\(\int \frac {(e+f x)^3 \cosh (c+d x)}{(a+b \sinh (c+d x))^3} \, dx\) [329]

Optimal result

Integrand size = 26, antiderivative size = 631 \[ \int \frac {(e+f x)^3 \cosh (c+d x)}{(a+b \sinh (c+d x))^3} \, dx=-\frac {3 f (e+f x)^2}{2 b \left (a^2+b^2\right ) d^2}+\frac {3 f^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d^3}+\frac {3 a f (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{2 b \left (a^2+b^2\right )^{3/2} d^2}+\frac {3 f^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d^3}-\frac {3 a f (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{2 b \left (a^2+b^2\right )^{3/2} d^2}+\frac {3 f^3 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d^4}+\frac {3 a f^2 (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^3}+\frac {3 f^3 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d^4}-\frac {3 a f^2 (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^3}-\frac {3 a f^3 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^4}+\frac {3 a f^3 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2} d^4}-\frac {(e+f x)^3}{2 b d (a+b \sinh (c+d x))^2}-\frac {3 f (e+f x)^2 \cosh (c+d x)}{2 \left (a^2+b^2\right ) d^2 (a+b \sinh (c+d x))} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(5753\) vs. \(2(631)=1262\).

Time = 13.58 (sec) , antiderivative size = 5753, normalized size of antiderivative = 9.12 \[ \int \frac {(e+f x)^3 \cosh (c+d x)}{(a+b \sinh (c+d x))^3} \, dx=\text {Result too large to show} \] Input:

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

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 3.14 (sec) , antiderivative size = 561, normalized size of antiderivative = 0.89, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {5987, 3042, 3805, 3042, 3803, 25, 2694, 27, 2620, 3011, 2720, 6095, 2620, 2715, 2838, 7143}

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.

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

\(\Big \downarrow \) 5987

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3805

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3803

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2694

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 6095

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

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

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 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 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

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 3011

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) 
*(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + 
b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F]))   Int[(f + g*x)^( 
m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e 
, f, g, n}, x] && GtQ[m, 0]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

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 3805

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

rule 5987

Int[Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.)*((a_) + (b_.)*Sinh[ 
(c_.) + (d_.)*(x_)])^(n_.), x_Symbol] :> Simp[(e + f*x)^m*((a + b*Sinh[c + 
d*x])^(n + 1)/(b*d*(n + 1))), x] - Simp[f*(m/(b*d*(n + 1)))   Int[(e + f*x) 
^(m - 1)*(a + b*Sinh[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, 
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 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 11757 vs. \(2 (575) = 1150\).

Time = 0.32 (sec) , antiderivative size = 11757, normalized size of antiderivative = 18.63 \[ \int \frac {(e+f x)^3 \cosh (c+d x)}{(a+b \sinh (c+d x))^3} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

