\(\int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) [373]

Optimal result

Integrand size = 36, antiderivative size = 792 \[ \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {a^2 (e+f x)^2}{4 b^3 d}-\frac {3 (e+f x)^2}{32 b d}-\frac {a^2 \left (a^2+b^2\right ) (e+f x)^3}{3 b^5 f}+\frac {2 a^3 f (e+f x) \cosh (c+d x)}{b^4 d^2}+\frac {4 a f (e+f x) \cosh (c+d x)}{3 b^2 d^2}+\frac {3 f^2 \cosh ^2(c+d x)}{32 b d^3}+\frac {2 a f (e+f x) \cosh ^3(c+d x)}{9 b^2 d^2}+\frac {f^2 \cosh ^4(c+d x)}{32 b d^3}+\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 b d}+\frac {a^2 \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^5 d}+\frac {a^2 \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^5 d}+\frac {2 a^2 \left (a^2+b^2\right ) f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^5 d^2}+\frac {2 a^2 \left (a^2+b^2\right ) f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^5 d^2}-\frac {2 a^2 \left (a^2+b^2\right ) f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^5 d^3}-\frac {2 a^2 \left (a^2+b^2\right ) f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^5 d^3}-\frac {2 a^3 f^2 \sinh (c+d x)}{b^4 d^3}-\frac {14 a f^2 \sinh (c+d x)}{9 b^2 d^3}-\frac {a^3 (e+f x)^2 \sinh (c+d x)}{b^4 d}-\frac {2 a (e+f x)^2 \sinh (c+d x)}{3 b^2 d}-\frac {a^2 f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^3 d^2}-\frac {3 f (e+f x) \cosh (c+d x) \sinh (c+d x)}{16 b d^2}-\frac {a (e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 b^2 d}-\frac {f (e+f x) \cosh ^3(c+d x) \sinh (c+d x)}{8 b d^2}+\frac {a^2 f^2 \sinh ^2(c+d x)}{4 b^3 d^3}+\frac {a^2 (e+f x)^2 \sinh ^2(c+d x)}{2 b^3 d}-\frac {2 a f^2 \sinh ^3(c+d x)}{27 b^2 d^3} \] Output:

2*a^2*(a^2+b^2)*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^5 
/d^2+2*a^2*(a^2+b^2)*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)) 
)/b^5/d^2-14/9*a*f^2*sinh(d*x+c)/b^2/d^3+a^2*(a^2+b^2)*(f*x+e)^2*ln(1+b*ex 
p(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^5/d+a^2*(a^2+b^2)*(f*x+e)^2*ln(1+b*exp(d*x 
+c)/(a-(a^2+b^2)^(1/2)))/b^5/d-2*a^2*(a^2+b^2)*f^2*polylog(3,-b*exp(d*x+c) 
/(a+(a^2+b^2)^(1/2)))/b^5/d^3-2*a^2*(a^2+b^2)*f^2*polylog(3,-b*exp(d*x+c)/ 
(a-(a^2+b^2)^(1/2)))/b^5/d^3+1/4*a^2*(f*x+e)^2/b^3/d+3/32*f^2*cosh(d*x+c)^ 
2/b/d^3+1/32*f^2*cosh(d*x+c)^4/b/d^3+1/4*(f*x+e)^2*cosh(d*x+c)^4/b/d-2*a^3 
*f^2*sinh(d*x+c)/b^4/d^3+1/4*a^2*f^2*sinh(d*x+c)^2/b^3/d^3+1/2*a^2*(f*x+e) 
^2*sinh(d*x+c)^2/b^3/d-2/27*a*f^2*sinh(d*x+c)^3/b^2/d^3-1/3*a^2*(a^2+b^2)* 
(f*x+e)^3/b^5/f+4/3*a*f*(f*x+e)*cosh(d*x+c)/b^2/d^2-3/16*f*(f*x+e)*cosh(d* 
x+c)*sinh(d*x+c)/b/d^2-a^3*(f*x+e)^2*sinh(d*x+c)/b^4/d-2/3*a*(f*x+e)^2*sin 
h(d*x+c)/b^2/d-3/32*(f*x+e)^2/b/d+2*a^3*f*(f*x+e)*cosh(d*x+c)/b^4/d^2+2/9* 
a*f*(f*x+e)*cosh(d*x+c)^3/b^2/d^2-1/3*a*(f*x+e)^2*cosh(d*x+c)^2*sinh(d*x+c 
)/b^2/d-1/8*f*(f*x+e)*cosh(d*x+c)^3*sinh(d*x+c)/b/d^2-1/2*a^2*f*(f*x+e)*co 
sh(d*x+c)*sinh(d*x+c)/b^3/d^2
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(5113\) vs. \(2(792)=1584\).

