\(\int \frac {(c+d x)^3}{(a+a \sin (e+f x))^2} \, dx\) [112]

Optimal result

Integrand size = 20, antiderivative size = 309 \[ \int \frac {(c+d x)^3}{(a+a \sin (e+f x))^2} \, dx=-\frac {i (c+d x)^3}{3 a^2 f}-\frac {2 d^2 (c+d x) \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{a^2 f^3}-\frac {(c+d x)^3 \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{3 a^2 f}-\frac {d (c+d x)^2 \csc ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{2 a^2 f^2}-\frac {(c+d x)^3 \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \csc ^2\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{6 a^2 f}+\frac {2 d (c+d x)^2 \log \left (1-i e^{i (e+f x)}\right )}{a^2 f^2}+\frac {4 d^3 \log \left (\sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )\right )}{a^2 f^4}-\frac {4 i d^2 (c+d x) \operatorname {PolyLog}\left (2,i e^{i (e+f x)}\right )}{a^2 f^3}+\frac {4 d^3 \operatorname {PolyLog}\left (3,i e^{i (e+f x)}\right )}{a^2 f^4} \] Output:

-1/3*I*(d*x+c)^3/a^2/f-2*d^2*(d*x+c)*cot(1/2*e+1/4*Pi+1/2*f*x)/a^2/f^3-1/3 
*(d*x+c)^3*cot(1/2*e+1/4*Pi+1/2*f*x)/a^2/f-1/2*d*(d*x+c)^2*csc(1/2*e+1/4*P 
i+1/2*f*x)^2/a^2/f^2-1/6*(d*x+c)^3*cot(1/2*e+1/4*Pi+1/2*f*x)*csc(1/2*e+1/4 
*Pi+1/2*f*x)^2/a^2/f+2*d*(d*x+c)^2*ln(1-I*exp(I*(f*x+e)))/a^2/f^2+4*d^3*ln 
(sin(1/2*e+1/4*Pi+1/2*f*x))/a^2/f^4-4*I*d^2*(d*x+c)*polylog(2,I*exp(I*(f*x 
+e)))/a^2/f^3+4*d^3*polylog(3,I*exp(I*(f*x+e)))/a^2/f^4
 

Mathematica [A] (verified)

Time = 2.28 (sec) , antiderivative size = 257, normalized size of antiderivative = 0.83 \[ \int \frac {(c+d x)^3}{(a+a \sin (e+f x))^2} \, dx=\frac {-2 i f (c+d x)^3+12 d (c+d x)^2 \log \left (1-i e^{i (e+f x)}\right )+\frac {24 d^3 \log \left (\cos \left (\frac {1}{4} (2 e-\pi +2 f x)\right )\right )}{f^2}+\frac {24 d^2 \left (-i f (c+d x) \operatorname {PolyLog}\left (2,i e^{i (e+f x)}\right )+d \operatorname {PolyLog}\left (3,i e^{i (e+f x)}\right )\right )}{f^2}-3 d (c+d x)^2 \sec ^2\left (\frac {1}{4} (2 e-\pi +2 f x)\right )+\frac {12 d^2 (c+d x) \tan \left (\frac {1}{4} (2 e-\pi +2 f x)\right )}{f}+2 f (c+d x)^3 \tan \left (\frac {1}{4} (2 e-\pi +2 f x)\right )+f (c+d x)^3 \sec ^2\left (\frac {1}{4} (2 e-\pi +2 f x)\right ) \tan \left (\frac {1}{4} (2 e-\pi +2 f x)\right )}{6 a^2 f^2} \] Input:

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

Output:

((-2*I)*f*(c + d*x)^3 + 12*d*(c + d*x)^2*Log[1 - I*E^(I*(e + f*x))] + (24* 
d^3*Log[Cos[(2*e - Pi + 2*f*x)/4]])/f^2 + (24*d^2*((-I)*f*(c + d*x)*PolyLo 
g[2, I*E^(I*(e + f*x))] + d*PolyLog[3, I*E^(I*(e + f*x))]))/f^2 - 3*d*(c + 
 d*x)^2*Sec[(2*e - Pi + 2*f*x)/4]^2 + (12*d^2*(c + d*x)*Tan[(2*e - Pi + 2* 
f*x)/4])/f + 2*f*(c + d*x)^3*Tan[(2*e - Pi + 2*f*x)/4] + f*(c + d*x)^3*Sec 
[(2*e - Pi + 2*f*x)/4]^2*Tan[(2*e - Pi + 2*f*x)/4])/(6*a^2*f^2)
 

Rubi [A] (verified)

Time = 1.39 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.06, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {3042, 3799, 3042, 4674, 3042, 4672, 3042, 25, 3956, 4202, 2620, 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.

