\(\int \frac {(c+d x)^3}{(a+a \cosh (e+f x))^2} \, dx\) [116]

Optimal result

Integrand size = 20, antiderivative size = 255 \[ \int \frac {(c+d x)^3}{(a+a \cosh (e+f x))^2} \, dx=\frac {(c+d x)^3}{3 a^2 f}-\frac {2 d (c+d x)^2 \log \left (1+e^{e+f x}\right )}{a^2 f^2}+\frac {4 d^3 \log \left (\cosh \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{a^2 f^4}-\frac {4 d^2 (c+d x) \operatorname {PolyLog}\left (2,-e^{e+f x}\right )}{a^2 f^3}+\frac {4 d^3 \operatorname {PolyLog}\left (3,-e^{e+f x}\right )}{a^2 f^4}+\frac {d (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{2 a^2 f^2}-\frac {2 d^2 (c+d x) \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{a^2 f^3}+\frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^3 \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f} \] Output:

1/3*(d*x+c)^3/a^2/f-2*d*(d*x+c)^2*ln(1+exp(f*x+e))/a^2/f^2+4*d^3*ln(cosh(1 
/2*f*x+1/2*e))/a^2/f^4-4*d^2*(d*x+c)*polylog(2,-exp(f*x+e))/a^2/f^3+4*d^3* 
polylog(3,-exp(f*x+e))/a^2/f^4+1/2*d*(d*x+c)^2*sech(1/2*f*x+1/2*e)^2/a^2/f 
^2-2*d^2*(d*x+c)*tanh(1/2*f*x+1/2*e)/a^2/f^3+1/3*(d*x+c)^3*tanh(1/2*f*x+1/ 
2*e)/a^2/f+1/6*(d*x+c)^3*sech(1/2*f*x+1/2*e)^2*tanh(1/2*f*x+1/2*e)/a^2/f
 

Mathematica [A] (verified)

Time = 1.89 (sec) , antiderivative size = 506, normalized size of antiderivative = 1.98 \[ \int \frac {(c+d x)^3}{(a+a \cosh (e+f x))^2} \, dx=\frac {\cosh \left (\frac {1}{2} (e+f x)\right ) \left (-\frac {8 d \cosh ^3\left (\frac {1}{2} (e+f x)\right ) \left (6 d^2 e^e f x-3 c^2 e^e f^3 x+3 c d f^3 x^2+d^2 f^3 x^3+6 c d f^2 x \log \left (1+e^{-e-f x}\right )+6 c d e^e f^2 x \log \left (1+e^{-e-f x}\right )+3 d^2 f^2 x^2 \log \left (1+e^{-e-f x}\right )+3 d^2 e^e f^2 x^2 \log \left (1+e^{-e-f x}\right )-6 d^2 \log \left (1+e^{e+f x}\right )-6 d^2 e^e \log \left (1+e^{e+f x}\right )+3 c^2 f^2 \log \left (1+e^{e+f x}\right )+3 c^2 e^e f^2 \log \left (1+e^{e+f x}\right )-6 d \left (1+e^e\right ) f (c+d x) \operatorname {PolyLog}\left (2,-e^{-e-f x}\right )-6 d^2 \left (1+e^e\right ) \operatorname {PolyLog}\left (3,-e^{-e-f x}\right )\right )}{\left (1+e^e\right ) f}+(c+d x) \text {sech}\left (\frac {e}{2}\right ) \left (3 d f (c+d x) \cosh \left (\frac {f x}{2}\right )+3 d f (c+d x) \cosh \left (e+\frac {f x}{2}\right )-12 d^2 \sinh \left (\frac {f x}{2}\right )+3 c^2 f^2 \sinh \left (\frac {f x}{2}\right )+6 c d f^2 x \sinh \left (\frac {f x}{2}\right )+3 d^2 f^2 x^2 \sinh \left (\frac {f x}{2}\right )+6 d^2 \sinh \left (e+\frac {f x}{2}\right )-6 d^2 \sinh \left (e+\frac {3 f x}{2}\right )+c^2 f^2 \sinh \left (e+\frac {3 f x}{2}\right )+2 c d f^2 x \sinh \left (e+\frac {3 f x}{2}\right )+d^2 f^2 x^2 \sinh \left (e+\frac {3 f x}{2}\right )\right )\right )}{3 a^2 f^3 (1+\cosh (e+f x))^2} \] Input:

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

Output:

