^2-b^2}}-\frac {i b \int \frac {e^{i \left (d x^2+c\right )} x^4}{a-i b e^{i \left (d x^2+c\right )}-\sqrt {a^2-b^2}}dx^2}{2 \sqrt {a^2-b^2}}\right )}{a^2-b^2}+\frac {b x^4 \cos \left (c+d x^2\right )}{d \left (a^2-b^2\right ) \left (a+b \sin \left (c+d x^2\right )\right )}\right )\)

\(\Big \downarrow \) 2620

\(\displaystyle \frac {1}{2} \left (-\frac {2 b \int \frac {x^2 \cos \left (d x^2+c\right )}{a+b \sin \left (d x^2+c\right )}dx^2}{d \left (a^2-b^2\right )}+\frac {2 a \left (\frac {i b \left (\frac {x^4 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{\sqrt {a^2-b^2}+a}\right )}{b d}-\frac {2 \int x^2 \log \left (1-\frac {i b e^{i \left (d x^2+c\right )}}{a+\sqrt {a^2-b^2}}\right )dx^2}{b d}\right )}{2 \sqrt {a^2-b^2}}-\frac {i b \left (\frac {x^4 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{b d}-\frac {2 \int x^2 \log \left (1-\frac {i b e^{i \left (d x^2+c\right )}}{a-\sqrt {a^2-b^2}}\right )dx^2}{b d}\right )}{2 \sqrt {a^2-b^2}}\right )}{a^2-b^2}+\frac {b x^4 \cos \left (c+d x^2\right )}{d \left (a^2-b^2\right ) \left (a+b \sin \left (c+d x^2\right )\right )}\right )\)

\(\Big \downarrow \) 3011

\(\displaystyle \frac {1}{2} \left (\frac {2 a \left (\frac {i b \left (\frac {x^4 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{\sqrt {a^2-b^2}+a}\right )}{b d}-\frac {2 \left (\frac {i x^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i \left (d x^2+c\right )}}{a+\sqrt {a^2-b^2}}\right )}{d}-\frac {i \int \operatorname {PolyLog}\left (2,\frac {i b e^{i \left (d x^2+c\right )}}{a+\sqrt {a^2-b^2}}\right )dx^2}{d}\right )}{b d}\right )}{2 \sqrt {a^2-b^2}}-\frac {i b \left (\frac {x^4 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{b d}-\frac {2 \left (\frac {i x^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i \left (d x^2+c\right )}}{a-\sqrt {a^2-b^2}}\right )}{d}-\frac {i \int \operatorname {PolyLog}\left (2,\frac {i b e^{i \left (d x^2+c\right )}}{a-\sqrt {a^2-b^2}}\right )dx^2}{d}\right )}{b d}\right )}{2 \sqrt {a^2-b^2}}\right )}{a^2-b^2}-\frac {2 b \int \frac {x^2 \cos \left (d x^2+c\right )}{a+b \sin \left (d x^2+c\right )}dx^2}{d \left (a^2-b^2\right )}+\frac {b x^4 \cos \left (c+d x^2\right )}{d \left (a^2-b^2\right ) \left (a+b \sin \left (c+d x^2\right )\right )}\right )\)

\(\Big \downarrow \) 2720

\(\displaystyle \frac {1}{2} \left (\frac {2 a \left (\frac {i b \left (\frac {x^4 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{\sqrt {a^2-b^2}+a}\right )}{b d}-\frac {2 \left (\frac {i x^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i \left (d x^2+c\right )}}{a+\sqrt {a^2-b^2}}\right )}{d}-\frac {\int \frac {\operatorname {PolyLog}\left (2,\frac {i b e^{i \left (d x^2+c\right )}}{a+\sqrt {a^2-b^2}}\right )}{x^2}de^{i \left (d x^2+c\right )}}{d^2}\right )}{b d}\right )}{2 \sqrt {a^2-b^2}}-\frac {i b \left (\frac {x^4 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{b d}-\frac {2 \left (\frac {i x^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i \left (d x^2+c\right )}}{a-\sqrt {a^2-b^2}}\right )}{d}-\frac {\int \frac {\operatorname {PolyLog}\left (2,\frac {i b e^{i \left (d x^2+c\right )}}{a-\sqrt {a^2-b^2}}\right )}{x^2}de^{i \left (d x^2+c\right )}}{d^2}\right )}{b d}\right )}{2 \sqrt {a^2-b^2}}\right )}{a^2-b^2}-\frac {2 b \int \frac {x^2 \cos \left (d x^2+c\right )}{a+b \sin \left (d x^2+c\right )}dx^2}{d \left (a^2-b^2\right )}+\frac {b x^4 \cos \left (c+d x^2\right )}{d \left (a^2-b^2\right ) \left (a+b \sin \left (c+d x^2\right )\right )}\right )\)

