\(\int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) [299]

Optimal result

Integrand size = 28, antiderivative size = 642 \[ \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {3 f^3 x}{8 b d^3}+\frac {(e+f x)^3}{4 b d}-\frac {\left (a^2+b^2\right ) (e+f x)^4}{4 b^3 f}+\frac {6 a f^3 \cosh (c+d x)}{b^2 d^4}+\frac {3 a f (e+f x)^2 \cosh (c+d x)}{b^2 d^2}+\frac {\left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {\left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {3 \left (a^2+b^2\right ) f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {3 \left (a^2+b^2\right ) f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^2}-\frac {6 \left (a^2+b^2\right ) f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^3}-\frac {6 \left (a^2+b^2\right ) f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^3}+\frac {6 \left (a^2+b^2\right ) f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^4}+\frac {6 \left (a^2+b^2\right ) f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^4}-\frac {6 a f^2 (e+f x) \sinh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^3 \sinh (c+d x)}{b^2 d}-\frac {3 f^3 \cosh (c+d x) \sinh (c+d x)}{8 b d^4}-\frac {3 f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 b d^2}+\frac {3 f^2 (e+f x) \sinh ^2(c+d x)}{4 b d^3}+\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 b d} \] Output:

3/8*f^3*x/b/d^3+1/4*(f*x+e)^3/b/d-1/4*(a^2+b^2)*(f*x+e)^4/b^3/f+6*a*f^3*co 
sh(d*x+c)/b^2/d^4+3*a*f*(f*x+e)^2*cosh(d*x+c)/b^2/d^2+(a^2+b^2)*(f*x+e)^3* 
ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^3/d+(a^2+b^2)*(f*x+e)^3*ln(1+b*ex 
p(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^3/d+3*(a^2+b^2)*f*(f*x+e)^2*polylog(2,-b*e 
xp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^3/d^2+3*(a^2+b^2)*f*(f*x+e)^2*polylog(2,- 
b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^3/d^2-6*(a^2+b^2)*f^2*(f*x+e)*polylog( 
3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^3/d^3-6*(a^2+b^2)*f^2*(f*x+e)*polyl 
og(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^3/d^3+6*(a^2+b^2)*f^3*polylog(4, 
-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^3/d^4+6*(a^2+b^2)*f^3*polylog(4,-b*ex 
p(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^3/d^4-6*a*f^2*(f*x+e)*sinh(d*x+c)/b^2/d^3- 
a*(f*x+e)^3*sinh(d*x+c)/b^2/d-3/8*f^3*cosh(d*x+c)*sinh(d*x+c)/b/d^4-3/4*f* 
(f*x+e)^2*cosh(d*x+c)*sinh(d*x+c)/b/d^2+3/4*f^2*(f*x+e)*sinh(d*x+c)^2/b/d^ 
3+1/2*(f*x+e)^3*sinh(d*x+c)^2/b/d
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1977\) vs. \(2(642)=1284\).

Time = 8.34 (sec) , antiderivative size = 1977, normalized size of antiderivative = 3.08 \[ \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \] Input:

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

Output:

(32*(a^2 + b^2)*e^3*x*Coth[c] + 48*(a^2 + b^2)*e^2*f*x^2*Coth[c] + 32*(a^2 
 + b^2)*e*f^2*x^3*Coth[c] + 8*(a^2 + b^2)*f^3*x^4*Coth[c] - (16*(a^2 + b^2 
)*(4*e^3*E^(2*c)*x + 6*e^2*E^(2*c)*f*x^2 + 4*e*E^(2*c)*f^2*x^3 + E^(2*c)*f 
^3*x^4 + (4*a*Sqrt[a^2 + b^2]*e^3*ArcTan[(a + b*E^(c + d*x))/Sqrt[-a^2 - b 
^2]])/(Sqrt[-(a^2 + b^2)^2]*d) + (4*a*Sqrt[-a^2 - b^2]*e^3*ArcTanh[(a + b* 
E^(c + d*x))/Sqrt[a^2 + b^2]])/(Sqrt[-(a^2 + b^2)^2]*d) - (2*e^3*E^(2*c)*L 
og[b - 2*a*E^(c + d*x) - b*E^(2*(c + d*x))])/d + (2*e^3*Log[2*a*E^(c + d*x 
) + b*(-1 + E^(2*(c + d*x)))])/d + (6*e^2*f*x*Log[1 + (b*E^(2*c + d*x))/(a 
*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (6*e^2*E^(2*c)*f*x*Log[1 + (b*E^(2 
*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/d + (6*e*f^2*x^2*Log[1 + 
(b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (6*e*E^(2*c)*f 
^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/d + 
 (2*f^3*x^3*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])] 
)/d - (2*E^(2*c)*f^3*x^3*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^ 
2)*E^(2*c)])])/d + (6*e^2*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 
 + b^2)*E^(2*c)])])/d - (6*e^2*E^(2*c)*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^ 
c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d + (6*e*f^2*x^2*Log[1 + (b*E^(2*c + d*x) 
)/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (6*e*E^(2*c)*f^2*x^2*Log[1 + ( 
b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d + (2*f^3*x^3*Log[ 
1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (2*E^(2...
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 3.67 (sec) , antiderivative size = 591, normalized size of antiderivative = 0.92, number of steps used = 30, number of rules used = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.036, Rules used = {6099, 3042, 3777, 26, 3042, 26, 3777, 3042, 3777, 26, 3042, 26, 3118, 5969, 3042, 25, 3792, 17, 25, 3042, 25, 3115, 24, 6095, 2620, 3011, 7163, 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.

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

\(\Big \downarrow \) 6099

\(\displaystyle \frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \int (e+f x)^3 \cosh (c+d x)dx}{b^2}+\frac {\int (e+f x)^3 \cosh (c+d x) \sinh (c+d x)dx}{b}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

\(\displaystyle \frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \left (\frac {(e+f x)^3 \sinh (c+d x)}{d}-\frac {3 i f \int -i (e+f x)^2 \sinh (c+d x)dx}{d}\right )}{b^2}+\frac {\int (e+f x)^3 \cosh (c+d x) \sinh (c+d x)dx}{b}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \left (\frac {(e+f x)^3 \sinh (c+d x)}{d}-\frac {3 f \int (e+f x)^2 \sinh (c+d x)dx}{d}\right )}{b^2}+\frac {\int (e+f x)^3 \cosh (c+d x) \sinh (c+d x)dx}{b}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3777

\(\displaystyle \frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \left (\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \int (e+f x) \cosh (c+d x)dx}{d}\right )}{d}\right )}{b^2}+\frac {\int (e+f x)^3 \cosh (c+d x) \sinh (c+d x)dx}{b}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

\(\displaystyle \frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \left (\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {i f \int -i \sinh (c+d x)dx}{d}\right )}{d}\right )}{d}\right )}{b^2}+\frac {\int (e+f x)^3 \cosh (c+d x) \sinh (c+d x)dx}{b}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \left (\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int \sinh (c+d x)dx}{d}\right )}{d}\right )}{d}\right )}{b^2}+\frac {\int (e+f x)^3 \cosh (c+d x) \sinh (c+d x)dx}{b}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3118

\(\displaystyle \frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}+\frac {\int (e+f x)^3 \cosh (c+d x) \sinh (c+d x)dx}{b}-\frac {a \left (\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b^2}\)

\(\Big \downarrow \) 5969

\(\displaystyle \frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}+\frac {\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 d}-\frac {3 f \int (e+f x)^2 \sinh ^2(c+d x)dx}{2 d}}{b}-\frac {a \left (\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b^2}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3792

\(\displaystyle \frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}+\frac {\frac {3 f \left (\frac {f^2 \int -\sinh ^2(c+d x)dx}{2 d^2}+\frac {1}{2} \int (e+f x)^2dx+\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d}+\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 d}}{b}-\frac {a \left (\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b^2}\)

