\(\int \frac {(e+f x)^2 \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx\) [204]

Optimal result

Integrand size = 28, antiderivative size = 327 \[ \int \frac {(e+f x)^2 \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {2 i (e+f x)^2}{a d}+\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{a d}-\frac {(e+f x)^2 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {(e+f x)^2 \cot (c+d x)}{a d}+\frac {4 f (e+f x) \log \left (1-i e^{i (c+d x)}\right )}{a d^2}+\frac {2 f (e+f x) \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}-\frac {2 i f (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^2}-\frac {4 i f^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{a d^3}+\frac {2 i f (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^2}-\frac {i f^2 \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^3}+\frac {2 f^2 \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a d^3}-\frac {2 f^2 \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{a d^3} \] Output:

-2*I*(f*x+e)^2/a/d+2*(f*x+e)^2*arctanh(exp(I*(d*x+c)))/a/d-(f*x+e)^2*cot(1 
/2*c+1/4*Pi+1/2*d*x)/a/d-(f*x+e)^2*cot(d*x+c)/a/d+4*f*(f*x+e)*ln(1-I*exp(I 
*(d*x+c)))/a/d^2+2*f*(f*x+e)*ln(1-exp(2*I*(d*x+c)))/a/d^2-2*I*f*(f*x+e)*po 
lylog(2,-exp(I*(d*x+c)))/a/d^2-4*I*f^2*polylog(2,I*exp(I*(d*x+c)))/a/d^3+2 
*I*f*(f*x+e)*polylog(2,exp(I*(d*x+c)))/a/d^2-I*f^2*polylog(2,exp(2*I*(d*x+ 
c)))/a/d^3+2*f^2*polylog(3,-exp(I*(d*x+c)))/a/d^3-2*f^2*polylog(3,exp(I*(d 
*x+c)))/a/d^3
 

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(709\) vs. \(2(327)=654\).

Time = 9.57 (sec) , antiderivative size = 709, normalized size of antiderivative = 2.17 \[ \int \frac {(e+f x)^2 \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {i d^2 e (d e-2 f) x-i d^2 e (d e+2 f) x-\frac {2 i d^2 (e+f x)^2}{-1+e^{2 i c}}-2 d (d e-f) f x \log \left (1-e^{-i (c+d x)}\right )-d^2 f^2 x^2 \log \left (1-e^{-i (c+d x)}\right )+2 d f (d e+f) x \log \left (1+e^{-i (c+d x)}\right )+d^2 f^2 x^2 \log \left (1+e^{-i (c+d x)}\right )-d e (d e-2 f) \log \left (1-e^{i (c+d x)}\right )+d e (d e+2 f) \log \left (1+e^{i (c+d x)}\right )+2 i f (d e+f) \operatorname {PolyLog}\left (2,-e^{-i (c+d x)}\right )+2 i d f^2 x \operatorname {PolyLog}\left (2,-e^{-i (c+d x)}\right )-2 i (d e-f) f \operatorname {PolyLog}\left (2,e^{-i (c+d x)}\right )-2 i d f^2 x \operatorname {PolyLog}\left (2,e^{-i (c+d x)}\right )+2 f^2 \operatorname {PolyLog}\left (3,-e^{-i (c+d x)}\right )-2 f^2 \operatorname {PolyLog}\left (3,e^{-i (c+d x)}\right )}{a d^3}-\frac {4 f (\cos (c)+i \sin (c)) \left (\frac {(e+f x)^2 (\cos (c)-i \sin (c))}{2 f}-\frac {(e+f x) \log (1+i \cos (c+d x)+\sin (c+d x)) (1+i \cos (c)+\sin (c))}{d}+\frac {f \operatorname {PolyLog}(2,-i \cos (c+d x)-\sin (c+d x)) (\cos (c)-i (1+\sin (c)))}{d^2}\right )}{a d (\cos (c)+i (1+\sin (c)))}+\frac {\csc \left (\frac {c}{2}\right ) \csc \left (\frac {c}{2}+\frac {d x}{2}\right ) \left (e^2 \sin \left (\frac {d x}{2}\right )+2 e f x \sin \left (\frac {d x}{2}\right )+f^2 x^2 \sin \left (\frac {d x}{2}\right )\right )}{2 a d}+\frac {\sec \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}+\frac {d x}{2}\right ) \left (e^2 \sin \left (\frac {d x}{2}\right )+2 e f x \sin \left (\frac {d x}{2}\right )+f^2 x^2 \sin \left (\frac {d x}{2}\right )\right )}{2 a d}+\frac {2 \left (e^2 \sin \left (\frac {d x}{2}\right )+2 e f x \sin \left (\frac {d x}{2}\right )+f^2 x^2 \sin \left (\frac {d x}{2}\right )\right )}{a d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )} \] Input:

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

Output:

(I*d^2*e*(d*e - 2*f)*x - I*d^2*e*(d*e + 2*f)*x - ((2*I)*d^2*(e + f*x)^2)/( 
-1 + E^((2*I)*c)) - 2*d*(d*e - f)*f*x*Log[1 - E^((-I)*(c + d*x))] - d^2*f^ 
2*x^2*Log[1 - E^((-I)*(c + d*x))] + 2*d*f*(d*e + f)*x*Log[1 + E^((-I)*(c + 
 d*x))] + d^2*f^2*x^2*Log[1 + E^((-I)*(c + d*x))] - d*e*(d*e - 2*f)*Log[1 
- E^(I*(c + d*x))] + d*e*(d*e + 2*f)*Log[1 + E^(I*(c + d*x))] + (2*I)*f*(d 
*e + f)*PolyLog[2, -E^((-I)*(c + d*x))] + (2*I)*d*f^2*x*PolyLog[2, -E^((-I 
)*(c + d*x))] - (2*I)*(d*e - f)*f*PolyLog[2, E^((-I)*(c + d*x))] - (2*I)*d 
*f^2*x*PolyLog[2, E^((-I)*(c + d*x))] + 2*f^2*PolyLog[3, -E^((-I)*(c + d*x 
))] - 2*f^2*PolyLog[3, E^((-I)*(c + d*x))])/(a*d^3) - (4*f*(Cos[c] + I*Sin 
[c])*(((e + f*x)^2*(Cos[c] - I*Sin[c]))/(2*f) - ((e + f*x)*Log[1 + I*Cos[c 
 + d*x] + Sin[c + d*x]]*(1 + I*Cos[c] + Sin[c]))/d + (f*PolyLog[2, (-I)*Co 
s[c + d*x] - Sin[c + d*x]]*(Cos[c] - I*(1 + Sin[c])))/d^2))/(a*d*(Cos[c] + 
 I*(1 + Sin[c]))) + (Csc[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)/2] + 2*e*f*x*Sin[(d*x)/2] + f^2*x^2*Sin[(d*x)/2]))/(2 
*a*d) + (2*(e^2*Sin[(d*x)/2] + 2*e*f*x*Sin[(d*x)/2] + f^2*x^2*Sin[(d*x)/2] 
))/(a*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]))
 

Rubi [A] (verified)

Time = 2.63 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.15, number of steps used = 25, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {5046, 3042, 4672, 3042, 25, 4202, 2620, 2715, 2838, 5046, 3042, 3799, 3042, 4671, 3011, 2720, 4672, 3042, 25, 4202, 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.

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

\(\Big \downarrow \) 5046

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4672

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 4202

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 5046

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3799

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4671

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 4672

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 4202

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 7143

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

Input:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 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 3799

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.) 
, x_Symbol] :> Simp[(2*a)^n   Int[(c + d*x)^m*Sin[(1/2)*(e + Pi*(a/(2*b))) 
+ f*(x/2)]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2 - b^ 
2, 0] && IntegerQ[n] && (GtQ[n, 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 [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 983 vs. \(2 (296 ) = 592\).

Time = 1.46 (sec) , antiderivative size = 984, normalized size of antiderivative = 3.01

method	result	size
risch	\(\text {Expression too large to display}\)	\(984\)

Input:

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

Output:

-2*(-2*x^2*f^2+I*exp(I*(d*x+c))*f^2*x^2-4*e*f*x+2*I*exp(I*(d*x+c))*e*f*x-2 
*e^2+I*exp(I*(d*x+c))*e^2+exp(2*I*(d*x+c))*x^2*f^2+2*exp(2*I*(d*x+c))*e*f* 
x+exp(2*I*(d*x+c))*e^2)/(exp(2*I*(d*x+c))-1)/(exp(I*(d*x+c))+I)/a/d-1/d/a* 
e^2*ln(exp(I*(d*x+c))-1)+1/d/a*e^2*ln(exp(I*(d*x+c))+1)-4*I*f^2*polylog(2, 
I*exp(I*(d*x+c)))/a/d^3+4/a/d^3*f^2*ln(1-I*exp(I*(d*x+c)))*c+8/a/d^3*f^2*c 
*ln(exp(I*(d*x+c)))-8/a/d^2*f*e*ln(exp(I*(d*x+c)))+4/a/d^2*f^2*ln(1-I*exp( 
I*(d*x+c)))*x+2/d^2/a*f^2*ln(exp(I*(d*x+c))+1)*x-2/d^3/a*c*f^2*ln(exp(I*(d 
*x+c))-1)-2/d^3/a*c*f^2*ln(exp(2*I*(d*x+c))+1)+2/d^2/a*e*f*ln(exp(I*(d*x+c 
))-1)+2/d^2/a*e*f*ln(exp(2*I*(d*x+c))+1)+2/d^2/a*e*f*ln(exp(I*(d*x+c))+1)- 
2*I/d^3/a*f^2*polylog(2,-exp(I*(d*x+c)))-4*I/d/a*f^2*x^2-4*I/d^3/a*f^2*c^2 
-2/d/a*e*f*ln(1-exp(I*(d*x+c)))*x+2/d/a*e*f*ln(exp(I*(d*x+c))+1)*x-2/d^2/a 
*e*f*ln(1-exp(I*(d*x+c)))*c+2/d^2/a*c*e*f*ln(exp(I*(d*x+c))-1)+2/d^2/a*f^2 
*ln(1-exp(I*(d*x+c)))*x+2/d^3/a*f^2*ln(1-exp(I*(d*x+c)))*c+2*f^2*polylog(3 
,-exp(I*(d*x+c)))/a/d^3-2*f^2*polylog(3,exp(I*(d*x+c)))/a/d^3+1/d^3/a*f^2* 
ln(1-exp(I*(d*x+c)))*c^2+1/d/a*f^2*ln(exp(I*(d*x+c))+1)*x^2-1/d/a*f^2*ln(1 
-exp(I*(d*x+c)))*x^2-1/d^3/a*c^2*f^2*ln(exp(I*(d*x+c))-1)-2*I*f^2*polylog( 
2,exp(I*(d*x+c)))/a/d^3-2*I/d^2/a*e*f*polylog(2,-exp(I*(d*x+c)))+2*I/d^2/a 
*e*f*polylog(2,exp(I*(d*x+c)))-4*I/d^2/a*e*f*arctan(exp(I*(d*x+c)))-2*I/d^ 
2/a*f^2*polylog(2,-exp(I*(d*x+c)))*x+2*I/d^2/a*f^2*polylog(2,exp(I*(d*x+c) 
))*x-8*I/d^2/a*f^2*c*x+4*I/d^3/a*f^2*c*arctan(exp(I*(d*x+c)))
 

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. 2539 vs. \(2 (285) = 570\).