Time = 15.35 (sec) , antiderivative size = 5113, normalized size of antiderivative = 6.46 \[ \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Result too large to show} \] Input:

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

Output:

Result too large to show
 

Rubi [F]

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.

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

\(\Big \downarrow \) 6113

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

\(\Big \downarrow \) 5970

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3791

\(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \int (e+f x) \cosh ^2(c+d x)dx-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )^2dx-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}\)

\(\Big \downarrow \) 3791

\(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (\frac {1}{2} \int (e+f x)dx-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\)

\(\Big \downarrow \) 17

\(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\)

\(\Big \downarrow \) 6113

\(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (\frac {\int (e+f x)^2 \cosh ^3(c+d x)dx}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )^3dx}{b}\right )}{b}\)

\(\Big \downarrow \) 3792

\(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (\frac {\frac {2 f^2 \int \cosh ^3(c+d x)dx}{9 d^2}+\frac {2}{3} \int (e+f x)^2 \cosh (c+d x)dx-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2 f^2 \int \sin \left (i c+i d x+\frac {\pi }{2}\right )^3dx}{9 d^2}+\frac {2}{3} \int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )dx-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\)

\(\Big \downarrow \) 3113

\(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2 i f^2 \int \left (\sinh ^2(c+d x)+1\right )d(-i \sinh (c+d x))}{9 d^3}+\frac {2}{3} \int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )dx-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2}{3} \int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )dx+\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\)

\(\Big \downarrow \) 3777

\(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 i f \int -i (e+f x) \sinh (c+d x)dx}{d}\right )+\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 f \int (e+f x) \sinh (c+d x)dx}{d}\right )+\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 f \int -i (e+f x) \sin (i c+i d x)dx}{d}\right )+\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \int (e+f x) \sin (i c+i d x)dx}{d}\right )+\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\)

\(\Big \downarrow \) 3777

\(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \cosh (c+d x)dx}{d}\right )}{d}\right )+\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{d}\right )+\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\)

\(\Big \downarrow \) 3117

\(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\)

\(\Big \downarrow \) 6099

\(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \int (e+f x)^2 \cosh (c+d x)dx}{b^2}+\frac {\int (e+f x)^2 \cosh (c+d x) \sinh (c+d x)dx}{b}\right )}{b}+\frac {\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (\frac {\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{b^2}+\frac {\int (e+f x)^2 \cosh (c+d x) \sinh (c+d x)dx}{b}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 3777

\(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (\frac {\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 i f \int -i (e+f x) \sinh (c+d x)dx}{d}\right )}{b^2}+\frac {\int (e+f x)^2 \cosh (c+d x) \sinh (c+d x)dx}{b}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 f \int (e+f x) \sinh (c+d x)dx}{d}\right )}{b^2}+\frac {\int (e+f x)^2 \cosh (c+d x) \sinh (c+d x)dx}{b}\right )}{b}+\frac {\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (\frac {\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 f \int -i (e+f x) \sin (i c+i d x)dx}{d}\right )}{b^2}+\frac {\int (e+f x)^2 \cosh (c+d x) \sinh (c+d x)dx}{b}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (\frac {\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \int (e+f x) \sin (i c+i d x)dx}{d}\right )}{b^2}+\frac {\int (e+f x)^2 \cosh (c+d x) \sinh (c+d x)dx}{b}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 3777

\(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (\frac {\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \cosh (c+d x)dx}{d}\right )}{d}\right )}{b^2}+\frac {\int (e+f x)^2 \cosh (c+d x) \sinh (c+d x)dx}{b}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (\frac {\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{d}\right )}{b^2}+\frac {\int (e+f x)^2 \cosh (c+d x) \sinh (c+d x)dx}{b}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 3117

