\(\int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) [396]

Optimal result

Integrand size = 36, antiderivative size = 1022 \[ \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {3 a^2 f (e+f x)^2}{8 b^3 d^2}+\frac {a^4 (e+f x)^4}{4 b^5 f}+\frac {a^2 (e+f x)^4}{8 b^3 f}-\frac {(e+f x)^4}{32 b f}-\frac {6 a^3 f^2 (e+f x) \cosh (c+d x)}{b^4 d^3}-\frac {4 a f^2 (e+f x) \cosh (c+d x)}{3 b^2 d^3}-\frac {a^3 (e+f x)^3 \cosh (c+d x)}{b^4 d}-\frac {3 a^2 f^3 \cosh ^2(c+d x)}{8 b^3 d^4}-\frac {3 a^2 f (e+f x)^2 \cosh ^2(c+d x)}{4 b^3 d^2}-\frac {2 a f^2 (e+f x) \cosh ^3(c+d x)}{9 b^2 d^3}-\frac {a (e+f x)^3 \cosh ^3(c+d x)}{3 b^2 d}-\frac {3 f^3 \cosh (4 c+4 d x)}{1024 b d^4}-\frac {3 f (e+f x)^2 \cosh (4 c+4 d x)}{128 b d^2}-\frac {a^3 \sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^5 d}+\frac {a^3 \sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^5 d}-\frac {3 a^3 \sqrt {a^2+b^2} f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^5 d^2}+\frac {3 a^3 \sqrt {a^2+b^2} f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^5 d^2}+\frac {6 a^3 \sqrt {a^2+b^2} f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^5 d^3}-\frac {6 a^3 \sqrt {a^2+b^2} f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^5 d^3}-\frac {6 a^3 \sqrt {a^2+b^2} f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^5 d^4}+\frac {6 a^3 \sqrt {a^2+b^2} f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^5 d^4}+\frac {6 a^3 f^3 \sinh (c+d x)}{b^4 d^4}+\frac {14 a f^3 \sinh (c+d x)}{9 b^2 d^4}+\frac {3 a^3 f (e+f x)^2 \sinh (c+d x)}{b^4 d^2}+\frac {2 a f (e+f x)^2 \sinh (c+d x)}{3 b^2 d^2}+\frac {3 a^2 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 b^3 d^3}+\frac {a^2 (e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b^3 d}+\frac {a f (e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 b^2 d^2}+\frac {2 a f^3 \sinh ^3(c+d x)}{27 b^2 d^4}+\frac {3 f^2 (e+f x) \sinh (4 c+4 d x)}{256 b d^3}+\frac {(e+f x)^3 \sinh (4 c+4 d x)}{32 b d} \] Output:

-6*a^3*(a^2+b^2)^(1/2)*f^2*(f*x+e)*polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1 
/2)))/b^5/d^3+6*a^3*(a^2+b^2)^(1/2)*f^2*(f*x+e)*polylog(3,-b*exp(d*x+c)/(a 
-(a^2+b^2)^(1/2)))/b^5/d^3+3*a^3*(a^2+b^2)^(1/2)*f*(f*x+e)^2*polylog(2,-b* 
exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^5/d^2-3*a^3*(a^2+b^2)^(1/2)*f*(f*x+e)^2* 
polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^5/d^2+3*a^3*f*(f*x+e)^2*sin 
h(d*x+c)/b^4/d^2-3/4*a^2*f*(f*x+e)^2*cosh(d*x+c)^2/b^3/d^2-2/9*a*f^2*(f*x+ 
e)*cosh(d*x+c)^3/b^2/d^3+1/2*a^2*(f*x+e)^3*cosh(d*x+c)*sinh(d*x+c)/b^3/d-6 
*a^3*f^2*(f*x+e)*cosh(d*x+c)/b^4/d^3+3/4*a^2*f^2*(f*x+e)*cosh(d*x+c)*sinh( 
d*x+c)/b^3/d^3+14/9*a*f^3*sinh(d*x+c)/b^2/d^4+3/256*f^2*(f*x+e)*sinh(4*d*x 
+4*c)/b/d^3+6*a^3*f^3*sinh(d*x+c)/b^4/d^4+2/27*a*f^3*sinh(d*x+c)^3/b^2/d^4 
-3/128*f*(f*x+e)^2*cosh(4*d*x+4*c)/b/d^2-3/8*a^2*f^3*cosh(d*x+c)^2/b^3/d^4 
-1/3*a*(f*x+e)^3*cosh(d*x+c)^3/b^2/d+3/8*a^2*f*(f*x+e)^2/b^3/d^2-a^3*(f*x+ 
e)^3*cosh(d*x+c)/b^4/d+1/3*a*f*(f*x+e)^2*cosh(d*x+c)^2*sinh(d*x+c)/b^2/d^2 
+1/32*(f*x+e)^3*sinh(4*d*x+4*c)/b/d-3/1024*f^3*cosh(4*d*x+4*c)/b/d^4+1/4*a 
^4*(f*x+e)^4/b^5/f+6*a^3*(a^2+b^2)^(1/2)*f^3*polylog(4,-b*exp(d*x+c)/(a+(a 
^2+b^2)^(1/2)))/b^5/d^4-6*a^3*(a^2+b^2)^(1/2)*f^3*polylog(4,-b*exp(d*x+c)/ 
(a-(a^2+b^2)^(1/2)))/b^5/d^4+a^3*(a^2+b^2)^(1/2)*(f*x+e)^3*ln(1+b*exp(d*x+ 
c)/(a+(a^2+b^2)^(1/2)))/b^5/d-a^3*(a^2+b^2)^(1/2)*(f*x+e)^3*ln(1+b*exp(d*x 
+c)/(a-(a^2+b^2)^(1/2)))/b^5/d-4/3*a*f^2*(f*x+e)*cosh(d*x+c)/b^2/d^3+2/3*a 
*f*(f*x+e)^2*sinh(d*x+c)/b^2/d^2-1/32*(f*x+e)^4/b/f+1/8*a^2*(f*x+e)^4/b...
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(5984\) vs. \(2(1022)=2044\).

