\(\int \frac {(e+f x)^2 \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) [229]

Optimal result

Integrand size = 28, antiderivative size = 407 \[ \int \frac {(e+f x)^2 \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {a (e+f x)^3}{3 b^2 f}+\frac {2 f^2 \cosh (c+d x)}{b d^3}+\frac {(e+f x)^2 \cosh (c+d x)}{b d}+\frac {a^2 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 \sqrt {a^2+b^2} d}-\frac {a^2 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 \sqrt {a^2+b^2} d}+\frac {2 a^2 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 \sqrt {a^2+b^2} d^2}-\frac {2 a^2 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 \sqrt {a^2+b^2} d^2}-\frac {2 a^2 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 \sqrt {a^2+b^2} d^3}+\frac {2 a^2 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 \sqrt {a^2+b^2} d^3}-\frac {2 f (e+f x) \sinh (c+d x)}{b d^2} \] Output:

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

Mathematica [A] (verified)

Time = 1.69 (sec) , antiderivative size = 453, normalized size of antiderivative = 1.11 \[ \int \frac {(e+f x)^2 \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {-a x \left (3 e^2+3 e f x+f^2 x^2\right )+\frac {3 a^2 \left (-2 d^2 e^2 \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )+2 d^2 e f x \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+d^2 f^2 x^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-2 d^2 e f x \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-d^2 f^2 x^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+2 d f (e+f x) \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-2 d f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-2 f^2 \operatorname {PolyLog}\left (3,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+2 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{\sqrt {a^2+b^2} d^3}+\frac {3 b \cosh (d x) \left (\left (2 f^2+d^2 (e+f x)^2\right ) \cosh (c)-2 d f (e+f x) \sinh (c)\right )}{d^3}+\frac {3 b \left (-2 d f (e+f x) \cosh (c)+\left (2 f^2+d^2 (e+f x)^2\right ) \sinh (c)\right ) \sinh (d x)}{d^3}}{3 b^2} \] Input:

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

Output:

(-(a*x*(3*e^2 + 3*e*f*x + f^2*x^2)) + (3*a^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 - 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^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*f*(e + f*x)*PolyLo 
g[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]))]))/(Sqrt[a^2 + b^2]*d^3) + (3*b*Cosh[d*x]*((2*f^2 + d^2*(e + 
f*x)^2)*Cosh[c] - 2*d*f*(e + f*x)*Sinh[c]))/d^3 + (3*b*(-2*d*f*(e + f*x)*C 
osh[c] + (2*f^2 + d^2*(e + f*x)^2)*Sinh[c])*Sinh[d*x])/d^3)/(3*b^2)
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 2.21 (sec) , antiderivative size = 376, normalized size of antiderivative = 0.92, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6091, 3042, 26, 3777, 3042, 3777, 26, 3042, 26, 3118, 6091, 17, 3042, 3803, 25, 2694, 27, 2620, 3011, 2720, 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.

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

\(\Big \downarrow \) 6091

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3118

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

\(\Big \downarrow \) 6091

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

\(\Big \downarrow \) 17

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3803

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2694

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

-((a*((e + f*x)^3/(3*b*f) + (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 + Sqrt[a^2 + b^2]))])/d^2))/(b*d)))/(2*Sqrt[a^2 + b^2 
])))/b))/b) - (I*((I*(e + f*x)^2*Cosh[c + d*x])/d - ((2*I)*f*(-((f*Cosh[c 
+ d*x])/d^2) + ((e + f*x)*Sinh[c + d*x])/d))/d))/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 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 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 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 3118

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[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 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 6091

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

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

Input:

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

Output:

int((f*x+e)^2*sinh(d*x+c)^2/(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. 1689 vs. \(2 (373) = 746\).

