\(\int \frac {(e+f x)^2 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx\) [426]

Optimal result

Integrand size = 32, antiderivative size = 462 \[ \int \frac {(e+f x)^2 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {(e+f x)^3}{3 b f}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {\sqrt {a^2+b^2} (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b d}+\frac {\sqrt {a^2+b^2} (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b d}-\frac {2 f (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}+\frac {2 f (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}-\frac {2 \sqrt {a^2+b^2} f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b d^2}+\frac {2 \sqrt {a^2+b^2} f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b d^2}+\frac {2 f^2 \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}-\frac {2 f^2 \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}+\frac {2 \sqrt {a^2+b^2} f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b d^3}-\frac {2 \sqrt {a^2+b^2} f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b d^3} \] Output:

1/3*(f*x+e)^3/b/f-2*(f*x+e)^2*arctanh(exp(d*x+c))/a/d-(a^2+b^2)^(1/2)*(f*x 
+e)^2*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a/b/d+(a^2+b^2)^(1/2)*(f*x+e) 
^2*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a/b/d-2*f*(f*x+e)*polylog(2,-exp 
(d*x+c))/a/d^2+2*f*(f*x+e)*polylog(2,exp(d*x+c))/a/d^2-2*(a^2+b^2)^(1/2)*f 
*(f*x+e)*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a/b/d^2+2*(a^2+b^2)^ 
(1/2)*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a/b/d^2+2*f^2 
*polylog(3,-exp(d*x+c))/a/d^3-2*f^2*polylog(3,exp(d*x+c))/a/d^3+2*(a^2+b^2 
)^(1/2)*f^2*polylog(3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a/b/d^3-2*(a^2+b^ 
2)^(1/2)*f^2*polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a/b/d^3
 

Mathematica [A] (verified)

Time = 0.66 (sec) , antiderivative size = 490, normalized size of antiderivative = 1.06 \[ \int \frac {(e+f x)^2 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {x \left (3 e^2+3 e f x+f^2 x^2\right )}{3 b}+\frac {(e+f x)^2 \log \left (1-e^{c+d x}\right )-(e+f x)^2 \log \left (1+e^{c+d x}\right )-\frac {2 f \left (d (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )-f \operatorname {PolyLog}\left (3,-e^{c+d x}\right )\right )}{d^2}+\frac {2 f \left (d (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )-f \operatorname {PolyLog}\left (3,e^{c+d x}\right )\right )}{d^2}}{a d}-\frac {\sqrt {a^2+b^2} \left (-2 d^2 e^2 \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )+2 d^2 e f x \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+d^2 f^2 x^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-2 d^2 e f x \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-d^2 f^2 x^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+2 d f (e+f x) \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-2 d f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-2 f^2 \operatorname {PolyLog}\left (3,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+2 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{a b d^3} \] Input:

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

Output:

(x*(3*e^2 + 3*e*f*x + f^2*x^2))/(3*b) + ((e + f*x)^2*Log[1 - E^(c + d*x)] 
- (e + f*x)^2*Log[1 + E^(c + d*x)] - (2*f*(d*(e + f*x)*PolyLog[2, -E^(c + 
d*x)] - f*PolyLog[3, -E^(c + d*x)]))/d^2 + (2*f*(d*(e + f*x)*PolyLog[2, E^ 
(c + d*x)] - f*PolyLog[3, E^(c + d*x)]))/d^2)/(a*d) - (Sqrt[a^2 + b^2]*(-2 
*d^2*e^2*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]] + 2*d^2*e*f*x*Log[1 
+ (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + d^2*f^2*x^2*Log[1 + (b*E^(c + d 
*x))/(a - Sqrt[a^2 + b^2])] - 2*d^2*e*f*x*Log[1 + (b*E^(c + d*x))/(a + Sqr 
t[a^2 + b^2])] - d^2*f^2*x^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]) 
] + 2*d*f*(e + f*x)*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] - 2 
*d*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))] - 2*f^ 
2*PolyLog[3, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + 2*f^2*PolyLog[3, -( 
(b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]))/(a*b*d^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.

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

\(\Big \downarrow \) 6119

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

\(\Big \downarrow \) 5973

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3118

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 3011

\(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-\frac {2 i f \left (\frac {f \int \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{a}\)

\(\Big \downarrow \) 2720

\(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {i \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{a}\)

\(\Big \downarrow \) 6099

\(\displaystyle -\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \int (e+f x)^2dx}{b^2}+\frac {\int (e+f x)^2 \sinh (c+d x)dx}{b}\right )}{a}+\frac {i \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{a}\)

\(\Big \downarrow \) 17

\(\displaystyle -\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a+b \sinh (c+d x)}dx}{b^2}+\frac {\int (e+f x)^2 \sinh (c+d x)dx}{b}-\frac {a (e+f x)^3}{3 b^2 f}\right )}{a}+\frac {i \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {i \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a-i b \sin (i c+i d x)}dx}{b^2}+\frac {\int -i (e+f x)^2 \sin (i c+i d x)dx}{b}-\frac {a (e+f x)^3}{3 b^2 f}\right )}{a}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {i \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \int (e+f x)^2 \sin (i c+i d x)dx}{b}-\frac {a (e+f x)^3}{3 b^2 f}\right )}{a}\)

\(\Big \downarrow \) 3777

\(\displaystyle \frac {i \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \int (e+f x) \cosh (c+d x)dx}{d}\right )}{b}-\frac {a (e+f x)^3}{3 b^2 f}\right )}{a}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