Time = 20.26 (sec) , antiderivative size = 5984, normalized size of antiderivative = 5.86 \[ \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Result too large to show} \] Input:

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

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

\(\Big \downarrow \) 6113

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

\(\Big \downarrow \) 5971

\(\displaystyle \frac {\int \left (\frac {1}{8} (e+f x)^3 \cosh (4 c+4 d x)-\frac {1}{8} (e+f x)^3\right )dx}{b}-\frac {a \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {3 f^3 \cosh (4 c+4 d x)}{1024 d^4}+\frac {3 f^2 (e+f x) \sinh (4 c+4 d x)}{256 d^3}-\frac {3 f (e+f x)^2 \cosh (4 c+4 d x)}{128 d^2}+\frac {(e+f x)^3 \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^4}{32 f}}{b}-\frac {a \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\)

\(\Big \downarrow \) 6113

\(\displaystyle \frac {-\frac {3 f^3 \cosh (4 c+4 d x)}{1024 d^4}+\frac {3 f^2 (e+f x) \sinh (4 c+4 d x)}{256 d^3}-\frac {3 f (e+f x)^2 \cosh (4 c+4 d x)}{128 d^2}+\frac {(e+f x)^3 \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^4}{32 f}}{b}-\frac {a \left (\frac {\int (e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)dx}{b}-\frac {a \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\)

\(\Big \downarrow \) 5970

\(\displaystyle \frac {-\frac {3 f^3 \cosh (4 c+4 d x)}{1024 d^4}+\frac {3 f^2 (e+f x) \sinh (4 c+4 d x)}{256 d^3}-\frac {3 f (e+f x)^2 \cosh (4 c+4 d x)}{128 d^2}+\frac {(e+f x)^3 \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^4}{32 f}}{b}-\frac {a \left (\frac {\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 d}-\frac {f \int (e+f x)^2 \cosh ^3(c+d x)dx}{d}}{b}-\frac {a \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {3 f^3 \cosh (4 c+4 d x)}{1024 d^4}+\frac {3 f^2 (e+f x) \sinh (4 c+4 d x)}{256 d^3}-\frac {3 f (e+f x)^2 \cosh (4 c+4 d x)}{128 d^2}+\frac {(e+f x)^3 \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^4}{32 f}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 d}-\frac {f \int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )^3dx}{d}}{b}\right )}{b}\)

