3.5.2 \(\int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) [402]

3.5.2.1 Optimal result

Integrand size = 36, antiderivative size = 1049 \[ \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {a^3 e f x}{2 b^4 d}+\frac {3 a e f x}{16 b^2 d}-\frac {a^3 f^2 x^2}{4 b^4 d}+\frac {3 a f^2 x^2}{32 b^2 d}+\frac {a^3 \left (a^2+b^2\right ) (e+f x)^3}{3 b^6 f}-\frac {2 a^4 f (e+f x) \cosh (c+d x)}{b^5 d^2}-\frac {4 a^2 f (e+f x) \cosh (c+d x)}{3 b^3 d^2}+\frac {f (e+f x) \cosh (c+d x)}{4 b d^2}-\frac {3 a f^2 \cosh ^2(c+d x)}{32 b^2 d^3}-\frac {2 a^2 f (e+f x) \cosh ^3(c+d x)}{9 b^3 d^2}-\frac {a f^2 \cosh ^4(c+d x)}{32 b^2 d^3}-\frac {a (e+f x)^2 \cosh ^4(c+d x)}{4 b^2 d}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 b d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 b d^2}-\frac {a^3 \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^6 d}-\frac {a^3 \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^6 d}-\frac {2 a^3 \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^6 d^2}-\frac {2 a^3 \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^6 d^2}+\frac {2 a^3 \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^6 d^3}+\frac {2 a^3 \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^6 d^3}+\frac {2 a^4 f^2 \sinh (c+d x)}{b^5 d^3}+\frac {14 a^2 f^2 \sinh (c+d x)}{9 b^3 d^3}-\frac {f^2 \sinh (c+d x)}{4 b d^3}+\frac {a^4 (e+f x)^2 \sinh (c+d x)}{b^5 d}+\frac {2 a^2 (e+f x)^2 \sinh (c+d x)}{3 b^3 d}-\frac {(e+f x)^2 \sinh (c+d x)}{8 b d}+\frac {a^3 f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^4 d^2}+\frac {3 a f (e+f x) \cosh (c+d x) \sinh (c+d x)}{16 b^2 d^2}+\frac {a^2 (e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 b^3 d}+\frac {a f (e+f x) \cosh ^3(c+d x) \sinh (c+d x)}{8 b^2 d^2}-\frac {a^3 f^2 \sinh ^2(c+d x)}{4 b^4 d^3}-\frac {a^3 (e+f x)^2 \sinh ^2(c+d x)}{2 b^4 d}+\frac {2 a^2 f^2 \sinh ^3(c+d x)}{27 b^3 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 b d^3}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 b d}+\frac {f^2 \sinh (5 c+5 d x)}{1000 b d^3}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 b d} \]

