3.2.46 \(\int \frac {x^3}{(a+i a \sinh (c+d x))^{5/2}} \, dx\) [146]

3.2.46.1 Optimal result

Integrand size = 21, antiderivative size = 1016 \[ \int \frac {x^3}{(a+i a \sinh (c+d x))^{5/2}} \, dx=-\frac {1}{a^2 d^4 \sqrt {a+i a \sinh (c+d x)}}+\frac {9 x^2}{8 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}-\frac {10 i x \text {arctanh}\left (e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right ) \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a^2 d^3 \sqrt {a+i a \sinh (c+d x)}}+\frac {3 i x^3 \text {arctanh}\left (e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right ) \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{8 a^2 d \sqrt {a+i a \sinh (c+d x)}}-\frac {10 i \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (2,-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{a^2 d^4 \sqrt {a+i a \sinh (c+d x)}}+\frac {9 i x^2 \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (2,-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{8 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}+\frac {10 i \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (2,e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{a^2 d^4 \sqrt {a+i a \sinh (c+d x)}}-\frac {9 i x^2 \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (2,e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{8 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}-\frac {9 i x \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (3,-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{2 a^2 d^3 \sqrt {a+i a \sinh (c+d x)}}+\frac {9 i x \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (3,e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{2 a^2 d^3 \sqrt {a+i a \sinh (c+d x)}}+\frac {9 i \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (4,-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{a^2 d^4 \sqrt {a+i a \sinh (c+d x)}}-\frac {9 i \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \operatorname {PolyLog}\left (4,e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{a^2 d^4 \sqrt {a+i a \sinh (c+d x)}}+\frac {x^2 \text {sech}^2\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{4 a^2 d^2 \sqrt {a+i a \sinh (c+d x)}}-\frac {x \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{2 a^2 d^3 \sqrt {a+i a \sinh (c+d x)}}+\frac {3 x^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{16 a^2 d \sqrt {a+i a \sinh (c+d x)}}+\frac {x^3 \text {sech}^2\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{8 a^2 d \sqrt {a+i a \sinh (c+d x)}} \]

output

-1/a^2/d^4/(a+I*a*sinh(d*x+c))^(1/2)+9/8*x^2/a^2/d^2/(a+I*a*sinh(d*x+c))^( 
1/2)+10*I*x*arctanh(exp(1/2*c+3/4*I*Pi+1/2*d*x))*cosh(1/2*c+1/4*I*Pi+1/2*d 
*x)/a^2/d^3/(a+I*a*sinh(d*x+c))^(1/2)+9/2*I*x*cosh(1/2*c+1/4*I*Pi+1/2*d*x) 
*polylog(3,-exp(1/2*c+3/4*I*Pi+1/2*d*x))/a^2/d^3/(a+I*a*sinh(d*x+c))^(1/2) 
+9/8*I*x^2*cosh(1/2*c+1/4*I*Pi+1/2*d*x)*polylog(2,exp(1/2*c+3/4*I*Pi+1/2*d 
*x))/a^2/d^2/(a+I*a*sinh(d*x+c))^(1/2)-9*I*cosh(1/2*c+1/4*I*Pi+1/2*d*x)*po 
lylog(4,-exp(1/2*c+3/4*I*Pi+1/2*d*x))/a^2/d^4/(a+I*a*sinh(d*x+c))^(1/2)-10 
*I*cosh(1/2*c+1/4*I*Pi+1/2*d*x)*polylog(2,exp(1/2*c+3/4*I*Pi+1/2*d*x))/a^2 
/d^4/(a+I*a*sinh(d*x+c))^(1/2)-3/8*I*x^3*arctanh(exp(1/2*c+3/4*I*Pi+1/2*d* 
x))*cosh(1/2*c+1/4*I*Pi+1/2*d*x)/a^2/d/(a+I*a*sinh(d*x+c))^(1/2)+10*I*cosh 
(1/2*c+1/4*I*Pi+1/2*d*x)*polylog(2,-exp(1/2*c+3/4*I*Pi+1/2*d*x))/a^2/d^4/( 
a+I*a*sinh(d*x+c))^(1/2)+9*I*cosh(1/2*c+1/4*I*Pi+1/2*d*x)*polylog(4,exp(1/ 
2*c+3/4*I*Pi+1/2*d*x))/a^2/d^4/(a+I*a*sinh(d*x+c))^(1/2)-9/2*I*x*cosh(1/2* 
c+1/4*I*Pi+1/2*d*x)*polylog(3,exp(1/2*c+3/4*I*Pi+1/2*d*x))/a^2/d^3/(a+I*a* 
sinh(d*x+c))^(1/2)-9/8*I*x^2*cosh(1/2*c+1/4*I*Pi+1/2*d*x)*polylog(2,-exp(1 
/2*c+3/4*I*Pi+1/2*d*x))/a^2/d^2/(a+I*a*sinh(d*x+c))^(1/2)+1/4*x^2*sech(1/2 
*c+1/4*I*Pi+1/2*d*x)^2/a^2/d^2/(a+I*a*sinh(d*x+c))^(1/2)-1/2*x*tanh(1/2*c+ 
1/4*I*Pi+1/2*d*x)/a^2/d^3/(a+I*a*sinh(d*x+c))^(1/2)+3/16*x^3*tanh(1/2*c+1/ 
4*I*Pi+1/2*d*x)/a^2/d/(a+I*a*sinh(d*x+c))^(1/2)+1/8*x^3*sech(1/2*c+1/4*I*P 
i+1/2*d*x)^2*tanh(1/2*c+1/4*I*Pi+1/2*d*x)/a^2/d/(a+I*a*sinh(d*x+c))^(1/...
 