Time = 0.16 (sec) , antiderivative size = 2539, normalized size of antiderivative = 7.76 \[ \int \frac {(e+f x)^2 \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/2*(2*d^2*f^2*x^2 + 4*d^2*e*f*x + 2*d^2*e^2 - 4*(d^2*f^2*x^2 + 2*d^2*e*f 
*x + d^2*e^2)*cos(d*x + c)^2 - 2*(d^2*f^2*x^2 + 2*d^2*e*f*x + d^2*e^2)*cos 
(d*x + c) - 2*(-I*d*f^2*x - I*d*e*f + (I*d*f^2*x + I*d*e*f - I*f^2)*cos(d* 
x + c)^2 + I*f^2 + (-I*d*f^2*x - I*d*e*f + I*f^2 + (-I*d*f^2*x - I*d*e*f + 
 I*f^2)*cos(d*x + c))*sin(d*x + c))*dilog(cos(d*x + c) + I*sin(d*x + c)) - 
 2*(I*d*f^2*x + I*d*e*f + (-I*d*f^2*x - I*d*e*f + I*f^2)*cos(d*x + c)^2 - 
I*f^2 + (I*d*f^2*x + I*d*e*f - I*f^2 + (I*d*f^2*x + I*d*e*f - I*f^2)*cos(d 
*x + c))*sin(d*x + c))*dilog(cos(d*x + c) - I*sin(d*x + c)) - 4*(-I*f^2*co 
s(d*x + c)^2 + I*f^2 + (I*f^2*cos(d*x + c) + I*f^2)*sin(d*x + c))*dilog(I* 
cos(d*x + c) - sin(d*x + c)) - 4*(I*f^2*cos(d*x + c)^2 - I*f^2 + (-I*f^2*c 
os(d*x + c) - I*f^2)*sin(d*x + c))*dilog(-I*cos(d*x + c) - sin(d*x + c)) - 
 2*(-I*d*f^2*x - I*d*e*f + (I*d*f^2*x + I*d*e*f + I*f^2)*cos(d*x + c)^2 - 
I*f^2 + (-I*d*f^2*x - I*d*e*f - I*f^2 + (-I*d*f^2*x - I*d*e*f - I*f^2)*cos 
(d*x + c))*sin(d*x + c))*dilog(-cos(d*x + c) + I*sin(d*x + c)) - 2*(I*d*f^ 
2*x + I*d*e*f + (-I*d*f^2*x - I*d*e*f - I*f^2)*cos(d*x + c)^2 + I*f^2 + (I 
*d*f^2*x + I*d*e*f + I*f^2 + (I*d*f^2*x + I*d*e*f + I*f^2)*cos(d*x + c))*s 
in(d*x + c))*dilog(-cos(d*x + c) - I*sin(d*x + c)) + (d^2*f^2*x^2 + d^2*e^ 
2 + 2*d*e*f - (d^2*f^2*x^2 + d^2*e^2 + 2*d*e*f + 2*(d^2*e*f + d*f^2)*x)*co 
s(d*x + c)^2 + 2*(d^2*e*f + d*f^2)*x + (d^2*f^2*x^2 + d^2*e^2 + 2*d*e*f + 
2*(d^2*e*f + d*f^2)*x + (d^2*f^2*x^2 + d^2*e^2 + 2*d*e*f + 2*(d^2*e*f +...
 

Sympy [F]

\[ \int \frac {(e+f x)^2 \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\int \frac {e^{2} \csc ^{2}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {f^{2} x^{2} \csc ^{2}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {2 e f x \csc ^{2}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \] Input:

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

Output:

(Integral(e**2*csc(c + d*x)**2/(sin(c + d*x) + 1), x) + Integral(f**2*x**2 
*csc(c + d*x)**2/(sin(c + d*x) + 1), x) + Integral(2*e*f*x*csc(c + d*x)**2 
/(sin(c + d*x) + 1), x))/a
 

Maxima [F(-2)]

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

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

Output:

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un 
defined.
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

\text{Hanged}
 

Reduce [F]

\[ \int \frac {(e+f x)^2 \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {2 \left (\int \frac {\csc \left (d x +c \right )^{2} x^{2}}{\sin \left (d x +c \right )+1}d x \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} d \,f^{2}+2 \left (\int \frac {\csc \left (d x +c \right )^{2} x^{2}}{\sin \left (d x +c \right )+1}d x \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) d \,f^{2}+4 \left (\int \frac {\csc \left (d x +c \right )^{2} x}{\sin \left (d x +c \right )+1}d x \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} d e f +4 \left (\int \frac {\csc \left (d x +c \right )^{2} x}{\sin \left (d x +c \right )+1}d x \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) d e f -2 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} e^{2}-2 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) e^{2}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} e^{2}+6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} e^{2}-e^{2}}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )} \] Input:

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

Output:

(2*int((csc(c + d*x)**2*x**2)/(sin(c + d*x) + 1),x)*tan((c + d*x)/2)**2*d* 
f**2 + 2*int((csc(c + d*x)**2*x**2)/(sin(c + d*x) + 1),x)*tan((c + d*x)/2) 
*d*f**2 + 4*int((csc(c + d*x)**2*x)/(sin(c + d*x) + 1),x)*tan((c + d*x)/2) 
**2*d*e*f + 4*int((csc(c + d*x)**2*x)/(sin(c + d*x) + 1),x)*tan((c + d*x)/ 
2)*d*e*f - 2*log(tan((c + d*x)/2))*tan((c + d*x)/2)**2*e**2 - 2*log(tan((c 
 + d*x)/2))*tan((c + d*x)/2)*e**2 + tan((c + d*x)/2)**3*e**2 + 6*tan((c + 
d*x)/2)**2*e**2 - e**2)/(2*tan((c + d*x)/2)*a*d*(tan((c + d*x)/2) + 1))