\(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (\frac {\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}+\frac {\int (e+f x)^2 \cosh (c+d x) \sinh (c+d x)dx}{b}-\frac {a \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{b^2}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 5969

\(\displaystyle \frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (\frac {\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}+\frac {\frac {(e+f x)^2 \sinh ^2(c+d x)}{2 d}-\frac {f \int (e+f x) \sinh ^2(c+d x)dx}{d}}{b}-\frac {a \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{b^2}\right )}{b}\right )}{b}\)

Input:

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

Output:

$Aborted
 

Maple [F]

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

Input:

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

Output:

int((f*x+e)^2*cosh(d*x+c)^3*sinh(d*x+c)^2/(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. 7645 vs. \(2 (738) = 1476\).

Time = 0.22 (sec) , antiderivative size = 7645, normalized size of antiderivative = 9.65 \[ \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

-1/192*e^2*((8*a*b^2*e^(-d*x - c) - 3*b^3 - 12*(2*a^2*b + b^3)*e^(-2*d*x - 
 2*c) + 24*(4*a^3 + 3*a*b^2)*e^(-3*d*x - 3*c))*e^(4*d*x + 4*c)/(b^4*d) - 1 
92*(a^4 + a^2*b^2)*(d*x + c)/(b^5*d) - (8*a*b^2*e^(-3*d*x - 3*c) + 3*b^3*e 
^(-4*d*x - 4*c) + 24*(4*a^3 + 3*a*b^2)*e^(-d*x - c) + 12*(2*a^2*b + b^3)*e 
^(-2*d*x - 2*c))/(b^4*d) - 192*(a^4 + a^2*b^2)*log(-2*a*e^(-d*x - c) + b*e 
^(-2*d*x - 2*c) - b)/(b^5*d)) + 1/13824*(4608*(a^4*d^3*f^2*e^(4*c) + a^2*b 
^2*d^3*f^2*e^(4*c))*x^3 + 13824*(a^4*d^3*e*f*e^(4*c) + a^2*b^2*d^3*e*f*e^( 
4*c))*x^2 + 27*(8*b^4*d^2*f^2*x^2*e^(8*c) + 4*(4*d^2*e*f - d*f^2)*b^4*x*e^ 
(8*c) - (4*d*e*f - f^2)*b^4*e^(8*c))*e^(4*d*x) - 64*(9*a*b^3*d^2*f^2*x^2*e 
^(7*c) + 6*(3*d^2*e*f - d*f^2)*a*b^3*x*e^(7*c) - 2*(3*d*e*f - f^2)*a*b^3*e 
^(7*c))*e^(3*d*x) - 432*(2*(2*d*e*f - f^2)*a^2*b^2*e^(6*c) + (2*d*e*f - f^ 
2)*b^4*e^(6*c) - 2*(2*a^2*b^2*d^2*f^2*e^(6*c) + b^4*d^2*f^2*e^(6*c))*x^2 - 
 2*(2*(2*d^2*e*f - d*f^2)*a^2*b^2*e^(6*c) + (2*d^2*e*f - d*f^2)*b^4*e^(6*c 
))*x)*e^(2*d*x) + 1728*(8*(d*e*f - f^2)*a^3*b*e^(5*c) + 6*(d*e*f - f^2)*a* 
b^3*e^(5*c) - (4*a^3*b*d^2*f^2*e^(5*c) + 3*a*b^3*d^2*f^2*e^(5*c))*x^2 - 2* 
(4*(d^2*e*f - d*f^2)*a^3*b*e^(5*c) + 3*(d^2*e*f - d*f^2)*a*b^3*e^(5*c))*x) 
*e^(d*x) + 1728*(8*(d*e*f + f^2)*a^3*b*e^(3*c) + 6*(d*e*f + f^2)*a*b^3*e^( 
3*c) + (4*a^3*b*d^2*f^2*e^(3*c) + 3*a*b^3*d^2*f^2*e^(3*c))*x^2 + 2*(4*(d^2 
*e*f + d*f^2)*a^3*b*e^(3*c) + 3*(d^2*e*f + d*f^2)*a*b^3*e^(3*c))*x)*e^(-d* 
x) + 432*(2*(2*d*e*f + f^2)*a^2*b^2*e^(2*c) + (2*d*e*f + f^2)*b^4*e^(2*...
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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