3.2.46.2 Mathematica [A] (verified)

Time = 2.99 (sec) , antiderivative size = 1200, normalized size of antiderivative = 1.18 \[ \int \frac {x^3}{(a+i a \sinh (c+d x))^{5/2}} \, dx =\text {Too large to display} \]

input

Integrate[x^3/(a + I*a*Sinh[c + d*x])^(5/2),x]
 

output

((Cosh[(c + d*x)/2] + I*Sinh[(c + d*x)/2])*(-48*Cosh[(c + d*x)/2] + (8*I)* 
c*Cosh[(c + d*x)/2] + 70*c^2*Cosh[(c + d*x)/2] - (11*I)*c^3*Cosh[(c + d*x) 
/2] - (8*I)*(c + d*x)*Cosh[(c + d*x)/2] - 140*c*(c + d*x)*Cosh[(c + d*x)/2 
] + (33*I)*c^2*(c + d*x)*Cosh[(c + d*x)/2] + 70*(c + d*x)^2*Cosh[(c + d*x) 
/2] - (33*I)*c*(c + d*x)^2*Cosh[(c + d*x)/2] + (11*I)*(c + d*x)^3*Cosh[(c 
+ d*x)/2] + 16*Cosh[(3*(c + d*x))/2] + (8*I)*c*Cosh[(3*(c + d*x))/2] - 18* 
c^2*Cosh[(3*(c + d*x))/2] - (3*I)*c^3*Cosh[(3*(c + d*x))/2] - (8*I)*(c + d 
*x)*Cosh[(3*(c + d*x))/2] + 36*c*(c + d*x)*Cosh[(3*(c + d*x))/2] + (9*I)*c 
^2*(c + d*x)*Cosh[(3*(c + d*x))/2] - 18*(c + d*x)^2*Cosh[(3*(c + d*x))/2] 
- (9*I)*c*(c + d*x)^2*Cosh[(3*(c + d*x))/2] + (3*I)*(c + d*x)^3*Cosh[(3*(c 
 + d*x))/2] + (1 - I)*(-1)^(3/4)*(-160*c*ArcTanh[(-1)^(3/4)*E^((c + d*x)/2 
)] + 6*c^3*ArcTanh[(-1)^(3/4)*E^((c + d*x)/2)] - 80*c*Log[1 - (-1)^(3/4)*E 
^((c + d*x)/2)] + 3*c^3*Log[1 - (-1)^(3/4)*E^((c + d*x)/2)] - 80*d*x*Log[1 
 - (-1)^(3/4)*E^((c + d*x)/2)] + 3*d^3*x^3*Log[1 - (-1)^(3/4)*E^((c + d*x) 
/2)] + 80*c*Log[1 + (-1)^(3/4)*E^((c + d*x)/2)] - 3*c^3*Log[1 + (-1)^(3/4) 
*E^((c + d*x)/2)] + 80*d*x*Log[1 + (-1)^(3/4)*E^((c + d*x)/2)] - 3*d^3*x^3 
*Log[1 + (-1)^(3/4)*E^((c + d*x)/2)] - 2*(-80 + 9*d^2*x^2)*PolyLog[2, -((- 
1)^(3/4)*E^((c + d*x)/2))] + 2*(-80 + 9*d^2*x^2)*PolyLog[2, (-1)^(3/4)*E^( 
(c + d*x)/2)] + 72*d*x*PolyLog[3, -((-1)^(3/4)*E^((c + d*x)/2))] - 72*d*x* 
PolyLog[3, (-1)^(3/4)*E^((c + d*x)/2)] - 144*PolyLog[4, -((-1)^(3/4)*E^...
 