\(\Big \downarrow \) 17

\(\displaystyle \frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}+\frac {\frac {3 f \left (\frac {f^2 \int -\sinh ^2(c+d x)dx}{2 d^2}+\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{2 d}+\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 d}}{b}-\frac {a \left (\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}+\frac {\frac {3 f \left (-\frac {f^2 \int \sinh ^2(c+d x)dx}{2 d^2}+\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{2 d}+\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 d}}{b}-\frac {a \left (\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}+\frac {\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 d}+\frac {3 f \left (-\frac {f^2 \int -\sin (i c+i d x)^2dx}{2 d^2}+\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{2 d}}{b}-\frac {a \left (\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}+\frac {\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 d}+\frac {3 f \left (\frac {f^2 \int \sin (i c+i d x)^2dx}{2 d^2}+\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{2 d}}{b}-\frac {a \left (\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b^2}\)

\(\Big \downarrow \) 3115

\(\displaystyle \frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}+\frac {\frac {3 f \left (\frac {f^2 \left (\frac {\int 1dx}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d^2}+\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{2 d}+\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 d}}{b}-\frac {a \left (\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b^2}\)

\(\Big \downarrow \) 24

\(\displaystyle \frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \left (\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b^2}+\frac {\frac {3 f \left (\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{2 d}+\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 d}}{b}\)

\(\Big \downarrow \) 6095

\(\displaystyle \frac {\left (a^2+b^2\right ) \left (\int \frac {e^{c+d x} (e+f x)^3}{a+b e^{c+d x}-\sqrt {a^2+b^2}}dx+\int \frac {e^{c+d x} (e+f x)^3}{a+b e^{c+d x}+\sqrt {a^2+b^2}}dx-\frac {(e+f x)^4}{4 b f}\right )}{b^2}-\frac {a \left (\frac {(e+f x)^3 \sinh (c+d x)}{d}+\frac {3 i f \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b^2}+\frac {\frac {3 f \left (\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{2 d}+\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 d}}{b}\)

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 7163

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

((a^2 + b^2)*(-1/4*(e + f*x)^4/(b*f) + ((e + f*x)^3*Log[1 + (b*E^(c + d*x) 
)/(a - Sqrt[a^2 + b^2])])/(b*d) + ((e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a 
+ Sqrt[a^2 + b^2])])/(b*d) - (3*f*(-(((e + f*x)^2*PolyLog[2, -((b*E^(c + d 
*x))/(a - Sqrt[a^2 + b^2]))])/d) + (2*f*(((e + f*x)*PolyLog[3, -((b*E^(c + 
 d*x))/(a - Sqrt[a^2 + b^2]))])/d - (f*PolyLog[4, -((b*E^(c + d*x))/(a - S 
qrt[a^2 + b^2]))])/d^2))/d))/(b*d) - (3*f*(-(((e + f*x)^2*PolyLog[2, -((b* 
E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/d) + (2*f*(((e + f*x)*PolyLog[3, -(( 
b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/d - (f*PolyLog[4, -((b*E^(c + d*x) 
)/(a + Sqrt[a^2 + b^2]))])/d^2))/d))/(b*d)))/b^2 - (a*(((e + f*x)^3*Sinh[c 
 + d*x])/d + ((3*I)*f*((I*(e + f*x)^2*Cosh[c + d*x])/d - ((2*I)*f*(-((f*Co 
sh[c + d*x])/d^2) + ((e + f*x)*Sinh[c + d*x])/d))/d))/d))/b^2 + (((e + f*x 
)^3*Sinh[c + d*x]^2)/(2*d) + (3*f*((e + f*x)^3/(6*f) - ((e + f*x)^2*Cosh[c 
 + d*x]*Sinh[c + d*x])/(2*d) + (f*(e + f*x)*Sinh[c + d*x]^2)/(2*d^2) + (f^ 
2*(x/2 - (Cosh[c + d*x]*Sinh[c + d*x])/(2*d)))/(2*d^2)))/(2*d))/b
 

Defintions of rubi rules used

rule 17

Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 
)/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
 