output

1/4*f*(f*x+e)*cosh(d*x+c)/b/d^2+3/32*a*f^2*x^2/b^2/d+14/9*a^2*f^2*sinh(d*x 
+c)/b^3/d^3+2/3*a^2*(f*x+e)^2*sinh(d*x+c)/b^3/d-1/8*(f*x+e)^2*sinh(d*x+c)/ 
b/d-1/2*a^3*e*f*x/b^4/d-2*a^4*f*(f*x+e)*cosh(d*x+c)/b^5/d^2+1/216*f^2*sinh 
(3*d*x+3*c)/b/d^3+1/48*(f*x+e)^2*sinh(3*d*x+3*c)/b/d+1/1000*f^2*sinh(5*d*x 
+5*c)/b/d^3+1/80*(f*x+e)^2*sinh(5*d*x+5*c)/b/d+1/2*a^3*f*(f*x+e)*cosh(d*x+ 
c)*sinh(d*x+c)/b^4/d^2+1/8*a*f*(f*x+e)*cosh(d*x+c)^3*sinh(d*x+c)/b^2/d^2-2 
*a^3*(a^2+b^2)*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^6/ 
d^2-2*a^3*(a^2+b^2)*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2))) 
/b^6/d^2-a^3*(a^2+b^2)*(f*x+e)^2*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^ 
6/d-a^3*(a^2+b^2)*(f*x+e)^2*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^6/d+a 
^4*(f*x+e)^2*sinh(d*x+c)/b^5/d+3/16*a*f*(f*x+e)*cosh(d*x+c)*sinh(d*x+c)/b^ 
2/d^2-1/4*f^2*sinh(d*x+c)/b/d^3-2/9*a^2*f*(f*x+e)*cosh(d*x+c)^3/b^3/d^2+1/ 
3*a^2*(f*x+e)^2*cosh(d*x+c)^2*sinh(d*x+c)/b^3/d+2*a^3*(a^2+b^2)*f^2*polylo 
g(3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^6/d^3+2*a^3*(a^2+b^2)*f^2*polylog 
(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^6/d^3+3/16*a*e*f*x/b^2/d-4/3*a^2*f 
*(f*x+e)*cosh(d*x+c)/b^3/d^2-1/4*a^3*f^2*x^2/b^4/d+1/3*a^3*(a^2+b^2)*(f*x+ 
e)^3/b^6/f-3/32*a*f^2*cosh(d*x+c)^2/b^2/d^3-1/32*a*f^2*cosh(d*x+c)^4/b^2/d 
^3-1/4*a*(f*x+e)^2*cosh(d*x+c)^4/b^2/d-1/72*f*(f*x+e)*cosh(3*d*x+3*c)/b/d^ 
2-1/200*f*(f*x+e)*cosh(5*d*x+5*c)/b/d^2+2*a^4*f^2*sinh(d*x+c)/b^5/d^3-1/4* 
a^3*f^2*sinh(d*x+c)^2/b^4/d^3-1/2*a^3*(f*x+e)^2*sinh(d*x+c)^2/b^4/d+2/2...
 

3.5.2.2 Mathematica [A] (verified)

Time = 9.73 (sec) , antiderivative size = 1811, normalized size of antiderivative = 1.73 \[ \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \]

input

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

output

((-8*a^3*(a^2 + b^2)*e^2*x*Coth[c])/b^6 - (8*a^3*(a^2 + b^2)*e*f*x^2*Coth[ 
c])/b^6 - (8*a^3*(a^2 + b^2)*f^2*x^3*Coth[c])/(3*b^6) + (8*a^3*(a^2 + b^2) 
*(6*e^2*E^(2*c)*x + 6*e*E^(2*c)*f*x^2 + 2*E^(2*c)*f^2*x^3 + (6*a*Sqrt[a^2 
+ b^2]*e^2*ArcTan[(a + b*E^(c + d*x))/Sqrt[-a^2 - b^2]])/(Sqrt[-(a^2 + b^2 
)^2]*d) + (6*a*Sqrt[-(a^2 + b^2)^2]*e^2*E^(2*c)*ArcTan[(a + b*E^(c + d*x)) 
/Sqrt[-a^2 - b^2]])/((a^2 + b^2)^(3/2)*d) - (6*a*Sqrt[-(a^2 + b^2)^2]*e^2* 
ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]])/((-a^2 - b^2)^(3/2)*d) + (6* 
a*Sqrt[-(a^2 + b^2)^2]*e^2*E^(2*c)*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + 
b^2]])/((-a^2 - b^2)^(3/2)*d) + (3*e^2*Log[2*a*E^(c + d*x) + b*(-1 + E^(2* 
(c + d*x)))])/d - (3*e^2*E^(2*c)*Log[2*a*E^(c + d*x) + b*(-1 + E^(2*(c + d 
*x)))])/d + (6*e*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E 
^(2*c)])])/d - (6*e*E^(2*c)*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a 
^2 + b^2)*E^(2*c)])])/d + (3*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sq 
rt[(a^2 + b^2)*E^(2*c)])])/d - (3*E^(2*c)*f^2*x^2*Log[1 + (b*E^(2*c + d*x) 
)/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/d + (6*e*f*x*Log[1 + (b*E^(2*c + d 
*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (6*e*E^(2*c)*f*x*Log[1 + (b 
*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d + (3*f^2*x^2*Log[1 
 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (3*E^(2*c)* 
f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d 
- (6*(-1 + E^(2*c))*f*(e + f*x)*PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c -...
 