3.2.46.3 Rubi [A] (verified)

Time = 2.71 (sec) , antiderivative size = 737, normalized size of antiderivative = 0.73, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.905, Rules used = {3042, 3800, 3042, 4674, 3042, 4673, 3042, 4670, 2715, 2838, 4674, 3042, 4670, 2715, 2838, 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.

\(\displaystyle \int \frac {x^3}{(a+i a \sinh (c+d x))^{5/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {x^3}{(a+a \sin (i c+i d x))^{5/2}}dx\)

\(\Big \downarrow \) 3800

\(\displaystyle \frac {\cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \int x^3 \text {sech}^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx}{4 a^2 \sqrt {a+i a \sinh (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \int x^3 \csc \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^5dx}{4 a^2 \sqrt {a+i a \sinh (c+d x)}}\)

\(\Big \downarrow \) 4674

\(\displaystyle \frac {\cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \left (-\frac {2 \int x \text {sech}^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx}{d^2}+\frac {3}{4} \int x^3 \text {sech}^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx+\frac {x^2 \text {sech}^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}+\frac {x^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \text {sech}^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{2 d}\right )}{4 a^2 \sqrt {a+i a \sinh (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \left (-\frac {2 \int x \csc \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^3dx}{d^2}+\frac {3}{4} \int x^3 \csc \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^3dx+\frac {x^2 \text {sech}^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}+\frac {x^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \text {sech}^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{2 d}\right )}{4 a^2 \sqrt {a+i a \sinh (c+d x)}}\)

\(\Big \downarrow \) 4673

\(\displaystyle \frac {\cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \left (-\frac {2 \left (\frac {1}{2} \int x \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx+\frac {2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}+\frac {x \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )}{d^2}+\frac {3}{4} \int x^3 \csc \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^3dx+\frac {x^2 \text {sech}^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}+\frac {x^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \text {sech}^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{2 d}\right )}{4 a^2 \sqrt {a+i a \sinh (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \left (-\frac {2 \left (\frac {1}{2} \int x \csc \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )dx+\frac {2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}+\frac {x \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )}{d^2}+\frac {3}{4} \int x^3 \csc \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^3dx+\frac {x^2 \text {sech}^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}+\frac {x^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \text {sech}^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{2 d}\right )}{4 a^2 \sqrt {a+i a \sinh (c+d x)}}\)

\(\Big \downarrow \) 4670

\(\displaystyle \frac {\cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \left (-\frac {2 \left (\frac {1}{2} \left (\frac {2 i \int \log \left (1-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )dx}{d}-\frac {2 i \int \log \left (1+e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )dx}{d}+\frac {4 i x \text {arctanh}\left (e^{\frac {d x}{2}+\frac {1}{4} (2 c-i \pi )}\right )}{d}\right )+\frac {2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}+\frac {x \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )}{d^2}+\frac {3}{4} \int x^3 \csc \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^3dx+\frac {x^2 \text {sech}^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}+\frac {x^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \text {sech}^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{2 d}\right )}{4 a^2 \sqrt {a+i a \sinh (c+d x)}}\)

\(\Big \downarrow \) 2715

\(\displaystyle \frac {\cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \left (-\frac {2 \left (\frac {1}{2} \left (\frac {4 i \int e^{\frac {1}{4} (i \pi -2 c)-\frac {d x}{2}} \log \left (1-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )de^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}}{d^2}-\frac {4 i \int e^{\frac {1}{4} (i \pi -2 c)-\frac {d x}{2}} \log \left (1+e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )de^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}}{d^2}+\frac {4 i x \text {arctanh}\left (e^{\frac {d x}{2}+\frac {1}{4} (2 c-i \pi )}\right )}{d}\right )+\frac {2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}+\frac {x \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )}{d^2}+\frac {3}{4} \int x^3 \csc \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^3dx+\frac {x^2 \text {sech}^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}+\frac {x^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \text {sech}^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{2 d}\right )}{4 a^2 \sqrt {a+i a \sinh (c+d x)}}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {\cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \left (\frac {3}{4} \int x^3 \csc \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^3dx-\frac {2 \left (\frac {1}{2} \left (\frac {4 i x \text {arctanh}\left (e^{\frac {d x}{2}+\frac {1}{4} (2 c-i \pi )}\right )}{d}+\frac {4 i \operatorname {PolyLog}\left (2,-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{d^2}-\frac {4 i \operatorname {PolyLog}\left (2,e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{d^2}\right )+\frac {2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}+\frac {x \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )}{d^2}+\frac {x^2 \text {sech}^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}+\frac {x^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \text {sech}^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{2 d}\right )}{4 a^2 \sqrt {a+i a \sinh (c+d x)}}\)

