\(\int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx\) [338]

Optimal result

Integrand size = 34, antiderivative size = 683 \[ \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {3 f (e+f x)^2}{8 b d^2}+\frac {a^2 (e+f x)^4}{4 b^3 f}+\frac {(e+f x)^4}{8 b f}-\frac {6 a f^2 (e+f x) \cosh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^3 \cosh (c+d x)}{b^2 d}-\frac {3 f^3 \cosh ^2(c+d x)}{8 b d^4}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 b d^2}-\frac {a \sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {a \sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d}-\frac {3 a \sqrt {a^2+b^2} f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {3 a \sqrt {a^2+b^2} f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {6 a \sqrt {a^2+b^2} f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^3}-\frac {6 a \sqrt {a^2+b^2} f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^3}-\frac {6 a \sqrt {a^2+b^2} f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^4}+\frac {6 a \sqrt {a^2+b^2} f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^4}+\frac {6 a f^3 \sinh (c+d x)}{b^2 d^4}+\frac {3 a f (e+f x)^2 \sinh (c+d x)}{b^2 d^2}+\frac {3 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b d} \] Output:

3/8*f*(f*x+e)^2/b/d^2+1/4*a^2*(f*x+e)^4/b^3/f+1/8*(f*x+e)^4/b/f-6*a*f^2*(f 
*x+e)*cosh(d*x+c)/b^2/d^3-a*(f*x+e)^3*cosh(d*x+c)/b^2/d-3/8*f^3*cosh(d*x+c 
)^2/b/d^4-3/4*f*(f*x+e)^2*cosh(d*x+c)^2/b/d^2-a*(a^2+b^2)^(1/2)*(f*x+e)^3* 
ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^3/d+a*(a^2+b^2)^(1/2)*(f*x+e)^3*l 
n(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^3/d-3*a*(a^2+b^2)^(1/2)*f*(f*x+e)^ 
2*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^3/d^2+3*a*(a^2+b^2)^(1/2) 
*f*(f*x+e)^2*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^3/d^2+6*a*(a^2 
+b^2)^(1/2)*f^2*(f*x+e)*polylog(3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^3/d 
^3-6*a*(a^2+b^2)^(1/2)*f^2*(f*x+e)*polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1 
/2)))/b^3/d^3-6*a*(a^2+b^2)^(1/2)*f^3*polylog(4,-b*exp(d*x+c)/(a-(a^2+b^2) 
^(1/2)))/b^3/d^4+6*a*(a^2+b^2)^(1/2)*f^3*polylog(4,-b*exp(d*x+c)/(a+(a^2+b 
^2)^(1/2)))/b^3/d^4+6*a*f^3*sinh(d*x+c)/b^2/d^4+3*a*f*(f*x+e)^2*sinh(d*x+c 
)/b^2/d^2+3/4*f^2*(f*x+e)*cosh(d*x+c)*sinh(d*x+c)/b/d^3+1/2*(f*x+e)^3*cosh 
(d*x+c)*sinh(d*x+c)/b/d
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(1971\) vs. \(2(683)=1366\).

Time = 8.90 (sec) , antiderivative size = 1971, normalized size of antiderivative = 2.89 \[ \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx =\text {Too large to display} \] Input:

Integrate[((e + f*x)^3*Cosh[c + d*x]^2*Sinh[c + d*x])/(a + b*Sinh[c + d*x] 
),x]
 

Output:

(4*b^2*e^3*(c/d + x - (2*a*ArcTan[(b - a*Tanh[(c + d*x)/2])/Sqrt[-a^2 - b^ 
2]])/(Sqrt[-a^2 - b^2]*d)) + 6*b^2*e^2*f*(x^2 - (2*a*(d*x*(Log[1 + (b*E^(c 
 + d*x))/(a - Sqrt[a^2 + b^2])] - Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + 
b^2])]) + PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] - PolyLog[2, 
-((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]))/(Sqrt[a^2 + b^2]*d^2)) + 4*b^2 
*e*f^2*(x^3 - (3*a*(d^2*x^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] 
 - d^2*x^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + 2*d*x*PolyLog[ 
2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] - 2*d*x*PolyLog[2, -((b*E^(c + 
d*x))/(a + Sqrt[a^2 + b^2]))] - 2*PolyLog[3, (b*E^(c + d*x))/(-a + Sqrt[a^ 
2 + b^2])] + 2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]))/(Sqr 
t[a^2 + b^2]*d^3)) + b^2*f^3*(x^4 - (4*a*(d^3*x^3*Log[1 + (b*E^(c + d*x))/ 
(a - Sqrt[a^2 + b^2])] - d^3*x^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b 
^2])] + 3*d^2*x^2*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] - 3*d 
^2*x^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))] - 6*d*x*PolyLo 
g[3, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + 6*d*x*PolyLog[3, -((b*E^(c 
+ d*x))/(a + Sqrt[a^2 + b^2]))] + 6*PolyLog[4, (b*E^(c + d*x))/(-a + Sqrt[ 
a^2 + b^2])] - 6*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]))/(S 
qrt[a^2 + b^2]*d^4)) + 2*e*f^2*(2*(4*a^2 + b^2)*x^3 - (6*a*(4*a^2 + 3*b^2) 
*(d^2*x^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] - d^2*x^2*Log[1 + 
 (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + 2*d*x*PolyLog[2, (b*E^(c + d*...
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 3.88 (sec) , antiderivative size = 641, normalized size of antiderivative = 0.94, number of steps used = 30, number of rules used = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.853, Rules used = {6113, 3042, 3792, 17, 3042, 3791, 17, 6099, 17, 3042, 26, 3777, 3042, 3777, 26, 3042, 26, 3777, 3042, 3117, 3803, 25, 2694, 27, 2620, 3011, 7163, 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.

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

\(\Big \downarrow \) 6113

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3792

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

\(\Big \downarrow \) 17

\(\displaystyle \frac {\frac {3 f^2 \int (e+f x) \cosh ^2(c+d x)dx}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {a \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}+\frac {\frac {3 f^2 \int (e+f x) \sin \left (i c+i d x+\frac {\pi }{2}\right )^2dx}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}\)

\(\Big \downarrow \) 3791

\(\displaystyle \frac {\frac {3 f^2 \left (\frac {1}{2} \int (e+f x)dx-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\)

\(\Big \downarrow \) 17

\(\displaystyle \frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)}dx}{b}\)

\(\Big \downarrow \) 6099

\(\displaystyle \frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{a+b \sinh (c+d x)}dx}{b^2}-\frac {a \int (e+f x)^3dx}{b^2}+\frac {\int (e+f x)^3 \sinh (c+d x)dx}{b}\right )}{b}\)

\(\Big \downarrow \) 17

\(\displaystyle \frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{a+b \sinh (c+d x)}dx}{b^2}+\frac {\int (e+f x)^3 \sinh (c+d x)dx}{b}-\frac {a (e+f x)^4}{4 b^2 f}\right )}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{a-i b \sin (i c+i d x)}dx}{b^2}+\frac {\int -i (e+f x)^3 \sin (i c+i d x)dx}{b}-\frac {a (e+f x)^4}{4 b^2 f}\right )}{b}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \int (e+f x)^3 \sin (i c+i d x)dx}{b}-\frac {a (e+f x)^4}{4 b^2 f}\right )}{b}\)

\(\Big \downarrow \) 3777

\(\displaystyle \frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \int (e+f x)^2 \cosh (c+d x)dx}{d}\right )}{b}-\frac {a (e+f x)^4}{4 b^2 f}\right )}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \int (e+f x)^2 \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{b}-\frac {a (e+f x)^4}{4 b^2 f}\right )}{b}\)