\(\displaystyle \int \frac {(c+d x)^3}{(a \sin (e+f x)+a)^2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(c+d x)^3}{(a \sin (e+f x)+a)^2}dx\)

\(\Big \downarrow \) 3799

\(\displaystyle \frac {\int (c+d x)^3 \csc ^4\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )dx}{4 a^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int (c+d x)^3 \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )^4dx}{4 a^2}\)

\(\Big \downarrow \) 4674

\(\displaystyle \frac {\frac {4 d^2 \int (c+d x) \csc ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )dx}{f^2}+\frac {2}{3} \int (c+d x)^3 \csc ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )dx-\frac {2 d (c+d x)^2 \csc ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f^2}-\frac {2 (c+d x)^3 \cot \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \csc ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f}}{4 a^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {4 d^2 \int (c+d x) \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )^2dx}{f^2}+\frac {2}{3} \int (c+d x)^3 \csc \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )^2dx-\frac {2 d (c+d x)^2 \csc ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f^2}-\frac {2 (c+d x)^3 \cot \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \csc ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f}}{4 a^2}\)

\(\Big \downarrow \) 4672

\(\displaystyle \frac {\frac {4 d^2 \left (\frac {2 d \int \cot \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )dx}{f}-\frac {2 (c+d x) \cot \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}\right )}{f^2}+\frac {2}{3} \left (\frac {6 d \int (c+d x)^2 \cot \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )dx}{f}-\frac {2 (c+d x)^3 \cot \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}\right )-\frac {2 d (c+d x)^2 \csc ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f^2}-\frac {2 (c+d x)^3 \cot \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \csc ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f}}{4 a^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {4 d^2 \left (\frac {2 d \int -\tan \left (\frac {e}{2}+\frac {f x}{2}+\frac {3 \pi }{4}\right )dx}{f}-\frac {2 (c+d x) \cot \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}\right )}{f^2}+\frac {2}{3} \left (\frac {6 d \int -(c+d x)^2 \tan \left (\frac {e}{2}+\frac {f x}{2}+\frac {3 \pi }{4}\right )dx}{f}-\frac {2 (c+d x)^3 \cot \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}\right )-\frac {2 d (c+d x)^2 \csc ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f^2}-\frac {2 (c+d x)^3 \cot \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \csc ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f}}{4 a^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {4 d^2 \left (-\frac {2 d \int \tan \left (\frac {1}{4} (2 e+3 \pi )+\frac {f x}{2}\right )dx}{f}-\frac {2 (c+d x) \cot \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}\right )}{f^2}+\frac {2}{3} \left (-\frac {6 d \int (c+d x)^2 \tan \left (\frac {1}{4} (2 e+3 \pi )+\frac {f x}{2}\right )dx}{f}-\frac {2 (c+d x)^3 \cot \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}\right )-\frac {2 d (c+d x)^2 \csc ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f^2}-\frac {2 (c+d x)^3 \cot \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \csc ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f}}{4 a^2}\)

\(\Big \downarrow \) 3956

\(\displaystyle \frac {\frac {2}{3} \left (-\frac {6 d \int (c+d x)^2 \tan \left (\frac {1}{4} (2 e+3 \pi )+\frac {f x}{2}\right )dx}{f}-\frac {2 (c+d x)^3 \cot \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}\right )+\frac {4 d^2 \left (\frac {4 d \log \left (-\cos \left (\frac {e}{2}+\frac {f x}{2}-\frac {\pi }{4}\right )\right )}{f^2}-\frac {2 (c+d x) \cot \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}\right )}{f^2}-\frac {2 d (c+d x)^2 \csc ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f^2}-\frac {2 (c+d x)^3 \cot \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \csc ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f}}{4 a^2}\)

\(\Big \downarrow \) 4202

\(\displaystyle \frac {\frac {2}{3} \left (-\frac {2 (c+d x)^3 \cot \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}-\frac {6 d \left (\frac {i (c+d x)^3}{3 d}-2 i \int \frac {e^{\frac {1}{2} i (2 e+2 f x+3 \pi )} (c+d x)^2}{1+e^{\frac {1}{2} i (2 e+2 f x+3 \pi )}}dx\right )}{f}\right )+\frac {4 d^2 \left (\frac {4 d \log \left (-\cos \left (\frac {e}{2}+\frac {f x}{2}-\frac {\pi }{4}\right )\right )}{f^2}-\frac {2 (c+d x) \cot \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}\right )}{f^2}-\frac {2 d (c+d x)^2 \csc ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f^2}-\frac {2 (c+d x)^3 \cot \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \csc ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f}}{4 a^2}\)

