\(\int \frac {(e+f x) \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) [369]

Optimal result

Integrand size = 34, antiderivative size = 395 \[ \int \frac {(e+f x) \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {a^3 (e+f x)^2}{2 b^4 f}-\frac {a (e+f x)^2}{4 b^2 f}+\frac {a^2 (e+f x) \cosh (c+d x)}{b^3 d}+\frac {a f \cosh ^2(c+d x)}{4 b^2 d^2}+\frac {(e+f x) \cosh ^3(c+d x)}{3 b d}+\frac {a^2 \sqrt {a^2+b^2} (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d}-\frac {a^2 \sqrt {a^2+b^2} (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d}+\frac {a^2 \sqrt {a^2+b^2} f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^2}-\frac {a^2 \sqrt {a^2+b^2} f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^2}-\frac {a^2 f \sinh (c+d x)}{b^3 d^2}-\frac {f \sinh (c+d x)}{3 b d^2}-\frac {a (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^2 d}-\frac {f \sinh ^3(c+d x)}{9 b d^2} \] Output:

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

Mathematica [A] (verified)

Time = 2.27 (sec) , antiderivative size = 626, normalized size of antiderivative = 1.58 \[ \int \frac {(e+f x) \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {72 a^3 c d e+36 a b^2 c d e-36 a^3 c^2 f-18 a b^2 c^2 f+72 a^3 d^2 e x+36 a b^2 d^2 e x+36 a^3 d^2 f x^2+18 a b^2 d^2 f x^2+144 a^2 \sqrt {a^2+b^2} d e \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )-144 a^2 \sqrt {a^2+b^2} c f \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )-72 a^2 b d e \cosh (c+d x)-18 b^3 d e \cosh (c+d x)-72 a^2 b d f x \cosh (c+d x)-18 b^3 d f x \cosh (c+d x)-9 a b^2 f \cosh (2 (c+d x))-6 b^3 d e \cosh (3 (c+d x))-6 b^3 d f x \cosh (3 (c+d x))-72 a^2 \sqrt {a^2+b^2} c f \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-72 a^2 \sqrt {a^2+b^2} d f x \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+72 a^2 \sqrt {a^2+b^2} c f \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+72 a^2 \sqrt {a^2+b^2} d f x \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-72 a^2 \sqrt {a^2+b^2} f \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+72 a^2 \sqrt {a^2+b^2} f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+72 a^2 b f \sinh (c+d x)+18 b^3 f \sinh (c+d x)+18 a b^2 d e \sinh (2 (c+d x))+18 a b^2 d f x \sinh (2 (c+d x))+2 b^3 f \sinh (3 (c+d x))}{72 b^4 d^2} \] Input:

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

Output:

-1/72*(72*a^3*c*d*e + 36*a*b^2*c*d*e - 36*a^3*c^2*f - 18*a*b^2*c^2*f + 72* 
a^3*d^2*e*x + 36*a*b^2*d^2*e*x + 36*a^3*d^2*f*x^2 + 18*a*b^2*d^2*f*x^2 + 1 
44*a^2*Sqrt[a^2 + b^2]*d*e*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]] - 
144*a^2*Sqrt[a^2 + b^2]*c*f*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]] - 
 72*a^2*b*d*e*Cosh[c + d*x] - 18*b^3*d*e*Cosh[c + d*x] - 72*a^2*b*d*f*x*Co 
sh[c + d*x] - 18*b^3*d*f*x*Cosh[c + d*x] - 9*a*b^2*f*Cosh[2*(c + d*x)] - 6 
*b^3*d*e*Cosh[3*(c + d*x)] - 6*b^3*d*f*x*Cosh[3*(c + d*x)] - 72*a^2*Sqrt[a 
^2 + b^2]*c*f*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] - 72*a^2*Sqrt 
[a^2 + b^2]*d*f*x*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + 72*a^2* 
Sqrt[a^2 + b^2]*c*f*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + 72*a^ 
2*Sqrt[a^2 + b^2]*d*f*x*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] - 7 
2*a^2*Sqrt[a^2 + b^2]*f*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] 
 + 72*a^2*Sqrt[a^2 + b^2]*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b 
^2]))] + 72*a^2*b*f*Sinh[c + d*x] + 18*b^3*f*Sinh[c + d*x] + 18*a*b^2*d*e* 
Sinh[2*(c + d*x)] + 18*a*b^2*d*f*x*Sinh[2*(c + d*x)] + 2*b^3*f*Sinh[3*(c + 
 d*x)])/(b^4*d^2)
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 2.49 (sec) , antiderivative size = 391, normalized size of antiderivative = 0.99, number of steps used = 24, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.676, Rules used = {6113, 5970, 3042, 3113, 2009, 6113, 3042, 3791, 17, 6099, 17, 3042, 26, 3777, 3042, 3117, 3803, 25, 2694, 27, 2620, 2715, 2838}

