\(\int \frac {(c+d x)^2}{(a+b \sinh (e+f x))^2} \, dx\) [174]

Optimal result

Integrand size = 20, antiderivative size = 549 \[ \int \frac {(c+d x)^2}{(a+b \sinh (e+f x))^2} \, dx=-\frac {(c+d x)^2}{\left (a^2+b^2\right ) f}+\frac {2 d (c+d x) \log \left (1+\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) f^2}+\frac {a (c+d x)^2 \log \left (1+\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} f}+\frac {2 d (c+d x) \log \left (1+\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) f^2}-\frac {a (c+d x)^2 \log \left (1+\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} f}+\frac {2 d^2 \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) f^3}+\frac {2 a d (c+d x) \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} f^2}+\frac {2 d^2 \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) f^3}-\frac {2 a d (c+d x) \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} f^2}-\frac {2 a d^2 \operatorname {PolyLog}\left (3,-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} f^3}+\frac {2 a d^2 \operatorname {PolyLog}\left (3,-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} f^3}-\frac {b (c+d x)^2 \cosh (e+f x)}{\left (a^2+b^2\right ) f (a+b \sinh (e+f x))} \] Output:

-(d*x+c)^2/(a^2+b^2)/f+2*d*(d*x+c)*ln(1+b*exp(f*x+e)/(a-(a^2+b^2)^(1/2)))/ 
(a^2+b^2)/f^2+a*(d*x+c)^2*ln(1+b*exp(f*x+e)/(a-(a^2+b^2)^(1/2)))/(a^2+b^2) 
^(3/2)/f+2*d*(d*x+c)*ln(1+b*exp(f*x+e)/(a+(a^2+b^2)^(1/2)))/(a^2+b^2)/f^2- 
a*(d*x+c)^2*ln(1+b*exp(f*x+e)/(a+(a^2+b^2)^(1/2)))/(a^2+b^2)^(3/2)/f+2*d^2 
*polylog(2,-b*exp(f*x+e)/(a-(a^2+b^2)^(1/2)))/(a^2+b^2)/f^3+2*a*d*(d*x+c)* 
polylog(2,-b*exp(f*x+e)/(a-(a^2+b^2)^(1/2)))/(a^2+b^2)^(3/2)/f^2+2*d^2*pol 
ylog(2,-b*exp(f*x+e)/(a+(a^2+b^2)^(1/2)))/(a^2+b^2)/f^3-2*a*d*(d*x+c)*poly 
log(2,-b*exp(f*x+e)/(a+(a^2+b^2)^(1/2)))/(a^2+b^2)^(3/2)/f^2-2*a*d^2*polyl 
og(3,-b*exp(f*x+e)/(a-(a^2+b^2)^(1/2)))/(a^2+b^2)^(3/2)/f^3+2*a*d^2*polylo 
g(3,-b*exp(f*x+e)/(a+(a^2+b^2)^(1/2)))/(a^2+b^2)^(3/2)/f^3-b*(d*x+c)^2*cos 
h(f*x+e)/(a^2+b^2)/f/(a+b*sinh(f*x+e))
 

Mathematica [A] (verified)

Time = 0.96 (sec) , antiderivative size = 428, normalized size of antiderivative = 0.78 \[ \int \frac {(c+d x)^2}{(a+b \sinh (e+f x))^2} \, dx=\frac {-f^2 (c+d x)^2+2 d f (c+d x) \log \left (1+\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )+2 d f (c+d x) \log \left (1+\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )+2 d^2 \operatorname {PolyLog}\left (2,\frac {b e^{e+f x}}{-a+\sqrt {a^2+b^2}}\right )+2 d^2 \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )-\frac {a \left (-f^2 (c+d x)^2 \log \left (1+\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )+f^2 (c+d x)^2 \log \left (1+\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )-2 d f (c+d x) \operatorname {PolyLog}\left (2,\frac {b e^{e+f x}}{-a+\sqrt {a^2+b^2}}\right )+2 d f (c+d x) \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )+2 d^2 \operatorname {PolyLog}\left (3,\frac {b e^{e+f x}}{-a+\sqrt {a^2+b^2}}\right )-2 d^2 \operatorname {PolyLog}\left (3,-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )\right )}{\sqrt {a^2+b^2}}-\frac {b f^2 (c+d x)^2 \cosh (e+f x)}{a+b \sinh (e+f x)}}{\left (a^2+b^2\right ) f^3} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 2.70 (sec) , antiderivative size = 520, normalized size of antiderivative = 0.95, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3805