\(\Big \downarrow \) 5030

\(\displaystyle \frac {1}{2} \left (\frac {2 a \left (\frac {i b \left (\frac {x^4 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{\sqrt {a^2-b^2}+a}\right )}{b d}-\frac {2 \left (\frac {i x^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i \left (d x^2+c\right )}}{a+\sqrt {a^2-b^2}}\right )}{d}-\frac {\int \frac {\operatorname {PolyLog}\left (2,\frac {i b e^{i \left (d x^2+c\right )}}{a+\sqrt {a^2-b^2}}\right )}{x^2}de^{i \left (d x^2+c\right )}}{d^2}\right )}{b d}\right )}{2 \sqrt {a^2-b^2}}-\frac {i b \left (\frac {x^4 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{b d}-\frac {2 \left (\frac {i x^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i \left (d x^2+c\right )}}{a-\sqrt {a^2-b^2}}\right )}{d}-\frac {\int \frac {\operatorname {PolyLog}\left (2,\frac {i b e^{i \left (d x^2+c\right )}}{a-\sqrt {a^2-b^2}}\right )}{x^2}de^{i \left (d x^2+c\right )}}{d^2}\right )}{b d}\right )}{2 \sqrt {a^2-b^2}}\right )}{a^2-b^2}-\frac {2 b \left (\int \frac {e^{i \left (d x^2+c\right )} x^2}{a-i b e^{i \left (d x^2+c\right )}-\sqrt {a^2-b^2}}dx^2+\int \frac {e^{i \left (d x^2+c\right )} x^2}{a-i b e^{i \left (d x^2+c\right )}+\sqrt {a^2-b^2}}dx^2-\frac {i x^4}{2 b}\right )}{d \left (a^2-b^2\right )}+\frac {b x^4 \cos \left (c+d x^2\right )}{d \left (a^2-b^2\right ) \left (a+b \sin \left (c+d x^2\right )\right )}\right )\)

\(\Big \downarrow \) 2620

\(\displaystyle \frac {1}{2} \left (\frac {2 a \left (\frac {i b \left (\frac {x^4 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{\sqrt {a^2-b^2}+a}\right )}{b d}-\frac {2 \left (\frac {i x^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i \left (d x^2+c\right )}}{a+\sqrt {a^2-b^2}}\right )}{d}-\frac {\int \frac {\operatorname {PolyLog}\left (2,\frac {i b e^{i \left (d x^2+c\right )}}{a+\sqrt {a^2-b^2}}\right )}{x^2}de^{i \left (d x^2+c\right )}}{d^2}\right )}{b d}\right )}{2 \sqrt {a^2-b^2}}-\frac {i b \left (\frac {x^4 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{b d}-\frac {2 \left (\frac {i x^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i \left (d x^2+c\right )}}{a-\sqrt {a^2-b^2}}\right )}{d}-\frac {\int \frac {\operatorname {PolyLog}\left (2,\frac {i b e^{i \left (d x^2+c\right )}}{a-\sqrt {a^2-b^2}}\right )}{x^2}de^{i \left (d x^2+c\right )}}{d^2}\right )}{b d}\right )}{2 \sqrt {a^2-b^2}}\right )}{a^2-b^2}-\frac {2 b \left (-\frac {\int \log \left (1-\frac {i b e^{i \left (d x^2+c\right )}}{a-\sqrt {a^2-b^2}}\right )dx^2}{b d}-\frac {\int \log \left (1-\frac {i b e^{i \left (d x^2+c\right )}}{a+\sqrt {a^2-b^2}}\right )dx^2}{b d}+\frac {x^2 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{b d}+\frac {x^2 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{\sqrt {a^2-b^2}+a}\right )}{b d}-\frac {i x^4}{2 b}\right )}{d \left (a^2-b^2\right )}+\frac {b x^4 \cos \left (c+d x^2\right )}{d \left (a^2-b^2\right ) \left (a+b \sin \left (c+d x^2\right )\right )}\right )\)