3.5.2.3 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.

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

\(\Big \downarrow \) 6113

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

\(\Big \downarrow \) 5971

\(\displaystyle \frac {\int \left (-\frac {1}{8} \cosh (c+d x) (e+f x)^2+\frac {1}{16} \cosh (3 c+3 d x) (e+f x)^2+\frac {1}{16} \cosh (5 c+5 d x) (e+f x)^2\right )dx}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {f^2 \sinh (c+d x)}{4 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 d^3}+\frac {f^2 \sinh (5 c+5 d x)}{1000 d^3}+\frac {f (e+f x) \cosh (c+d x)}{4 d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 d^2}-\frac {(e+f x)^2 \sinh (c+d x)}{8 d}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\)

\(\Big \downarrow \) 6113

\(\displaystyle \frac {-\frac {f^2 \sinh (c+d x)}{4 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 d^3}+\frac {f^2 \sinh (5 c+5 d x)}{1000 d^3}+\frac {f (e+f x) \cosh (c+d x)}{4 d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 d^2}-\frac {(e+f x)^2 \sinh (c+d x)}{8 d}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\int (e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)dx}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\)

\(\Big \downarrow \) 5970

\(\displaystyle \frac {-\frac {f^2 \sinh (c+d x)}{4 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 d^3}+\frac {f^2 \sinh (5 c+5 d x)}{1000 d^3}+\frac {f (e+f x) \cosh (c+d x)}{4 d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 d^2}-\frac {(e+f x)^2 \sinh (c+d x)}{8 d}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \int (e+f x) \cosh ^4(c+d x)dx}{2 d}}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {f^2 \sinh (c+d x)}{4 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 d^3}+\frac {f^2 \sinh (5 c+5 d x)}{1000 d^3}+\frac {f (e+f x) \cosh (c+d x)}{4 d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 d^2}-\frac {(e+f x)^2 \sinh (c+d x)}{8 d}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )^4dx}{2 d}}{b}\right )}{b}\)

\(\Big \downarrow \) 3791

\(\displaystyle \frac {-\frac {f^2 \sinh (c+d x)}{4 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 d^3}+\frac {f^2 \sinh (5 c+5 d x)}{1000 d^3}+\frac {f (e+f x) \cosh (c+d x)}{4 d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 d^2}-\frac {(e+f x)^2 \sinh (c+d x)}{8 d}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \int (e+f x) \cosh ^2(c+d x)dx-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {f^2 \sinh (c+d x)}{4 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 d^3}+\frac {f^2 \sinh (5 c+5 d x)}{1000 d^3}+\frac {f (e+f x) \cosh (c+d x)}{4 d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 d^2}-\frac {(e+f x)^2 \sinh (c+d x)}{8 d}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )^2dx-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}\right )}{b}\)

\(\Big \downarrow \) 3791

\(\displaystyle \frac {-\frac {f^2 \sinh (c+d x)}{4 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 d^3}+\frac {f^2 \sinh (5 c+5 d x)}{1000 d^3}+\frac {f (e+f x) \cosh (c+d x)}{4 d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 d^2}-\frac {(e+f x)^2 \sinh (c+d x)}{8 d}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (\frac {1}{2} \int (e+f x)dx-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\)

\(\Big \downarrow \) 17

\(\displaystyle \frac {-\frac {f^2 \sinh (c+d x)}{4 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 d^3}+\frac {f^2 \sinh (5 c+5 d x)}{1000 d^3}+\frac {f (e+f x) \cosh (c+d x)}{4 d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 d^2}-\frac {(e+f x)^2 \sinh (c+d x)}{8 d}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\)