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3803

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2694

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 6095

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

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

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 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]

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

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3957 vs. \(2 (503) = 1006\).

Time = 0.17 (sec) , antiderivative size = 3957, normalized size of antiderivative = 7.21 \[ \int \frac {(c+d x)^2}{(a+b \sinh (e+f x))^2} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(c+d x)^2}{(a+b \sinh (e+f x))^2} \, dx=\text {too large to display} \] Input:

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

Output:

(4*e**(2*e + 2*f*x)*sqrt(a**2 + b**2)*atan((e**(e + f*x)*b*i + a*i)/sqrt(a 
**2 + b**2))*a**3*b*c*d*f*i + 2*e**(2*e + 2*f*x)*sqrt(a**2 + b**2)*atan((e 
**(e + f*x)*b*i + a*i)/sqrt(a**2 + b**2))*a**3*b*d**2*i + 2*e**(2*e + 2*f* 
x)*sqrt(a**2 + b**2)*atan((e**(e + f*x)*b*i + a*i)/sqrt(a**2 + b**2))*a*b* 
*3*c**2*f**2*i + 4*e**(2*e + 2*f*x)*sqrt(a**2 + b**2)*atan((e**(e + f*x)*b 
*i + a*i)/sqrt(a**2 + b**2))*a*b**3*c*d*f*i + 2*e**(2*e + 2*f*x)*sqrt(a**2 
 + b**2)*atan((e**(e + f*x)*b*i + a*i)/sqrt(a**2 + b**2))*a*b**3*d**2*i + 
8*e**(e + f*x)*sqrt(a**2 + b**2)*atan((e**(e + f*x)*b*i + a*i)/sqrt(a**2 + 
 b**2))*a**4*c*d*f*i + 4*e**(e + f*x)*sqrt(a**2 + b**2)*atan((e**(e + f*x) 
*b*i + a*i)/sqrt(a**2 + b**2))*a**4*d**2*i + 4*e**(e + f*x)*sqrt(a**2 + b* 
*2)*atan((e**(e + f*x)*b*i + a*i)/sqrt(a**2 + b**2))*a**2*b**2*c**2*f**2*i 
 + 8*e**(e + f*x)*sqrt(a**2 + b**2)*atan((e**(e + f*x)*b*i + a*i)/sqrt(a** 
2 + b**2))*a**2*b**2*c*d*f*i + 4*e**(e + f*x)*sqrt(a**2 + b**2)*atan((e**( 
e + f*x)*b*i + a*i)/sqrt(a**2 + b**2))*a**2*b**2*d**2*i - 4*sqrt(a**2 + b* 
*2)*atan((e**(e + f*x)*b*i + a*i)/sqrt(a**2 + b**2))*a**3*b*c*d*f*i - 2*sq 
rt(a**2 + b**2)*atan((e**(e + f*x)*b*i + a*i)/sqrt(a**2 + b**2))*a**3*b*d* 
*2*i - 2*sqrt(a**2 + b**2)*atan((e**(e + f*x)*b*i + a*i)/sqrt(a**2 + b**2) 
)*a*b**3*c**2*f**2*i - 4*sqrt(a**2 + b**2)*atan((e**(e + f*x)*b*i + a*i)/s 
qrt(a**2 + b**2))*a*b**3*c*d*f*i - 2*sqrt(a**2 + b**2)*atan((e**(e + f*x)* 
b*i + a*i)/sqrt(a**2 + b**2))*a*b**3*d**2*i - 4*e**(3*e + 2*f*x)*int((e...