\(\Big \downarrow \) 2715

\(\displaystyle \frac {1}{2} \left (\frac {2 a \left (\frac {i b \left (\frac {x^4 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{\sqrt {a^2-b^2}+a}\right )}{b d}-\frac {2 \left (\frac {i x^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i \left (d x^2+c\right )}}{a+\sqrt {a^2-b^2}}\right )}{d}-\frac {\int \frac {\operatorname {PolyLog}\left (2,\frac {i b e^{i \left (d x^2+c\right )}}{a+\sqrt {a^2-b^2}}\right )}{x^2}de^{i \left (d x^2+c\right )}}{d^2}\right )}{b d}\right )}{2 \sqrt {a^2-b^2}}-\frac {i b \left (\frac {x^4 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{b d}-\frac {2 \left (\frac {i x^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i \left (d x^2+c\right )}}{a-\sqrt {a^2-b^2}}\right )}{d}-\frac {\int \frac {\operatorname {PolyLog}\left (2,\frac {i b e^{i \left (d x^2+c\right )}}{a-\sqrt {a^2-b^2}}\right )}{x^2}de^{i \left (d x^2+c\right )}}{d^2}\right )}{b d}\right )}{2 \sqrt {a^2-b^2}}\right )}{a^2-b^2}-\frac {2 b \left (\frac {i \int \frac {\log \left (1-\frac {i b e^{i \left (d x^2+c\right )}}{a-\sqrt {a^2-b^2}}\right )}{x^2}de^{i \left (d x^2+c\right )}}{b d^2}+\frac {i \int \frac {\log \left (1-\frac {i b e^{i \left (d x^2+c\right )}}{a+\sqrt {a^2-b^2}}\right )}{x^2}de^{i \left (d x^2+c\right )}}{b d^2}+\frac {x^2 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{b d}+\frac {x^2 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{\sqrt {a^2-b^2}+a}\right )}{b d}-\frac {i x^4}{2 b}\right )}{d \left (a^2-b^2\right )}+\frac {b x^4 \cos \left (c+d x^2\right )}{d \left (a^2-b^2\right ) \left (a+b \sin \left (c+d x^2\right )\right )}\right )\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {1}{2} \left (\frac {2 a \left (\frac {i b \left (\frac {x^4 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{\sqrt {a^2-b^2}+a}\right )}{b d}-\frac {2 \left (\frac {i x^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i \left (d x^2+c\right )}}{a+\sqrt {a^2-b^2}}\right )}{d}-\frac {\int \frac {\operatorname {PolyLog}\left (2,\frac {i b e^{i \left (d x^2+c\right )}}{a+\sqrt {a^2-b^2}}\right )}{x^2}de^{i \left (d x^2+c\right )}}{d^2}\right )}{b d}\right )}{2 \sqrt {a^2-b^2}}-\frac {i b \left (\frac {x^4 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{b d}-\frac {2 \left (\frac {i x^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i \left (d x^2+c\right )}}{a-\sqrt {a^2-b^2}}\right )}{d}-\frac {\int \frac {\operatorname {PolyLog}\left (2,\frac {i b e^{i \left (d x^2+c\right )}}{a-\sqrt {a^2-b^2}}\right )}{x^2}de^{i \left (d x^2+c\right )}}{d^2}\right )}{b d}\right )}{2 \sqrt {a^2-b^2}}\right )}{a^2-b^2}-\frac {2 b \left (-\frac {i \operatorname {PolyLog}\left (2,\frac {i b e^{i \left (d x^2+c\right )}}{a-\sqrt {a^2-b^2}}\right )}{b d^2}-\frac {i \operatorname {PolyLog}\left (2,\frac {i b e^{i \left (d x^2+c\right )}}{a+\sqrt {a^2-b^2}}\right )}{b d^2}+\frac {x^2 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{b d}+\frac {x^2 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{\sqrt {a^2-b^2}+a}\right )}{b d}-\frac {i x^4}{2 b}\right )}{d \left (a^2-b^2\right )}+\frac {b x^4 \cos \left (c+d x^2\right )}{d \left (a^2-b^2\right ) \left (a+b \sin \left (c+d x^2\right )\right )}\right )\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {1}{2} \left (-\frac {2 b \left (-\frac {i \operatorname {PolyLog}\left (2,\frac {i b e^{i \left (d x^2+c\right )}}{a-\sqrt {a^2-b^2}}\right )}{b d^2}-\frac {i \operatorname {PolyLog}\left (2,\frac {i b e^{i \left (d x^2+c\right )}}{a+\sqrt {a^2-b^2}}\right )}{b d^2}+\frac {x^2 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{b d}+\frac {x^2 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{\sqrt {a^2-b^2}+a}\right )}{b d}-\frac {i x^4}{2 b}\right )}{d \left (a^2-b^2\right )}+\frac {2 a \left (\frac {i b \left (\frac {x^4 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{\sqrt {a^2-b^2}+a}\right )}{b d}-\frac {2 \left (\frac {i x^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i \left (d x^2+c\right )}}{a+\sqrt {a^2-b^2}}\right )}{d}-\frac {\operatorname {PolyLog}\left (3,\frac {i b e^{i \left (d x^2+c\right )}}{a+\sqrt {a^2-b^2}}\right )}{d^2}\right )}{b d}\right )}{2 \sqrt {a^2-b^2}}-\frac {i b \left (\frac {x^4 \log \left (1-\frac {i b e^{i \left (c+d x^2\right )}}{a-\sqrt {a^2-b^2}}\right )}{b d}-\frac {2 \left (\frac {i x^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i \left (d x^2+c\right )}}{a-\sqrt {a^2-b^2}}\right )}{d}-\frac {\operatorname {PolyLog}\left (3,\frac {i b e^{i \left (d x^2+c\right )}}{a-\sqrt {a^2-b^2}}\right )}{d^2}\right )}{b d}\right )}{2 \sqrt {a^2-b^2}}\right )}{a^2-b^2}+\frac {b x^4 \cos \left (c+d x^2\right )}{d \left (a^2-b^2\right ) \left (a+b \sin \left (c+d x^2\right )\right )}\right )\)