\(\Big \downarrow \) 6113

\(\displaystyle \frac {-\frac {f^2 \sinh (c+d x)}{4 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 d^3}+\frac {f^2 \sinh (5 c+5 d x)}{1000 d^3}+\frac {f (e+f x) \cosh (c+d x)}{4 d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 d^2}-\frac {(e+f x)^2 \sinh (c+d x)}{8 d}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (\frac {\int (e+f x)^2 \cosh ^3(c+d x)dx}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {f^2 \sinh (c+d x)}{4 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 d^3}+\frac {f^2 \sinh (5 c+5 d x)}{1000 d^3}+\frac {f (e+f x) \cosh (c+d x)}{4 d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 d^2}-\frac {(e+f x)^2 \sinh (c+d x)}{8 d}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )^3dx}{b}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 3792

\(\displaystyle \frac {-\frac {f^2 \sinh (c+d x)}{4 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 d^3}+\frac {f^2 \sinh (5 c+5 d x)}{1000 d^3}+\frac {f (e+f x) \cosh (c+d x)}{4 d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 d^2}-\frac {(e+f x)^2 \sinh (c+d x)}{8 d}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (\frac {\frac {2 f^2 \int \cosh ^3(c+d x)dx}{9 d^2}+\frac {2}{3} \int (e+f x)^2 \cosh (c+d x)dx-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {f^2 \sinh (c+d x)}{4 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 d^3}+\frac {f^2 \sinh (5 c+5 d x)}{1000 d^3}+\frac {f (e+f x) \cosh (c+d x)}{4 d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 d^2}-\frac {(e+f x)^2 \sinh (c+d x)}{8 d}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2 f^2 \int \sin \left (i c+i d x+\frac {\pi }{2}\right )^3dx}{9 d^2}+\frac {2}{3} \int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )dx-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 3113

\(\displaystyle \frac {-\frac {f^2 \sinh (c+d x)}{4 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 d^3}+\frac {f^2 \sinh (5 c+5 d x)}{1000 d^3}+\frac {f (e+f x) \cosh (c+d x)}{4 d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 d^2}-\frac {(e+f x)^2 \sinh (c+d x)}{8 d}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2 i f^2 \int \left (\sinh ^2(c+d x)+1\right )d(-i \sinh (c+d x))}{9 d^3}+\frac {2}{3} \int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )dx-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {f^2 \sinh (c+d x)}{4 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 d^3}+\frac {f^2 \sinh (5 c+5 d x)}{1000 d^3}+\frac {f (e+f x) \cosh (c+d x)}{4 d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 d^2}-\frac {(e+f x)^2 \sinh (c+d x)}{8 d}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2}{3} \int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )dx+\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 3777

\(\displaystyle \frac {-\frac {f^2 \sinh (c+d x)}{4 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 d^3}+\frac {f^2 \sinh (5 c+5 d x)}{1000 d^3}+\frac {f (e+f x) \cosh (c+d x)}{4 d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 d^2}-\frac {(e+f x)^2 \sinh (c+d x)}{8 d}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 i f \int -i (e+f x) \sinh (c+d x)dx}{d}\right )+\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {-\frac {f^2 \sinh (c+d x)}{4 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 d^3}+\frac {f^2 \sinh (5 c+5 d x)}{1000 d^3}+\frac {f (e+f x) \cosh (c+d x)}{4 d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 d^2}-\frac {(e+f x)^2 \sinh (c+d x)}{8 d}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 f \int (e+f x) \sinh (c+d x)dx}{d}\right )+\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {f^2 \sinh (c+d x)}{4 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 d^3}+\frac {f^2 \sinh (5 c+5 d x)}{1000 d^3}+\frac {f (e+f x) \cosh (c+d x)}{4 d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 d^2}-\frac {(e+f x)^2 \sinh (c+d x)}{8 d}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 f \int -i (e+f x) \sin (i c+i d x)dx}{d}\right )+\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {-\frac {f^2 \sinh (c+d x)}{4 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 d^3}+\frac {f^2 \sinh (5 c+5 d x)}{1000 d^3}+\frac {f (e+f x) \cosh (c+d x)}{4 d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 d^2}-\frac {(e+f x)^2 \sinh (c+d x)}{8 d}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \int (e+f x) \sin (i c+i d x)dx}{d}\right )+\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 3777

