\(\int \frac {(e+f x)^2 \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx\) [237]

Optimal result

Integrand size = 28, antiderivative size = 639 \[ \int \frac {(e+f x)^2 \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {i (e+f x)^2}{a d}+\frac {2 b (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{a^2 d}-\frac {(e+f x)^2 \cot (c+d x)}{a d}-\frac {i b^2 (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d}+\frac {i b^2 (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d}+\frac {2 f (e+f x) \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}-\frac {2 i b f (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {2 i b f (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a^2 d^2}-\frac {2 b^2 f (e+f x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d^2}+\frac {2 b^2 f (e+f x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d^2}-\frac {i f^2 \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^3}+\frac {2 b f^2 \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a^2 d^3}-\frac {2 b f^2 \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{a^2 d^3}-\frac {2 i b^2 f^2 \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d^3}+\frac {2 i b^2 f^2 \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d^3} \] Output:

-I*(f*x+e)^2/a/d+2*b*(f*x+e)^2*arctanh(exp(I*(d*x+c)))/a^2/d-(f*x+e)^2*cot 
(d*x+c)/a/d-I*b^2*(f*x+e)^2*ln(1-I*b*exp(I*(d*x+c))/(a-(a^2-b^2)^(1/2)))/a 
^2/(a^2-b^2)^(1/2)/d+I*b^2*(f*x+e)^2*ln(1-I*b*exp(I*(d*x+c))/(a+(a^2-b^2)^ 
(1/2)))/a^2/(a^2-b^2)^(1/2)/d+2*f*(f*x+e)*ln(1-exp(2*I*(d*x+c)))/a/d^2-2*I 
*b*f*(f*x+e)*polylog(2,-exp(I*(d*x+c)))/a^2/d^2+2*I*b*f*(f*x+e)*polylog(2, 
exp(I*(d*x+c)))/a^2/d^2-2*b^2*f*(f*x+e)*polylog(2,I*b*exp(I*(d*x+c))/(a-(a 
^2-b^2)^(1/2)))/a^2/(a^2-b^2)^(1/2)/d^2+2*b^2*f*(f*x+e)*polylog(2,I*b*exp( 
I*(d*x+c))/(a+(a^2-b^2)^(1/2)))/a^2/(a^2-b^2)^(1/2)/d^2-I*f^2*polylog(2,ex 
p(2*I*(d*x+c)))/a/d^3+2*b*f^2*polylog(3,-exp(I*(d*x+c)))/a^2/d^3-2*b*f^2*p 
olylog(3,exp(I*(d*x+c)))/a^2/d^3-2*I*b^2*f^2*polylog(3,I*b*exp(I*(d*x+c))/ 
(a-(a^2-b^2)^(1/2)))/a^2/(a^2-b^2)^(1/2)/d^3+2*I*b^2*f^2*polylog(3,I*b*exp 
(I*(d*x+c))/(a+(a^2-b^2)^(1/2)))/a^2/(a^2-b^2)^(1/2)/d^3
 

Mathematica [A] (warning: unable to verify)

Time = 10.90 (sec) , antiderivative size = 933, normalized size of antiderivative = 1.46 \[ \int \frac {(e+f x)^2 \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx =\text {Too large to display} \] Input:

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

Output:

(I*d^2*e*(b*d*e - 2*a*f)*x - I*d^2*e*(b*d*e + 2*a*f)*x - ((2*I)*a*d^2*(e + 
 f*x)^2)/(-1 + E^((2*I)*c)) - 2*d*f*(b*d*e - a*f)*x*Log[1 - E^((-I)*(c + d 
*x))] - b*d^2*f^2*x^2*Log[1 - E^((-I)*(c + d*x))] + 2*d*f*(b*d*e + a*f)*x* 
Log[1 + E^((-I)*(c + d*x))] + b*d^2*f^2*x^2*Log[1 + E^((-I)*(c + d*x))] - 
d*e*(b*d*e - 2*a*f)*Log[1 - E^(I*(c + d*x))] + d*e*(b*d*e + 2*a*f)*Log[1 + 
 E^(I*(c + d*x))] + (2*I)*f*(b*d*e + a*f)*PolyLog[2, -E^((-I)*(c + d*x))] 
+ (2*I)*b*d*f^2*x*PolyLog[2, -E^((-I)*(c + d*x))] + (2*I)*f*(-(b*d*e) + a* 
f)*PolyLog[2, E^((-I)*(c + d*x))] - (2*I)*b*d*f^2*x*PolyLog[2, E^((-I)*(c 
+ d*x))] + 2*b*f^2*PolyLog[3, -E^((-I)*(c + d*x))] - 2*b*f^2*PolyLog[3, E^ 
((-I)*(c + d*x))])/(a^2*d^3) + (I*b^2*(-2*Sqrt[a^2 - b^2]*d*f*(e + f*x)*Po 
lyLog[2, (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] + 2*Sqrt[a^2 - b 
^2]*d*f*(e + f*x)*PolyLog[2, -((b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 + b^2] 
))] - I*(d^2*(2*Sqrt[-a^2 + b^2]*e^2*ArcTan[(I*a + b*E^(I*(c + d*x)))/Sqrt 
[a^2 - b^2]] + Sqrt[a^2 - b^2]*f*x*(2*e + f*x)*(Log[1 - (b*E^(I*(c + d*x)) 
)/((-I)*a + Sqrt[-a^2 + b^2])] - Log[1 + (b*E^(I*(c + d*x)))/(I*a + Sqrt[- 
a^2 + b^2])])) + 2*Sqrt[a^2 - b^2]*f^2*PolyLog[3, (b*E^(I*(c + d*x)))/((-I 
)*a + Sqrt[-a^2 + b^2])] - 2*Sqrt[a^2 - b^2]*f^2*PolyLog[3, -((b*E^(I*(c + 
 d*x)))/(I*a + Sqrt[-a^2 + b^2]))])))/(a^2*Sqrt[-(a^2 - b^2)^2]*d^3) + (Cs 
c[c/2]*Csc[c/2 + (d*x)/2]*(e^2*Sin[(d*x)/2] + 2*e*f*x*Sin[(d*x)/2] + f^2*x 
^2*Sin[(d*x)/2]))/(2*a*d) + (Sec[c/2]*Sec[c/2 + (d*x)/2]*(e^2*Sin[(d*x)...
 