\(\Big \downarrow \) 2620

\(\displaystyle \frac {\frac {2}{3} \left (-\frac {2 (c+d x)^3 \cot \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}-\frac {6 d \left (\frac {i (c+d x)^3}{3 d}-2 i \left (\frac {2 i d \int (c+d x) \log \left (1+e^{\frac {1}{2} i (2 e+2 f x+3 \pi )}\right )dx}{f}-\frac {i (c+d x)^2 \log \left (1+e^{\frac {1}{2} i (2 e+2 f x+3 \pi )}\right )}{f}\right )\right )}{f}\right )+\frac {4 d^2 \left (\frac {4 d \log \left (-\cos \left (\frac {e}{2}+\frac {f x}{2}-\frac {\pi }{4}\right )\right )}{f^2}-\frac {2 (c+d x) \cot \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}\right )}{f^2}-\frac {2 d (c+d x)^2 \csc ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f^2}-\frac {2 (c+d x)^3 \cot \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \csc ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f}}{4 a^2}\)

\(\Big \downarrow \) 3011

\(\displaystyle \frac {\frac {2}{3} \left (-\frac {2 (c+d x)^3 \cot \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}-\frac {6 d \left (\frac {i (c+d x)^3}{3 d}-2 i \left (\frac {2 i d \left (\frac {i (c+d x) \operatorname {PolyLog}\left (2,-e^{\frac {1}{2} i (2 e+2 f x+3 \pi )}\right )}{f}-\frac {i d \int \operatorname {PolyLog}\left (2,-e^{\frac {1}{2} i (2 e+2 f x+3 \pi )}\right )dx}{f}\right )}{f}-\frac {i (c+d x)^2 \log \left (1+e^{\frac {1}{2} i (2 e+2 f x+3 \pi )}\right )}{f}\right )\right )}{f}\right )+\frac {4 d^2 \left (\frac {4 d \log \left (-\cos \left (\frac {e}{2}+\frac {f x}{2}-\frac {\pi }{4}\right )\right )}{f^2}-\frac {2 (c+d x) \cot \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}\right )}{f^2}-\frac {2 d (c+d x)^2 \csc ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f^2}-\frac {2 (c+d x)^3 \cot \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \csc ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f}}{4 a^2}\)

\(\Big \downarrow \) 2720

\(\displaystyle \frac {\frac {2}{3} \left (-\frac {2 (c+d x)^3 \cot \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}-\frac {6 d \left (\frac {i (c+d x)^3}{3 d}-2 i \left (\frac {2 i d \left (\frac {i (c+d x) \operatorname {PolyLog}\left (2,-e^{\frac {1}{2} i (2 e+2 f x+3 \pi )}\right )}{f}-\frac {d \int e^{-\frac {1}{2} i (2 e+2 f x+3 \pi )} \operatorname {PolyLog}\left (2,-e^{\frac {1}{2} i (2 e+2 f x+3 \pi )}\right )de^{\frac {1}{2} i (2 e+2 f x+3 \pi )}}{f^2}\right )}{f}-\frac {i (c+d x)^2 \log \left (1+e^{\frac {1}{2} i (2 e+2 f x+3 \pi )}\right )}{f}\right )\right )}{f}\right )+\frac {4 d^2 \left (\frac {4 d \log \left (-\cos \left (\frac {e}{2}+\frac {f x}{2}-\frac {\pi }{4}\right )\right )}{f^2}-\frac {2 (c+d x) \cot \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}\right )}{f^2}-\frac {2 d (c+d x)^2 \csc ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f^2}-\frac {2 (c+d x)^3 \cot \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \csc ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f}}{4 a^2}\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {\frac {4 d^2 \left (\frac {4 d \log \left (-\cos \left (\frac {e}{2}+\frac {f x}{2}-\frac {\pi }{4}\right )\right )}{f^2}-\frac {2 (c+d x) \cot \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}\right )}{f^2}+\frac {2}{3} \left (-\frac {2 (c+d x)^3 \cot \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f}-\frac {6 d \left (\frac {i (c+d x)^3}{3 d}-2 i \left (\frac {2 i d \left (\frac {i (c+d x) \operatorname {PolyLog}\left (2,-e^{\frac {1}{2} i (2 e+2 f x+3 \pi )}\right )}{f}-\frac {d \operatorname {PolyLog}\left (3,-e^{\frac {1}{2} i (2 e+2 f x+3 \pi )}\right )}{f^2}\right )}{f}-\frac {i (c+d x)^2 \log \left (1+e^{\frac {1}{2} i (2 e+2 f x+3 \pi )}\right )}{f}\right )\right )}{f}\right )-\frac {2 d (c+d x)^2 \csc ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{f^2}-\frac {2 (c+d x)^3 \cot \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right ) \csc ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{3 f}}{4 a^2}\)