\(\displaystyle \frac {-\frac {f^2 \sinh (c+d x)}{4 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 d^3}+\frac {f^2 \sinh (5 c+5 d x)}{1000 d^3}+\frac {f (e+f x) \cosh (c+d x)}{4 d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 d^2}-\frac {(e+f x)^2 \sinh (c+d x)}{8 d}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \cosh (c+d x)dx}{d}\right )}{d}\right )+\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {f^2 \sinh (c+d x)}{4 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 d^3}+\frac {f^2 \sinh (5 c+5 d x)}{1000 d^3}+\frac {f (e+f x) \cosh (c+d x)}{4 d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 d^2}-\frac {(e+f x)^2 \sinh (c+d x)}{8 d}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{d}\right )+\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 3117

\(\displaystyle \frac {-\frac {f^2 \sinh (c+d x)}{4 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 d^3}+\frac {f^2 \sinh (5 c+5 d x)}{1000 d^3}+\frac {f (e+f x) \cosh (c+d x)}{4 d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 d^2}-\frac {(e+f x)^2 \sinh (c+d x)}{8 d}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 6099

\(\displaystyle \frac {-\frac {f^2 \sinh (c+d x)}{4 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 d^3}+\frac {f^2 \sinh (5 c+5 d x)}{1000 d^3}+\frac {f (e+f x) \cosh (c+d x)}{4 d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 d^2}-\frac {(e+f x)^2 \sinh (c+d x)}{8 d}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \int (e+f x)^2 \cosh (c+d x)dx}{b^2}+\frac {\int (e+f x)^2 \cosh (c+d x) \sinh (c+d x)dx}{b}\right )}{b}+\frac {\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {f^2 \sinh (c+d x)}{4 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 d^3}+\frac {f^2 \sinh (5 c+5 d x)}{1000 d^3}+\frac {f (e+f x) \cosh (c+d x)}{4 d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 d^2}-\frac {(e+f x)^2 \sinh (c+d x)}{8 d}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (\frac {\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{b^2}+\frac {\int (e+f x)^2 \cosh (c+d x) \sinh (c+d x)dx}{b}\right )}{b}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 3777

\(\displaystyle \frac {-\frac {f^2 \sinh (c+d x)}{4 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 d^3}+\frac {f^2 \sinh (5 c+5 d x)}{1000 d^3}+\frac {f (e+f x) \cosh (c+d x)}{4 d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 d^2}-\frac {(e+f x)^2 \sinh (c+d x)}{8 d}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (\frac {\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 i f \int -i (e+f x) \sinh (c+d x)dx}{d}\right )}{b^2}+\frac {\int (e+f x)^2 \cosh (c+d x) \sinh (c+d x)dx}{b}\right )}{b}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {-\frac {f^2 \sinh (c+d x)}{4 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 d^3}+\frac {f^2 \sinh (5 c+5 d x)}{1000 d^3}+\frac {f (e+f x) \cosh (c+d x)}{4 d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 d^2}-\frac {(e+f x)^2 \sinh (c+d x)}{8 d}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 f \int (e+f x) \sinh (c+d x)dx}{d}\right )}{b^2}+\frac {\int (e+f x)^2 \cosh (c+d x) \sinh (c+d x)dx}{b}\right )}{b}+\frac {\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {f^2 \sinh (c+d x)}{4 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 d^3}+\frac {f^2 \sinh (5 c+5 d x)}{1000 d^3}+\frac {f (e+f x) \cosh (c+d x)}{4 d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 d^2}-\frac {(e+f x)^2 \sinh (c+d x)}{8 d}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (\frac {\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 f \int -i (e+f x) \sin (i c+i d x)dx}{d}\right )}{b^2}+\frac {\int (e+f x)^2 \cosh (c+d x) \sinh (c+d x)dx}{b}\right )}{b}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {-\frac {f^2 \sinh (c+d x)}{4 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 d^3}+\frac {f^2 \sinh (5 c+5 d x)}{1000 d^3}+\frac {f (e+f x) \cosh (c+d x)}{4 d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 d^2}-\frac {(e+f x)^2 \sinh (c+d x)}{8 d}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (\frac {\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \int (e+f x) \sin (i c+i d x)dx}{d}\right )}{b^2}+\frac {\int (e+f x)^2 \cosh (c+d x) \sinh (c+d x)dx}{b}\right )}{b}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 3777