\(\Big \downarrow \) 3777

\(\displaystyle \frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 i f \int -i (e+f x) \sinh (c+d x)dx}{d}\right )}{d}\right )}{b}-\frac {a (e+f x)^4}{4 b^2 f}\right )}{b}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 f \int (e+f x) \sinh (c+d x)dx}{d}\right )}{d}\right )}{b}-\frac {a (e+f x)^4}{4 b^2 f}\right )}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}-\frac {2 f \int -i (e+f x) \sin (i c+i d x)dx}{d}\right )}{d}\right )}{b}-\frac {a (e+f x)^4}{4 b^2 f}\right )}{b}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \int (e+f x) \sin (i c+i d x)dx}{d}\right )}{d}\right )}{b}-\frac {a (e+f x)^4}{4 b^2 f}\right )}{b}\)

\(\Big \downarrow \) 3777

\(\displaystyle \frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \cosh (c+d x)dx}{d}\right )}{d}\right )}{d}\right )}{b}-\frac {a (e+f x)^4}{4 b^2 f}\right )}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \int \sin \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}\right )}{d}\right )}{d}\right )}{b}-\frac {a (e+f x)^4}{4 b^2 f}\right )}{b}\)

\(\Big \downarrow \) 3117

\(\displaystyle \frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{a-i b \sin (i c+i d x)}dx}{b^2}-\frac {a (e+f x)^4}{4 b^2 f}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 3803

\(\displaystyle \frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (\frac {2 \left (a^2+b^2\right ) \int -\frac {e^{c+d x} (e+f x)^3}{-2 e^{c+d x} a-b e^{2 (c+d x)}+b}dx}{b^2}-\frac {a (e+f x)^4}{4 b^2 f}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (-\frac {2 \left (a^2+b^2\right ) \int \frac {e^{c+d x} (e+f x)^3}{-2 e^{c+d x} a-b e^{2 (c+d x)}+b}dx}{b^2}-\frac {a (e+f x)^4}{4 b^2 f}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 2694

\(\displaystyle \frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (-\frac {2 \left (a^2+b^2\right ) \left (\frac {b \int -\frac {e^{c+d x} (e+f x)^3}{2 \left (a+b e^{c+d x}-\sqrt {a^2+b^2}\right )}dx}{\sqrt {a^2+b^2}}-\frac {b \int -\frac {e^{c+d x} (e+f x)^3}{2 \left (a+b e^{c+d x}+\sqrt {a^2+b^2}\right )}dx}{\sqrt {a^2+b^2}}\right )}{b^2}-\frac {a (e+f x)^4}{4 b^2 f}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (-\frac {2 \left (a^2+b^2\right ) \left (\frac {b \int \frac {e^{c+d x} (e+f x)^3}{a+b e^{c+d x}+\sqrt {a^2+b^2}}dx}{2 \sqrt {a^2+b^2}}-\frac {b \int \frac {e^{c+d x} (e+f x)^3}{a+b e^{c+d x}-\sqrt {a^2+b^2}}dx}{2 \sqrt {a^2+b^2}}\right )}{b^2}-\frac {a (e+f x)^4}{4 b^2 f}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 2620

\(\displaystyle \frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (-\frac {2 \left (a^2+b^2\right ) \left (\frac {b \left (\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {3 f \int (e+f x)^2 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )dx}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {3 f \int (e+f x)^2 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )dx}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{b^2}-\frac {a (e+f x)^4}{4 b^2 f}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 3011

\(\displaystyle \frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (-\frac {2 \left (a^2+b^2\right ) \left (\frac {b \left (\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {3 f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {3 f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{b^2}-\frac {a (e+f x)^4}{4 b^2 f}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 7163

\(\displaystyle \frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (-\frac {2 \left (a^2+b^2\right ) \left (\frac {b \left (\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}-\frac {f \int \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )dx}{d}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}-\frac {f \int \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )dx}{d}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{b^2}-\frac {a (e+f x)^4}{4 b^2 f}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 2720