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.

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

\(\Big \downarrow \) 6113

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

\(\Big \downarrow \) 5970

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3113

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 6113

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3791

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

\(\Big \downarrow \) 17

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

\(\Big \downarrow \) 6099

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

\(\Big \downarrow \) 17

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3117

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

\(\Big \downarrow \) 3803

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2694

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

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

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 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 2694

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.) 
*(F_)^(v_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[2*(c/q)   Int 
[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Simp[2*(c/q)   Int[(f + g*x) 
^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[ 
v, 2*u] && LinearQ[u, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[m, 0]
 

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

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 3803

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

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 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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1127\) vs. \(2(361)=722\).

Time = 19.55 (sec) , antiderivative size = 1128, normalized size of antiderivative = 2.86

method	result	size
risch	\(\text {Expression too large to display}\)	\(1128\)

Input:

int((f*x+e)*cosh(d*x+c)^2*sinh(d*x+c)^2/(a+b*sinh(d*x+c)),x,method=_RETURN 
VERBOSE)
 

Output:

-1/16*a*(2*d*f*x+2*d*e-f)/b^2/d^2*exp(2*d*x+2*c)+1/8*(4*a^2+b^2)*(d*f*x+d* 
e+f)/b^3/d^2*exp(-d*x-c)+1/16*a*(2*d*f*x+2*d*e+f)/b^2/d^2*exp(-2*d*x-2*c)+ 
1/8*(4*a^2*d*f*x+b^2*d*f*x+4*a^2*d*e+b^2*d*e-4*a^2*f-b^2*f)/b^3/d^2*exp(d* 
x+c)+1/72*(3*d*f*x+3*d*e-f)/b/d^2*exp(3*d*x+3*c)+1/72*(3*d*f*x+3*d*e+f)/b/ 
d^2*exp(-3*d*x-3*c)-1/2*a^3/b^4*f*x^2-1/4*a/b^2*f*x^2-a^3/b^4*e*x-1/2*a/b^ 
2*e*x+1/d^2/b^2*a^2*f/(a^2+b^2)^(1/2)*dilog((-b*exp(d*x+c)+(a^2+b^2)^(1/2) 
-a)/(-a+(a^2+b^2)^(1/2)))+2/d^2*a^4/b^4*f*c/(a^2+b^2)^(1/2)*arctanh(1/2*(2 
*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+1/d*a^4/b^4*f/(a^2+b^2)^(1/2)*ln((-b*e 
xp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x-1/d*a^4/b^4*f/(a^2+b^ 
2)^(1/2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*x+1/d^2* 
a^4/b^4*f/(a^2+b^2)^(1/2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^ 
2)^(1/2)))*c-1/d^2*a^4/b^4*f/(a^2+b^2)^(1/2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1 
/2)+a)/(a+(a^2+b^2)^(1/2)))*c+1/d^2*a^2/b^2*f/(a^2+b^2)^(1/2)*ln((-b*exp(d 
*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*c-1/d/b^2*a^2*f/(a^2+b^2)^( 
1/2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*x-1/d^2/b^2* 
a^2*f/(a^2+b^2)^(1/2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/ 
2)))*c+2/d^2/b^2*a^2*f*c/(a^2+b^2)^(1/2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/ 
(a^2+b^2)^(1/2))+1/d*a^2/b^2*f/(a^2+b^2)^(1/2)*ln((-b*exp(d*x+c)+(a^2+b^2) 
^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x-2/d*a^4/b^4*e/(a^2+b^2)^(1/2)*arctanh(1/ 
2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-2/d*a^2/b^2*e/(a^2+b^2)^(1/2)*a...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2195 vs. \(2 (359) = 718\).

