3.1.17 \(\int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \text {csch}(c+d x)} \, dx\) [17]

3.1.17.1 Optimal result

Integrand size = 26, antiderivative size = 448 \[ \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \text {csch}(c+d x)} \, dx=\frac {b (e+f x)^4}{4 a^2 f}-\frac {6 f^3 \cosh (c+d x)}{a d^4}-\frac {3 f (e+f x)^2 \cosh (c+d x)}{a d^2}-\frac {b (e+f x)^3 \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^2 d}-\frac {b (e+f x)^3 \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^2 d}-\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^2 d^2}-\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^2 d^2}+\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^2 d^3}+\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^2 d^3}-\frac {6 b f^3 \operatorname {PolyLog}\left (4,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^2 d^4}-\frac {6 b f^3 \operatorname {PolyLog}\left (4,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^2 d^4}+\frac {6 f^2 (e+f x) \sinh (c+d x)}{a d^3}+\frac {(e+f x)^3 \sinh (c+d x)}{a d} \]

output

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

3.1.17.2 Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(1792\) vs. \(2(448)=896\).

Time = 20.20 (sec) , antiderivative size = 1792, normalized size of antiderivative = 4.00 \[ \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \text {csch}(c+d x)} \, dx =\text {Too large to display} \]

input

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

output

(e*f^2*Csch[c + d*x]*((4*b*x^3)/(-1 + E^(2*c)) - 2*b*x^3*Coth[c] - (6*a^2* 
b*(d^2*x^2*Log[1 + ((b - Sqrt[a^2 + b^2])*E^(-c - d*x))/a] - 2*d*x*PolyLog 
[2, ((-b + Sqrt[a^2 + b^2])*E^(-c - d*x))/a] - 2*PolyLog[3, ((-b + Sqrt[a^ 
2 + b^2])*E^(-c - d*x))/a]))/(Sqrt[a^2 + b^2]*(-b + Sqrt[a^2 + b^2])*d^3) 
- (6*a^2*b*(d^2*x^2*Log[1 + ((b + Sqrt[a^2 + b^2])*E^(-c - d*x))/a] - 2*d* 
x*PolyLog[2, -(((b + Sqrt[a^2 + b^2])*E^(-c - d*x))/a)] - 2*PolyLog[3, -(( 
(b + Sqrt[a^2 + b^2])*E^(-c - d*x))/a)]))/(Sqrt[a^2 + b^2]*(b + Sqrt[a^2 + 
 b^2])*d^3) + (6*b^2*(d^2*x^2*Log[1 + (a*E^(c + d*x))/(b - Sqrt[a^2 + b^2] 
)] + 2*d*x*PolyLog[2, (a*E^(c + d*x))/(-b + Sqrt[a^2 + b^2])] - 2*PolyLog[ 
3, (a*E^(c + d*x))/(-b + Sqrt[a^2 + b^2])]))/(Sqrt[a^2 + b^2]*d^3) - (6*b^ 
2*(d^2*x^2*Log[1 + (a*E^(c + d*x))/(b + Sqrt[a^2 + b^2])] + 2*d*x*PolyLog[ 
2, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))] - 2*PolyLog[3, -((a*E^(c + d* 
x))/(b + Sqrt[a^2 + b^2]))]))/(Sqrt[a^2 + b^2]*d^3) + (6*a*Cosh[d*x]*(-2*d 
*x*Cosh[c] + (2 + d^2*x^2)*Sinh[c]))/d^3 + (6*a*((2 + d^2*x^2)*Cosh[c] - 2 
*d*x*Sinh[c])*Sinh[d*x])/d^3)*(b + a*Sinh[c + d*x]))/(2*a^2*(a + b*Csch[c 
+ d*x])) + (f^3*Csch[c + d*x]*((b*(x^4 - (2*a^2*(-1 + E^(2*c))*(d^3*x^3*Lo 
g[1 + ((b - Sqrt[a^2 + b^2])*E^(-c - d*x))/a] - 3*d^2*x^2*PolyLog[2, ((-b 
+ Sqrt[a^2 + b^2])*E^(-c - d*x))/a] - 6*d*x*PolyLog[3, ((-b + Sqrt[a^2 + b 
^2])*E^(-c - d*x))/a] - 6*PolyLog[4, ((-b + Sqrt[a^2 + b^2])*E^(-c - d*x)) 
/a]))/(Sqrt[a^2 + b^2]*(-b + Sqrt[a^2 + b^2])*d^4) - (2*a^2*(-1 + E^(2*...
 

3.1.17.3 Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 2.45 (sec) , antiderivative size = 452, normalized size of antiderivative = 1.01, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.769, Rules used = {6128, 6113, 3042, 3777, 26, 3042, 26, 3777, 3042, 3777, 26, 3042, 26, 3118, 6095, 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.

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

\(\Big \downarrow \) 6128

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

\(\Big \downarrow \) 6113

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3118

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

\(\Big \downarrow \) 6095

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 7163

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

input

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

output

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

3.1.17.3.1 Defintions of rubi rules used

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 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 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 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 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 6128

Int[(((e_.) + (f_.)*(x_))^(m_.)*(F_)[(c_.) + (d_.)*(x_)]^(n_.))/(Csch[(c_.) 
 + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Int[(e + f*x)^m*Sinh[c + d*x]*(F 
[c + d*x]^n/(b + a*Sinh[c + d*x])), x] /; FreeQ[{a, b, c, d, e, f}, x] && H 
yperbolicQ[F] && IntegersQ[m, n]
 

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]
 

3.1.17.4 Maple [F]

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

input

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

output

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

3.1.17.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1976 vs. \(2 (420) = 840\).