Time = 0.11 (sec) , antiderivative size = 1689, normalized size of antiderivative = 4.15 \[ \int \frac {(e+f x)^2 \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:

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

Output:

1/6*(3*(a^2*b + b^3)*d^2*f^2*x^2 + 3*(a^2*b + b^3)*d^2*e^2 + 6*(a^2*b + b^ 
3)*d*e*f + 6*(a^2*b + b^3)*f^2 + 3*((a^2*b + b^3)*d^2*f^2*x^2 + (a^2*b + b 
^3)*d^2*e^2 - 2*(a^2*b + b^3)*d*e*f + 2*(a^2*b + b^3)*f^2 + 2*((a^2*b + b^ 
3)*d^2*e*f - (a^2*b + b^3)*d*f^2)*x)*cosh(d*x + c)^2 + 3*((a^2*b + b^3)*d^ 
2*f^2*x^2 + (a^2*b + b^3)*d^2*e^2 - 2*(a^2*b + b^3)*d*e*f + 2*(a^2*b + b^3 
)*f^2 + 2*((a^2*b + b^3)*d^2*e*f - (a^2*b + b^3)*d*f^2)*x)*sinh(d*x + c)^2 
 + 12*((a^2*b*d*f^2*x + a^2*b*d*e*f)*cosh(d*x + c) + (a^2*b*d*f^2*x + a^2* 
b*d*e*f)*sinh(d*x + c))*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) - 12*((a^2*b*d*f^2*x + a^2*b*d*e*f)*cosh(d*x + c) + (a^2*b*d*f^ 
2*x + a^2*b*d*e*f)*sinh(d*x + c))*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) - 6*((a^2*b*d^2*e^2 - 2*a^2*b*c*d*e*f + a^2*b*c^2*f^2 
)*cosh(d*x + c) + (a^2*b*d^2*e^2 - 2*a^2*b*c*d*e*f + a^2*b*c^2*f^2)*sinh(d 
*x + c))*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) + 6*((a^2*b*d^2*e^2 - 2*a^2*b*c*d*e*f + 
a^2*b*c^2*f^2)*cosh(d*x + c) + (a^2*b*d^2*e^2 - 2*a^2*b*c*d*e*f + a^2*b*c^ 
2*f^2)*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2)*log(2*b*cosh(d*x + c) + 2*b*si 
nh(d*x + c) - 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) + 6*((a^2*b*d^2*f^2*x^2 + 2 
*a^2*b*d^2*e*f*x + 2*a^2*b*c*d*e*f - a^2*b*c^2*f^2)*cosh(d*x + c) + (a^...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