Time = 0.16 (sec) , antiderivative size = 2195, normalized size of antiderivative = 5.56 \[ \int \frac {(e+f x) \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:

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

Output:

1/144*(2*(3*b^3*d*f*x + 3*b^3*d*e - b^3*f)*cosh(d*x + c)^6 + 2*(3*b^3*d*f* 
x + 3*b^3*d*e - b^3*f)*sinh(d*x + c)^6 + 6*b^3*d*f*x - 9*(2*a*b^2*d*f*x + 
2*a*b^2*d*e - a*b^2*f)*cosh(d*x + c)^5 - 3*(6*a*b^2*d*f*x + 6*a*b^2*d*e - 
3*a*b^2*f - 4*(3*b^3*d*f*x + 3*b^3*d*e - b^3*f)*cosh(d*x + c))*sinh(d*x + 
c)^5 + 6*b^3*d*e + 18*((4*a^2*b + b^3)*d*f*x + (4*a^2*b + b^3)*d*e - (4*a^ 
2*b + b^3)*f)*cosh(d*x + c)^4 + 3*(6*(4*a^2*b + b^3)*d*f*x + 6*(4*a^2*b + 
b^3)*d*e + 10*(3*b^3*d*f*x + 3*b^3*d*e - b^3*f)*cosh(d*x + c)^2 - 6*(4*a^2 
*b + b^3)*f - 15*(2*a*b^2*d*f*x + 2*a*b^2*d*e - a*b^2*f)*cosh(d*x + c))*si 
nh(d*x + c)^4 + 2*b^3*f - 36*((2*a^3 + a*b^2)*d^2*f*x^2 + 2*(2*a^3 + a*b^2 
)*d^2*e*x)*cosh(d*x + c)^3 - 2*(18*(2*a^3 + a*b^2)*d^2*f*x^2 + 36*(2*a^3 + 
 a*b^2)*d^2*e*x - 20*(3*b^3*d*f*x + 3*b^3*d*e - b^3*f)*cosh(d*x + c)^3 + 4 
5*(2*a*b^2*d*f*x + 2*a*b^2*d*e - a*b^2*f)*cosh(d*x + c)^2 - 36*((4*a^2*b + 
 b^3)*d*f*x + (4*a^2*b + b^3)*d*e - (4*a^2*b + b^3)*f)*cosh(d*x + c))*sinh 
(d*x + c)^3 + 18*((4*a^2*b + b^3)*d*f*x + (4*a^2*b + b^3)*d*e + (4*a^2*b + 
 b^3)*f)*cosh(d*x + c)^2 + 6*(5*(3*b^3*d*f*x + 3*b^3*d*e - b^3*f)*cosh(d*x 
 + c)^4 + 3*(4*a^2*b + b^3)*d*f*x - 15*(2*a*b^2*d*f*x + 2*a*b^2*d*e - a*b^ 
2*f)*cosh(d*x + c)^3 + 3*(4*a^2*b + b^3)*d*e + 18*((4*a^2*b + b^3)*d*f*x + 
 (4*a^2*b + b^3)*d*e - (4*a^2*b + b^3)*f)*cosh(d*x + c)^2 + 3*(4*a^2*b + b 
^3)*f - 18*((2*a^3 + a*b^2)*d^2*f*x^2 + 2*(2*a^3 + a*b^2)*d^2*e*x)*cosh(d* 
x + c))*sinh(d*x + c)^2 + 144*(a^2*b*f*cosh(d*x + c)^3 + 3*a^2*b*f*cosh...
 