Time = 0.30 (sec) , antiderivative size = 1976, normalized size of antiderivative = 4.41 \[ \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \text {csch}(c+d x)} \, dx=\text {Too large to display} \]

input

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

output

-1/4*(2*a*d^3*f^3*x^3 + 2*a*d^3*e^3 + 6*a*d^2*e^2*f + 12*a*d*e*f^2 + 12*a* 
f^3 + 6*(a*d^3*e*f^2 + a*d^2*f^3)*x^2 - 2*(a*d^3*f^3*x^3 + a*d^3*e^3 - 3*a 
*d^2*e^2*f + 6*a*d*e*f^2 - 6*a*f^3 + 3*(a*d^3*e*f^2 - a*d^2*f^3)*x^2 + 3*( 
a*d^3*e^2*f - 2*a*d^2*e*f^2 + 2*a*d*f^3)*x)*cosh(d*x + c)^2 - 2*(a*d^3*f^3 
*x^3 + a*d^3*e^3 - 3*a*d^2*e^2*f + 6*a*d*e*f^2 - 6*a*f^3 + 3*(a*d^3*e*f^2 
- a*d^2*f^3)*x^2 + 3*(a*d^3*e^2*f - 2*a*d^2*e*f^2 + 2*a*d*f^3)*x)*sinh(d*x 
 + c)^2 + 6*(a*d^3*e^2*f + 2*a*d^2*e*f^2 + 2*a*d*f^3)*x - (b*d^4*f^3*x^4 + 
 4*b*d^4*e*f^2*x^3 + 6*b*d^4*e^2*f*x^2 + 4*b*d^4*e^3*x + 8*b*c*d^3*e^3 - 1 
2*b*c^2*d^2*e^2*f + 8*b*c^3*d*e*f^2 - 2*b*c^4*f^3)*cosh(d*x + c) + 12*((b* 
d^2*f^3*x^2 + 2*b*d^2*e*f^2*x + b*d^2*e^2*f)*cosh(d*x + c) + (b*d^2*f^3*x^ 
2 + 2*b*d^2*e*f^2*x + b*d^2*e^2*f)*sinh(d*x + c))*dilog((b*cosh(d*x + c) + 
 b*sinh(d*x + c) + (a*cosh(d*x + c) + a*sinh(d*x + c))*sqrt((a^2 + b^2)/a^ 
2) - a)/a + 1) + 12*((b*d^2*f^3*x^2 + 2*b*d^2*e*f^2*x + b*d^2*e^2*f)*cosh( 
d*x + c) + (b*d^2*f^3*x^2 + 2*b*d^2*e*f^2*x + b*d^2*e^2*f)*sinh(d*x + c))* 
dilog((b*cosh(d*x + c) + b*sinh(d*x + c) - (a*cosh(d*x + c) + a*sinh(d*x + 
 c))*sqrt((a^2 + b^2)/a^2) - a)/a + 1) + 4*((b*d^3*e^3 - 3*b*c*d^2*e^2*f + 
 3*b*c^2*d*e*f^2 - b*c^3*f^3)*cosh(d*x + c) + (b*d^3*e^3 - 3*b*c*d^2*e^2*f 
 + 3*b*c^2*d*e*f^2 - b*c^3*f^3)*sinh(d*x + c))*log(2*a*cosh(d*x + c) + 2*a 
*sinh(d*x + c) + 2*a*sqrt((a^2 + b^2)/a^2) + 2*b) + 4*((b*d^3*e^3 - 3*b*c* 
d^2*e^2*f + 3*b*c^2*d*e*f^2 - b*c^3*f^3)*cosh(d*x + c) + (b*d^3*e^3 - 3...
 

3.1.17.6 Sympy [F]

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

input

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

output

Integral((e + f*x)**3*cosh(c + d*x)/(a + b*csch(c + d*x)), x)
 

3.1.17.7 Maxima [F]

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

input

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

output

-1/2*e^3*(2*(d*x + c)*b/(a^2*d) - e^(d*x + c)/(a*d) + e^(-d*x - c)/(a*d) + 
 2*b*log(-2*b*e^(-d*x - c) + a*e^(-2*d*x - 2*c) - a)/(a^2*d)) - 1/4*(b*d^4 
*f^3*x^4*e^c + 4*b*d^4*e*f^2*x^3*e^c + 6*b*d^4*e^2*f*x^2*e^c - 2*(a*d^3*f^ 
3*x^3*e^(2*c) + 3*(d^3*e*f^2 - d^2*f^3)*a*x^2*e^(2*c) + 3*(d^3*e^2*f - 2*d 
^2*e*f^2 + 2*d*f^3)*a*x*e^(2*c) - 3*(d^2*e^2*f - 2*d*e*f^2 + 2*f^3)*a*e^(2 
*c))*e^(d*x) + 2*(a*d^3*f^3*x^3 + 3*(d^3*e*f^2 + d^2*f^3)*a*x^2 + 3*(d^3*e 
^2*f + 2*d^2*e*f^2 + 2*d*f^3)*a*x + 3*(d^2*e^2*f + 2*d*e*f^2 + 2*f^3)*a)*e 
^(-d*x))*e^(-c)/(a^2*d^4) + integrate(-2*(a*b*f^3*x^3 + 3*a*b*e*f^2*x^2 + 
3*a*b*e^2*f*x - (b^2*f^3*x^3*e^c + 3*b^2*e*f^2*x^2*e^c + 3*b^2*e^2*f*x*e^c 
)*e^(d*x))/(a^3*e^(2*d*x + 2*c) + 2*a^2*b*e^(d*x + c) - a^3), x)
 

3.1.17.8 Giac [F]

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

input

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

output

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

3.1.17.9 Mupad [F(-1)]

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

input

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

output

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