1/2*e^2*(2*a^2*log((b*e^(-d*x - c) - a - sqrt(a^2 + b^2))/(b*e^(-d*x - c) 
- a + sqrt(a^2 + b^2)))/(sqrt(a^2 + b^2)*b^2*d) - 2*(d*x + c)*a/(b^2*d) + 
e^(d*x + c)/(b*d) + e^(-d*x - c)/(b*d)) - 1/6*(2*a*d^3*f^2*x^3*e^c + 6*a*d 
^3*e*f*x^2*e^c - 3*(b*d^2*f^2*x^2*e^(2*c) + 2*(d^2*e*f - d*f^2)*b*x*e^(2*c 
) - 2*(d*e*f - f^2)*b*e^(2*c))*e^(d*x) - 3*(b*d^2*f^2*x^2 + 2*(d^2*e*f + d 
*f^2)*b*x + 2*(d*e*f + f^2)*b)*e^(-d*x))*e^(-c)/(b^2*d^3) + integrate(2*(a 
^2*f^2*x^2*e^c + 2*a^2*e*f*x*e^c)*e^(d*x)/(b^3*e^(2*d*x + 2*c) + 2*a*b^2*e 
^(d*x + c) - b^3), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(e+f x)^2 \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {6 \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{d x +c} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) a^{2} d^{2} e^{2} i +3 \cosh \left (d x +c \right ) a^{2} b \,d^{2} e^{2}+6 \cosh \left (d x +c \right ) a^{2} b \,d^{2} e f x +3 \cosh \left (d x +c \right ) a^{2} b \,d^{2} f^{2} x^{2}+6 \cosh \left (d x +c \right ) a^{2} b \,f^{2}+3 \cosh \left (d x +c \right ) b^{3} d^{2} e^{2}+6 \cosh \left (d x +c \right ) b^{3} d^{2} e f x +3 \cosh \left (d x +c \right ) b^{3} d^{2} f^{2} x^{2}+6 \cosh \left (d x +c \right ) b^{3} f^{2}+6 e^{c} \left (\int \frac {e^{d x} x^{2}}{e^{2 d x +2 c} b +2 e^{d x +c} a -b}d x \right ) a^{4} d^{3} f^{2}+6 e^{c} \left (\int \frac {e^{d x} x^{2}}{e^{2 d x +2 c} b +2 e^{d x +c} a -b}d x \right ) a^{2} b^{2} d^{3} f^{2}+12 e^{c} \left (\int \frac {e^{d x} x}{e^{2 d x +2 c} b +2 e^{d x +c} a -b}d x \right ) a^{4} d^{3} e f +12 e^{c} \left (\int \frac {e^{d x} x}{e^{2 d x +2 c} b +2 e^{d x +c} a -b}d x \right ) a^{2} b^{2} d^{3} e f -6 \sinh \left (d x +c \right ) a^{2} b d e f -6 \sinh \left (d x +c \right ) a^{2} b d \,f^{2} x -6 \sinh \left (d x +c \right ) b^{3} d e f -6 \sinh \left (d x +c \right ) b^{3} d \,f^{2} x -3 a^{3} d^{3} e^{2} x -3 a^{3} d^{3} e f \,x^{2}-a^{3} d^{3} f^{2} x^{3}-3 a \,b^{2} d^{3} e^{2} x -3 a \,b^{2} d^{3} e f \,x^{2}-a \,b^{2} d^{3} f^{2} x^{3}}{3 b^{2} d^{3} \left (a^{2}+b^{2}\right )} \] Input:

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

Output:

(6*sqrt(a**2 + b**2)*atan((e**(c + d*x)*b*i + a*i)/sqrt(a**2 + b**2))*a**2 
*d**2*e**2*i + 3*cosh(c + d*x)*a**2*b*d**2*e**2 + 6*cosh(c + d*x)*a**2*b*d 
**2*e*f*x + 3*cosh(c + d*x)*a**2*b*d**2*f**2*x**2 + 6*cosh(c + d*x)*a**2*b 
*f**2 + 3*cosh(c + d*x)*b**3*d**2*e**2 + 6*cosh(c + d*x)*b**3*d**2*e*f*x + 
 3*cosh(c + d*x)*b**3*d**2*f**2*x**2 + 6*cosh(c + d*x)*b**3*f**2 + 6*e**c* 
int((e**(d*x)*x**2)/(e**(2*c + 2*d*x)*b + 2*e**(c + d*x)*a - b),x)*a**4*d* 
*3*f**2 + 6*e**c*int((e**(d*x)*x**2)/(e**(2*c + 2*d*x)*b + 2*e**(c + d*x)* 
a - b),x)*a**2*b**2*d**3*f**2 + 12*e**c*int((e**(d*x)*x)/(e**(2*c + 2*d*x) 
*b + 2*e**(c + d*x)*a - b),x)*a**4*d**3*e*f + 12*e**c*int((e**(d*x)*x)/(e* 
*(2*c + 2*d*x)*b + 2*e**(c + d*x)*a - b),x)*a**2*b**2*d**3*e*f - 6*sinh(c 
+ d*x)*a**2*b*d*e*f - 6*sinh(c + d*x)*a**2*b*d*f**2*x - 6*sinh(c + d*x)*b* 
*3*d*e*f - 6*sinh(c + d*x)*b**3*d*f**2*x - 3*a**3*d**3*e**2*x - 3*a**3*d** 
3*e*f*x**2 - a**3*d**3*f**2*x**3 - 3*a*b**2*d**3*e**2*x - 3*a*b**2*d**3*e* 
f*x**2 - a*b**2*d**3*f**2*x**3)/(3*b**2*d**3*(a**2 + b**2))