(Cosh[(e + f*x)/2]*((-8*d*Cosh[(e + f*x)/2]^3*(6*d^2*E^e*f*x - 3*c^2*E^e*f 
^3*x + 3*c*d*f^3*x^2 + d^2*f^3*x^3 + 6*c*d*f^2*x*Log[1 + E^(-e - f*x)] + 6 
*c*d*E^e*f^2*x*Log[1 + E^(-e - f*x)] + 3*d^2*f^2*x^2*Log[1 + E^(-e - f*x)] 
 + 3*d^2*E^e*f^2*x^2*Log[1 + E^(-e - f*x)] - 6*d^2*Log[1 + E^(e + f*x)] - 
6*d^2*E^e*Log[1 + E^(e + f*x)] + 3*c^2*f^2*Log[1 + E^(e + f*x)] + 3*c^2*E^ 
e*f^2*Log[1 + E^(e + f*x)] - 6*d*(1 + E^e)*f*(c + d*x)*PolyLog[2, -E^(-e - 
 f*x)] - 6*d^2*(1 + E^e)*PolyLog[3, -E^(-e - f*x)]))/((1 + E^e)*f) + (c + 
d*x)*Sech[e/2]*(3*d*f*(c + d*x)*Cosh[(f*x)/2] + 3*d*f*(c + d*x)*Cosh[e + ( 
f*x)/2] - 12*d^2*Sinh[(f*x)/2] + 3*c^2*f^2*Sinh[(f*x)/2] + 6*c*d*f^2*x*Sin 
h[(f*x)/2] + 3*d^2*f^2*x^2*Sinh[(f*x)/2] + 6*d^2*Sinh[e + (f*x)/2] - 6*d^2 
*Sinh[e + (3*f*x)/2] + c^2*f^2*Sinh[e + (3*f*x)/2] + 2*c*d*f^2*x*Sinh[e + 
(3*f*x)/2] + d^2*f^2*x^2*Sinh[e + (3*f*x)/2])))/(3*a^2*f^3*(1 + Cosh[e + f 
*x])^2)
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.40 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {3042, 3799, 3042, 4674, 3042, 4672, 26, 3042, 26, 3956, 4201, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3799

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4674

\(\displaystyle \frac {-\frac {4 d^2 \int (c+d x) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )dx}{f^2}+\frac {2}{3} \int (c+d x)^3 \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )dx+\frac {2 d (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{f^2}+\frac {2 (c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f}}{4 a^2}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4672

\(\displaystyle \frac {-\frac {4 d^2 \left (\frac {2 (c+d x) \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{f}-\frac {2 i d \int -i \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )dx}{f}\right )}{f^2}+\frac {2}{3} \left (\frac {2 (c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{f}-\frac {6 i d \int -i (c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )dx}{f}\right )+\frac {2 d (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{f^2}+\frac {2 (c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f}}{4 a^2}\)

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {4 d^2 \left (\frac {2 (c+d x) \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{f}-\frac {2 d \int -i \tan \left (\frac {i e}{2}+\frac {i f x}{2}\right )dx}{f}\right )}{f^2}+\frac {2}{3} \left (\frac {2 (c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{f}-\frac {6 d \int -i (c+d x)^2 \tan \left (\frac {i e}{2}+\frac {i f x}{2}\right )dx}{f}\right )+\frac {2 d (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{f^2}+\frac {2 (c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f}}{4 a^2}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {-\frac {4 d^2 \left (\frac {2 (c+d x) \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{f}+\frac {2 i d \int \tan \left (\frac {i e}{2}+\frac {i f x}{2}\right )dx}{f}\right )}{f^2}+\frac {2}{3} \left (\frac {2 (c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{f}+\frac {6 i d \int (c+d x)^2 \tan \left (\frac {i e}{2}+\frac {i f x}{2}\right )dx}{f}\right )+\frac {2 d (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{f^2}+\frac {2 (c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f}}{4 a^2}\)

\(\Big \downarrow \) 3956

\(\displaystyle \frac {\frac {2}{3} \left (\frac {2 (c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{f}+\frac {6 i d \int (c+d x)^2 \tan \left (\frac {i e}{2}+\frac {i f x}{2}\right )dx}{f}\right )-\frac {4 d^2 \left (\frac {2 (c+d x) \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{f}-\frac {4 d \log \left (\cosh \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{f^2}\right )}{f^2}+\frac {2 d (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{f^2}+\frac {2 (c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f}}{4 a^2}\)

