3.4.91 \(\int \frac {(e+f x)^3 \cosh (c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) [391]

3.4.91.1 Optimal result

Integrand size = 34, antiderivative size = 792 \[ \int \frac {(e+f x)^3 \cosh (c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {3 a f^3 x}{8 b^2 d^3}-\frac {a (e+f x)^3}{4 b^2 d}+\frac {a^3 (e+f x)^4}{4 b^4 f}-\frac {6 a^2 f^3 \cosh (c+d x)}{b^3 d^4}+\frac {14 f^3 \cosh (c+d x)}{9 b d^4}-\frac {3 a^2 f (e+f x)^2 \cosh (c+d x)}{b^3 d^2}+\frac {2 f (e+f x)^2 \cosh (c+d x)}{3 b d^2}-\frac {2 f^3 \cosh ^3(c+d x)}{27 b d^4}-\frac {a^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d}-\frac {a^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d}-\frac {3 a^3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^2}-\frac {3 a^3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^2}+\frac {6 a^3 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^3}+\frac {6 a^3 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^3}-\frac {6 a^3 f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^4}-\frac {6 a^3 f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^4}+\frac {6 a^2 f^2 (e+f x) \sinh (c+d x)}{b^3 d^3}-\frac {4 f^2 (e+f x) \sinh (c+d x)}{3 b d^3}+\frac {a^2 (e+f x)^3 \sinh (c+d x)}{b^3 d}+\frac {3 a f^3 \cosh (c+d x) \sinh (c+d x)}{8 b^2 d^4}+\frac {3 a f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 b^2 d^2}-\frac {3 a f^2 (e+f x) \sinh ^2(c+d x)}{4 b^2 d^3}-\frac {a (e+f x)^3 \sinh ^2(c+d x)}{2 b^2 d}-\frac {f (e+f x)^2 \cosh (c+d x) \sinh ^2(c+d x)}{3 b d^2}+\frac {2 f^2 (e+f x) \sinh ^3(c+d x)}{9 b d^3}+\frac {(e+f x)^3 \sinh ^3(c+d x)}{3 b d} \]

output

-1/3*f*(f*x+e)^2*cosh(d*x+c)*sinh(d*x+c)^2/b/d^2-3*a^3*f*(f*x+e)^2*polylog 
(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^4/d^2-3*a^3*f*(f*x+e)^2*polylog(2, 
-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^4/d^2+6*a^3*f^2*(f*x+e)*polylog(3,-b* 
exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^4/d^3+6*a^3*f^2*(f*x+e)*polylog(3,-b*exp 
(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^4/d^3-3*a^2*f*(f*x+e)^2*cosh(d*x+c)/b^3/d^2 
+2/3*f*(f*x+e)^2*cosh(d*x+c)/b/d^2-4/3*f^2*(f*x+e)*sinh(d*x+c)/b/d^3-3/8*a 
*f^3*x/b^2/d^3-6*a^2*f^3*cosh(d*x+c)/b^3/d^4-1/2*a*(f*x+e)^3*sinh(d*x+c)^2 
/b^2/d+1/3*(f*x+e)^3*sinh(d*x+c)^3/b/d+a^2*(f*x+e)^3*sinh(d*x+c)/b^3/d-a^3 
*(f*x+e)^3*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^4/d-a^3*(f*x+e)^3*ln(1 
+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^4/d+1/4*a^3*(f*x+e)^4/b^4/f+2/9*f^2*( 
f*x+e)*sinh(d*x+c)^3/b/d^3-6*a^3*f^3*polylog(4,-b*exp(d*x+c)/(a-(a^2+b^2)^ 
(1/2)))/b^4/d^4-6*a^3*f^3*polylog(4,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^4 
/d^4-1/4*a*(f*x+e)^3/b^2/d-2/27*f^3*cosh(d*x+c)^3/b/d^4+3/4*a*f*(f*x+e)^2* 
cosh(d*x+c)*sinh(d*x+c)/b^2/d^2+14/9*f^3*cosh(d*x+c)/b/d^4+6*a^2*f^2*(f*x+ 
e)*sinh(d*x+c)/b^3/d^3+3/8*a*f^3*cosh(d*x+c)*sinh(d*x+c)/b^2/d^4-3/4*a*f^2 
*(f*x+e)*sinh(d*x+c)^2/b^2/d^3
 

3.4.91.2 Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(5656\) vs. \(2(792)=1584\).

Time = 28.49 (sec) , antiderivative size = 5656, normalized size of antiderivative = 7.14 \[ \int \frac {(e+f x)^3 \cosh (c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Result too large to show} \]

input

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

output

Result too large to show
 

3.4.91.3 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.

