\(\int x^3 (a+i a \sinh (c+d x))^{5/2} \, dx\) [130]

Optimal result

Integrand size = 21, antiderivative size = 638 \[ \int x^3 (a+i a \sinh (c+d x))^{5/2} \, dx=-\frac {265216 a^2 \sqrt {a+i a \sinh (c+d x)}}{1125 d^4}-\frac {128 a^2 x^2 \sqrt {a+i a \sinh (c+d x)}}{5 d^2}-\frac {17408 a^2 \cosh ^2\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}}{3375 d^4}-\frac {64 a^2 x^2 \cosh ^2\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}}{15 d^2}-\frac {384 a^2 \cosh ^4\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}}{625 d^4}-\frac {48 a^2 x^2 \cosh ^4\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}}{25 d^2}+\frac {8704 a^2 x \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sinh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}}{1125 d^3}+\frac {32 a^2 x^3 \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sinh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}}{15 d}+\frac {192 a^2 x \cosh ^3\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sinh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}}{125 d^3}+\frac {8 a^2 x^3 \cosh ^3\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sinh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}}{5 d}+\frac {132608 a^2 x \sqrt {a+i a \sinh (c+d x)} \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{1125 d^3}+\frac {64 a^2 x^3 \sqrt {a+i a \sinh (c+d x)} \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{15 d} \] Output:

-265216/1125*a^2*(a+I*a*sinh(d*x+c))^(1/2)/d^4-128/5*a^2*x^2*(a+I*a*sinh(d 
*x+c))^(1/2)/d^2-17408/3375*a^2*cosh(1/2*c+1/4*I*Pi+1/2*d*x)^2*(a+I*a*sinh 
(d*x+c))^(1/2)/d^4-64/15*a^2*x^2*cosh(1/2*c+1/4*I*Pi+1/2*d*x)^2*(a+I*a*sin 
h(d*x+c))^(1/2)/d^2-384/625*a^2*cosh(1/2*c+1/4*I*Pi+1/2*d*x)^4*(a+I*a*sinh 
(d*x+c))^(1/2)/d^4-48/25*a^2*x^2*cosh(1/2*c+1/4*I*Pi+1/2*d*x)^4*(a+I*a*sin 
h(d*x+c))^(1/2)/d^2+8704/1125*a^2*x*cosh(1/2*c+1/4*I*Pi+1/2*d*x)*sinh(1/2* 
c+1/4*I*Pi+1/2*d*x)*(a+I*a*sinh(d*x+c))^(1/2)/d^3+32/15*a^2*x^3*cosh(1/2*c 
+1/4*I*Pi+1/2*d*x)*sinh(1/2*c+1/4*I*Pi+1/2*d*x)*(a+I*a*sinh(d*x+c))^(1/2)/ 
d+192/125*a^2*x*cosh(1/2*c+1/4*I*Pi+1/2*d*x)^3*sinh(1/2*c+1/4*I*Pi+1/2*d*x 
)*(a+I*a*sinh(d*x+c))^(1/2)/d^3+8/5*a^2*x^3*cosh(1/2*c+1/4*I*Pi+1/2*d*x)^3 
*sinh(1/2*c+1/4*I*Pi+1/2*d*x)*(a+I*a*sinh(d*x+c))^(1/2)/d+132608/1125*a^2* 
x*(a+I*a*sinh(d*x+c))^(1/2)*tanh(1/2*c+1/4*I*Pi+1/2*d*x)/d^3+64/15*a^2*x^3 
*(a+I*a*sinh(d*x+c))^(1/2)*tanh(1/2*c+1/4*I*Pi+1/2*d*x)/d
 

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2918\) vs. \(2(638)=1276\).