\(\Big \downarrow \) 3792

\(\displaystyle \frac {-\frac {3 f^3 \cosh (4 c+4 d x)}{1024 d^4}+\frac {3 f^2 (e+f x) \sinh (4 c+4 d x)}{256 d^3}-\frac {3 f (e+f x)^2 \cosh (4 c+4 d x)}{128 d^2}+\frac {(e+f x)^3 \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^4}{32 f}}{b}-\frac {a \left (\frac {\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 d}-\frac {f \left (\frac {2 f^2 \int \cosh ^3(c+d x)dx}{9 d^2}+\frac {2}{3} \int (e+f x)^2 \cosh (c+d x)dx-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{d}}{b}-\frac {a \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {3 f^3 \cosh (4 c+4 d x)}{1024 d^4}+\frac {3 f^2 (e+f x) \sinh (4 c+4 d x)}{256 d^3}-\frac {3 f (e+f x)^2 \cosh (4 c+4 d x)}{128 d^2}+\frac {(e+f x)^3 \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^4}{32 f}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 d}-\frac {f \left (\frac {2 f^2 \int \sin \left (i c+i d x+\frac {\pi }{2}\right )^3dx}{9 d^2}+\frac {2}{3} \int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )dx-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{d}}{b}\right )}{b}\)

\(\Big \downarrow \) 3113

\(\displaystyle \frac {-\frac {3 f^3 \cosh (4 c+4 d x)}{1024 d^4}+\frac {3 f^2 (e+f x) \sinh (4 c+4 d x)}{256 d^3}-\frac {3 f (e+f x)^2 \cosh (4 c+4 d x)}{128 d^2}+\frac {(e+f x)^3 \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^4}{32 f}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 d}-\frac {f \left (\frac {2 i f^2 \int \left (\sinh ^2(c+d x)+1\right )d(-i \sinh (c+d x))}{9 d^3}+\frac {2}{3} \int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )dx-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{d}}{b}\right )}{b}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {3 f^3 \cosh (4 c+4 d x)}{1024 d^4}+\frac {3 f^2 (e+f x) \sinh (4 c+4 d x)}{256 d^3}-\frac {3 f (e+f x)^2 \cosh (4 c+4 d x)}{128 d^2}+\frac {(e+f x)^3 \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^4}{32 f}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 d}-\frac {f \left (\frac {2}{3} \int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )dx+\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{d}}{b}\right )}{b}\)

\(\Big \downarrow \) 3777

\(\displaystyle \frac {-\frac {3 f^3 \cosh (4 c+4 d x)}{1024 d^4}+\frac {3 f^2 (e+f x) \sinh (4 c+4 d x)}{256 d^3}-\frac {3 f (e+f x)^2 \cosh (4 c+4 d x)}{128 d^2}+\frac {(e+f x)^3 \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^4}{32 f}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 d}-\frac {f \left (\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 i f \int -i (e+f x) \sinh (c+d x)dx}{d}\right )+\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{d}}{b}\right )}{b}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {-\frac {3 f^3 \cosh (4 c+4 d x)}{1024 d^4}+\frac {3 f^2 (e+f x) \sinh (4 c+4 d x)}{256 d^3}-\frac {3 f (e+f x)^2 \cosh (4 c+4 d x)}{128 d^2}+\frac {(e+f x)^3 \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^4}{32 f}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 d}-\frac {f \left (\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 f \int (e+f x) \sinh (c+d x)dx}{d}\right )+\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{d}}{b}\right )}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {3 f^3 \cosh (4 c+4 d x)}{1024 d^4}+\frac {3 f^2 (e+f x) \sinh (4 c+4 d x)}{256 d^3}-\frac {3 f (e+f x)^2 \cosh (4 c+4 d x)}{128 d^2}+\frac {(e+f x)^3 \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^4}{32 f}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 d}-\frac {f \left (\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 f \int -i (e+f x) \sin (i c+i d x)dx}{d}\right )+\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{d}}{b}\right )}{b}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {-\frac {3 f^3 \cosh (4 c+4 d x)}{1024 d^4}+\frac {3 f^2 (e+f x) \sinh (4 c+4 d x)}{256 d^3}-\frac {3 f (e+f x)^2 \cosh (4 c+4 d x)}{128 d^2}+\frac {(e+f x)^3 \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^4}{32 f}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 d}-\frac {f \left (\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \int (e+f x) \sin (i c+i d x)dx}{d}\right )+\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{d}}{b}\right )}{b}\)