Input:

Int[x^5/(a + b*Sin[c + d*x^2])^2,x]
 

Output:

((-2*b*(((-1/2*I)*x^4)/b + (x^2*Log[1 - (I*b*E^(I*(c + d*x^2)))/(a - Sqrt[ 
a^2 - b^2])])/(b*d) + (x^2*Log[1 - (I*b*E^(I*(c + d*x^2)))/(a + Sqrt[a^2 - 
 b^2])])/(b*d) - (I*PolyLog[2, (I*b*E^(I*(c + d*x^2)))/(a - Sqrt[a^2 - b^2 
])])/(b*d^2) - (I*PolyLog[2, (I*b*E^(I*(c + d*x^2)))/(a + Sqrt[a^2 - b^2]) 
])/(b*d^2)))/((a^2 - b^2)*d) + (2*a*(((-1/2*I)*b*((x^4*Log[1 - (I*b*E^(I*( 
c + d*x^2)))/(a - Sqrt[a^2 - b^2])])/(b*d) - (2*((I*x^2*PolyLog[2, (I*b*E^ 
(I*(c + d*x^2)))/(a - Sqrt[a^2 - b^2])])/d - PolyLog[3, (I*b*E^(I*(c + d*x 
^2)))/(a - Sqrt[a^2 - b^2])]/d^2))/(b*d)))/Sqrt[a^2 - b^2] + ((I/2)*b*((x^ 
4*Log[1 - (I*b*E^(I*(c + d*x^2)))/(a + Sqrt[a^2 - b^2])])/(b*d) - (2*((I*x 
^2*PolyLog[2, (I*b*E^(I*(c + d*x^2)))/(a + Sqrt[a^2 - b^2])])/d - PolyLog[ 
3, (I*b*E^(I*(c + d*x^2)))/(a + Sqrt[a^2 - b^2])]/d^2))/(b*d)))/Sqrt[a^2 - 
 b^2]))/(a^2 - b^2) + (b*x^4*Cos[c + d*x^2])/((a^2 - b^2)*d*(a + b*Sin[c + 
 d*x^2])))/2
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2620