\(\Big \downarrow \) 4201

\(\displaystyle \frac {\frac {2}{3} \left (\frac {2 (c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{f}+\frac {6 i d \left (2 i \int \frac {e^{e+f x} (c+d x)^2}{1+e^{e+f x}}dx-\frac {i (c+d x)^3}{3 d}\right )}{f}\right )-\frac {4 d^2 \left (\frac {2 (c+d x) \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{f}-\frac {4 d \log \left (\cosh \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{f^2}\right )}{f^2}+\frac {2 d (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{f^2}+\frac {2 (c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f}}{4 a^2}\)

\(\Big \downarrow \) 2620

\(\displaystyle \frac {\frac {2}{3} \left (\frac {2 (c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{f}+\frac {6 i d \left (2 i \left (\frac {(c+d x)^2 \log \left (e^{e+f x}+1\right )}{f}-\frac {2 d \int (c+d x) \log \left (1+e^{e+f x}\right )dx}{f}\right )-\frac {i (c+d x)^3}{3 d}\right )}{f}\right )-\frac {4 d^2 \left (\frac {2 (c+d x) \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{f}-\frac {4 d \log \left (\cosh \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{f^2}\right )}{f^2}+\frac {2 d (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{f^2}+\frac {2 (c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f}}{4 a^2}\)

\(\Big \downarrow \) 3011

\(\displaystyle \frac {\frac {2}{3} \left (\frac {2 (c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{f}+\frac {6 i d \left (2 i \left (\frac {(c+d x)^2 \log \left (e^{e+f x}+1\right )}{f}-\frac {2 d \left (\frac {d \int \operatorname {PolyLog}\left (2,-e^{e+f x}\right )dx}{f}-\frac {(c+d x) \operatorname {PolyLog}\left (2,-e^{e+f x}\right )}{f}\right )}{f}\right )-\frac {i (c+d x)^3}{3 d}\right )}{f}\right )-\frac {4 d^2 \left (\frac {2 (c+d x) \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{f}-\frac {4 d \log \left (\cosh \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{f^2}\right )}{f^2}+\frac {2 d (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{f^2}+\frac {2 (c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f}}{4 a^2}\)

\(\Big \downarrow \) 2720

\(\displaystyle \frac {\frac {2}{3} \left (\frac {2 (c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{f}+\frac {6 i d \left (2 i \left (\frac {(c+d x)^2 \log \left (e^{e+f x}+1\right )}{f}-\frac {2 d \left (\frac {d \int e^{-e-f x} \operatorname {PolyLog}\left (2,-e^{e+f x}\right )de^{e+f x}}{f^2}-\frac {(c+d x) \operatorname {PolyLog}\left (2,-e^{e+f x}\right )}{f}\right )}{f}\right )-\frac {i (c+d x)^3}{3 d}\right )}{f}\right )-\frac {4 d^2 \left (\frac {2 (c+d x) \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{f}-\frac {4 d \log \left (\cosh \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{f^2}\right )}{f^2}+\frac {2 d (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{f^2}+\frac {2 (c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f}}{4 a^2}\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {-\frac {4 d^2 \left (\frac {2 (c+d x) \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{f}-\frac {4 d \log \left (\cosh \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{f^2}\right )}{f^2}+\frac {2}{3} \left (\frac {2 (c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{f}+\frac {6 i d \left (2 i \left (\frac {(c+d x)^2 \log \left (e^{e+f x}+1\right )}{f}-\frac {2 d \left (\frac {d \operatorname {PolyLog}\left (3,-e^{e+f x}\right )}{f^2}-\frac {(c+d x) \operatorname {PolyLog}\left (2,-e^{e+f x}\right )}{f}\right )}{f}\right )-\frac {i (c+d x)^3}{3 d}\right )}{f}\right )+\frac {2 d (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{f^2}+\frac {2 (c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 f}}{4 a^2}\)

Input:

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

Output:

((2*d*(c + d*x)^2*Sech[e/2 + (f*x)/2]^2)/f^2 + (2*(c + d*x)^3*Sech[e/2 + ( 
f*x)/2]^2*Tanh[e/2 + (f*x)/2])/(3*f) - (4*d^2*((-4*d*Log[Cosh[e/2 + (f*x)/ 
2]])/f^2 + (2*(c + d*x)*Tanh[e/2 + (f*x)/2])/f))/f^2 + (2*(((6*I)*d*(((-1/ 
3*I)*(c + d*x)^3)/d + (2*I)*(((c + d*x)^2*Log[1 + E^(e + f*x)])/f - (2*d*( 
-(((c + d*x)*PolyLog[2, -E^(e + f*x)])/f) + (d*PolyLog[3, -E^(e + f*x)])/f 
^2))/f)))/f + (2*(c + d*x)^3*Tanh[e/2 + (f*x)/2])/f))/3)/(4*a^2)
 

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

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 4201

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(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*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x], x] 
 /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[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)

Leaf count of result is larger than twice the leaf count of optimal. \(599\) vs. \(2(220)=440\).