Input:

Int[(c + d*x)^3/(a + a*Sin[e + f*x])^2,x]
 

Output:

((-2*d*(c + d*x)^2*Csc[e/2 + Pi/4 + (f*x)/2]^2)/f^2 - (2*(c + d*x)^3*Cot[e 
/2 + Pi/4 + (f*x)/2]*Csc[e/2 + Pi/4 + (f*x)/2]^2)/(3*f) + (4*d^2*((-2*(c + 
 d*x)*Cot[e/2 + Pi/4 + (f*x)/2])/f + (4*d*Log[-Cos[e/2 - Pi/4 + (f*x)/2]]) 
/f^2))/f^2 + (2*((-2*(c + d*x)^3*Cot[e/2 + Pi/4 + (f*x)/2])/f - (6*d*(((I/ 
3)*(c + d*x)^3)/d - (2*I)*(((-I)*(c + d*x)^2*Log[1 + E^((I/2)*(2*e + 3*Pi 
+ 2*f*x))])/f + ((2*I)*d*((I*(c + d*x)*PolyLog[2, -E^((I/2)*(2*e + 3*Pi + 
2*f*x))])/f - (d*PolyLog[3, -E^((I/2)*(2*e + 3*Pi + 2*f*x))])/f^2))/f)))/f 
))/3)/(4*a^2)
 

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 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 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 3956

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d 
*x], x]]/d, x] /; FreeQ[{c, d}, x]
 

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 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 4674

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_))^(m_), x_Symbo 
l] :> Simp[(-b^2)*(c + d*x)^m*Cot[e + f*x]*((b*Csc[e + f*x])^(n - 2)/(f*(n 
- 1))), x] + (-Simp[b^2*d*m*(c + d*x)^(m - 1)*((b*Csc[e + f*x])^(n - 2)/(f^ 
2*(n - 1)*(n - 2))), x] + Simp[b^2*d^2*m*((m - 1)/(f^2*(n - 1)*(n - 2))) 
Int[(c + d*x)^(m - 2)*(b*Csc[e + f*x])^(n - 2), x], x] + Simp[b^2*((n - 2)/ 
(n - 1))   Int[(c + d*x)^m*(b*Csc[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c 
, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]
 

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. 894 vs. \(2 (254 ) = 508\).

Time = 4.15 (sec) , antiderivative size = 895, normalized size of antiderivative = 2.90