\(\Big \downarrow \) 4674

\(\displaystyle \frac {\cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \left (\frac {3}{4} \left (-\frac {12 \int x \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx}{d^2}+\frac {1}{2} \int x^3 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx+\frac {6 x^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}+\frac {x^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )-\frac {2 \left (\frac {1}{2} \left (\frac {4 i x \text {arctanh}\left (e^{\frac {d x}{2}+\frac {1}{4} (2 c-i \pi )}\right )}{d}+\frac {4 i \operatorname {PolyLog}\left (2,-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{d^2}-\frac {4 i \operatorname {PolyLog}\left (2,e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{d^2}\right )+\frac {2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}+\frac {x \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )}{d^2}+\frac {x^2 \text {sech}^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}+\frac {x^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \text {sech}^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{2 d}\right )}{4 a^2 \sqrt {a+i a \sinh (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \left (\frac {3}{4} \left (-\frac {12 \int x \csc \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )dx}{d^2}+\frac {1}{2} \int x^3 \csc \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )dx+\frac {6 x^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}+\frac {x^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )-\frac {2 \left (\frac {1}{2} \left (\frac {4 i x \text {arctanh}\left (e^{\frac {d x}{2}+\frac {1}{4} (2 c-i \pi )}\right )}{d}+\frac {4 i \operatorname {PolyLog}\left (2,-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{d^2}-\frac {4 i \operatorname {PolyLog}\left (2,e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{d^2}\right )+\frac {2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}+\frac {x \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )}{d^2}+\frac {x^2 \text {sech}^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}+\frac {x^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \text {sech}^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{2 d}\right )}{4 a^2 \sqrt {a+i a \sinh (c+d x)}}\)

\(\Big \downarrow \) 4670

\(\displaystyle \frac {\cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \left (\frac {3}{4} \left (-\frac {12 \left (\frac {2 i \int \log \left (1-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )dx}{d}-\frac {2 i \int \log \left (1+e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )dx}{d}+\frac {4 i x \text {arctanh}\left (e^{\frac {d x}{2}+\frac {1}{4} (2 c-i \pi )}\right )}{d}\right )}{d^2}+\frac {1}{2} \left (\frac {6 i \int x^2 \log \left (1-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )dx}{d}-\frac {6 i \int x^2 \log \left (1+e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )dx}{d}+\frac {4 i x^3 \text {arctanh}\left (e^{\frac {d x}{2}+\frac {1}{4} (2 c-i \pi )}\right )}{d}\right )+\frac {6 x^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}+\frac {x^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )-\frac {2 \left (\frac {1}{2} \left (\frac {4 i x \text {arctanh}\left (e^{\frac {d x}{2}+\frac {1}{4} (2 c-i \pi )}\right )}{d}+\frac {4 i \operatorname {PolyLog}\left (2,-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{d^2}-\frac {4 i \operatorname {PolyLog}\left (2,e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{d^2}\right )+\frac {2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}+\frac {x \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )}{d^2}+\frac {x^2 \text {sech}^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}+\frac {x^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \text {sech}^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{2 d}\right )}{4 a^2 \sqrt {a+i a \sinh (c+d x)}}\)