3*a*d*f^3*integrate(x^2*e^(d*x + c)/(a^2*b^2*d^2*e^(2*d*x + 2*c) + b^4*d^2 
*e^(2*d*x + 2*c) + 2*a^3*b*d^2*e^(d*x + c) + 2*a*b^3*d^2*e^(d*x + c) - a^2 
*b^2*d^2 - b^4*d^2), x) + 6*a*d*e*f^2*integrate(x*e^(d*x + c)/(a^2*b^2*d^2 
*e^(2*d*x + 2*c) + b^4*d^2*e^(2*d*x + 2*c) + 2*a^3*b*d^2*e^(d*x + c) + 2*a 
*b^3*d^2*e^(d*x + c) - a^2*b^2*d^2 - b^4*d^2), x) + 3*b*e*f^2*(a*log((b*e^ 
(d*x + c) + a - sqrt(a^2 + b^2))/(b*e^(d*x + c) + a + sqrt(a^2 + b^2)))/(( 
a^2*b^2 + b^4)*sqrt(a^2 + b^2)*d^3) - 2*(d*x + c)/((a^2*b^2 + b^4)*d^3) + 
log(b*e^(2*d*x + 2*c) + 2*a*e^(d*x + c) - b)/((a^2*b^2 + b^4)*d^3)) - 6*a* 
f^3*integrate(x*e^(d*x + c)/(a^2*b^2*d^2*e^(2*d*x + 2*c) + b^4*d^2*e^(2*d* 
x + 2*c) + 2*a^3*b*d^2*e^(d*x + c) + 2*a*b^3*d^2*e^(d*x + c) - a^2*b^2*d^2 
 - b^4*d^2), x) + 6*b*f^3*integrate(x/(a^2*b^2*d^2*e^(2*d*x + 2*c) + b^4*d 
^2*e^(2*d*x + 2*c) + 2*a^3*b*d^2*e^(d*x + c) + 2*a*b^3*d^2*e^(d*x + c) - a 
^2*b^2*d^2 - b^4*d^2), x) + 3/2*e^2*f*(2*(a*b*e^(3*d*x + 3*c) - 3*a*b*e^(d 
*x + c) + b^2 + (2*a^2*e^(2*c) - b^2*e^(2*c) - 2*(a^2*d*e^(2*c) + b^2*d*e^ 
(2*c))*x)*e^(2*d*x))/(a^2*b^3*d^2 + b^5*d^2 + (a^2*b^3*d^2*e^(4*c) + b^5*d 
^2*e^(4*c))*e^(4*d*x) + 4*(a^3*b^2*d^2*e^(3*c) + a*b^4*d^2*e^(3*c))*e^(3*d 
*x) + 2*(2*a^4*b*d^2*e^(2*c) + a^2*b^3*d^2*e^(2*c) - b^5*d^2*e^(2*c))*e^(2 
*d*x) - 4*(a^3*b^2*d^2*e^c + a*b^4*d^2*e^c)*e^(d*x)) + a*log((b*e^(d*x + 2 
*c) + a*e^c - sqrt(a^2 + b^2)*e^c)/(b*e^(d*x + 2*c) + a*e^c + sqrt(a^2 + b 
^2)*e^c))/((a^2*b + b^3)*sqrt(a^2 + b^2)*d^2)) - 2*e^3*e^(-2*d*x - 2*c)...
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(192*e**(4*c + 4*d*x)*sqrt(a**2 + b**2)*atan((e**(c + d*x)*b*i + a*i)/sqrt 
(a**2 + b**2))*a**5*b**2*d*e*f**2*i + 16*e**(4*c + 4*d*x)*sqrt(a**2 + b**2 
)*atan((e**(c + d*x)*b*i + a*i)/sqrt(a**2 + b**2))*a**5*b**2*f**3*i + 576* 
e**(4*c + 4*d*x)*sqrt(a**2 + b**2)*atan((e**(c + d*x)*b*i + a*i)/sqrt(a**2 
 + b**2))*a**3*b**4*d*e*f**2*i + 240*e**(4*c + 4*d*x)*sqrt(a**2 + b**2)*at 
an((e**(c + d*x)*b*i + a*i)/sqrt(a**2 + b**2))*a**3*b**4*f**3*i + 144*e**( 
4*c + 4*d*x)*sqrt(a**2 + b**2)*atan((e**(c + d*x)*b*i + a*i)/sqrt(a**2 + b 
**2))*a*b**6*d**2*e**2*f*i + 384*e**(4*c + 4*d*x)*sqrt(a**2 + b**2)*atan(( 
e**(c + d*x)*b*i + a*i)/sqrt(a**2 + b**2))*a*b**6*d*e*f**2*i + 236*e**(4*c 
 + 4*d*x)*sqrt(a**2 + b**2)*atan((e**(c + d*x)*b*i + a*i)/sqrt(a**2 + b**2 
))*a*b**6*f**3*i + 768*e**(3*c + 3*d*x)*sqrt(a**2 + b**2)*atan((e**(c + d* 
x)*b*i + a*i)/sqrt(a**2 + b**2))*a**6*b*d*e*f**2*i + 64*e**(3*c + 3*d*x)*s 
qrt(a**2 + b**2)*atan((e**(c + d*x)*b*i + a*i)/sqrt(a**2 + b**2))*a**6*b*f 
**3*i + 2304*e**(3*c + 3*d*x)*sqrt(a**2 + b**2)*atan((e**(c + d*x)*b*i + a 
*i)/sqrt(a**2 + b**2))*a**4*b**3*d*e*f**2*i + 960*e**(3*c + 3*d*x)*sqrt(a* 
*2 + b**2)*atan((e**(c + d*x)*b*i + a*i)/sqrt(a**2 + b**2))*a**4*b**3*f**3 
*i + 576*e**(3*c + 3*d*x)*sqrt(a**2 + b**2)*atan((e**(c + d*x)*b*i + a*i)/ 
sqrt(a**2 + b**2))*a**2*b**5*d**2*e**2*f*i + 1536*e**(3*c + 3*d*x)*sqrt(a* 
*2 + b**2)*atan((e**(c + d*x)*b*i + a*i)/sqrt(a**2 + b**2))*a**2*b**5*d*e* 
f**2*i + 944*e**(3*c + 3*d*x)*sqrt(a**2 + b**2)*atan((e**(c + d*x)*b*i ...