method	result	size
risch	\(\frac {2 d^{3} \ln \left (1-i {\mathrm e}^{i \left (f x +e \right )}\right ) x^{2}}{a^{2} f^{2}}-\frac {2 e^{2} d^{3} \ln \left (1-i {\mathrm e}^{i \left (f x +e \right )}\right )}{a^{2} f^{4}}-\frac {2 i d^{3} x^{3}}{3 a^{2} f}-\frac {4 i d^{3} \arctan \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{a^{2} f^{4}}+\frac {4 i d^{3} e^{3}}{3 a^{2} f^{4}}-\frac {2 i \left (6 i f c \,d^{2} x \,{\mathrm e}^{i \left (f x +e \right )}+3 d^{3} f^{2} x^{3} {\mathrm e}^{i \left (f x +e \right )}+6 i c \,d^{2}+i d^{3} f^{2} x^{3}+9 c \,d^{2} f^{2} x^{2} {\mathrm e}^{i \left (f x +e \right )}+3 f \,d^{3} x^{2} {\mathrm e}^{2 i \left (f x +e \right )}+3 i c^{2} d \,f^{2} x -6 i c \,d^{2} {\mathrm e}^{2 i \left (f x +e \right )}+i c^{3} f^{2}+9 c^{2} d \,f^{2} x \,{\mathrm e}^{i \left (f x +e \right )}+6 f c \,d^{2} x \,{\mathrm e}^{2 i \left (f x +e \right )}-6 i d^{3} x \,{\mathrm e}^{2 i \left (f x +e \right )}+6 i d^{3} x +3 i f \,d^{3} x^{2} {\mathrm e}^{i \left (f x +e \right )}+3 c^{3} f^{2} {\mathrm e}^{i \left (f x +e \right )}+3 f \,c^{2} d \,{\mathrm e}^{2 i \left (f x +e \right )}+3 i c \,d^{2} f^{2} x^{2}+12 d^{3} x \,{\mathrm e}^{i \left (f x +e \right )}+3 i f \,c^{2} d \,{\mathrm e}^{i \left (f x +e \right )}+12 c \,d^{2} {\mathrm e}^{i \left (f x +e \right )}\right )}{3 \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{3} f^{3} a^{2}}+\frac {4 i c \,d^{2} e \arctan \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{a^{2} f^{3}}-\frac {4 i c \,d^{2} e x}{a^{2} f^{2}}-\frac {4 i d^{3} \operatorname {polylog}\left (2, i {\mathrm e}^{i \left (f x +e \right )}\right ) x}{a^{2} f^{3}}-\frac {2 i c^{2} d \arctan \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{a^{2} f^{2}}-\frac {4 i c \,d^{2} \operatorname {polylog}\left (2, i {\mathrm e}^{i \left (f x +e \right )}\right )}{a^{2} f^{3}}-\frac {2 i c \,d^{2} e^{2}}{a^{2} f^{3}}+\frac {c^{2} d \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}{a^{2} f^{2}}-\frac {2 c^{2} d \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{a^{2} f^{2}}-\frac {2 d^{3} e^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{a^{2} f^{4}}+\frac {d^{3} e^{2} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}{a^{2} f^{4}}+\frac {4 c \,d^{2} \ln \left (1-i {\mathrm e}^{i \left (f x +e \right )}\right ) x}{a^{2} f^{2}}+\frac {4 c \,d^{2} \ln \left (1-i {\mathrm e}^{i \left (f x +e \right )}\right ) e}{a^{2} f^{3}}+\frac {4 c \,d^{2} e \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{a^{2} f^{3}}-\frac {2 c \,d^{2} e \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}{a^{2} f^{3}}-\frac {2 i c \,d^{2} x^{2}}{a^{2} f}-\frac {2 i d^{3} e^{2} \arctan \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{a^{2} f^{4}}+\frac {2 i d^{3} e^{2} x}{a^{2} f^{3}}+\frac {4 d^{3} \operatorname {polylog}\left (3, i {\mathrm e}^{i \left (f x +e \right )}\right )}{a^{2} f^{4}}+\frac {2 d^{3} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}{a^{2} f^{4}}-\frac {4 d^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{a^{2} f^{4}}\)	\(895\)

Input:

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

Output:

2/a^2/f^2*d^3*ln(1-I*exp(I*(f*x+e)))*x^2+1/a^2/f^2*c^2*d*ln(exp(2*I*(f*x+e 
))+1)-2/a^2/f^2*c^2*d*ln(exp(I*(f*x+e)))-2/a^2/f^4*d^3*e^2*ln(exp(I*(f*x+e 
)))+1/a^2/f^4*d^3*e^2*ln(exp(2*I*(f*x+e))+1)-2/a^2/f^4*e^2*d^3*ln(1-I*exp( 
I*(f*x+e)))-2/3*I/a^2/f*d^3*x^3-4*I/a^2/f^4*d^3*arctan(exp(I*(f*x+e)))+4/3 
*I/a^2/f^4*d^3*e^3+4/a^2/f^2*c*d^2*ln(1-I*exp(I*(f*x+e)))*x+4/a^2/f^3*c*d^ 
2*ln(1-I*exp(I*(f*x+e)))*e+4/a^2/f^3*c*d^2*e*ln(exp(I*(f*x+e)))-2/a^2/f^3* 
c*d^2*e*ln(exp(2*I*(f*x+e))+1)-2/3*I*(6*I*f*c*d^2*x*exp(I*(f*x+e))+3*d^3*f 
^2*x^3*exp(I*(f*x+e))+6*I*c*d^2+I*d^3*f^2*x^3+9*c*d^2*f^2*x^2*exp(I*(f*x+e 
))+3*f*d^3*x^2*exp(2*I*(f*x+e))+3*I*c^2*d*f^2*x-6*I*c*d^2*exp(2*I*(f*x+e)) 
+I*c^3*f^2+9*c^2*d*f^2*x*exp(I*(f*x+e))+6*f*c*d^2*x*exp(2*I*(f*x+e))-6*I*d 
^3*x*exp(2*I*(f*x+e))+6*I*d^3*x+3*I*f*d^3*x^2*exp(I*(f*x+e))+3*c^3*f^2*exp 
(I*(f*x+e))+3*f*c^2*d*exp(2*I*(f*x+e))+3*I*c*d^2*f^2*x^2+12*d^3*x*exp(I*(f 
*x+e))+3*I*f*c^2*d*exp(I*(f*x+e))+12*c*d^2*exp(I*(f*x+e)))/(exp(I*(f*x+e)) 
+I)^3/f^3/a^2-2*I/a^2/f*c*d^2*x^2-2*I/a^2/f^4*d^3*e^2*arctan(exp(I*(f*x+e) 
))+2*I/a^2/f^3*d^3*e^2*x-4*I/a^2/f^3*d^3*polylog(2,I*exp(I*(f*x+e)))*x-2*I 
/a^2/f^2*c^2*d*arctan(exp(I*(f*x+e)))-4*I/a^2/f^3*c*d^2*polylog(2,I*exp(I* 
(f*x+e)))-2*I/a^2/f^3*c*d^2*e^2+4*I/a^2/f^3*c*d^2*e*arctan(exp(I*(f*x+e))) 
-4*I/a^2/f^2*c*d^2*e*x+4*d^3*polylog(3,I*exp(I*(f*x+e)))/a^2/f^4+2/a^2/f^4 
*d^3*ln(exp(2*I*(f*x+e))+1)-4/a^2/f^4*d^3*ln(exp(I*(f*x+e)))
 

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. 1708 vs. \(2 (248) = 496\).

Time = 0.15 (sec) , antiderivative size = 1708, normalized size of antiderivative = 5.53 \[ \int \frac {(c+d x)^3}{(a+a \sin (e+f x))^2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/3*(d^3*f^3*x^3 + c^3*f^3 + 3*c^2*d*f^2 + 3*(c*d^2*f^3 + d^3*f^2)*x^2 + ( 
d^3*f^3*x^3 + 3*c*d^2*f^3*x^2 + c^3*f^3 + 6*c*d^2*f + 3*(c^2*d*f^3 + 2*d^3 
*f)*x)*cos(f*x + e)^2 + 3*(c^2*d*f^3 + 2*c*d^2*f^2)*x + (2*d^3*f^3*x^3 + 2 
*c^3*f^3 + 3*c^2*d*f^2 + 6*c*d^2*f + 3*(2*c*d^2*f^3 + d^3*f^2)*x^2 + 6*(c^ 
2*d*f^3 + c*d^2*f^2 + d^3*f)*x)*cos(f*x + e) + 6*(2*I*d^3*f*x + 2*I*c*d^2* 
f + (-I*d^3*f*x - I*c*d^2*f)*cos(f*x + e)^2 + (I*d^3*f*x + I*c*d^2*f)*cos( 
f*x + e) + (2*I*d^3*f*x + 2*I*c*d^2*f + (I*d^3*f*x + I*c*d^2*f)*cos(f*x + 
e))*sin(f*x + e))*dilog(I*cos(f*x + e) - sin(f*x + e)) + 6*(-2*I*d^3*f*x - 
 2*I*c*d^2*f + (I*d^3*f*x + I*c*d^2*f)*cos(f*x + e)^2 + (-I*d^3*f*x - I*c* 
d^2*f)*cos(f*x + e) + (-2*I*d^3*f*x - 2*I*c*d^2*f + (-I*d^3*f*x - I*c*d^2* 
f)*cos(f*x + e))*sin(f*x + e))*dilog(-I*cos(f*x + e) - sin(f*x + e)) - 3*( 
2*d^3*e^2 - 4*c*d^2*e*f + 2*c^2*d*f^2 + 4*d^3 - (d^3*e^2 - 2*c*d^2*e*f + c 
^2*d*f^2 + 2*d^3)*cos(f*x + e)^2 + (d^3*e^2 - 2*c*d^2*e*f + c^2*d*f^2 + 2* 
d^3)*cos(f*x + e) + (2*d^3*e^2 - 4*c*d^2*e*f + 2*c^2*d*f^2 + 4*d^3 + (d^3* 
e^2 - 2*c*d^2*e*f + c^2*d*f^2 + 2*d^3)*cos(f*x + e))*sin(f*x + e))*log(cos 
(f*x + e) + I*sin(f*x + e) + I) - 3*(2*d^3*f^2*x^2 + 4*c*d^2*f^2*x - 2*d^3 
*e^2 + 4*c*d^2*e*f - (d^3*f^2*x^2 + 2*c*d^2*f^2*x - d^3*e^2 + 2*c*d^2*e*f) 
*cos(f*x + e)^2 + (d^3*f^2*x^2 + 2*c*d^2*f^2*x - d^3*e^2 + 2*c*d^2*e*f)*co 
s(f*x + e) + (2*d^3*f^2*x^2 + 4*c*d^2*f^2*x - 2*d^3*e^2 + 4*c*d^2*e*f + (d 
^3*f^2*x^2 + 2*c*d^2*f^2*x - d^3*e^2 + 2*c*d^2*e*f)*cos(f*x + e))*sin(f...
 