Time = 15.39 (sec) , antiderivative size = 2918, normalized size of antiderivative = 4.57 \[ \int x^3 (a+i a \sinh (c+d x))^{5/2} \, dx=\text {Result too large to show} \] Input:

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

Output:

(2*(((-1/135000 - I/135000)*Cosh[5*(c/2 + (d*x)/2)])/d^3 + ((1/135000 + I/ 
135000)*Sinh[5*(c/2 + (d*x)/2)])/d^3)*(1296*I - (3240*I)*c + (4050*I)*c^2 
- (3375*I)*c^3 + (6480*I)*(c/2 + (d*x)/2) - (16200*I)*c*(c/2 + (d*x)/2) + 
(20250*I)*c^2*(c/2 + (d*x)/2) + (16200*I)*(c/2 + (d*x)/2)^2 - (40500*I)*c* 
(c/2 + (d*x)/2)^2 + (27000*I)*(c/2 + (d*x)/2)^3 - 50000*Cosh[2*(c/2 + (d*x 
)/2)] + 75000*c*Cosh[2*(c/2 + (d*x)/2)] - 56250*c^2*Cosh[2*(c/2 + (d*x)/2) 
] + 28125*c^3*Cosh[2*(c/2 + (d*x)/2)] - 150000*(c/2 + (d*x)/2)*Cosh[2*(c/2 
 + (d*x)/2)] + 225000*c*(c/2 + (d*x)/2)*Cosh[2*(c/2 + (d*x)/2)] - 168750*c 
^2*(c/2 + (d*x)/2)*Cosh[2*(c/2 + (d*x)/2)] - 225000*(c/2 + (d*x)/2)^2*Cosh 
[2*(c/2 + (d*x)/2)] + 337500*c*(c/2 + (d*x)/2)^2*Cosh[2*(c/2 + (d*x)/2)] - 
 225000*(c/2 + (d*x)/2)^3*Cosh[2*(c/2 + (d*x)/2)] - (8100000*I)*Cosh[4*(c/ 
2 + (d*x)/2)] + (4050000*I)*c*Cosh[4*(c/2 + (d*x)/2)] - (1012500*I)*c^2*Co 
sh[4*(c/2 + (d*x)/2)] + (168750*I)*c^3*Cosh[4*(c/2 + (d*x)/2)] - (8100000* 
I)*(c/2 + (d*x)/2)*Cosh[4*(c/2 + (d*x)/2)] + (4050000*I)*c*(c/2 + (d*x)/2) 
*Cosh[4*(c/2 + (d*x)/2)] - (1012500*I)*c^2*(c/2 + (d*x)/2)*Cosh[4*(c/2 + ( 
d*x)/2)] - (4050000*I)*(c/2 + (d*x)/2)^2*Cosh[4*(c/2 + (d*x)/2)] + (202500 
0*I)*c*(c/2 + (d*x)/2)^2*Cosh[4*(c/2 + (d*x)/2)] - (1350000*I)*(c/2 + (d*x 
)/2)^3*Cosh[4*(c/2 + (d*x)/2)] + 8100000*Cosh[6*(c/2 + (d*x)/2)] + 4050000 
*c*Cosh[6*(c/2 + (d*x)/2)] + 1012500*c^2*Cosh[6*(c/2 + (d*x)/2)] + 168750* 
c^3*Cosh[6*(c/2 + (d*x)/2)] - 8100000*(c/2 + (d*x)/2)*Cosh[6*(c/2 + (d*...
 

Rubi [F]

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3800

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3792

\(\displaystyle 4 a^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \left (\frac {24 \int x \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx}{25 d^2}+\frac {4}{5} \int x^3 \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx-\frac {12 x^2 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 4 a^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \left (\frac {24 \int x \sin \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^5dx}{25 d^2}+\frac {4}{5} \int x^3 \sin \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^3dx-\frac {12 x^2 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )\)

\(\Big \downarrow \) 3791

\(\displaystyle 4 a^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \left (\frac {24 \left (\frac {4}{5} \int x \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx-\frac {4 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )}{25 d^2}+\frac {4}{5} \int x^3 \sin \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^3dx-\frac {12 x^2 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 4 a^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \left (\frac {24 \left (\frac {4}{5} \int x \sin \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^3dx-\frac {4 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )}{25 d^2}+\frac {4}{5} \int x^3 \sin \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^3dx-\frac {12 x^2 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )\)

