Integrand size = 21, antiderivative size = 1016 \[ \int \frac {x^3}{(a+i a \sinh (c+d x))^{5/2}} \, dx =\text {Too large to display} \] 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)-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)+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)-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)+9/8*I*x^2*cosh(1/2*c+1/4*I*Pi+1/2*d*x)*po lylog(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)+9/2 *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)+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)-10*I*co sh(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)+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/...
Time = 2.90 (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^...
Time = 2.81 (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]])
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]
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]]]
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]
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]
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])
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]
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]
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]
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]
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]
\[\int \frac {x^{3}}{\left (a +i a \sinh \left (d x +c \right )\right )^{\frac {5}{2}}}d x\]
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)
\[ \int \frac {x^3}{(a+i a \sinh (c+d x))^{5/2}} \, dx=\int { \frac {x^{3}}{{\left (i \, a \sinh \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \] 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)
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
\[ \int \frac {x^3}{(a+i a \sinh (c+d x))^{5/2}} \, dx=\int { \frac {x^{3}}{{\left (i \, a \sinh \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \] 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)
\[ \int \frac {x^3}{(a+i a \sinh (c+d x))^{5/2}} \, dx=\int { \frac {x^{3}}{{\left (i \, a \sinh \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \] 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)
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)
\[ \int \frac {x^3}{(a+i a \sinh (c+d x))^{5/2}} \, dx=-\frac {\int \frac {x^{3}}{\sqrt {\sinh \left (d x +c \right ) i +1}\, \sinh \left (d x +c \right )^{2}-2 \sqrt {\sinh \left (d x +c \right ) i +1}\, \sinh \left (d x +c \right ) i -\sqrt {\sinh \left (d x +c \right ) i +1}}d x}{\sqrt {a}\, a^{2}} \] Input:
int(x^3/(a+I*a*sinh(d*x+c))^(5/2),x)
Output:
( - int(x**3/(sqrt(sinh(c + d*x)*i + 1)*sinh(c + d*x)**2 - 2*sqrt(sinh(c + d*x)*i + 1)*sinh(c + d*x)*i - sqrt(sinh(c + d*x)*i + 1)),x))/(sqrt(a)*a** 2)