Sympy [F]

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

integrate((d*x+c)**3/(a+a*sin(f*x+e))**2,x)
 

Output:

(Integral(c**3/(sin(e + f*x)**2 + 2*sin(e + f*x) + 1), x) + Integral(d**3* 
x**3/(sin(e + f*x)**2 + 2*sin(e + f*x) + 1), x) + Integral(3*c*d**2*x**2/( 
sin(e + f*x)**2 + 2*sin(e + f*x) + 1), x) + Integral(3*c**2*d*x/(sin(e + f 
*x)**2 + 2*sin(e + f*x) + 1), x))/a**2
 

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3593 vs. \(2 (248) = 496\).

Time = 0.63 (sec) , antiderivative size = 3593, normalized size of antiderivative = 11.63 \[ \int \frac {(c+d x)^3}{(a+a \sin (e+f x))^2} \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)^3/(a+a*sin(f*x+e))^2,x, algorithm="maxima")
 

Output:

-1/3*(6*c*d^2*e^2*(3*sin(f*x + e)/(cos(f*x + e) + 1) + 3*sin(f*x + e)^2/(c 
os(f*x + e) + 1)^2 + 2)/(a^2*f^2 + 3*a^2*f^2*sin(f*x + e)/(cos(f*x + e) + 
1) + 3*a^2*f^2*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + a^2*f^2*sin(f*x + e)^ 
3/(cos(f*x + e) + 1)^3) + 6*(2*(f*x + 3*(f*x + e)*sin(f*x + e) + e + cos(f 
*x + e) + sin(2*f*x + 2*e))*cos(3*f*x + 3*e) - 2*(9*(f*x + e)*cos(f*x + e) 
 - 6*sin(f*x + e) - 1)*cos(2*f*x + 2*e) - 6*cos(2*f*x + 2*e)^2 - 6*cos(f*x 
 + e)^2 - (6*(cos(f*x + e) + sin(2*f*x + 2*e))*cos(3*f*x + 3*e) - cos(3*f* 
x + 3*e)^2 + 6*(3*sin(f*x + e) + 1)*cos(2*f*x + 2*e) - 9*cos(2*f*x + 2*e)^ 
2 - 9*cos(f*x + e)^2 - 2*(3*cos(2*f*x + 2*e) - 3*sin(f*x + e) - 1)*sin(3*f 
*x + 3*e) - sin(3*f*x + 3*e)^2 - 18*cos(f*x + e)*sin(2*f*x + 2*e) - 9*sin( 
2*f*x + 2*e)^2 - 9*sin(f*x + e)^2 - 6*sin(f*x + e) - 1)*log(cos(f*x + e)^2 
 + sin(f*x + e)^2 + 2*sin(f*x + e) + 1) - 2*(3*(f*x + e)*cos(f*x + e) + co 
s(2*f*x + 2*e) - sin(f*x + e))*sin(3*f*x + 3*e) - 6*(f*x + 3*(f*x + e)*sin 
(f*x + e) + e + 2*cos(f*x + e))*sin(2*f*x + 2*e) - 6*sin(2*f*x + 2*e)^2 - 
6*sin(f*x + e)^2 - 2*sin(f*x + e))*c*d^2*e/(a^2*f^2*cos(3*f*x + 3*e)^2 + 9 
*a^2*f^2*cos(2*f*x + 2*e)^2 + 9*a^2*f^2*cos(f*x + e)^2 + a^2*f^2*sin(3*f*x 
 + 3*e)^2 + 18*a^2*f^2*cos(f*x + e)*sin(2*f*x + 2*e) + 9*a^2*f^2*sin(2*f*x 
 + 2*e)^2 + 9*a^2*f^2*sin(f*x + e)^2 + 6*a^2*f^2*sin(f*x + e) + a^2*f^2 - 
6*(a^2*f^2*cos(f*x + e) + a^2*f^2*sin(2*f*x + 2*e))*cos(3*f*x + 3*e) - 6*( 
3*a^2*f^2*sin(f*x + e) + a^2*f^2)*cos(2*f*x + 2*e) + 2*(3*a^2*f^2*cos(2...
 