Rubi [A] (verified)

Time = 3.22 (sec) , antiderivative size = 604, normalized size of antiderivative = 0.95, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {5046, 3042, 4672, 3042, 25, 4202, 2620, 2715, 2838, 5046, 3042, 3804, 2694, 27, 2620, 3011, 2720, 4671, 3011, 2720, 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.

\(\displaystyle \int \frac {(e+f x)^2 \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx\)

\(\Big \downarrow \) 5046

\(\displaystyle \frac {\int (e+f x)^2 \csc ^2(c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^2 \csc (c+d x)}{a+b \sin (c+d x)}dx}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int (e+f x)^2 \csc (c+d x)^2dx}{a}-\frac {b \int \frac {(e+f x)^2 \csc (c+d x)}{a+b \sin (c+d x)}dx}{a}\)

\(\Big \downarrow \) 4672

\(\displaystyle \frac {\frac {2 f \int (e+f x) \cot (c+d x)dx}{d}-\frac {(e+f x)^2 \cot (c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x)^2 \csc (c+d x)}{a+b \sin (c+d x)}dx}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {2 f \int -\left ((e+f x) \tan \left (c+d x+\frac {\pi }{2}\right )\right )dx}{d}-\frac {(e+f x)^2 \cot (c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x)^2 \csc (c+d x)}{a+b \sin (c+d x)}dx}{a}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {-\frac {2 f \int (e+f x) \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx}{d}-\frac {(e+f x)^2 \cot (c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x)^2 \csc (c+d x)}{a+b \sin (c+d x)}dx}{a}\)

\(\Big \downarrow \) 4202

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

\(\Big \downarrow \) 2620

\(\displaystyle -\frac {b \int \frac {(e+f x)^2 \csc (c+d x)}{a+b \sin (c+d x)}dx}{a}+\frac {-\frac {(e+f x)^2 \cot (c+d x)}{d}-\frac {2 f \left (\frac {i (e+f x)^2}{2 f}-2 i \left (\frac {i f \int \log \left (1+e^{i (2 c+2 d x+\pi )}\right )dx}{2 d}-\frac {i (e+f x) \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )\right )}{d}}{a}\)

\(\Big \downarrow \) 2715

\(\displaystyle -\frac {b \int \frac {(e+f x)^2 \csc (c+d x)}{a+b \sin (c+d x)}dx}{a}+\frac {-\frac {(e+f x)^2 \cot (c+d x)}{d}-\frac {2 f \left (\frac {i (e+f x)^2}{2 f}-2 i \left (\frac {f \int e^{-i (2 c+2 d x+\pi )} \log \left (1+e^{i (2 c+2 d x+\pi )}\right )de^{i (2 c+2 d x+\pi )}}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )\right )}{d}}{a}\)

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 5046

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3804

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

\(\Big \downarrow \) 2694

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 4671

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

(-(((e + f*x)^2*Cot[c + d*x])/d) - (2*f*(((I/2)*(e + f*x)^2)/f - (2*I)*((( 
-1/2*I)*(e + f*x)*Log[1 + E^(I*(2*c + Pi + 2*d*x))])/d - (f*PolyLog[2, -E^ 
(I*(2*c + Pi + 2*d*x))])/(4*d^2))))/d)/a - (b*(((-2*(e + f*x)^2*ArcTanh[E^ 
(I*(c + d*x))])/d + (2*f*((I*(e + f*x)*PolyLog[2, -E^(I*(c + d*x))])/d - ( 
f*PolyLog[3, -E^(I*(c + d*x))])/d^2))/d - (2*f*((I*(e + f*x)*PolyLog[2, E^ 
(I*(c + d*x))])/d - (f*PolyLog[3, E^(I*(c + d*x))])/d^2))/d)/a - (2*b*(((- 
1/2*I)*b*(((e + f*x)^2*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2]) 
])/(b*d) - (2*f*((I*(e + f*x)*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a 
^2 - b^2])])/d - (f*PolyLog[3, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2]) 
])/d^2))/(b*d)))/Sqrt[a^2 - b^2] + ((I/2)*b*(((e + f*x)^2*Log[1 - (I*b*E^( 
I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*d) - (2*f*((I*(e + f*x)*PolyLog[2 
, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/d - (f*PolyLog[3, (I*b*E^( 
I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/d^2))/(b*d)))/Sqrt[a^2 - b^2]))/a))/ 
a
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 4202