Time = 0.82 (sec) , antiderivative size = 600, normalized size of antiderivative = 2.35

method	result	size
risch	\(-\frac {2 \left (3 d^{3} f^{2} x^{3} {\mathrm e}^{f x +e}+9 c \,d^{2} f^{2} x^{2} {\mathrm e}^{f x +e}+d^{3} f^{2} x^{3}-3 d^{3} f \,x^{2} {\mathrm e}^{2 f x +2 e}+9 c^{2} d \,f^{2} x \,{\mathrm e}^{f x +e}+3 c \,d^{2} f^{2} x^{2}-6 c \,d^{2} f x \,{\mathrm e}^{2 f x +2 e}-3 d^{3} f \,x^{2} {\mathrm e}^{f x +e}+3 c^{3} f^{2} {\mathrm e}^{f x +e}+3 c^{2} d \,f^{2} x -3 c^{2} d f \,{\mathrm e}^{2 f x +2 e}-6 c \,d^{2} f x \,{\mathrm e}^{f x +e}-6 d^{3} x \,{\mathrm e}^{2 f x +2 e}+c^{3} f^{2}-3 c^{2} d f \,{\mathrm e}^{f x +e}-6 c \,d^{2} {\mathrm e}^{2 f x +2 e}-12 d^{3} x \,{\mathrm e}^{f x +e}-12 c \,d^{2} {\mathrm e}^{f x +e}-6 d^{3} x -6 d^{2} c \right )}{3 f^{3} a^{2} \left ({\mathrm e}^{f x +e}+1\right )^{3}}-\frac {4 d^{3} \operatorname {polylog}\left (2, -{\mathrm e}^{f x +e}\right ) x}{a^{2} f^{3}}+\frac {2 d^{2} c \,e^{2}}{a^{2} f^{3}}-\frac {4 d^{2} c \operatorname {polylog}\left (2, -{\mathrm e}^{f x +e}\right )}{a^{2} f^{3}}+\frac {2 d^{3} x^{3}}{3 a^{2} f}+\frac {2 d \,c^{2} \ln \left ({\mathrm e}^{f x +e}\right )}{a^{2} f^{2}}-\frac {2 d \,c^{2} \ln \left ({\mathrm e}^{f x +e}+1\right )}{a^{2} f^{2}}+\frac {2 d^{3} e^{2} \ln \left ({\mathrm e}^{f x +e}\right )}{a^{2} f^{4}}-\frac {4 d^{3} e^{3}}{3 a^{2} f^{4}}+\frac {4 d^{3} \operatorname {polylog}\left (3, -{\mathrm e}^{f x +e}\right )}{a^{2} f^{4}}-\frac {4 d^{3} \ln \left ({\mathrm e}^{f x +e}\right )}{a^{2} f^{4}}+\frac {4 d^{3} \ln \left ({\mathrm e}^{f x +e}+1\right )}{a^{2} f^{4}}+\frac {2 d^{2} c \,x^{2}}{a^{2} f}-\frac {4 d^{2} c e \ln \left ({\mathrm e}^{f x +e}\right )}{a^{2} f^{3}}+\frac {4 d^{2} c e x}{a^{2} f^{2}}-\frac {4 d^{2} c \ln \left ({\mathrm e}^{f x +e}+1\right ) x}{a^{2} f^{2}}-\frac {2 d^{3} e^{2} x}{a^{2} f^{3}}-\frac {2 d^{3} \ln \left ({\mathrm e}^{f x +e}+1\right ) x^{2}}{a^{2} f^{2}}\)	\(600\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1863 vs. \(2 (219) = 438\).