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

\(\Big \downarrow \) 6113

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

\(\Big \downarrow \) 5969

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3792

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3113

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3118

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

\(\Big \downarrow \) 6113

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

\(\Big \downarrow \) 5969

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3792

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

\(\Big \downarrow \) 17

\(\displaystyle -\frac {a \left (\frac {\frac {3 f \left (\frac {f^2 \int -\sinh ^2(c+d x)dx}{2 d^2}+\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{2 d}+\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 d}}{b}-\frac {a \int \frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}+\frac {\frac {(e+f x)^3 \sinh ^3(c+d x)}{3 d}-\frac {i f \left (\frac {2 i f^2 \left (\cosh (c+d x)-\frac {1}{3} \cosh ^3(c+d x)\right )}{9 d^3}+\frac {2 i f (e+f x) \sinh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )-\frac {i (e+f x)^2 \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{d}}{b}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {a \left (\frac {\frac {3 f \left (-\frac {f^2 \int \sinh ^2(c+d x)dx}{2 d^2}+\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{2 d}+\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 d}}{b}-\frac {a \int \frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}+\frac {\frac {(e+f x)^3 \sinh ^3(c+d x)}{3 d}-\frac {i f \left (\frac {2 i f^2 \left (\cosh (c+d x)-\frac {1}{3} \cosh ^3(c+d x)\right )}{9 d^3}+\frac {2 i f (e+f x) \sinh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )-\frac {i (e+f x)^2 \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{d}}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {(e+f x)^3 \sinh ^3(c+d x)}{3 d}-\frac {i f \left (\frac {2 i f^2 \left (\cosh (c+d x)-\frac {1}{3} \cosh ^3(c+d x)\right )}{9 d^3}+\frac {2 i f (e+f x) \sinh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )-\frac {i (e+f x)^2 \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 d}+\frac {3 f \left (-\frac {f^2 \int -\sin (i c+i d x)^2dx}{2 d^2}+\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{2 d}}{b}\right )}{b}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {(e+f x)^3 \sinh ^3(c+d x)}{3 d}-\frac {i f \left (\frac {2 i f^2 \left (\cosh (c+d x)-\frac {1}{3} \cosh ^3(c+d x)\right )}{9 d^3}+\frac {2 i f (e+f x) \sinh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )-\frac {i (e+f x)^2 \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 d}+\frac {3 f \left (\frac {f^2 \int \sin (i c+i d x)^2dx}{2 d^2}+\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{2 d}}{b}\right )}{b}\)

\(\Big \downarrow \) 3115

\(\displaystyle -\frac {a \left (\frac {\frac {3 f \left (\frac {f^2 \left (\frac {\int 1dx}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d^2}+\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{2 d}+\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 d}}{b}-\frac {a \int \frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}+\frac {\frac {(e+f x)^3 \sinh ^3(c+d x)}{3 d}-\frac {i f \left (\frac {2 i f^2 \left (\cosh (c+d x)-\frac {1}{3} \cosh ^3(c+d x)\right )}{9 d^3}+\frac {2 i f (e+f x) \sinh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )-\frac {i (e+f x)^2 \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{d}}{b}\)

\(\Big \downarrow \) 24

\(\displaystyle -\frac {a \left (\frac {\frac {3 f \left (\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{2 d}+\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 d}}{b}-\frac {a \int \frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}+\frac {\frac {(e+f x)^3 \sinh ^3(c+d x)}{3 d}-\frac {i f \left (\frac {2 i f^2 \left (\cosh (c+d x)-\frac {1}{3} \cosh ^3(c+d x)\right )}{9 d^3}+\frac {2 i f (e+f x) \sinh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )-\frac {i (e+f x)^2 \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{d}}{b}\)

\(\Big \downarrow \) 6113

\(\displaystyle -\frac {a \left (\frac {\frac {3 f \left (\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{2 d}+\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 d}}{b}-\frac {a \left (\frac {\int (e+f x)^3 \cosh (c+d x)dx}{b}-\frac {a \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\right )}{b}+\frac {\frac {(e+f x)^3 \sinh ^3(c+d x)}{3 d}-\frac {i f \left (\frac {2 i f^2 \left (\cosh (c+d x)-\frac {1}{3} \cosh ^3(c+d x)\right )}{9 d^3}+\frac {2 i f (e+f x) \sinh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )-\frac {i (e+f x)^2 \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{d}}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {(e+f x)^3 \sinh ^3(c+d x)}{3 d}-\frac {i f \left (\frac {2 i f^2 \left (\cosh (c+d x)-\frac {1}{3} \cosh ^3(c+d x)\right )}{9 d^3}+\frac {2 i f (e+f x) \sinh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )-\frac {i (e+f x)^2 \sinh ^2(c+d x) \cosh (c+d x)}{3 d}\right )}{d}}{b}-\frac {a \left (\frac {\frac {3 f \left (\frac {f (e+f x) \sinh ^2(c+d x)}{2 d^2}+\frac {f^2 \left (\frac {x}{2}-\frac {\sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d^2}-\frac {(e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{2 d}+\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\int (e+f x)^3 \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{b}\right )}{b}\right )}{b}\)

input

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

output

$Aborted
 

3.4.91.3.1 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 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3113