Sympy [F(-1)]

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

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

Output:

Timed out
                                                                                    
                                                                                    
 

Maxima [F]

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

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

Output:

1/144*(288*(a^4*e^c + a^2*b^2*e^c)*integrate(x*e^(d*x)/(b^5*e^(2*d*x + 2*c 
) + 2*a*b^4*e^(d*x + c) - b^5), x) - (36*(2*a^3*d^2*e^(3*c) + a*b^2*d^2*e^ 
(3*c))*x^2 - 2*(3*b^3*d*x*e^(6*c) - b^3*e^(6*c))*e^(3*d*x) + 9*(2*a*b^2*d* 
x*e^(5*c) - a*b^2*e^(5*c))*e^(2*d*x) + 18*(4*a^2*b*e^(4*c) + b^3*e^(4*c) - 
 (4*a^2*b*d*e^(4*c) + b^3*d*e^(4*c))*x)*e^(d*x) - 18*(4*a^2*b*e^(2*c) + b^ 
3*e^(2*c) + (4*a^2*b*d*e^(2*c) + b^3*d*e^(2*c))*x)*e^(-d*x) - 9*(2*a*b^2*d 
*x*e^c + a*b^2*e^c)*e^(-2*d*x) - 2*(3*b^3*d*x + b^3)*e^(-3*d*x))*e^(-3*c)/ 
(b^4*d^2))*f + 1/24*e*(24*sqrt(a^2 + b^2)*a^2*log((b*e^(-d*x - c) - a - sq 
rt(a^2 + b^2))/(b*e^(-d*x - c) - a + sqrt(a^2 + b^2)))/(b^4*d) - (3*a*b*e^ 
(-d*x - c) - b^2 - 3*(4*a^2 + b^2)*e^(-2*d*x - 2*c))*e^(3*d*x + 3*c)/(b^3* 
d) - 12*(2*a^3 + a*b^2)*(d*x + c)/(b^4*d) + (3*a*b*e^(-2*d*x - 2*c) + b^2* 
e^(-3*d*x - 3*c) + 3*(4*a^2 + b^2)*e^(-d*x - c))/(b^3*d))
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(e+f x) \cosh ^2(c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {72 e^{d x +c} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{d x +c} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) a^{2} b d e i -18 e^{d x +c} \cosh \left (d x +c \right ) \sinh \left (d x +c \right ) a \,b^{3} d f x +8 e^{d x +c} \sinh \left (d x +c \right )^{3} b^{4} f +18 a^{2} b^{2} d e -18 e^{2 d x +2 c} a^{2} b^{2} f +18 e^{2 d x +2 c} a^{2} b^{2} d f x -54 a^{2} b^{2} f -18 e^{d x +c} \cosh \left (d x +c \right )^{2} a \,b^{3} d^{2} e x -9 e^{d x +c} \cosh \left (d x +c \right )^{2} a \,b^{3} d^{2} f \,x^{2}-18 e^{d x +c} \cosh \left (d x +c \right ) \sinh \left (d x +c \right ) a \,b^{3} d e +18 e^{d x +c} \sinh \left (d x +c \right )^{2} a \,b^{3} d^{2} e x +9 e^{d x +c} \sinh \left (d x +c \right )^{2} a \,b^{3} d^{2} f \,x^{2}-54 a^{2} b^{2} d f x -72 a^{4} d f x +12 e^{d x +c} \cosh \left (d x +c \right )^{3} b^{4} d e -12 e^{d x +c} \cosh \left (d x +c \right )^{2} \sinh \left (d x +c \right ) b^{4} f +9 e^{d x +c} \cosh \left (d x +c \right )^{2} a \,b^{3} f -72 a^{4} f +12 e^{d x +c} \cosh \left (d x +c \right )^{3} b^{4} d f x -36 e^{d x +c} a^{3} b \,d^{2} e x -18 e^{d x +c} a^{3} b \,d^{2} f \,x^{2}-144 e^{d x +c} \left (\int \frac {x}{e^{2 d x +2 c} b +2 e^{d x +c} a -b}d x \right ) a^{3} b^{2} d^{2} f +72 e^{d x} \left (\int \frac {x}{e^{3 d x +2 c} b +2 e^{2 d x +c} a -e^{d x} b}d x \right ) a^{4} b \,d^{2} f +72 e^{d x} \left (\int \frac {x}{e^{3 d x +2 c} b +2 e^{2 d x +c} a -e^{d x} b}d x \right ) a^{2} b^{3} d^{2} f -144 e^{d x +c} \left (\int \frac {x}{e^{2 d x +2 c} b +2 e^{d x +c} a -b}d x \right ) a^{5} d^{2} f +18 e^{2 d x +2 c} a^{2} b^{2} d e}{36 e^{d x +c} b^{5} d^{2}} \] Input:

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

Output:

(72*e**(c + d*x)*sqrt(a**2 + b**2)*atan((e**(c + d*x)*b*i + a*i)/sqrt(a**2 
 + b**2))*a**2*b*d*e*i + 12*e**(c + d*x)*cosh(c + d*x)**3*b**4*d*e + 12*e* 
*(c + d*x)*cosh(c + d*x)**3*b**4*d*f*x - 12*e**(c + d*x)*cosh(c + d*x)**2* 
sinh(c + d*x)*b**4*f - 18*e**(c + d*x)*cosh(c + d*x)**2*a*b**3*d**2*e*x - 
9*e**(c + d*x)*cosh(c + d*x)**2*a*b**3*d**2*f*x**2 + 9*e**(c + d*x)*cosh(c 
 + d*x)**2*a*b**3*f - 18*e**(c + d*x)*cosh(c + d*x)*sinh(c + d*x)*a*b**3*d 
*e - 18*e**(c + d*x)*cosh(c + d*x)*sinh(c + d*x)*a*b**3*d*f*x + 18*e**(2*c 
 + 2*d*x)*a**2*b**2*d*e + 18*e**(2*c + 2*d*x)*a**2*b**2*d*f*x - 18*e**(2*c 
 + 2*d*x)*a**2*b**2*f - 144*e**(c + d*x)*int(x/(e**(2*c + 2*d*x)*b + 2*e** 
(c + d*x)*a - b),x)*a**5*d**2*f - 144*e**(c + d*x)*int(x/(e**(2*c + 2*d*x) 
*b + 2*e**(c + d*x)*a - b),x)*a**3*b**2*d**2*f + 8*e**(c + d*x)*sinh(c + d 
*x)**3*b**4*f + 18*e**(c + d*x)*sinh(c + d*x)**2*a*b**3*d**2*e*x + 9*e**(c 
 + d*x)*sinh(c + d*x)**2*a*b**3*d**2*f*x**2 - 36*e**(c + d*x)*a**3*b*d**2* 
e*x - 18*e**(c + d*x)*a**3*b*d**2*f*x**2 + 72*e**(d*x)*int(x/(e**(2*c + 3* 
d*x)*b + 2*e**(c + 2*d*x)*a - e**(d*x)*b),x)*a**4*b*d**2*f + 72*e**(d*x)*i 
nt(x/(e**(2*c + 3*d*x)*b + 2*e**(c + 2*d*x)*a - e**(d*x)*b),x)*a**2*b**3*d 
**2*f - 72*a**4*d*f*x - 72*a**4*f + 18*a**2*b**2*d*e - 54*a**2*b**2*d*f*x 
- 54*a**2*b**2*f)/(36*e**(c + d*x)*b**5*d**2)