\(\Big \downarrow \) 2715

\(\displaystyle \frac {\cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \left (\frac {3}{4} \left (-\frac {12 \left (\frac {4 i \int e^{\frac {1}{4} (i \pi -2 c)-\frac {d x}{2}} \log \left (1-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )de^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}}{d^2}-\frac {4 i \int e^{\frac {1}{4} (i \pi -2 c)-\frac {d x}{2}} \log \left (1+e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )de^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}}{d^2}+\frac {4 i x \text {arctanh}\left (e^{\frac {d x}{2}+\frac {1}{4} (2 c-i \pi )}\right )}{d}\right )}{d^2}+\frac {1}{2} \left (\frac {6 i \int x^2 \log \left (1-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )dx}{d}-\frac {6 i \int x^2 \log \left (1+e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )dx}{d}+\frac {4 i x^3 \text {arctanh}\left (e^{\frac {d x}{2}+\frac {1}{4} (2 c-i \pi )}\right )}{d}\right )+\frac {6 x^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}+\frac {x^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )-\frac {2 \left (\frac {1}{2} \left (\frac {4 i x \text {arctanh}\left (e^{\frac {d x}{2}+\frac {1}{4} (2 c-i \pi )}\right )}{d}+\frac {4 i \operatorname {PolyLog}\left (2,-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{d^2}-\frac {4 i \operatorname {PolyLog}\left (2,e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{d^2}\right )+\frac {2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}+\frac {x \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )}{d^2}+\frac {x^2 \text {sech}^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}+\frac {x^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \text {sech}^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{2 d}\right )}{4 a^2 \sqrt {a+i a \sinh (c+d x)}}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {\cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \left (\frac {3}{4} \left (\frac {1}{2} \left (\frac {6 i \int x^2 \log \left (1-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )dx}{d}-\frac {6 i \int x^2 \log \left (1+e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )dx}{d}+\frac {4 i x^3 \text {arctanh}\left (e^{\frac {d x}{2}+\frac {1}{4} (2 c-i \pi )}\right )}{d}\right )-\frac {12 \left (\frac {4 i x \text {arctanh}\left (e^{\frac {d x}{2}+\frac {1}{4} (2 c-i \pi )}\right )}{d}+\frac {4 i \operatorname {PolyLog}\left (2,-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{d^2}-\frac {4 i \operatorname {PolyLog}\left (2,e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{d^2}\right )}{d^2}+\frac {6 x^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}+\frac {x^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )-\frac {2 \left (\frac {1}{2} \left (\frac {4 i x \text {arctanh}\left (e^{\frac {d x}{2}+\frac {1}{4} (2 c-i \pi )}\right )}{d}+\frac {4 i \operatorname {PolyLog}\left (2,-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{d^2}-\frac {4 i \operatorname {PolyLog}\left (2,e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{d^2}\right )+\frac {2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}+\frac {x \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )}{d^2}+\frac {x^2 \text {sech}^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}+\frac {x^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \text {sech}^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{2 d}\right )}{4 a^2 \sqrt {a+i a \sinh (c+d x)}}\)

\(\Big \downarrow \) 3011

\(\displaystyle \frac {\cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \left (\frac {3}{4} \left (\frac {1}{2} \left (-\frac {6 i \left (\frac {4 \int x \operatorname {PolyLog}\left (2,-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )dx}{d}-\frac {2 x^2 \operatorname {PolyLog}\left (2,-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{d}\right )}{d}+\frac {6 i \left (\frac {4 \int x \operatorname {PolyLog}\left (2,e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )dx}{d}-\frac {2 x^2 \operatorname {PolyLog}\left (2,e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{d}\right )}{d}+\frac {4 i x^3 \text {arctanh}\left (e^{\frac {d x}{2}+\frac {1}{4} (2 c-i \pi )}\right )}{d}\right )-\frac {12 \left (\frac {4 i x \text {arctanh}\left (e^{\frac {d x}{2}+\frac {1}{4} (2 c-i \pi )}\right )}{d}+\frac {4 i \operatorname {PolyLog}\left (2,-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{d^2}-\frac {4 i \operatorname {PolyLog}\left (2,e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{d^2}\right )}{d^2}+\frac {6 x^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}+\frac {x^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )-\frac {2 \left (\frac {1}{2} \left (\frac {4 i x \text {arctanh}\left (e^{\frac {d x}{2}+\frac {1}{4} (2 c-i \pi )}\right )}{d}+\frac {4 i \operatorname {PolyLog}\left (2,-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{d^2}-\frac {4 i \operatorname {PolyLog}\left (2,e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{d^2}\right )+\frac {2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}+\frac {x \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )}{d^2}+\frac {x^2 \text {sech}^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}+\frac {x^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \text {sech}^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{2 d}\right )}{4 a^2 \sqrt {a+i a \sinh (c+d x)}}\)