Time = 0.11 (sec) , antiderivative size = 1863, normalized size of antiderivative = 7.31 \[ \int \frac {(c+d x)^3}{(a+a \cosh (e+f x))^2} \, dx=\text {Too large to display} \] Input:

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

Output:

2/3*(d^3*e^3 + 3*c^2*d*e*f^2 - c^3*f^3 - 6*d^3*e + (d^3*f^3*x^3 + 3*c*d^2* 
f^3*x^2 + d^3*e^3 - 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 - 6*d^3*e + 3*(c^2*d*f^3 
 - 2*d^3*f)*x)*cosh(f*x + e)^3 + (d^3*f^3*x^3 + 3*c*d^2*f^3*x^2 + d^3*e^3 
- 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 - 6*d^3*e + 3*(c^2*d*f^3 - 2*d^3*f)*x)*sin 
h(f*x + e)^3 + 3*(d^3*f^3*x^3 + d^3*e^3 - 6*d^3*e + (3*c^2*d*e + c^2*d)*f^ 
2 + (3*c*d^2*f^3 + d^3*f^2)*x^2 - (3*c*d^2*e^2 - 2*c*d^2)*f + (3*c^2*d*f^3 
 + 2*c*d^2*f^2 - 4*d^3*f)*x)*cosh(f*x + e)^2 + 3*(d^3*f^3*x^3 + d^3*e^3 - 
6*d^3*e + (3*c^2*d*e + c^2*d)*f^2 + (3*c*d^2*f^3 + d^3*f^2)*x^2 - (3*c*d^2 
*e^2 - 2*c*d^2)*f + (3*c^2*d*f^3 + 2*c*d^2*f^2 - 4*d^3*f)*x + (d^3*f^3*x^3 
 + 3*c*d^2*f^3*x^2 + d^3*e^3 - 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 - 6*d^3*e + 3 
*(c^2*d*f^3 - 2*d^3*f)*x)*cosh(f*x + e))*sinh(f*x + e)^2 - 3*(c*d^2*e^2 - 
2*c*d^2)*f + 3*(d^3*f^2*x^2 + d^3*e^3 - c^3*f^3 - 6*d^3*e + (3*c^2*d*e + c 
^2*d)*f^2 - (3*c*d^2*e^2 - 4*c*d^2)*f + 2*(c*d^2*f^2 - d^3*f)*x)*cosh(f*x 
+ e) - 6*(d^3*f*x + c*d^2*f + (d^3*f*x + c*d^2*f)*cosh(f*x + e)^3 + (d^3*f 
*x + c*d^2*f)*sinh(f*x + e)^3 + 3*(d^3*f*x + c*d^2*f)*cosh(f*x + e)^2 + 3* 
(d^3*f*x + c*d^2*f + (d^3*f*x + c*d^2*f)*cosh(f*x + e))*sinh(f*x + e)^2 + 
3*(d^3*f*x + c*d^2*f)*cosh(f*x + e) + 3*(d^3*f*x + c*d^2*f + (d^3*f*x + c* 
d^2*f)*cosh(f*x + e)^2 + 2*(d^3*f*x + c*d^2*f)*cosh(f*x + e))*sinh(f*x + e 
))*dilog(-cosh(f*x + e) - sinh(f*x + e)) - 3*(d^3*f^2*x^2 + 2*c*d^2*f^2*x 
+ c^2*d*f^2 + (d^3*f^2*x^2 + 2*c*d^2*f^2*x + c^2*d*f^2 - 2*d^3)*cosh(f*...
 

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 610 vs. \(2 (219) = 438\).

Time = 0.26 (sec) , antiderivative size = 610, normalized size of antiderivative = 2.39 \[ \int \frac {(c+d x)^3}{(a+a \cosh (e+f x))^2} \, dx =\text {Too large to display} \] Input:

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

Output:

2*c^2*d*((f*x*e^(3*f*x + 3*e) + (3*f*x*e^(2*e) + e^(2*e))*e^(2*f*x) + e^(f 
*x + e))/(a^2*f^2*e^(3*f*x + 3*e) + 3*a^2*f^2*e^(2*f*x + 2*e) + 3*a^2*f^2* 
e^(f*x + e) + a^2*f^2) - log((e^(f*x + e) + 1)*e^(-e))/(a^2*f^2)) + 2/3*c^ 
3*(3*e^(-f*x - e)/((3*a^2*e^(-f*x - e) + 3*a^2*e^(-2*f*x - 2*e) + a^2*e^(- 
3*f*x - 3*e) + a^2)*f) + 1/((3*a^2*e^(-f*x - e) + 3*a^2*e^(-2*f*x - 2*e) + 
 a^2*e^(-3*f*x - 3*e) + a^2)*f)) - 2/3*(d^3*f^2*x^3 + 3*c*d^2*f^2*x^2 - 6* 
d^3*x - 6*c*d^2 - 3*(d^3*f*x^2*e^(2*e) + 2*c*d^2*e^(2*e) + 2*(c*d^2*f*e^(2 
*e) + d^3*e^(2*e))*x)*e^(2*f*x) + 3*(d^3*f^2*x^3*e^e - 4*c*d^2*e^e + (3*c* 
d^2*f^2*e^e - d^3*f*e^e)*x^2 - 2*(c*d^2*f*e^e + 2*d^3*e^e)*x)*e^(f*x))/(a^ 
2*f^3*e^(3*f*x + 3*e) + 3*a^2*f^3*e^(2*f*x + 2*e) + 3*a^2*f^3*e^(f*x + e) 
+ a^2*f^3) - 4*(f*x*log(e^(f*x + e) + 1) + dilog(-e^(f*x + e)))*c*d^2/(a^2 
*f^3) - 4*d^3*x/(a^2*f^3) - 2*(f^2*x^2*log(e^(f*x + e) + 1) + 2*f*x*dilog( 
-e^(f*x + e)) - 2*polylog(3, -e^(f*x + e)))*d^3/(a^2*f^4) + 4*d^3*log(e^(f 
*x + e) + 1)/(a^2*f^4) + 2/3*(d^3*f^3*x^3 + 3*c*d^2*f^3*x^2)/(a^2*f^4)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(54*e**(3*e + 3*f*x)*int(x**2/(e**(4*e + 4*f*x) + 4*e**(3*e + 3*f*x) + 6*e 
**(2*e + 2*f*x) + 4*e**(e + f*x) + 1),x)*d**3*f**3 + 108*e**(3*e + 3*f*x)* 
int(x/(e**(4*e + 4*f*x) + 4*e**(3*e + 3*f*x) + 6*e**(2*e + 2*f*x) + 4*e**( 
e + f*x) + 1),x)*c*d**2*f**3 + 198*e**(3*e + 3*f*x)*int(x/(e**(4*e + 4*f*x 
) + 4*e**(3*e + 3*f*x) + 6*e**(2*e + 2*f*x) + 4*e**(e + f*x) + 1),x)*d**3* 
f**2 - 54*e**(3*e + 3*f*x)*log(e**(e + f*x) + 1)*c**2*d*f**2 - 198*e**(3*e 
 + 3*f*x)*log(e**(e + f*x) + 1)*c*d**2*f - 147*e**(3*e + 3*f*x)*log(e**(e 
+ f*x) + 1)*d**3 + 54*e**(3*e + 3*f*x)*c**2*d*f**3*x - 18*e**(3*e + 3*f*x) 
*c**2*d*f**2 + 198*e**(3*e + 3*f*x)*c*d**2*f**2*x - 66*e**(3*e + 3*f*x)*c* 
d**2*f + 147*e**(3*e + 3*f*x)*d**3*f*x - 49*e**(3*e + 3*f*x)*d**3 + 162*e* 
*(2*e + 2*f*x)*int(x**2/(e**(4*e + 4*f*x) + 4*e**(3*e + 3*f*x) + 6*e**(2*e 
 + 2*f*x) + 4*e**(e + f*x) + 1),x)*d**3*f**3 + 324*e**(2*e + 2*f*x)*int(x/ 
(e**(4*e + 4*f*x) + 4*e**(3*e + 3*f*x) + 6*e**(2*e + 2*f*x) + 4*e**(e + f* 
x) + 1),x)*c*d**2*f**3 + 594*e**(2*e + 2*f*x)*int(x/(e**(4*e + 4*f*x) + 4* 
e**(3*e + 3*f*x) + 6*e**(2*e + 2*f*x) + 4*e**(e + f*x) + 1),x)*d**3*f**2 - 
 162*e**(2*e + 2*f*x)*log(e**(e + f*x) + 1)*c**2*d*f**2 - 594*e**(2*e + 2* 
f*x)*log(e**(e + f*x) + 1)*c*d**2*f - 441*e**(2*e + 2*f*x)*log(e**(e + f*x 
) + 1)*d**3 + 162*e**(2*e + 2*f*x)*c**2*d*f**3*x + 594*e**(2*e + 2*f*x)*c* 
d**2*f**2*x + 441*e**(2*e + 2*f*x)*d**3*f*x + 162*e**(e + f*x)*int(x**2/(e 
**(4*e + 4*f*x) + 4*e**(3*e + 3*f*x) + 6*e**(2*e + 2*f*x) + 4*e**(e + f...