\(\Big \downarrow \) 3791

\(\displaystyle 4 a^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \left (\frac {24 \left (\frac {4}{5} \left (\frac {2}{3} \int x \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx-\frac {4 \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{9 d^2}+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )-\frac {4 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )}{25 d^2}+\frac {4}{5} \int x^3 \sin \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^3dx-\frac {12 x^2 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 4 a^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \left (\frac {24 \left (\frac {4}{5} \left (\frac {2}{3} \int x \sin \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )dx-\frac {4 \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{9 d^2}+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )-\frac {4 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )}{25 d^2}+\frac {4}{5} \int x^3 \sin \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^3dx-\frac {12 x^2 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )\)

\(\Big \downarrow \) 3777

\(\displaystyle 4 a^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \left (\frac {24 \left (\frac {4}{5} \left (\frac {2}{3} \left (\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {2 i \int -i \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx}{d}\right )-\frac {4 \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{9 d^2}+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )-\frac {4 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )}{25 d^2}+\frac {4}{5} \int x^3 \sin \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^3dx-\frac {12 x^2 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle 4 a^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \left (\frac {24 \left (\frac {4}{5} \left (\frac {2}{3} \left (\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {2 \int \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx}{d}\right )-\frac {4 \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{9 d^2}+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )-\frac {4 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )}{25 d^2}+\frac {4}{5} \int x^3 \sin \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^3dx-\frac {12 x^2 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 4 a^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \left (\frac {24 \left (\frac {4}{5} \left (\frac {2}{3} \left (\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {2 \int -i \sin \left (\frac {i c}{2}+\frac {i d x}{2}-\frac {\pi }{4}\right )dx}{d}\right )-\frac {4 \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{9 d^2}+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )-\frac {4 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )}{25 d^2}+\frac {4}{5} \int x^3 \sin \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^3dx-\frac {12 x^2 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle 4 a^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \left (\frac {24 \left (\frac {4}{5} \left (\frac {2}{3} \left (\frac {2 i \int \sin \left (\frac {i c}{2}+\frac {i d x}{2}-\frac {\pi }{4}\right )dx}{d}+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )-\frac {4 \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{9 d^2}+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )-\frac {4 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )}{25 d^2}+\frac {4}{5} \int x^3 \sin \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^3dx-\frac {12 x^2 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )\)

\(\Big \downarrow \) 3118

\(\displaystyle 4 a^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \left (\frac {4}{5} \int x^3 \sin \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^3dx-\frac {12 x^2 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {24 \left (-\frac {4 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {4}{5} \left (-\frac {4 \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{9 d^2}+\frac {2}{3} \left (\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {4 \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}\right )+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )}{25 d^2}+\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )\)

\(\Big \downarrow \) 3792

\(\displaystyle 4 a^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \left (\frac {4}{5} \left (\frac {8 \int x \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx}{3 d^2}+\frac {2}{3} \int x^3 \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx-\frac {4 x^2 \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d^2}+\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )-\frac {12 x^2 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {24 \left (-\frac {4 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {4}{5} \left (-\frac {4 \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{9 d^2}+\frac {2}{3} \left (\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {4 \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}\right )+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )}{25 d^2}+\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 4 a^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \left (\frac {4}{5} \left (\frac {8 \int x \sin \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^3dx}{3 d^2}+\frac {2}{3} \int x^3 \sin \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )dx-\frac {4 x^2 \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d^2}+\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )-\frac {12 x^2 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {24 \left (-\frac {4 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {4}{5} \left (-\frac {4 \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{9 d^2}+\frac {2}{3} \left (\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {4 \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}\right )+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )}{25 d^2}+\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )\)