\(\Big \downarrow \) 3777

\(\displaystyle \frac {-\frac {3 f^3 \cosh (4 c+4 d x)}{1024 d^4}+\frac {3 f^2 (e+f x) \sinh (4 c+4 d x)}{256 d^3}-\frac {3 f (e+f x)^2 \cosh (4 c+4 d x)}{128 d^2}+\frac {(e+f x)^3 \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^4}{32 f}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 d}-\frac {f \left (\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \cosh (c+d x)dx}{d}\right )}{d}\right )+\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{d}}{b}\right )}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {3 f^3 \cosh (4 c+4 d x)}{1024 d^4}+\frac {3 f^2 (e+f x) \sinh (4 c+4 d x)}{256 d^3}-\frac {3 f (e+f x)^2 \cosh (4 c+4 d x)}{128 d^2}+\frac {(e+f x)^3 \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^4}{32 f}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 d}-\frac {f \left (\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{d}\right )+\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{d}}{b}\right )}{b}\)

\(\Big \downarrow \) 3117

\(\displaystyle \frac {-\frac {3 f^3 \cosh (4 c+4 d x)}{1024 d^4}+\frac {3 f^2 (e+f x) \sinh (4 c+4 d x)}{256 d^3}-\frac {3 f (e+f x)^2 \cosh (4 c+4 d x)}{128 d^2}+\frac {(e+f x)^3 \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^4}{32 f}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 d}-\frac {f \left (\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{d}}{b}\right )}{b}\)

\(\Big \downarrow \) 6113

\(\displaystyle \frac {-\frac {3 f^3 \cosh (4 c+4 d x)}{1024 d^4}+\frac {3 f^2 (e+f x) \sinh (4 c+4 d x)}{256 d^3}-\frac {3 f (e+f x)^2 \cosh (4 c+4 d x)}{128 d^2}+\frac {(e+f x)^3 \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^4}{32 f}}{b}-\frac {a \left (-\frac {a \left (\frac {\int (e+f x)^3 \cosh ^2(c+d x)dx}{b}-\frac {a \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}+\frac {\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 d}-\frac {f \left (\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{d}}{b}\right )}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {3 f^3 \cosh (4 c+4 d x)}{1024 d^4}+\frac {3 f^2 (e+f x) \sinh (4 c+4 d x)}{256 d^3}-\frac {3 f (e+f x)^2 \cosh (4 c+4 d x)}{128 d^2}+\frac {(e+f x)^3 \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^4}{32 f}}{b}-\frac {a \left (\frac {\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 d}-\frac {f \left (\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\int (e+f x)^3 \sin \left (i c+i d x+\frac {\pi }{2}\right )^2dx}{b}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 3792

\(\displaystyle \frac {-\frac {3 f^3 \cosh (4 c+4 d x)}{1024 d^4}+\frac {3 f^2 (e+f x) \sinh (4 c+4 d x)}{256 d^3}-\frac {3 f (e+f x)^2 \cosh (4 c+4 d x)}{128 d^2}+\frac {(e+f x)^3 \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^4}{32 f}}{b}-\frac {a \left (-\frac {a \left (\frac {\frac {3 f^2 \int (e+f x) \cosh ^2(c+d x)dx}{2 d^2}+\frac {1}{2} \int (e+f x)^3dx-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}}{b}-\frac {a \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}+\frac {\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 d}-\frac {f \left (\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{d}}{b}\right )}{b}\)

\(\Big \downarrow \) 17

\(\displaystyle \frac {-\frac {3 f^3 \cosh (4 c+4 d x)}{1024 d^4}+\frac {3 f^2 (e+f x) \sinh (4 c+4 d x)}{256 d^3}-\frac {3 f (e+f x)^2 \cosh (4 c+4 d x)}{128 d^2}+\frac {(e+f x)^3 \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^4}{32 f}}{b}-\frac {a \left (-\frac {a \left (\frac {\frac {3 f^2 \int (e+f x) \cosh ^2(c+d x)dx}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}+\frac {\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 d}-\frac {f \left (\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{d}}{b}\right )}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {3 f^3 \cosh (4 c+4 d x)}{1024 d^4}+\frac {3 f^2 (e+f x) \sinh (4 c+4 d x)}{256 d^3}-\frac {3 f (e+f x)^2 \cosh (4 c+4 d x)}{128 d^2}+\frac {(e+f x)^3 \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^4}{32 f}}{b}-\frac {a \left (\frac {\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 d}-\frac {f \left (\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {3 f^2 \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )^2dx}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 3791

\(\displaystyle \frac {-\frac {3 f^3 \cosh (4 c+4 d x)}{1024 d^4}+\frac {3 f^2 (e+f x) \sinh (4 c+4 d x)}{256 d^3}-\frac {3 f (e+f x)^2 \cosh (4 c+4 d x)}{128 d^2}+\frac {(e+f x)^3 \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^4}{32 f}}{b}-\frac {a \left (-\frac {a \left (\frac {\frac {3 f^2 \left (\frac {1}{2} \int (e+f x)dx-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}+\frac {\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 d}-\frac {f \left (\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{d}}{b}\right )}{b}\)

