3.1.27 \(\int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx\) [27]

3.1.27.1 Optimal result

Integrand size = 28, antiderivative size = 636 \[ \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=-\frac {b e f x}{2 a^2 d}-\frac {b f^2 x^2}{4 a^2 d}+\frac {b \left (a^2+b^2\right ) (e+f x)^3}{3 a^4 f}-\frac {4 f (e+f x) \cosh (c+d x)}{3 a d^2}-\frac {2 b^2 f (e+f x) \cosh (c+d x)}{a^3 d^2}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 a d^2}-\frac {b \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d}-\frac {b \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d}-\frac {2 b \left (a^2+b^2\right ) f (e+f x) \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d^2}-\frac {2 b \left (a^2+b^2\right ) f (e+f x) \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d^2}+\frac {2 b \left (a^2+b^2\right ) f^2 \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d^3}+\frac {2 b \left (a^2+b^2\right ) f^2 \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d^3}+\frac {14 f^2 \sinh (c+d x)}{9 a d^3}+\frac {2 b^2 f^2 \sinh (c+d x)}{a^3 d^3}+\frac {2 (e+f x)^2 \sinh (c+d x)}{3 a d}+\frac {b^2 (e+f x)^2 \sinh (c+d x)}{a^3 d}+\frac {b f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 a^2 d^2}+\frac {(e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 a d}-\frac {b f^2 \sinh ^2(c+d x)}{4 a^2 d^3}-\frac {b (e+f x)^2 \sinh ^2(c+d x)}{2 a^2 d}+\frac {2 f^2 \sinh ^3(c+d x)}{27 a d^3} \]

output

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

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

Leaf count is larger than twice the leaf count of optimal. \(2129\) vs. \(2(636)=1272\).

Time = 16.80 (sec) , antiderivative size = 2129, normalized size of antiderivative = 3.35 \[ \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\text {Result too large to show} \]

input

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

output

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

3.1.27.3 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.

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

\(\Big \downarrow \) 6128

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

\(\Big \downarrow \) 6113

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3792

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3113

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3117

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

\(\Big \downarrow \) 6099

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3117

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

\(\Big \downarrow \) 5969

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3791

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

\(\Big \downarrow \) 17

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

\(\Big \downarrow \) 6095

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

\(\Big \downarrow \) 2620

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

input

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

output

$Aborted
 

3.1.27.3.1 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 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 2009

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

rule 2620

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

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 5969

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

rule 6095

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

rule 6128

Int[(((e_.) + (f_.)*(x_))^(m_.)*(F_)[(c_.) + (d_.)*(x_)]^(n_.))/(Csch[(c_.) 
 + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Int[(e + f*x)^m*Sinh[c + d*x]*(F 
[c + d*x]^n/(b + a*Sinh[c + d*x])), x] /; FreeQ[{a, b, c, d, e, f}, x] && H 
yperbolicQ[F] && IntegersQ[m, n]
 

3.1.27.4 Maple [F]

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

input

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

output

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

3.1.27.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4887 vs. \(2 (592) = 1184\).

Time = 0.34 (sec) , antiderivative size = 4887, normalized size of antiderivative = 7.68 \[ \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\text {Too large to display} \]

input

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

output

-1/432*(18*a^3*d^2*f^2*x^2 + 18*a^3*d^2*e^2 - 2*(9*a^3*d^2*f^2*x^2 + 9*a^3 
*d^2*e^2 - 6*a^3*d*e*f + 2*a^3*f^2 + 6*(3*a^3*d^2*e*f - a^3*d*f^2)*x)*cosh 
(d*x + c)^6 - 2*(9*a^3*d^2*f^2*x^2 + 9*a^3*d^2*e^2 - 6*a^3*d*e*f + 2*a^3*f 
^2 + 6*(3*a^3*d^2*e*f - a^3*d*f^2)*x)*sinh(d*x + c)^6 + 12*a^3*d*e*f + 27* 
(2*a^2*b*d^2*f^2*x^2 + 2*a^2*b*d^2*e^2 - 2*a^2*b*d*e*f + a^2*b*f^2 + 2*(2* 
a^2*b*d^2*e*f - a^2*b*d*f^2)*x)*cosh(d*x + c)^5 + 3*(18*a^2*b*d^2*f^2*x^2 
+ 18*a^2*b*d^2*e^2 - 18*a^2*b*d*e*f + 9*a^2*b*f^2 + 18*(2*a^2*b*d^2*e*f - 
a^2*b*d*f^2)*x - 4*(9*a^3*d^2*f^2*x^2 + 9*a^3*d^2*e^2 - 6*a^3*d*e*f + 2*a^ 
3*f^2 + 6*(3*a^3*d^2*e*f - a^3*d*f^2)*x)*cosh(d*x + c))*sinh(d*x + c)^5 + 
4*a^3*f^2 - 54*((3*a^3 + 4*a*b^2)*d^2*f^2*x^2 + (3*a^3 + 4*a*b^2)*d^2*e^2 
- 2*(3*a^3 + 4*a*b^2)*d*e*f + 2*(3*a^3 + 4*a*b^2)*f^2 + 2*((3*a^3 + 4*a*b^ 
2)*d^2*e*f - (3*a^3 + 4*a*b^2)*d*f^2)*x)*cosh(d*x + c)^4 - 3*(18*(3*a^3 + 
4*a*b^2)*d^2*f^2*x^2 + 18*(3*a^3 + 4*a*b^2)*d^2*e^2 - 36*(3*a^3 + 4*a*b^2) 
*d*e*f + 36*(3*a^3 + 4*a*b^2)*f^2 + 10*(9*a^3*d^2*f^2*x^2 + 9*a^3*d^2*e^2 
- 6*a^3*d*e*f + 2*a^3*f^2 + 6*(3*a^3*d^2*e*f - a^3*d*f^2)*x)*cosh(d*x + c) 
^2 + 36*((3*a^3 + 4*a*b^2)*d^2*e*f - (3*a^3 + 4*a*b^2)*d*f^2)*x - 45*(2*a^ 
2*b*d^2*f^2*x^2 + 2*a^2*b*d^2*e^2 - 2*a^2*b*d*e*f + a^2*b*f^2 + 2*(2*a^2*b 
*d^2*e*f - a^2*b*d*f^2)*x)*cosh(d*x + c))*sinh(d*x + c)^4 - 144*((a^2*b + 
b^3)*d^3*f^2*x^3 + 3*(a^2*b + b^3)*d^3*e*f*x^2 + 3*(a^2*b + b^3)*d^3*e^2*x 
 + 6*(a^2*b + b^3)*c*d^2*e^2 - 6*(a^2*b + b^3)*c^2*d*e*f + 2*(a^2*b + b...
 

3.1.27.6 Sympy [F]

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

input

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

output

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

3.1.27.7 Maxima [F]

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

input

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

output

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

3.1.27.8 Giac [F]

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

input

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

output

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

3.1.27.9 Mupad [F(-1)]

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

input

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

output

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