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1)   Subst[Int[Exp 
and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] 
 && IGtQ[(n - 1)/2, 0]
 

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 6113

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

3.4.91.4 Maple [F]

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

input

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

output

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

3.4.91.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 7020 vs. \(2 (738) = 1476\).

Time = 0.36 (sec) , antiderivative size = 7020, normalized size of antiderivative = 8.86 \[ \int \frac {(e+f x)^3 \cosh (c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]

input

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

output

Too large to include
 

3.4.91.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.4.91.7 Maxima [F]

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

input

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

output

-1/24*e^3*(24*(d*x + c)*a^3/(b^4*d) + 24*a^3*log(-2*a*e^(-d*x - c) + b*e^( 
-2*d*x - 2*c) - b)/(b^4*d) + (3*a*b*e^(-d*x - c) - b^2 - 3*(4*a^2 - b^2)*e 
^(-2*d*x - 2*c))*e^(3*d*x + 3*c)/(b^3*d) + (3*a*b*e^(-2*d*x - 2*c) + b^2*e 
^(-3*d*x - 3*c) + 3*(4*a^2 - b^2)*e^(-d*x - c))/(b^3*d)) - 1/864*(216*a^3* 
d^4*f^3*x^4*e^(3*c) + 864*a^3*d^4*e*f^2*x^3*e^(3*c) + 1296*a^3*d^4*e^2*f*x 
^2*e^(3*c) - 4*(9*b^3*d^3*f^3*x^3*e^(6*c) + 9*(3*d^3*e*f^2 - d^2*f^3)*b^3* 
x^2*e^(6*c) + 3*(9*d^3*e^2*f - 6*d^2*e*f^2 + 2*d*f^3)*b^3*x*e^(6*c) - (9*d 
^2*e^2*f - 6*d*e*f^2 + 2*f^3)*b^3*e^(6*c))*e^(3*d*x) + 27*(4*a*b^2*d^3*f^3 
*x^3*e^(5*c) + 6*(2*d^3*e*f^2 - d^2*f^3)*a*b^2*x^2*e^(5*c) + 6*(2*d^3*e^2* 
f - 2*d^2*e*f^2 + d*f^3)*a*b^2*x*e^(5*c) - 3*(2*d^2*e^2*f - 2*d*e*f^2 + f^ 
3)*a*b^2*e^(5*c))*e^(2*d*x) + 108*(12*(d^2*e^2*f - 2*d*e*f^2 + 2*f^3)*a^2* 
b*e^(4*c) - 3*(d^2*e^2*f - 2*d*e*f^2 + 2*f^3)*b^3*e^(4*c) - (4*a^2*b*d^3*f 
^3*e^(4*c) - b^3*d^3*f^3*e^(4*c))*x^3 - 3*(4*(d^3*e*f^2 - d^2*f^3)*a^2*b*e 
^(4*c) - (d^3*e*f^2 - d^2*f^3)*b^3*e^(4*c))*x^2 - 3*(4*(d^3*e^2*f - 2*d^2* 
e*f^2 + 2*d*f^3)*a^2*b*e^(4*c) - (d^3*e^2*f - 2*d^2*e*f^2 + 2*d*f^3)*b^3*e 
^(4*c))*x)*e^(d*x) + 108*(12*(d^2*e^2*f + 2*d*e*f^2 + 2*f^3)*a^2*b*e^(2*c) 
 - 3*(d^2*e^2*f + 2*d*e*f^2 + 2*f^3)*b^3*e^(2*c) + (4*a^2*b*d^3*f^3*e^(2*c 
) - b^3*d^3*f^3*e^(2*c))*x^3 + 3*(4*(d^3*e*f^2 + d^2*f^3)*a^2*b*e^(2*c) - 
(d^3*e*f^2 + d^2*f^3)*b^3*e^(2*c))*x^2 + 3*(4*(d^3*e^2*f + 2*d^2*e*f^2 + 2 
*d*f^3)*a^2*b*e^(2*c) - (d^3*e^2*f + 2*d^2*e*f^2 + 2*d*f^3)*b^3*e^(2*c)...
 

3.4.91.8 Giac [F]

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

input

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

output

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

3.4.91.9 Mupad [F(-1)]

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

input

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

output

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