\(\Big \downarrow \) 17

\(\displaystyle \frac {-\frac {3 f^3 \cosh (4 c+4 d x)}{1024 d^4}+\frac {3 f^2 (e+f x) \sinh (4 c+4 d x)}{256 d^3}-\frac {3 f (e+f x)^2 \cosh (4 c+4 d x)}{128 d^2}+\frac {(e+f x)^3 \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^4}{32 f}}{b}-\frac {a \left (-\frac {a \left (\frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}+\frac {\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 d}-\frac {f \left (\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{d}}{b}\right )}{b}\)

\(\Big \downarrow \) 6099

\(\displaystyle \frac {-\frac {3 f^3 \cosh (4 c+4 d x)}{1024 d^4}+\frac {3 f^2 (e+f x) \sinh (4 c+4 d x)}{256 d^3}-\frac {3 f (e+f x)^2 \cosh (4 c+4 d x)}{128 d^2}+\frac {(e+f x)^3 \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^4}{32 f}}{b}-\frac {a \left (-\frac {a \left (\frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \int (e+f x)^3dx}{b^2}+\frac {\int (e+f x)^3 \sinh (c+d x)dx}{b}\right )}{b}\right )}{b}+\frac {\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 d}-\frac {f \left (\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{d}}{b}\right )}{b}\)

\(\Big \downarrow \) 17

\(\displaystyle \frac {-\frac {3 f^3 \cosh (4 c+4 d x)}{1024 d^4}+\frac {3 f^2 (e+f x) \sinh (4 c+4 d x)}{256 d^3}-\frac {3 f (e+f x)^2 \cosh (4 c+4 d x)}{128 d^2}+\frac {(e+f x)^3 \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^4}{32 f}}{b}-\frac {a \left (-\frac {a \left (\frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{a+b \sinh (c+d x)}dx}{b^2}+\frac {\int (e+f x)^3 \sinh (c+d x)dx}{b}-\frac {a (e+f x)^4}{4 b^2 f}\right )}{b}\right )}{b}+\frac {\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 d}-\frac {f \left (\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{d}}{b}\right )}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {3 f^3 \cosh (4 c+4 d x)}{1024 d^4}+\frac {3 f^2 (e+f x) \sinh (4 c+4 d x)}{256 d^3}-\frac {3 f (e+f x)^2 \cosh (4 c+4 d x)}{128 d^2}+\frac {(e+f x)^3 \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^4}{32 f}}{b}-\frac {a \left (\frac {\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 d}-\frac {f \left (\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{d}}{b}-\frac {a \left (\frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{a-i b \sin (i c+i d x)}dx}{b^2}+\frac {\int -i (e+f x)^3 \sin (i c+i d x)dx}{b}-\frac {a (e+f x)^4}{4 b^2 f}\right )}{b}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {-\frac {3 f^3 \cosh (4 c+4 d x)}{1024 d^4}+\frac {3 f^2 (e+f x) \sinh (4 c+4 d x)}{256 d^3}-\frac {3 f (e+f x)^2 \cosh (4 c+4 d x)}{128 d^2}+\frac {(e+f x)^3 \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^4}{32 f}}{b}-\frac {a \left (\frac {\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 d}-\frac {f \left (\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{d}}{b}-\frac {a \left (\frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \int (e+f x)^3 \sin (i c+i d x)dx}{b}-\frac {a (e+f x)^4}{4 b^2 f}\right )}{b}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 3777

\(\displaystyle \frac {-\frac {3 f^3 \cosh (4 c+4 d x)}{1024 d^4}+\frac {3 f^2 (e+f x) \sinh (4 c+4 d x)}{256 d^3}-\frac {3 f (e+f x)^2 \cosh (4 c+4 d x)}{128 d^2}+\frac {(e+f x)^3 \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^4}{32 f}}{b}-\frac {a \left (\frac {\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 d}-\frac {f \left (\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{d}}{b}-\frac {a \left (\frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \int (e+f x)^2 \cosh (c+d x)dx}{d}\right )}{b}-\frac {a (e+f x)^4}{4 b^2 f}\right )}{b}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {3 f^3 \cosh (4 c+4 d x)}{1024 d^4}+\frac {3 f^2 (e+f x) \sinh (4 c+4 d x)}{256 d^3}-\frac {3 f (e+f x)^2 \cosh (4 c+4 d x)}{128 d^2}+\frac {(e+f x)^3 \sinh (4 c+4 d x)}{32 d}-\frac {(e+f x)^4}{32 f}}{b}-\frac {a \left (\frac {\frac {(e+f x)^3 \cosh ^3(c+d x)}{3 d}-\frac {f \left (\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}\right )}{d}}{b}-\frac {a \left (\frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{b}-\frac {a (e+f x)^4}{4 b^2 f}\right )}{b}\right )}{b}\right )}{b}\)