\(\displaystyle \frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (-\frac {2 \left (a^2+b^2\right ) \left (\frac {b \left (\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}-\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )de^{c+d x}}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}-\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )de^{c+d x}}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{b^2}-\frac {a (e+f x)^4}{4 b^2 f}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {\frac {3 f^2 \left (-\frac {f \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (-\frac {2 \left (a^2+b^2\right ) \left (\frac {b \left (\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}-\frac {f \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}-\frac {b \left (\frac {(e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}-\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}-\frac {f \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2+b^2}}\right )}{b^2}-\frac {a (e+f x)^4}{4 b^2 f}-\frac {i \left (\frac {i (e+f x)^3 \cosh (c+d x)}{d}-\frac {3 i f \left (\frac {(e+f x)^2 \sinh (c+d x)}{d}+\frac {2 i f \left (\frac {i (e+f x) \cosh (c+d x)}{d}-\frac {i f \sinh (c+d x)}{d^2}\right )}{d}\right )}{d}\right )}{b}\right )}{b}\)

Input:

Int[((e + f*x)^3*Cosh[c + d*x]^2*Sinh[c + d*x])/(a + b*Sinh[c + d*x]),x]
 

Output:

((e + f*x)^4/(8*f) - (3*f*(e + f*x)^2*Cosh[c + d*x]^2)/(4*d^2) + ((e + f*x 
)^3*Cosh[c + d*x]*Sinh[c + d*x])/(2*d) + (3*f^2*((e + f*x)^2/(4*f) - (f*Co 
sh[c + d*x]^2)/(4*d^2) + ((e + f*x)*Cosh[c + d*x]*Sinh[c + d*x])/(2*d)))/( 
2*d^2))/b - (a*(-1/4*(a*(e + f*x)^4)/(b^2*f) - (2*(a^2 + b^2)*(-1/2*(b*((( 
e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b*d) - (3*f*(- 
(((e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/d) + ( 
2*f*(((e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/d - 
(f*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/d^2))/d))/(b*d))) 
/Sqrt[a^2 + b^2] + (b*(((e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 
+ b^2])])/(b*d) - (3*f*(-(((e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + S 
qrt[a^2 + b^2]))])/d) + (2*f*(((e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + 
 Sqrt[a^2 + b^2]))])/d - (f*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b 
^2]))])/d^2))/d))/(b*d)))/(2*Sqrt[a^2 + b^2])))/b^2 - (I*((I*(e + f*x)^3*C 
osh[c + d*x])/d - ((3*I)*f*(((e + f*x)^2*Sinh[c + d*x])/d + ((2*I)*f*((I*( 
e + f*x)*Cosh[c + d*x])/d - (I*f*Sinh[c + d*x])/d^2))/d))/d))/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 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 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 3792

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo 
l] :> Simp[d*m*(c + d*x)^(m - 1)*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Sim 
p[b*(c + d*x)^m*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^ 
2*((n - 1)/n)   Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[d^2 
*m*((m - 1)/(f^2*n^2))   Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x]) 
/; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 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 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]
 

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]
 

rule 7163

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

Maple [F]

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

Input:

int((f*x+e)^3*cosh(d*x+c)^2*sinh(d*x+c)/(a+b*sinh(d*x+c)),x)
 

Output:

int((f*x+e)^3*cosh(d*x+c)^2*sinh(d*x+c)/(a+b*sinh(d*x+c)),x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3847 vs. \(2 (627) = 1254\).

Time = 0.19 (sec) , antiderivative size = 3847, normalized size of antiderivative = 5.63 \[ \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:

integrate((f*x+e)^3*cosh(d*x+c)^2*sinh(d*x+c)/(a+b*sinh(d*x+c)),x, algorit 
hm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \] Input:

integrate((f*x+e)**3*cosh(d*x+c)**2*sinh(d*x+c)/(a+b*sinh(d*x+c)),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )}{b \sinh \left (d x + c\right ) + a} \,d x } \] Input:

integrate((f*x+e)^3*cosh(d*x+c)^2*sinh(d*x+c)/(a+b*sinh(d*x+c)),x, algorit 
hm="maxima")
 

Output:

-1/8*e^3*((4*a*e^(-d*x - c) - b)*e^(2*d*x + 2*c)/(b^2*d) + 8*sqrt(a^2 + b^ 
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)))/(b^3*d) - 4*(2*a^2 + b^2)*(d*x + c)/(b^3*d) + (4*a*e^(-d*x - 
 c) + b*e^(-2*d*x - 2*c))/(b^2*d)) + 1/32*(4*(2*a^2*d^4*f^3*e^(2*c) + b^2* 
d^4*f^3*e^(2*c))*x^4 + 16*(2*a^2*d^4*e*f^2*e^(2*c) + b^2*d^4*e*f^2*e^(2*c) 
)*x^3 + 24*(2*a^2*d^4*e^2*f*e^(2*c) + b^2*d^4*e^2*f*e^(2*c))*x^2 + (4*b^2* 
d^3*f^3*x^3*e^(4*c) + 6*(2*d^3*e*f^2 - d^2*f^3)*b^2*x^2*e^(4*c) + 6*(2*d^3 
*e^2*f - 2*d^2*e*f^2 + d*f^3)*b^2*x*e^(4*c) - 3*(2*d^2*e^2*f - 2*d*e*f^2 + 
 f^3)*b^2*e^(4*c))*e^(2*d*x) - 16*(a*b*d^3*f^3*x^3*e^(3*c) + 3*(d^3*e*f^2 
- d^2*f^3)*a*b*x^2*e^(3*c) + 3*(d^3*e^2*f - 2*d^2*e*f^2 + 2*d*f^3)*a*b*x*e 
^(3*c) - 3*(d^2*e^2*f - 2*d*e*f^2 + 2*f^3)*a*b*e^(3*c))*e^(d*x) - 16*(a*b* 
d^3*f^3*x^3*e^c + 3*(d^3*e*f^2 + d^2*f^3)*a*b*x^2*e^c + 3*(d^3*e^2*f + 2*d 
^2*e*f^2 + 2*d*f^3)*a*b*x*e^c + 3*(d^2*e^2*f + 2*d*e*f^2 + 2*f^3)*a*b*e^c) 
*e^(-d*x) - (4*b^2*d^3*f^3*x^3 + 6*(2*d^3*e*f^2 + d^2*f^3)*b^2*x^2 + 6*(2* 
d^3*e^2*f + 2*d^2*e*f^2 + d*f^3)*b^2*x + 3*(2*d^2*e^2*f + 2*d*e*f^2 + f^3) 
*b^2)*e^(-2*d*x))*e^(-2*c)/(b^3*d^4) - integrate(2*((a^3*f^3*e^c + a*b^2*f 
^3*e^c)*x^3 + 3*(a^3*e*f^2*e^c + a*b^2*e*f^2*e^c)*x^2 + 3*(a^3*e^2*f*e^c + 
 a*b^2*e^2*f*e^c)*x)*e^(d*x)/(b^4*e^(2*d*x + 2*c) + 2*a*b^3*e^(d*x + c) - 
b^4), x)
 

Giac [F]

\[ \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )}{b \sinh \left (d x + c\right ) + a} \,d x } \] Input:

integrate((f*x+e)^3*cosh(d*x+c)^2*sinh(d*x+c)/(a+b*sinh(d*x+c)),x, algorit 
hm="giac")
 

Output:

integrate((f*x + e)^3*cosh(d*x + c)^2*sinh(d*x + c)/(b*sinh(d*x + c) + a), 
 x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^2\,\mathrm {sinh}\left (c+d\,x\right )\,{\left (e+f\,x\right )}^3}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \] Input:

int((cosh(c + d*x)^2*sinh(c + d*x)*(e + f*x)^3)/(a + b*sinh(c + d*x)),x)
 

Output:

int((cosh(c + d*x)^2*sinh(c + d*x)*(e + f*x)^3)/(a + b*sinh(c + d*x)), x)
 

Reduce [F]

\[ \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\left (f x +e \right )^{3} \cosh \left (d x +c \right )^{2} \sinh \left (d x +c \right )}{a +b \sinh \left (d x +c \right )}d x \] Input:

int((f*x+e)^3*cosh(d*x+c)^2*sinh(d*x+c)/(a+b*sinh(d*x+c)),x)
 

Output:

int((f*x+e)^3*cosh(d*x+c)^2*sinh(d*x+c)/(a+b*sinh(d*x+c)),x)