\(\int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx\) [344]

Optimal result

Integrand size = 34, antiderivative size = 623 \[ \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {a (e+f x)^2}{4 b^2 d}+\frac {a \left (a^2+b^2\right ) (e+f x)^3}{3 b^4 f}-\frac {2 a^2 f (e+f x) \cosh (c+d x)}{b^3 d^2}-\frac {4 f (e+f x) \cosh (c+d x)}{3 b d^2}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 b d^2}-\frac {a \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d}-\frac {a \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d}-\frac {2 a \left (a^2+b^2\right ) f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^2}-\frac {2 a \left (a^2+b^2\right ) f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^2}+\frac {2 a \left (a^2+b^2\right ) f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^3}+\frac {2 a \left (a^2+b^2\right ) f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^3}+\frac {2 a^2 f^2 \sinh (c+d x)}{b^3 d^3}+\frac {14 f^2 \sinh (c+d x)}{9 b d^3}+\frac {a^2 (e+f x)^2 \sinh (c+d x)}{b^3 d}+\frac {2 (e+f x)^2 \sinh (c+d x)}{3 b d}+\frac {a f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^2 d^2}+\frac {(e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 b d}-\frac {a f^2 \sinh ^2(c+d x)}{4 b^2 d^3}-\frac {a (e+f x)^2 \sinh ^2(c+d x)}{2 b^2 d}+\frac {2 f^2 \sinh ^3(c+d x)}{27 b d^3} \] Output:

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

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(1961\) vs. \(2(623)=1246\).

Time = 7.72 (sec) , antiderivative size = 1961, normalized size of antiderivative = 3.15 \[ \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \] Input:

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

Output:

(f^2*((4*a*x^3)/(-1 + E^(2*c)) - 2*a*x^3*Coth[c] - (6*a*b^2*(d^2*x^2*Log[1 
 + ((a - Sqrt[a^2 + b^2])*E^(-c - d*x))/b] - 2*d*x*PolyLog[2, ((-a + Sqrt[ 
a^2 + b^2])*E^(-c - d*x))/b] - 2*PolyLog[3, ((-a + Sqrt[a^2 + b^2])*E^(-c 
- d*x))/b]))/(Sqrt[a^2 + b^2]*(-a + Sqrt[a^2 + b^2])*d^3) - (6*a*b^2*(d^2* 
x^2*Log[1 + ((a + Sqrt[a^2 + b^2])*E^(-c - d*x))/b] - 2*d*x*PolyLog[2, -(( 
(a + Sqrt[a^2 + b^2])*E^(-c - d*x))/b)] - 2*PolyLog[3, -(((a + Sqrt[a^2 + 
b^2])*E^(-c - d*x))/b)]))/(Sqrt[a^2 + b^2]*(a + Sqrt[a^2 + b^2])*d^3) + (6 
*a^2*(d^2*x^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + 2*d*x*PolyL 
og[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] - 2*PolyLog[3, (b*E^(c + d*x 
))/(-a + Sqrt[a^2 + b^2])]))/(Sqrt[a^2 + b^2]*d^3) - (6*a^2*(d^2*x^2*Log[1 
 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + 2*d*x*PolyLog[2, -((b*E^(c + d 
*x))/(a + Sqrt[a^2 + b^2]))] - 2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^ 
2 + b^2]))]))/(Sqrt[a^2 + b^2]*d^3) + (6*b*Cosh[d*x]*(-2*d*x*Cosh[c] + (2 
+ d^2*x^2)*Sinh[c]))/d^3 + (6*b*((2 + d^2*x^2)*Cosh[c] - 2*d*x*Sinh[c])*Si 
nh[d*x])/d^3))/(12*b^2) - (e^2*((a*Log[a + b*Sinh[c + d*x]])/b^2 - Sinh[c 
+ d*x]/b))/(2*d) + (e*f*(-2*b*Cosh[c + d*x] - a*(2*c*(c + d*x) - (c + d*x) 
^2 + 2*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + 2*(c + d 
*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] - 2*c*Log[b - 2*a*E^(c 
+ d*x) - b*E^(2*(c + d*x))] + 2*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 
+ b^2])] + 2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]) + 2*...
 

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.

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

\(\Big \downarrow \) 6113

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3792

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3113

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3117

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

\(\Big \downarrow \) 6099

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3117

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

\(\Big \downarrow \) 5969

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3791

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

\(\Big \downarrow \) 17

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

\(\Big \downarrow \) 6095

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 3011

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

Input:

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

Output:

$Aborted
 

Maple [F]

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

Input:

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

Output:

int((f*x+e)^2*cosh(d*x+c)^3*sinh(d*x+c)/(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. 4887 vs. \(2 (581) = 1162\).

Time = 0.17 (sec) , antiderivative size = 4887, normalized size of antiderivative = 7.84 \[ \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