Input:

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

Output:

$Aborted
 

Maple [F]

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

Input:

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

Output:

int((f*x+e)^3*cosh(d*x+c)^2*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. 10658 vs. \(2 (942) = 1884\).

Time = 0.29 (sec) , antiderivative size = 10658, normalized size of antiderivative = 10.43 \[ \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

-1/192*e^3*(192*sqrt(a^2 + b^2)*a^3*log((b*e^(-d*x - c) - a - sqrt(a^2 + b 
^2))/(b*e^(-d*x - c) - a + sqrt(a^2 + b^2)))/(b^5*d) + (8*a*b^2*e^(-d*x - 
c) - 24*a^2*b*e^(-2*d*x - 2*c) - 3*b^3 + 24*(4*a^3 + a*b^2)*e^(-3*d*x - 3* 
c))*e^(4*d*x + 4*c)/(b^4*d) - 24*(8*a^4 + 4*a^2*b^2 - b^4)*(d*x + c)/(b^5* 
d) + (24*a^2*b*e^(-2*d*x - 2*c) + 8*a*b^2*e^(-3*d*x - 3*c) + 3*b^3*e^(-4*d 
*x - 4*c) + 24*(4*a^3 + a*b^2)*e^(-d*x - c))/(b^4*d)) + 1/55296*(1728*(8*a 
^4*d^4*f^3*e^(4*c) + 4*a^2*b^2*d^4*f^3*e^(4*c) - b^4*d^4*f^3*e^(4*c))*x^4 
+ 6912*(8*a^4*d^4*e*f^2*e^(4*c) + 4*a^2*b^2*d^4*e*f^2*e^(4*c) - b^4*d^4*e* 
f^2*e^(4*c))*x^3 + 10368*(8*a^4*d^4*e^2*f*e^(4*c) + 4*a^2*b^2*d^4*e^2*f*e^ 
(4*c) - b^4*d^4*e^2*f*e^(4*c))*x^2 + 27*(32*b^4*d^3*f^3*x^3*e^(8*c) + 24*( 
4*d^3*e*f^2 - d^2*f^3)*b^4*x^2*e^(8*c) + 12*(8*d^3*e^2*f - 4*d^2*e*f^2 + d 
*f^3)*b^4*x*e^(8*c) - 3*(8*d^2*e^2*f - 4*d*e*f^2 + f^3)*b^4*e^(8*c))*e^(4* 
d*x) - 256*(9*a*b^3*d^3*f^3*x^3*e^(7*c) + 9*(3*d^3*e*f^2 - d^2*f^3)*a*b^3* 
x^2*e^(7*c) + 3*(9*d^3*e^2*f - 6*d^2*e*f^2 + 2*d*f^3)*a*b^3*x*e^(7*c) - (9 
*d^2*e^2*f - 6*d*e*f^2 + 2*f^3)*a*b^3*e^(7*c))*e^(3*d*x) + 1728*(4*a^2*b^2 
*d^3*f^3*x^3*e^(6*c) + 6*(2*d^3*e*f^2 - d^2*f^3)*a^2*b^2*x^2*e^(6*c) + 6*( 
2*d^3*e^2*f - 2*d^2*e*f^2 + d*f^3)*a^2*b^2*x*e^(6*c) - 3*(2*d^2*e^2*f - 2* 
d*e*f^2 + f^3)*a^2*b^2*e^(6*c))*e^(2*d*x) + 6912*(12*(d^2*e^2*f - 2*d*e*f^ 
2 + 2*f^3)*a^3*b*e^(5*c) + 3*(d^2*e^2*f - 2*d*e*f^2 + 2*f^3)*a*b^3*e^(5*c) 
 - (4*a^3*b*d^3*f^3*e^(5*c) + a*b^3*d^3*f^3*e^(5*c))*x^3 - 3*(4*(d^3*e*...
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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