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

rule 25

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

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 3115

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* 
x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n)   Int[(b*Sin 
[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 
2*n]
 

rule 3118

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

rule 3777

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

rule 3792

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

rule 5969

Int[Cosh[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)* 
(x_)]^(n_.), x_Symbol] :> Simp[(c + d*x)^m*(Sinh[a + b*x]^(n + 1)/(b*(n + 1 
))), x] - Simp[d*(m/(b*(n + 1)))   Int[(c + d*x)^(m - 1)*Sinh[a + b*x]^(n + 
 1), x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
 

rule 6095

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

rule 6099

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

rule 7163

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

Maple [F]

\[\int \frac {\left (f x +e \right )^{3} \cosh \left (d x +c \right )^{3}}{a +b \sinh \left (d x +c \right )}d x\]

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4371 vs. \(2 (602) = 1204\).

Time = 0.17 (sec) , antiderivative size = 4371, normalized size of antiderivative = 6.81 \[ \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

-1/8*e^3*((4*a*e^(-d*x - c) - b)*e^(2*d*x + 2*c)/(b^2*d) - 8*(a^2 + b^2)*( 
d*x + c)/(b^3*d) - (4*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c))/(b^2*d) - 8*(a^ 
2 + b^2)*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/(b^3*d)) + 1/32*( 
8*(a^2*d^4*f^3*e^(2*c) + b^2*d^4*f^3*e^(2*c))*x^4 + 32*(a^2*d^4*e*f^2*e^(2 
*c) + b^2*d^4*e*f^2*e^(2*c))*x^3 + 48*(a^2*d^4*e^2*f*e^(2*c) + b^2*d^4*e^2 
*f*e^(2*c))*x^2 + (4*b^2*d^3*f^3*x^3*e^(4*c) + 6*(2*d^3*e*f^2 - d^2*f^3)*b 
^2*x^2*e^(4*c) + 6*(2*d^3*e^2*f - 2*d^2*e*f^2 + d*f^3)*b^2*x*e^(4*c) - 3*( 
2*d^2*e^2*f - 2*d*e*f^2 + f^3)*b^2*e^(4*c))*e^(2*d*x) - 16*(a*b*d^3*f^3*x^ 
3*e^(3*c) + 3*(d^3*e*f^2 - d^2*f^3)*a*b*x^2*e^(3*c) + 3*(d^3*e^2*f - 2*d^2 
*e*f^2 + 2*d*f^3)*a*b*x*e^(3*c) - 3*(d^2*e^2*f - 2*d*e*f^2 + 2*f^3)*a*b*e^ 
(3*c))*e^(d*x) + 16*(a*b*d^3*f^3*x^3*e^c + 3*(d^3*e*f^2 + d^2*f^3)*a*b*x^2 
*e^c + 3*(d^3*e^2*f + 2*d^2*e*f^2 + 2*d*f^3)*a*b*x*e^c + 3*(d^2*e^2*f + 2* 
d*e*f^2 + 2*f^3)*a*b*e^c)*e^(-d*x) + (4*b^2*d^3*f^3*x^3 + 6*(2*d^3*e*f^2 + 
 d^2*f^3)*b^2*x^2 + 6*(2*d^3*e^2*f + 2*d^2*e*f^2 + d*f^3)*b^2*x + 3*(2*d^2 
*e^2*f + 2*d*e*f^2 + f^3)*b^2)*e^(-2*d*x))*e^(-2*c)/(b^3*d^4) - integrate( 
-2*((a^2*b*f^3 + b^3*f^3)*x^3 + 3*(a^2*b*e*f^2 + b^3*e*f^2)*x^2 + 3*(a^2*b 
*e^2*f + b^3*e^2*f)*x - ((a^3*f^3*e^c + a*b^2*f^3*e^c)*x^3 + 3*(a^3*e*f^2* 
e^c + a*b^2*e*f^2*e^c)*x^2 + 3*(a^3*e^2*f*e^c + a*b^2*e^2*f*e^c)*x)*e^(d*x 
))/(b^4*e^(2*d*x + 2*c) + 2*a*b^3*e^(d*x + c) - b^4), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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