\(\displaystyle \frac {i \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {i f \int -i \sinh (c+d x)dx}{d}\right )}{d}\right )}{b}-\frac {a (e+f x)^3}{3 b^2 f}\right )}{a}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {i \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int \sinh (c+d x)dx}{d}\right )}{d}\right )}{b}-\frac {a (e+f x)^3}{3 b^2 f}\right )}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {i \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int -i \sin (i c+i d x)dx}{d}\right )}{d}\right )}{b}-\frac {a (e+f x)^3}{3 b^2 f}\right )}{a}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {i \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}+\frac {i f \int \sin (i c+i d x)dx}{d}\right )}{d}\right )}{b}-\frac {a (e+f x)^3}{3 b^2 f}\right )}{a}\)

\(\Big \downarrow \) 3118

\(\displaystyle \frac {i \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{a}-\frac {b \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {a (e+f x)^3}{3 b^2 f}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{b}\right )}{a}\)

\(\Big \downarrow \) 3803

\(\displaystyle \frac {i \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{a}-\frac {b \left (\frac {2 \left (a^2+b^2\right ) \int -\frac {e^{c+d x} (e+f x)^2}{-2 e^{c+d x} a-b e^{2 (c+d x)}+b}dx}{b^2}-\frac {a (e+f x)^3}{3 b^2 f}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{b}\right )}{a}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {i \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{a}-\frac {b \left (-\frac {2 \left (a^2+b^2\right ) \int \frac {e^{c+d x} (e+f x)^2}{-2 e^{c+d x} a-b e^{2 (c+d x)}+b}dx}{b^2}-\frac {a (e+f x)^3}{3 b^2 f}-\frac {i \left (\frac {i (e+f x)^2 \cosh (c+d x)}{d}-\frac {2 i f \left (\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}\right )}{d}\right )}{b}\right )}{a}\)

\(\Big \downarrow \) 2694

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2620

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

Input:

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

Output:

$Aborted
 

Maple [F]

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

Input:

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

Output:

int((f*x+e)^2*cosh(d*x+c)*coth(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. 992 vs. \(2 (421) = 842\).

Time = 0.12 (sec) , antiderivative size = 992, normalized size of antiderivative = 2.15 \[ \int \frac {(e+f x)^2 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:

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

Output:

1/3*(a*d^3*f^2*x^3 + 3*a*d^3*e*f*x^2 + 3*a*d^3*e^2*x + 6*b*f^2*sqrt((a^2 + 
 b^2)/b^2)*polylog(3, (a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c 
) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2))/b) - 6*b*f^2*sqrt((a^2 + b^2)/ 
b^2)*polylog(3, (a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b* 
sinh(d*x + c))*sqrt((a^2 + b^2)/b^2))/b) - 6*b*f^2*polylog(3, cosh(d*x + c 
) + sinh(d*x + c)) + 6*b*f^2*polylog(3, -cosh(d*x + c) - sinh(d*x + c)) - 
6*(b*d*f^2*x + b*d*e*f)*sqrt((a^2 + b^2)/b^2)*dilog((a*cosh(d*x + c) + a*s 
inh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - 
 b)/b + 1) + 6*(b*d*f^2*x + b*d*e*f)*sqrt((a^2 + b^2)/b^2)*dilog((a*cosh(d 
*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 
+ b^2)/b^2) - b)/b + 1) + 3*(b*d^2*e^2 - 2*b*c*d*e*f + b*c^2*f^2)*sqrt((a^ 
2 + b^2)/b^2)*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) + 2*b*sqrt((a^2 + 
b^2)/b^2) + 2*a) - 3*(b*d^2*e^2 - 2*b*c*d*e*f + b*c^2*f^2)*sqrt((a^2 + b^2 
)/b^2)*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) - 2*b*sqrt((a^2 + b^2)/b^ 
2) + 2*a) - 3*(b*d^2*f^2*x^2 + 2*b*d^2*e*f*x + 2*b*c*d*e*f - b*c^2*f^2)*sq 
rt((a^2 + b^2)/b^2)*log(-(a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x 
+ c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b) + 3*(b*d^2*f^2*x^2 + 
 2*b*d^2*e*f*x + 2*b*c*d*e*f - b*c^2*f^2)*sqrt((a^2 + b^2)/b^2)*log(-(a*co 
sh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt(( 
a^2 + b^2)/b^2) - b)/b) + 6*(b*d*f^2*x + b*d*e*f)*dilog(cosh(d*x + c) +...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

e^2*((d*x + c)/(b*d) - log(e^(-d*x - c) + 1)/(a*d) + log(e^(-d*x - c) - 1) 
/(a*d) - sqrt(a^2 + b^2)*log((b*e^(-d*x - c) - a - sqrt(a^2 + b^2))/(b*e^( 
-d*x - c) - a + sqrt(a^2 + b^2)))/(a*b*d)) + 1/3*(f^2*x^3 + 3*e*f*x^2)/b - 
 2*(d*x*log(e^(d*x + c) + 1) + dilog(-e^(d*x + c)))*e*f/(a*d^2) + 2*(d*x*l 
og(-e^(d*x + c) + 1) + dilog(e^(d*x + c)))*e*f/(a*d^2) - (d^2*x^2*log(e^(d 
*x + c) + 1) + 2*d*x*dilog(-e^(d*x + c)) - 2*polylog(3, -e^(d*x + c)))*f^2 
/(a*d^3) + (d^2*x^2*log(-e^(d*x + c) + 1) + 2*d*x*dilog(e^(d*x + c)) - 2*p 
olylog(3, e^(d*x + c)))*f^2/(a*d^3) - integrate(2*((a^2*f^2*e^c + b^2*f^2* 
e^c)*x^2 + 2*(a^2*e*f*e^c + b^2*e*f*e^c)*x)*e^(d*x)/(a*b^2*e^(2*d*x + 2*c) 
 + 2*a^2*b*e^(d*x + c) - a*b^2), x)
 

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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