-1/24*e^2*((3*a*b*e^(-d*x - c) - b^2 - 3*(4*a^2 + 3*b^2)*e^(-2*d*x - 2*c)) 
*e^(3*d*x + 3*c)/(b^3*d) + 24*(a^3 + a*b^2)*(d*x + c)/(b^4*d) + (3*a*b*e^( 
-2*d*x - 2*c) + b^2*e^(-3*d*x - 3*c) + 3*(4*a^2 + 3*b^2)*e^(-d*x - c))/(b^ 
3*d) + 24*(a^3 + a*b^2)*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/(b 
^4*d)) - 1/432*(144*(a^3*d^3*f^2*e^(3*c) + a*b^2*d^3*f^2*e^(3*c))*x^3 + 43 
2*(a^3*d^3*e*f*e^(3*c) + a*b^2*d^3*e*f*e^(3*c))*x^2 - 2*(9*b^3*d^2*f^2*x^2 
*e^(6*c) + 6*(3*d^2*e*f - d*f^2)*b^3*x*e^(6*c) - 2*(3*d*e*f - f^2)*b^3*e^( 
6*c))*e^(3*d*x) + 27*(2*a*b^2*d^2*f^2*x^2*e^(5*c) + 2*(2*d^2*e*f - d*f^2)* 
a*b^2*x*e^(5*c) - (2*d*e*f - f^2)*a*b^2*e^(5*c))*e^(2*d*x) + 54*(8*(d*e*f 
- f^2)*a^2*b*e^(4*c) + 6*(d*e*f - f^2)*b^3*e^(4*c) - (4*a^2*b*d^2*f^2*e^(4 
*c) + 3*b^3*d^2*f^2*e^(4*c))*x^2 - 2*(4*(d^2*e*f - d*f^2)*a^2*b*e^(4*c) + 
3*(d^2*e*f - d*f^2)*b^3*e^(4*c))*x)*e^(d*x) + 54*(8*(d*e*f + f^2)*a^2*b*e^ 
(2*c) + 6*(d*e*f + f^2)*b^3*e^(2*c) + (4*a^2*b*d^2*f^2*e^(2*c) + 3*b^3*d^2 
*f^2*e^(2*c))*x^2 + 2*(4*(d^2*e*f + d*f^2)*a^2*b*e^(2*c) + 3*(d^2*e*f + d* 
f^2)*b^3*e^(2*c))*x)*e^(-d*x) + 27*(2*a*b^2*d^2*f^2*x^2*e^c + 2*(2*d^2*e*f 
 + d*f^2)*a*b^2*x*e^c + (2*d*e*f + f^2)*a*b^2*e^c)*e^(-2*d*x) + 2*(9*b^3*d 
^2*f^2*x^2 + 6*(3*d^2*e*f + d*f^2)*b^3*x + 2*(3*d*e*f + f^2)*b^3)*e^(-3*d* 
x))*e^(-3*c)/(b^4*d^3) + integrate(-2*((a^3*b*f^2 + a*b^3*f^2)*x^2 + 2*(a^ 
3*b*e*f + a*b^3*e*f)*x - ((a^4*f^2*e^c + a^2*b^2*f^2*e^c)*x^2 + 2*(a^4*e*f 
*e^c + a^2*b^2*e*f*e^c)*x)*e^(d*x))/(b^5*e^(2*d*x + 2*c) + 2*a*b^4*e^(d...
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(18*e**(6*c + 6*d*x)*b**6*d**2*e**2 + 36*e**(6*c + 6*d*x)*b**6*d**2*e*f*x 
+ 18*e**(6*c + 6*d*x)*b**6*d**2*f**2*x**2 - 12*e**(6*c + 6*d*x)*b**6*d*e*f 
 - 12*e**(6*c + 6*d*x)*b**6*d*f**2*x + 4*e**(6*c + 6*d*x)*b**6*f**2 - 54*e 
**(5*c + 5*d*x)*a*b**5*d**2*e**2 - 108*e**(5*c + 5*d*x)*a*b**5*d**2*e*f*x 
- 54*e**(5*c + 5*d*x)*a*b**5*d**2*f**2*x**2 + 54*e**(5*c + 5*d*x)*a*b**5*d 
*e*f + 54*e**(5*c + 5*d*x)*a*b**5*d*f**2*x - 27*e**(5*c + 5*d*x)*a*b**5*f* 
*2 + 216*e**(4*c + 4*d*x)*a**2*b**4*d**2*e**2 + 432*e**(4*c + 4*d*x)*a**2* 
b**4*d**2*e*f*x + 216*e**(4*c + 4*d*x)*a**2*b**4*d**2*f**2*x**2 - 432*e**( 
4*c + 4*d*x)*a**2*b**4*d*e*f - 432*e**(4*c + 4*d*x)*a**2*b**4*d*f**2*x + 4 
32*e**(4*c + 4*d*x)*a**2*b**4*f**2 + 162*e**(4*c + 4*d*x)*b**6*d**2*e**2 + 
 324*e**(4*c + 4*d*x)*b**6*d**2*e*f*x + 162*e**(4*c + 4*d*x)*b**6*d**2*f** 
2*x**2 - 324*e**(4*c + 4*d*x)*b**6*d*e*f - 324*e**(4*c + 4*d*x)*b**6*d*f** 
2*x + 324*e**(4*c + 4*d*x)*b**6*f**2 + 3456*e**(3*c + 3*d*x)*int(x**2/(e** 
(5*c + 5*d*x)*b + 2*e**(4*c + 4*d*x)*a - e**(3*c + 3*d*x)*b),x)*a**6*b*d** 
3*f**2 + 6048*e**(3*c + 3*d*x)*int(x**2/(e**(5*c + 5*d*x)*b + 2*e**(4*c + 
4*d*x)*a - e**(3*c + 3*d*x)*b),x)*a**4*b**3*d**3*f**2 + 2592*e**(3*c + 3*d 
*x)*int(x**2/(e**(5*c + 5*d*x)*b + 2*e**(4*c + 4*d*x)*a - e**(3*c + 3*d*x) 
*b),x)*a**2*b**5*d**3*f**2 + 6912*e**(3*c + 3*d*x)*int(x/(e**(5*c + 5*d*x) 
*b + 2*e**(4*c + 4*d*x)*a - e**(3*c + 3*d*x)*b),x)*a**6*b*d**3*e*f + 12096 
*e**(3*c + 3*d*x)*int(x/(e**(5*c + 5*d*x)*b + 2*e**(4*c + 4*d*x)*a - e*...