\(\displaystyle \frac {-\frac {f^2 \sinh (c+d x)}{4 d^3}+\frac {f^2 \sinh (3 c+3 d x)}{216 d^3}+\frac {f^2 \sinh (5 c+5 d x)}{1000 d^3}+\frac {f (e+f x) \cosh (c+d x)}{4 d^2}-\frac {f (e+f x) \cosh (3 c+3 d x)}{72 d^2}-\frac {f (e+f x) \cosh (5 c+5 d x)}{200 d^2}-\frac {(e+f x)^2 \sinh (c+d x)}{8 d}+\frac {(e+f x)^2 \sinh (3 c+3 d x)}{48 d}+\frac {(e+f x)^2 \sinh (5 c+5 d x)}{80 d}}{b}-\frac {a \left (\frac {\frac {(e+f x)^2 \cosh ^4(c+d x)}{4 d}-\frac {f \left (\frac {3}{4} \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )-\frac {f \cosh ^4(c+d x)}{16 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh ^3(c+d x)}{4 d}\right )}{2 d}}{b}-\frac {a \left (\frac {\frac {2 i f^2 \left (-\frac {1}{3} i \sinh ^3(c+d x)-i \sinh (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 d^2}+\frac {2}{3} \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 d}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \cosh (c+d x)dx}{d}\right )}{d}\right )}{b^2}+\frac {\int (e+f x)^2 \cosh (c+d x) \sinh (c+d x)dx}{b}\right )}{b}\right )}{b}\right )}{b}\)

input

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

output

$Aborted
 

3.5.2.3.1 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 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 3117

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[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 3791

Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> 
 Simp[d*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Simp[b*(c + d*x)*Cos[e + f*x 
]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^2*((n - 1)/n)   Int[(c + d* 
x)*(b*Sin[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 
 1]
 

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 5970

Int[Cosh[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + 
(b_.)*(x_)], x_Symbol] :> Simp[(c + d*x)^m*(Cosh[a + b*x]^(n + 1)/(b*(n + 1 
))), x] - Simp[d*(m/(b*(n + 1)))   Int[(c + d*x)^(m - 1)*Cosh[a + b*x]^(n + 
 1), x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
 

rule 5971

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + 
(b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + 
b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & 
& IGtQ[p, 0]
 

rule 6099

Int[(Cosh[(c_.) + (d_.)*(x_)]^(n_)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_. 
)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[-a/b^2   Int[(e + f*x)^m*Cos 
h[c + d*x]^(n - 2), x], x] + (Simp[1/b   Int[(e + f*x)^m*Cosh[c + d*x]^(n - 
 2)*Sinh[c + d*x], x], x] + Simp[(a^2 + b^2)/b^2   Int[(e + f*x)^m*(Cosh[c 
+ d*x]^(n - 2)/(a + b*Sinh[c + d*x])), x], x]) /; FreeQ[{a, b, c, d, e, f}, 
 x] && IGtQ[n, 1] && NeQ[a^2 + b^2, 0] && IGtQ[m, 0]
 

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]
 

3.5.2.4 Maple [F]

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

input

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

output

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

3.5.2.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 11318 vs. \(2 (973) = 1946\).