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I 
*((c + d*x)^(m + 1)/(d*(m + 1))), x] - Simp[2*I   Int[(c + d*x)^m*(E^(2*I*( 
e + f*x))/(1 + E^(2*I*(e + f*x)))), x], x] /; FreeQ[{c, d, e, f}, x] && IGt 
Q[m, 0]
 

rule 4671

Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[- 
2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*x))]/f), x] + (-Simp[d*(m/f)   Int[(c + 
d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Simp[d*(m/f)   Int[(c + d*x 
)^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IG 
tQ[m, 0]
 

rule 4672

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp 
[(-(c + d*x)^m)*(Cot[e + f*x]/f), x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1) 
*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
 

rule 5046

Int[(Csc[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_. 
)*Sin[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[1/a   Int[(e + f*x)^m*Csc[c + 
 d*x]^n, x], x] - Simp[b/a   Int[(e + f*x)^m*(Csc[c + d*x]^(n - 1)/(a + b*S 
in[c + d*x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ 
[n, 0]
 

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]

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

Input:

int((f*x+e)^2*csc(d*x+c)^2/(a+b*sin(d*x+c)),x)
 

Output:

int((f*x+e)^2*csc(d*x+c)^2/(a+b*sin(d*x+c)),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. 2972 vs. \(2 (556) = 1112\).

Time = 0.34 (sec) , antiderivative size = 2972, normalized size of antiderivative = 4.65 \[ \int \frac {(e+f x)^2 \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Sympy [F]

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

integrate((f*x+e)**2*csc(d*x+c)**2/(a+b*sin(d*x+c)),x)
 

Output:

Integral((e + f*x)**2*csc(c + d*x)**2/(a + b*sin(c + d*x)), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(e+f x)^2 \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?` f 
or more de
 

Giac [F(-1)]

Timed out. \[ \int \frac {(e+f x)^2 \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(e+f x)^2 \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Hanged} \] Input:

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

Output:

\text{Hanged}
 

Reduce [F]

\[ \int \frac {(e+f x)^2 \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {2 \sqrt {a^{2}-b^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a +b}{\sqrt {a^{2}-b^{2}}}\right ) \sin \left (d x +c \right ) b^{2} e^{2}-\cos \left (d x +c \right ) a^{3} e^{2}+\cos \left (d x +c \right ) a \,b^{2} e^{2}+\left (\int \frac {\csc \left (d x +c \right )^{2} x^{2}}{\sin \left (d x +c \right ) b +a}d x \right ) \sin \left (d x +c \right ) a^{4} d \,f^{2}-\left (\int \frac {\csc \left (d x +c \right )^{2} x^{2}}{\sin \left (d x +c \right ) b +a}d x \right ) \sin \left (d x +c \right ) a^{2} b^{2} d \,f^{2}+2 \left (\int \frac {\csc \left (d x +c \right )^{2} x}{\sin \left (d x +c \right ) b +a}d x \right ) \sin \left (d x +c \right ) a^{4} d e f -2 \left (\int \frac {\csc \left (d x +c \right )^{2} x}{\sin \left (d x +c \right ) b +a}d x \right ) \sin \left (d x +c \right ) a^{2} b^{2} d e f -\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right ) a^{2} b \,e^{2}+\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right ) b^{3} e^{2}}{\sin \left (d x +c \right ) a^{2} d \left (a^{2}-b^{2}\right )} \] Input:

int((f*x+e)^2*csc(d*x+c)^2/(a+b*sin(d*x+c)),x)
 

Output:

(2*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin( 
c + d*x)*b**2*e**2 - cos(c + d*x)*a**3*e**2 + cos(c + d*x)*a*b**2*e**2 + i 
nt((csc(c + d*x)**2*x**2)/(sin(c + d*x)*b + a),x)*sin(c + d*x)*a**4*d*f**2 
 - int((csc(c + d*x)**2*x**2)/(sin(c + d*x)*b + a),x)*sin(c + d*x)*a**2*b* 
*2*d*f**2 + 2*int((csc(c + d*x)**2*x)/(sin(c + d*x)*b + a),x)*sin(c + d*x) 
*a**4*d*e*f - 2*int((csc(c + d*x)**2*x)/(sin(c + d*x)*b + a),x)*sin(c + d* 
x)*a**2*b**2*d*e*f - log(tan((c + d*x)/2))*sin(c + d*x)*a**2*b*e**2 + log( 
tan((c + d*x)/2))*sin(c + d*x)*b**3*e**2)/(sin(c + d*x)*a**2*d*(a**2 - b** 
2))