\(\Big \downarrow \) 7163

\(\displaystyle \frac {\cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \left (\frac {3}{4} \left (\frac {1}{2} \left (-\frac {6 i \left (\frac {4 \left (\frac {2 x \operatorname {PolyLog}\left (3,-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{d}-\frac {2 \int \operatorname {PolyLog}\left (3,-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )dx}{d}\right )}{d}-\frac {2 x^2 \operatorname {PolyLog}\left (2,-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{d}\right )}{d}+\frac {6 i \left (\frac {4 \left (\frac {2 x \operatorname {PolyLog}\left (3,e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{d}-\frac {2 \int \operatorname {PolyLog}\left (3,e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )dx}{d}\right )}{d}-\frac {2 x^2 \operatorname {PolyLog}\left (2,e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{d}\right )}{d}+\frac {4 i x^3 \text {arctanh}\left (e^{\frac {d x}{2}+\frac {1}{4} (2 c-i \pi )}\right )}{d}\right )-\frac {12 \left (\frac {4 i x \text {arctanh}\left (e^{\frac {d x}{2}+\frac {1}{4} (2 c-i \pi )}\right )}{d}+\frac {4 i \operatorname {PolyLog}\left (2,-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{d^2}-\frac {4 i \operatorname {PolyLog}\left (2,e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{d^2}\right )}{d^2}+\frac {6 x^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}+\frac {x^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )-\frac {2 \left (\frac {1}{2} \left (\frac {4 i x \text {arctanh}\left (e^{\frac {d x}{2}+\frac {1}{4} (2 c-i \pi )}\right )}{d}+\frac {4 i \operatorname {PolyLog}\left (2,-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{d^2}-\frac {4 i \operatorname {PolyLog}\left (2,e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{d^2}\right )+\frac {2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}+\frac {x \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )}{d^2}+\frac {x^2 \text {sech}^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}+\frac {x^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \text {sech}^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{2 d}\right )}{4 a^2 \sqrt {a+i a \sinh (c+d x)}}\)

\(\Big \downarrow \) 2720

\(\displaystyle \frac {\cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \left (\frac {x^2 \text {sech}^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}+\frac {x^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \text {sech}^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{2 d}-\frac {2 \left (\frac {1}{2} \left (\frac {4 i x \text {arctanh}\left (e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{d}+\frac {4 i \operatorname {PolyLog}\left (2,-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{d^2}-\frac {4 i \operatorname {PolyLog}\left (2,e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{d^2}\right )+\frac {2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}+\frac {x \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )}{d^2}+\frac {3}{4} \left (\frac {\text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) x^3}{d}+\frac {6 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) x^2}{d^2}-\frac {12 \left (\frac {4 i x \text {arctanh}\left (e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{d}+\frac {4 i \operatorname {PolyLog}\left (2,-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{d^2}-\frac {4 i \operatorname {PolyLog}\left (2,e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{d^2}\right )}{d^2}+\frac {1}{2} \left (\frac {4 i \text {arctanh}\left (e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right ) x^3}{d}-\frac {6 i \left (\frac {4 \left (\frac {2 x \operatorname {PolyLog}\left (3,-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{d}-\frac {4 \int e^{\frac {1}{4} (i \pi -2 c)-\frac {d x}{2}} \operatorname {PolyLog}\left (3,-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )de^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}}{d^2}\right )}{d}-\frac {2 x^2 \operatorname {PolyLog}\left (2,-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{d}\right )}{d}+\frac {6 i \left (\frac {4 \left (\frac {2 x \operatorname {PolyLog}\left (3,e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{d}-\frac {4 \int e^{\frac {1}{4} (i \pi -2 c)-\frac {d x}{2}} \operatorname {PolyLog}\left (3,e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )de^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}}{d^2}\right )}{d}-\frac {2 x^2 \operatorname {PolyLog}\left (2,e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{d}\right )}{d}\right )\right )\right )}{4 a^2 \sqrt {i \sinh (c+d x) a+a}}\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {\cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \left (\frac {3}{4} \left (\frac {1}{2} \left (\frac {4 i x^3 \text {arctanh}\left (e^{\frac {d x}{2}+\frac {1}{4} (2 c-i \pi )}\right )}{d}-\frac {6 i \left (\frac {4 \left (\frac {2 x \operatorname {PolyLog}\left (3,-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{d}-\frac {4 \operatorname {PolyLog}\left (4,-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{d^2}\right )}{d}-\frac {2 x^2 \operatorname {PolyLog}\left (2,-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{d}\right )}{d}+\frac {6 i \left (\frac {4 \left (\frac {2 x \operatorname {PolyLog}\left (3,e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{d}-\frac {4 \operatorname {PolyLog}\left (4,e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{d^2}\right )}{d}-\frac {2 x^2 \operatorname {PolyLog}\left (2,e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{d}\right )}{d}\right )-\frac {12 \left (\frac {4 i x \text {arctanh}\left (e^{\frac {d x}{2}+\frac {1}{4} (2 c-i \pi )}\right )}{d}+\frac {4 i \operatorname {PolyLog}\left (2,-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{d^2}-\frac {4 i \operatorname {PolyLog}\left (2,e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{d^2}\right )}{d^2}+\frac {6 x^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}+\frac {x^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )-\frac {2 \left (\frac {1}{2} \left (\frac {4 i x \text {arctanh}\left (e^{\frac {d x}{2}+\frac {1}{4} (2 c-i \pi )}\right )}{d}+\frac {4 i \operatorname {PolyLog}\left (2,-e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{d^2}-\frac {4 i \operatorname {PolyLog}\left (2,e^{\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}}\right )}{d^2}\right )+\frac {2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}+\frac {x \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )}{d^2}+\frac {x^2 \text {sech}^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}+\frac {x^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \text {sech}^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{2 d}\right )}{4 a^2 \sqrt {a+i a \sinh (c+d x)}}\)