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

rule 2694

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.) 
*(F_)^(v_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[2*(c/q)   Int 
[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Simp[2*(c/q)   Int[(f + g*x) 
^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[ 
v, 2*u] && LinearQ[u, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[m, 0]
 

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 3011

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) 
*(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + 
b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F]))   Int[(f + g*x)^( 
m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e 
, f, g, n}, x] && GtQ[m, 0]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3804

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Sy 
mbol] :> Simp[2   Int[(c + d*x)^m*(E^(I*(e + f*x))/(I*b + 2*a*E^(I*(e + f*x 
)) - I*b*E^(2*I*(e + f*x)))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ 
[a^2 - b^2, 0] && IGtQ[m, 0]
 

rule 3805

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^2, x_ 
Symbol] :> Simp[b*(c + d*x)^m*(Cos[e + f*x]/(f*(a^2 - b^2)*(a + b*Sin[e + f 
*x]))), x] + (Simp[a/(a^2 - b^2)   Int[(c + d*x)^m/(a + b*Sin[e + f*x]), x] 
, x] - Simp[b*d*(m/(f*(a^2 - b^2)))   Int[(c + d*x)^(m - 1)*(Cos[e + f*x]/( 
a + b*Sin[e + f*x])), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 - 
b^2, 0] && IGtQ[m, 0]
 

rule 3860

Int[(x_)^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol 
] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Sin[c + d*x])^ 
p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Simplify[ 
(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[ 
(m + 1)/n], 0]))
 

rule 5030

Int[(Cos[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin[ 
(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[(-I)*((e + f*x)^(m + 1)/(b*f*(m + 1 
))), x] + (Int[(e + f*x)^m*(E^(I*(c + d*x))/(a - Rt[a^2 - b^2, 2] - I*b*E^( 
I*(c + d*x)))), x] + Int[(e + f*x)^m*(E^(I*(c + d*x))/(a + Rt[a^2 - b^2, 2] 
 - I*b*E^(I*(c + d*x)))), x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] 
 && PosQ[a^2 - b^2]
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

Maple [F]

\[\int \frac {x^{5}}{{\left (a +b \sin \left (d \,x^{2}+c \right )\right )}^{2}}d x\]

Input:

int(x^5/(a+b*sin(d*x^2+c))^2,x)
 

Output:

int(x^5/(a+b*sin(d*x^2+c))^2,x)
 

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2469 vs. \(2 (565) = 1130\).

Time = 0.25 (sec) , antiderivative size = 2469, normalized size of antiderivative = 3.72 \[ \int \frac {x^5}{\left (a+b \sin \left (c+d x^2\right )\right )^2} \, dx=\text {Too large to display} \] Input:

integrate(x^5/(a+b*sin(d*x^2+c))^2,x, algorithm="fricas")
 

Output:

1/4*(2*(a^2*b - b^3)*d^2*x^4*cos(d*x^2 + c) + 2*(a*b^2*sin(d*x^2 + c) + a^ 
2*b)*sqrt(-(a^2 - b^2)/b^2)*polylog(3, -(I*a*cos(d*x^2 + c) + a*sin(d*x^2 
+ c) + (b*cos(d*x^2 + c) - I*b*sin(d*x^2 + c))*sqrt(-(a^2 - b^2)/b^2))/b) 
- 2*(a*b^2*sin(d*x^2 + c) + a^2*b)*sqrt(-(a^2 - b^2)/b^2)*polylog(3, -(I*a 
*cos(d*x^2 + c) + a*sin(d*x^2 + c) - (b*cos(d*x^2 + c) - I*b*sin(d*x^2 + c 
))*sqrt(-(a^2 - b^2)/b^2))/b) + 2*(a*b^2*sin(d*x^2 + c) + a^2*b)*sqrt(-(a^ 
2 - b^2)/b^2)*polylog(3, -(-I*a*cos(d*x^2 + c) + a*sin(d*x^2 + c) + (b*cos 
(d*x^2 + c) + I*b*sin(d*x^2 + c))*sqrt(-(a^2 - b^2)/b^2))/b) - 2*(a*b^2*si 
n(d*x^2 + c) + a^2*b)*sqrt(-(a^2 - b^2)/b^2)*polylog(3, -(-I*a*cos(d*x^2 + 
 c) + a*sin(d*x^2 + c) - (b*cos(d*x^2 + c) + I*b*sin(d*x^2 + c))*sqrt(-(a^ 
2 - b^2)/b^2))/b) - 2*(-I*a^3 + I*a*b^2 + (-I*a^2*b + I*b^3)*sin(d*x^2 + c 
) + (-I*a*b^2*d*x^2*sin(d*x^2 + c) - I*a^2*b*d*x^2)*sqrt(-(a^2 - b^2)/b^2) 
)*dilog((I*a*cos(d*x^2 + c) - a*sin(d*x^2 + c) + (b*cos(d*x^2 + c) + I*b*s 
in(d*x^2 + c))*sqrt(-(a^2 - b^2)/b^2) - b)/b + 1) - 2*(-I*a^3 + I*a*b^2 + 
(-I*a^2*b + I*b^3)*sin(d*x^2 + c) + (I*a*b^2*d*x^2*sin(d*x^2 + c) + I*a^2* 
b*d*x^2)*sqrt(-(a^2 - b^2)/b^2))*dilog((I*a*cos(d*x^2 + c) - a*sin(d*x^2 + 
 c) - (b*cos(d*x^2 + c) + I*b*sin(d*x^2 + c))*sqrt(-(a^2 - b^2)/b^2) - b)/ 
b + 1) - 2*(I*a^3 - I*a*b^2 + (I*a^2*b - I*b^3)*sin(d*x^2 + c) + (I*a*b^2* 
d*x^2*sin(d*x^2 + c) + I*a^2*b*d*x^2)*sqrt(-(a^2 - b^2)/b^2))*dilog((-I*a* 
cos(d*x^2 + c) - a*sin(d*x^2 + c) + (b*cos(d*x^2 + c) - I*b*sin(d*x^2 +...
 

Sympy [F]

\[ \int \frac {x^5}{\left (a+b \sin \left (c+d x^2\right )\right )^2} \, dx=\int \frac {x^{5}}{\left (a + b \sin {\left (c + d x^{2} \right )}\right )^{2}}\, dx \] Input:

integrate(x**5/(a+b*sin(d*x**2+c))**2,x)
 

Output:

Integral(x**5/(a + b*sin(c + d*x**2))**2, x)
 

Maxima [F]

\[ \int \frac {x^5}{\left (a+b \sin \left (c+d x^2\right )\right )^2} \, dx=\int { \frac {x^{5}}{{\left (b \sin \left (d x^{2} + c\right ) + a\right )}^{2}} \,d x } \] Input:

integrate(x^5/(a+b*sin(d*x^2+c))^2,x, algorithm="maxima")
 

Output:

(a*b*x^4*cos(2*d*x^2 + 2*c)*cos(d*x^2 + c) + a*b*x^4*cos(d*x^2 + c) + ((a^ 
2*b^2 - b^4)*d*cos(2*d*x^2 + 2*c)^2 + 4*(a^4 - a^2*b^2)*d*cos(d*x^2 + c)^2 
 + 4*(a^3*b - a*b^3)*d*cos(d*x^2 + c)*sin(2*d*x^2 + 2*c) + (a^2*b^2 - b^4) 
*d*sin(2*d*x^2 + 2*c)^2 + 4*(a^4 - a^2*b^2)*d*sin(d*x^2 + c)^2 + 4*(a^3*b 
- a*b^3)*d*sin(d*x^2 + c) + (a^2*b^2 - b^4)*d - 2*(2*(a^3*b - a*b^3)*d*sin 
(d*x^2 + c) + (a^2*b^2 - b^4)*d)*cos(2*d*x^2 + 2*c))*integrate(2*(2*a^2*d* 
x^5*cos(d*x^2 + c)^2 + 2*a^2*d*x^5*sin(d*x^2 + c)^2 + a*b*d*x^5*sin(d*x^2 
+ c) - 2*a*b*x^3*cos(d*x^2 + c) - (a*b*d*x^5*sin(d*x^2 + c) + 2*a*b*x^3*co 
s(d*x^2 + c))*cos(2*d*x^2 + 2*c) + (a*b*d*x^5*cos(d*x^2 + c) - 2*a*b*x^3*s 
in(d*x^2 + c) - 2*b^2*x^3)*sin(2*d*x^2 + 2*c))/((a^2*b^2 - b^4)*d*cos(2*d* 
x^2 + 2*c)^2 + 4*(a^4 - a^2*b^2)*d*cos(d*x^2 + c)^2 + 4*(a^3*b - a*b^3)*d* 
cos(d*x^2 + c)*sin(2*d*x^2 + 2*c) + (a^2*b^2 - b^4)*d*sin(2*d*x^2 + 2*c)^2 
 + 4*(a^4 - a^2*b^2)*d*sin(d*x^2 + c)^2 + 4*(a^3*b - a*b^3)*d*sin(d*x^2 + 
c) + (a^2*b^2 - b^4)*d - 2*(2*(a^3*b - a*b^3)*d*sin(d*x^2 + c) + (a^2*b^2 
- b^4)*d)*cos(2*d*x^2 + 2*c)), x) + (a*b*x^4*sin(d*x^2 + c) + b^2*x^4)*sin 
(2*d*x^2 + 2*c))/((a^2*b^2 - b^4)*d*cos(2*d*x^2 + 2*c)^2 + 4*(a^4 - a^2*b^ 
2)*d*cos(d*x^2 + c)^2 + 4*(a^3*b - a*b^3)*d*cos(d*x^2 + c)*sin(2*d*x^2 + 2 
*c) + (a^2*b^2 - b^4)*d*sin(2*d*x^2 + 2*c)^2 + 4*(a^4 - a^2*b^2)*d*sin(d*x 
^2 + c)^2 + 4*(a^3*b - a*b^3)*d*sin(d*x^2 + c) + (a^2*b^2 - b^4)*d - 2*(2* 
(a^3*b - a*b^3)*d*sin(d*x^2 + c) + (a^2*b^2 - b^4)*d)*cos(2*d*x^2 + 2*c...
 

Giac [F]

\[ \int \frac {x^5}{\left (a+b \sin \left (c+d x^2\right )\right )^2} \, dx=\int { \frac {x^{5}}{{\left (b \sin \left (d x^{2} + c\right ) + a\right )}^{2}} \,d x } \] Input:

integrate(x^5/(a+b*sin(d*x^2+c))^2,x, algorithm="giac")
 

Output:

integrate(x^5/(b*sin(d*x^2 + c) + a)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^5}{\left (a+b \sin \left (c+d x^2\right )\right )^2} \, dx=\int \frac {x^5}{{\left (a+b\,\sin \left (d\,x^2+c\right )\right )}^2} \,d x \] Input:

int(x^5/(a + b*sin(c + d*x^2))^2,x)
 

Output:

int(x^5/(a + b*sin(c + d*x^2))^2, x)
 

Reduce [F]

\[ \int \frac {x^5}{\left (a+b \sin \left (c+d x^2\right )\right )^2} \, dx=\int \frac {x^{5}}{\sin \left (d \,x^{2}+c \right )^{2} b^{2}+2 \sin \left (d \,x^{2}+c \right ) a b +a^{2}}d x \] Input:

int(x^5/(a+b*sin(d*x^2+c))^2,x)
 

Output:

int(x**5/(sin(c + d*x**2)**2*b**2 + 2*sin(c + d*x**2)*a*b + a**2),x)