Time = 0.41 (sec) , antiderivative size = 11318, normalized size of antiderivative = 10.79 \[ \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]

input

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

output

Too large to include
 

3.5.2.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.5.2.7 Maxima [F]

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

input

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

output

-1/960*e^2*((15*a*b^3*e^(-d*x - c) - 6*b^4 - 10*(4*a^2*b^2 + b^4)*e^(-2*d* 
x - 2*c) + 60*(2*a^3*b + a*b^3)*e^(-3*d*x - 3*c) - 60*(8*a^4 + 6*a^2*b^2 - 
 b^4)*e^(-4*d*x - 4*c))*e^(5*d*x + 5*c)/(b^5*d) + 960*(a^5 + a^3*b^2)*(d*x 
 + c)/(b^6*d) + (15*a*b^3*e^(-4*d*x - 4*c) + 6*b^4*e^(-5*d*x - 5*c) + 60*( 
8*a^4 + 6*a^2*b^2 - b^4)*e^(-d*x - c) + 60*(2*a^3*b + a*b^3)*e^(-2*d*x - 2 
*c) + 10*(4*a^2*b^2 + b^4)*e^(-3*d*x - 3*c))/(b^5*d) + 960*(a^5 + a^3*b^2) 
*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/(b^6*d)) - 1/1728000*(576 
000*(a^5*d^3*f^2*e^(5*c) + a^3*b^2*d^3*f^2*e^(5*c))*x^3 + 1728000*(a^5*d^3 
*e*f*e^(5*c) + a^3*b^2*d^3*e*f*e^(5*c))*x^2 - 432*(25*b^5*d^2*f^2*x^2*e^(1 
0*c) + 10*(5*d^2*e*f - d*f^2)*b^5*x*e^(10*c) - 2*(5*d*e*f - f^2)*b^5*e^(10 
*c))*e^(5*d*x) + 3375*(8*a*b^4*d^2*f^2*x^2*e^(9*c) + 4*(4*d^2*e*f - d*f^2) 
*a*b^4*x*e^(9*c) - (4*d*e*f - f^2)*a*b^4*e^(9*c))*e^(4*d*x) + 2000*(8*(3*d 
*e*f - f^2)*a^2*b^3*e^(8*c) + 2*(3*d*e*f - f^2)*b^5*e^(8*c) - 9*(4*a^2*b^3 
*d^2*f^2*e^(8*c) + b^5*d^2*f^2*e^(8*c))*x^2 - 6*(4*(3*d^2*e*f - d*f^2)*a^2 
*b^3*e^(8*c) + (3*d^2*e*f - d*f^2)*b^5*e^(8*c))*x)*e^(3*d*x) - 54000*(2*(2 
*d*e*f - f^2)*a^3*b^2*e^(7*c) + (2*d*e*f - f^2)*a*b^4*e^(7*c) - 2*(2*a^3*b 
^2*d^2*f^2*e^(7*c) + a*b^4*d^2*f^2*e^(7*c))*x^2 - 2*(2*(2*d^2*e*f - d*f^2) 
*a^3*b^2*e^(7*c) + (2*d^2*e*f - d*f^2)*a*b^4*e^(7*c))*x)*e^(2*d*x) + 10800 
0*(16*(d*e*f - f^2)*a^4*b*e^(6*c) + 12*(d*e*f - f^2)*a^2*b^3*e^(6*c) - 2*( 
d*e*f - f^2)*b^5*e^(6*c) - (8*a^4*b*d^2*f^2*e^(6*c) + 6*a^2*b^3*d^2*f^2...
 

3.5.2.8 Giac [F]

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

input

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

output

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

3.5.2.9 Mupad [F(-1)]

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

input

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

output

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