input

Int[x^3/(a + I*a*Sinh[c + d*x])^(5/2),x]
 

output

(Cosh[c/2 + (I/4)*Pi + (d*x)/2]*((x^2*Sech[c/2 + (I/4)*Pi + (d*x)/2]^3)/d^ 
2 + (x^3*Sech[c/2 + (I/4)*Pi + (d*x)/2]^3*Tanh[c/2 + (I/4)*Pi + (d*x)/2])/ 
(2*d) - (2*((((4*I)*x*ArcTanh[E^((2*c - I*Pi)/4 + (d*x)/2)])/d + ((4*I)*Po 
lyLog[2, -E^((2*c - I*Pi)/4 + (d*x)/2)])/d^2 - ((4*I)*PolyLog[2, E^((2*c - 
 I*Pi)/4 + (d*x)/2)])/d^2)/2 + (2*Sech[c/2 + (I/4)*Pi + (d*x)/2])/d^2 + (x 
*Sech[c/2 + (I/4)*Pi + (d*x)/2]*Tanh[c/2 + (I/4)*Pi + (d*x)/2])/d))/d^2 + 
(3*((-12*(((4*I)*x*ArcTanh[E^((2*c - I*Pi)/4 + (d*x)/2)])/d + ((4*I)*PolyL 
og[2, -E^((2*c - I*Pi)/4 + (d*x)/2)])/d^2 - ((4*I)*PolyLog[2, E^((2*c - I* 
Pi)/4 + (d*x)/2)])/d^2))/d^2 + (((4*I)*x^3*ArcTanh[E^((2*c - I*Pi)/4 + (d* 
x)/2)])/d - ((6*I)*((-2*x^2*PolyLog[2, -E^((2*c - I*Pi)/4 + (d*x)/2)])/d + 
 (4*((2*x*PolyLog[3, -E^((2*c - I*Pi)/4 + (d*x)/2)])/d - (4*PolyLog[4, -E^ 
((2*c - I*Pi)/4 + (d*x)/2)])/d^2))/d))/d + ((6*I)*((-2*x^2*PolyLog[2, E^(( 
2*c - I*Pi)/4 + (d*x)/2)])/d + (4*((2*x*PolyLog[3, E^((2*c - I*Pi)/4 + (d* 
x)/2)])/d - (4*PolyLog[4, E^((2*c - I*Pi)/4 + (d*x)/2)])/d^2))/d))/d)/2 + 
(6*x^2*Sech[c/2 + (I/4)*Pi + (d*x)/2])/d^2 + (x^3*Sech[c/2 + (I/4)*Pi + (d 
*x)/2]*Tanh[c/2 + (I/4)*Pi + (d*x)/2])/d))/4))/(4*a^2*Sqrt[a + I*a*Sinh[c 
+ d*x]])
 