\(\Big \downarrow \) 3777

\(\displaystyle 4 a^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \left (\frac {4}{5} \left (\frac {8 \int x \sin \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^3dx}{3 d^2}+\frac {2}{3} \left (\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {6 i \int -i x^2 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx}{d}\right )-\frac {4 x^2 \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d^2}+\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )-\frac {12 x^2 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {24 \left (-\frac {4 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {4}{5} \left (-\frac {4 \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{9 d^2}+\frac {2}{3} \left (\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {4 \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}\right )+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )}{25 d^2}+\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle 4 a^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \left (\frac {4}{5} \left (\frac {8 \int x \sin \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^3dx}{3 d^2}+\frac {2}{3} \left (\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {6 \int x^2 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx}{d}\right )-\frac {4 x^2 \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d^2}+\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )-\frac {12 x^2 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {24 \left (-\frac {4 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {4}{5} \left (-\frac {4 \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{9 d^2}+\frac {2}{3} \left (\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {4 \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}\right )+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )}{25 d^2}+\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 4 a^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \left (\frac {4}{5} \left (\frac {8 \int x \sin \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^3dx}{3 d^2}+\frac {2}{3} \left (\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {6 \int -i x^2 \sin \left (\frac {i c}{2}+\frac {i d x}{2}-\frac {\pi }{4}\right )dx}{d}\right )-\frac {4 x^2 \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d^2}+\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )-\frac {12 x^2 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {24 \left (-\frac {4 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {4}{5} \left (-\frac {4 \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{9 d^2}+\frac {2}{3} \left (\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {4 \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}\right )+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )}{25 d^2}+\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle 4 a^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \left (\frac {4}{5} \left (\frac {8 \int x \sin \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^3dx}{3 d^2}+\frac {2}{3} \left (\frac {6 i \int x^2 \sin \left (\frac {i c}{2}+\frac {i d x}{2}-\frac {\pi }{4}\right )dx}{d}+\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )-\frac {4 x^2 \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d^2}+\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )-\frac {12 x^2 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {24 \left (-\frac {4 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {4}{5} \left (-\frac {4 \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{9 d^2}+\frac {2}{3} \left (\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {4 \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}\right )+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )}{25 d^2}+\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )\)

\(\Big \downarrow \) 3777

\(\displaystyle 4 a^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \left (\frac {4}{5} \left (\frac {8 \int x \sin \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^3dx}{3 d^2}+\frac {2}{3} \left (\frac {6 i \left (\frac {2 i x^2 \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {4 i \int x \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx}{d}\right )}{d}+\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )-\frac {4 x^2 \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d^2}+\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )-\frac {12 x^2 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {24 \left (-\frac {4 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {4}{5} \left (-\frac {4 \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{9 d^2}+\frac {2}{3} \left (\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {4 \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}\right )+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )}{25 d^2}+\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 4 a^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \left (\frac {4}{5} \left (\frac {8 \int x \sin \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^3dx}{3 d^2}+\frac {2}{3} \left (\frac {6 i \left (\frac {2 i x^2 \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {4 i \int x \sin \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )dx}{d}\right )}{d}+\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )-\frac {4 x^2 \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d^2}+\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )-\frac {12 x^2 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {24 \left (-\frac {4 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {4}{5} \left (-\frac {4 \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{9 d^2}+\frac {2}{3} \left (\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {4 \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}\right )+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )}{25 d^2}+\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )\)

\(\Big \downarrow \) 3777

\(\displaystyle 4 a^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \left (\frac {4}{5} \left (\frac {8 \int x \sin \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^3dx}{3 d^2}+\frac {2}{3} \left (\frac {6 i \left (\frac {2 i x^2 \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {4 i \left (\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {2 i \int -i \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx}{d}\right )}{d}\right )}{d}+\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )-\frac {4 x^2 \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d^2}+\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )-\frac {12 x^2 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {24 \left (-\frac {4 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {4}{5} \left (-\frac {4 \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{9 d^2}+\frac {2}{3} \left (\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {4 \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}\right )+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )}{25 d^2}+\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle 4 a^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \left (\frac {4}{5} \left (\frac {8 \int x \sin \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^3dx}{3 d^2}+\frac {2}{3} \left (\frac {6 i \left (\frac {2 i x^2 \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {4 i \left (\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {2 \int \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx}{d}\right )}{d}\right )}{d}+\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )-\frac {4 x^2 \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d^2}+\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )-\frac {12 x^2 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {24 \left (-\frac {4 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {4}{5} \left (-\frac {4 \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{9 d^2}+\frac {2}{3} \left (\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {4 \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}\right )+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )}{25 d^2}+\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 4 a^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \left (\frac {4}{5} \left (\frac {8 \int x \sin \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^3dx}{3 d^2}+\frac {2}{3} \left (\frac {6 i \left (\frac {2 i x^2 \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {4 i \left (\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {2 \int -i \sin \left (\frac {i c}{2}+\frac {i d x}{2}-\frac {\pi }{4}\right )dx}{d}\right )}{d}\right )}{d}+\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )-\frac {4 x^2 \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d^2}+\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )-\frac {12 x^2 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {24 \left (-\frac {4 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {4}{5} \left (-\frac {4 \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{9 d^2}+\frac {2}{3} \left (\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {4 \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}\right )+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )}{25 d^2}+\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle 4 a^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \left (\frac {4}{5} \left (\frac {8 \int x \sin \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^3dx}{3 d^2}+\frac {2}{3} \left (\frac {6 i \left (\frac {2 i x^2 \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {4 i \left (\frac {2 i \int \sin \left (\frac {i c}{2}+\frac {i d x}{2}-\frac {\pi }{4}\right )dx}{d}+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )}{d}\right )}{d}+\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )-\frac {4 x^2 \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d^2}+\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )-\frac {12 x^2 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {24 \left (-\frac {4 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {4}{5} \left (-\frac {4 \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{9 d^2}+\frac {2}{3} \left (\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {4 \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}\right )+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )}{25 d^2}+\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )\)