Giac [F]

\[ \int \frac {(c+d x)^3}{(a+a \sin (e+f x))^2} \, dx=\int { \frac {{\left (d x + c\right )}^{3}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{2}} \,d x } \] Input:

integrate((d*x+c)^3/(a+a*sin(f*x+e))^2,x, algorithm="giac")
 

Output:

integrate((d*x + c)^3/(a*sin(f*x + e) + a)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x)^3}{(a+a \sin (e+f x))^2} \, dx=\text {Hanged} \] Input:

int((c + d*x)^3/(a + a*sin(e + f*x))^2,x)
 

Output:

\text{Hanged}
 

Reduce [F]

\[ \int \frac {(c+d x)^3}{(a+a \sin (e+f x))^2} \, dx=\text {too large to display} \] Input:

int((d*x+c)^3/(a+a*sin(f*x+e))^2,x)
 

Output:

(336*int(x**3/(tan((e + f*x)/2)**4 + 4*tan((e + f*x)/2)**3 + 6*tan((e + f* 
x)/2)**2 + 4*tan((e + f*x)/2) + 1),x)*tan((e + f*x)/2)**3*d**3*f**4 + 1008 
*int(x**3/(tan((e + f*x)/2)**4 + 4*tan((e + f*x)/2)**3 + 6*tan((e + f*x)/2 
)**2 + 4*tan((e + f*x)/2) + 1),x)*tan((e + f*x)/2)**2*d**3*f**4 + 1008*int 
(x**3/(tan((e + f*x)/2)**4 + 4*tan((e + f*x)/2)**3 + 6*tan((e + f*x)/2)**2 
 + 4*tan((e + f*x)/2) + 1),x)*tan((e + f*x)/2)*d**3*f**4 + 336*int(x**3/(t 
an((e + f*x)/2)**4 + 4*tan((e + f*x)/2)**3 + 6*tan((e + f*x)/2)**2 + 4*tan 
((e + f*x)/2) + 1),x)*d**3*f**4 + 1008*int(x**2/(tan((e + f*x)/2)**4 + 4*t 
an((e + f*x)/2)**3 + 6*tan((e + f*x)/2)**2 + 4*tan((e + f*x)/2) + 1),x)*ta 
n((e + f*x)/2)**3*c*d**2*f**4 - 864*int(x**2/(tan((e + f*x)/2)**4 + 4*tan( 
(e + f*x)/2)**3 + 6*tan((e + f*x)/2)**2 + 4*tan((e + f*x)/2) + 1),x)*tan(( 
e + f*x)/2)**3*d**3*f**3 + 3024*int(x**2/(tan((e + f*x)/2)**4 + 4*tan((e + 
 f*x)/2)**3 + 6*tan((e + f*x)/2)**2 + 4*tan((e + f*x)/2) + 1),x)*tan((e + 
f*x)/2)**2*c*d**2*f**4 - 2592*int(x**2/(tan((e + f*x)/2)**4 + 4*tan((e + f 
*x)/2)**3 + 6*tan((e + f*x)/2)**2 + 4*tan((e + f*x)/2) + 1),x)*tan((e + f* 
x)/2)**2*d**3*f**3 + 3024*int(x**2/(tan((e + f*x)/2)**4 + 4*tan((e + f*x)/ 
2)**3 + 6*tan((e + f*x)/2)**2 + 4*tan((e + f*x)/2) + 1),x)*tan((e + f*x)/2 
)*c*d**2*f**4 - 2592*int(x**2/(tan((e + f*x)/2)**4 + 4*tan((e + f*x)/2)**3 
 + 6*tan((e + f*x)/2)**2 + 4*tan((e + f*x)/2) + 1),x)*tan((e + f*x)/2)*d** 
3*f**3 + 1008*int(x**2/(tan((e + f*x)/2)**4 + 4*tan((e + f*x)/2)**3 + 6...