3.2.46.3.1 Defintions of rubi rules used

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 3800

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), 
 x_Symbol] :> Simp[(2*a)^IntPart[n]*((a + b*Sin[e + f*x])^FracPart[n]/Sin[e 
/2 + a*(Pi/(4*b)) + f*(x/2)]^(2*FracPart[n]))   Int[(c + d*x)^m*Sin[e/2 + a 
*(Pi/(4*b)) + f*(x/2)]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && 
EqQ[a^2 - b^2, 0] && IntegerQ[n + 1/2] && (GtQ[n, 0] || IGtQ[m, 0])
 

rule 4670

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x 
_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] 
 + (-Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x 
)], x], x] + Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e 
+ f*fz*x)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
 

rule 4673

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_)), x_Symbol] :> 
 Simp[(-b^2)*(c + d*x)*Cot[e + f*x]*((b*Csc[e + f*x])^(n - 2)/(f*(n - 1))), 
 x] + (-Simp[b^2*d*((b*Csc[e + f*x])^(n - 2)/(f^2*(n - 1)*(n - 2))), x] + S 
imp[b^2*((n - 2)/(n - 1))   Int[(c + d*x)*(b*Csc[e + f*x])^(n - 2), x], x]) 
 /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2]
 

rule 4674

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

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]
 

3.2.46.4 Maple [F]

\[\int \frac {x^{3}}{\left (a +i a \sinh \left (d x +c \right )\right )^{\frac {5}{2}}}d x\]

input

int(x^3/(a+I*a*sinh(d*x+c))^(5/2),x)
 

output

int(x^3/(a+I*a*sinh(d*x+c))^(5/2),x)
 

3.2.46.5 Fricas [F]

\[ \int \frac {x^3}{(a+i a \sinh (c+d x))^{5/2}} \, dx=\int { \frac {x^{3}}{{\left (i \, a \sinh \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]

input

integrate(x^3/(a+I*a*sinh(d*x+c))^(5/2),x, algorithm="fricas")
 

output

1/8*(8*(a^3*d^4*e^(4*d*x + 4*c) - 4*I*a^3*d^4*e^(3*d*x + 3*c) - 6*a^3*d^4* 
e^(2*d*x + 2*c) + 4*I*a^3*d^4*e^(d*x + c) + a^3*d^4)*integral(1/16*(-3*I*d 
^2*x^3 + 80*I*x)*sqrt(1/2*I*a*e^(-d*x - c))*e^(d*x + c)/(a^3*d^2*e^(d*x + 
c) - I*a^3*d^2), x) + ((-3*I*d^3*x^3 - 18*I*d^2*x^2 + 8*I*d*x + 16*I)*e^(4 
*d*x + 4*c) - (11*d^3*x^3 + 70*d^2*x^2 - 8*d*x - 48)*e^(3*d*x + 3*c) + (-1 
1*I*d^3*x^3 + 70*I*d^2*x^2 + 8*I*d*x - 48*I)*e^(2*d*x + 2*c) - (3*d^3*x^3 
- 18*d^2*x^2 - 8*d*x + 16)*e^(d*x + c))*sqrt(1/2*I*a*e^(-d*x - c)))/(a^3*d 
^4*e^(4*d*x + 4*c) - 4*I*a^3*d^4*e^(3*d*x + 3*c) - 6*a^3*d^4*e^(2*d*x + 2* 
c) + 4*I*a^3*d^4*e^(d*x + c) + a^3*d^4)
 

3.2.46.6 Sympy [F(-1)]

Timed out. \[ \int \frac {x^3}{(a+i a \sinh (c+d x))^{5/2}} \, dx=\text {Timed out} \]

input

integrate(x**3/(a+I*a*sinh(d*x+c))**(5/2),x)
 

output

Timed out
 

3.2.46.7 Maxima [F]

\[ \int \frac {x^3}{(a+i a \sinh (c+d x))^{5/2}} \, dx=\int { \frac {x^{3}}{{\left (i \, a \sinh \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]

input

integrate(x^3/(a+I*a*sinh(d*x+c))^(5/2),x, algorithm="maxima")
 

output

integrate(x^3/(I*a*sinh(d*x + c) + a)^(5/2), x)
 

3.2.46.8 Giac [F]

\[ \int \frac {x^3}{(a+i a \sinh (c+d x))^{5/2}} \, dx=\int { \frac {x^{3}}{{\left (i \, a \sinh \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]

input

integrate(x^3/(a+I*a*sinh(d*x+c))^(5/2),x, algorithm="giac")
 

output

integrate(x^3/(I*a*sinh(d*x + c) + a)^(5/2), x)
 

3.2.46.9 Mupad [F(-1)]

Timed out. \[ \int \frac {x^3}{(a+i a \sinh (c+d x))^{5/2}} \, dx=\int \frac {x^3}{{\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{5/2}} \,d x \]

input

int(x^3/(a + a*sinh(c + d*x)*1i)^(5/2),x)
 

output

int(x^3/(a + a*sinh(c + d*x)*1i)^(5/2), x)