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

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:

-3/8*a*f^3*x/b^2/d^3-1/2*a*(f*x+e)^3*sinh(d*x+c)^2/b^2/d-6*a^2*f^3*cosh(d* 
x+c)/b^3/d^4-1/3*f*(f*x+e)^2*cosh(d*x+c)*sinh(d*x+c)^2/b/d^2+6*a^2*f^2*(f* 
x+e)*sinh(d*x+c)/b^3/d^3-3/4*a*f^2*(f*x+e)*sinh(d*x+c)^2/b^2/d^3+3/8*a*f^3 
*cosh(d*x+c)*sinh(d*x+c)/b^2/d^4-3*a^2*f*(f*x+e)^2*cosh(d*x+c)/b^3/d^2+1/3 
*(f*x+e)^3*sinh(d*x+c)^3/b/d+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^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-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-6*a^3*f^3*pol 
ylog(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*e 
xp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^4/d^4+14/9*f^3*cosh(d*x+c)/b/d^4-2/27*f^3 
*cosh(d*x+c)^3/b/d^4-1/4*a*(f*x+e)^3/b^2/d+a^2*(f*x+e)^3*sinh(d*x+c)/b^3/d 
+2/9*f^2*(f*x+e)*sinh(d*x+c)^3/b/d^3+1/4*a^3*(f*x+e)^4/b^4/f+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+3/4*a*f*(f*x+e)^2*c 
osh(d*x+c)*sinh(d*x+c)/b^2/d^2
 

Mathematica [B] (warning: unable to verify)

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

Time = 27.26 (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
 

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
 

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)
 

Fricas [B] (verification not implemented)

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

Time = 0.20 (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
 

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
 

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)...
 

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)
 

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)
                                                                                    
                                                                                    
 

Reduce [F]

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