\(\Big \downarrow \) 3118

\(\displaystyle 4 a^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \left (\frac {4}{5} \left (\frac {8 \int x \sin \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^3dx}{3 d^2}-\frac {4 x^2 \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d^2}+\frac {2}{3} \left (\frac {6 i \left (\frac {2 i x^2 \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {4 i \left (\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {4 \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}\right )}{d}\right )}{d}+\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )+\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )-\frac {12 x^2 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {24 \left (-\frac {4 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {4}{5} \left (-\frac {4 \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{9 d^2}+\frac {2}{3} \left (\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {4 \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}\right )+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )}{25 d^2}+\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )\)

\(\Big \downarrow \) 3791

\(\displaystyle 4 a^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \left (\frac {4}{5} \left (\frac {8 \left (\frac {2}{3} \int x \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx-\frac {4 \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{9 d^2}+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )}{3 d^2}-\frac {4 x^2 \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d^2}+\frac {2}{3} \left (\frac {6 i \left (\frac {2 i x^2 \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {4 i \left (\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {4 \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}\right )}{d}\right )}{d}+\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )+\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )-\frac {12 x^2 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {24 \left (-\frac {4 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {4}{5} \left (-\frac {4 \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{9 d^2}+\frac {2}{3} \left (\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {4 \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}\right )+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )}{25 d^2}+\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 4 a^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \left (\frac {4}{5} \left (\frac {8 \left (\frac {2}{3} \int x \sin \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )dx-\frac {4 \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{9 d^2}+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )}{3 d^2}-\frac {4 x^2 \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d^2}+\frac {2}{3} \left (\frac {6 i \left (\frac {2 i x^2 \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {4 i \left (\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {4 \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}\right )}{d}\right )}{d}+\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )+\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )-\frac {12 x^2 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {24 \left (-\frac {4 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {4}{5} \left (-\frac {4 \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{9 d^2}+\frac {2}{3} \left (\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {4 \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}\right )+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )}{25 d^2}+\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )\)

\(\Big \downarrow \) 3777

\(\displaystyle 4 a^2 \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \left (\frac {4}{5} \left (\frac {8 \left (\frac {2}{3} \left (\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {2 i \int -i \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx}{d}\right )-\frac {4 \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{9 d^2}+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )}{3 d^2}-\frac {4 x^2 \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d^2}+\frac {2}{3} \left (\frac {6 i \left (\frac {2 i x^2 \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {4 i \left (\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {4 \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}\right )}{d}\right )}{d}+\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )+\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )-\frac {12 x^2 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {24 \left (-\frac {4 \cosh ^5\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{25 d^2}+\frac {4}{5} \left (-\frac {4 \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{9 d^2}+\frac {2}{3} \left (\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {4 \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d^2}\right )+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{3 d}\right )+\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )}{25 d^2}+\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{5 d}\right )\)

Input:

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

Output:

$Aborted
 

Maple [F]

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

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)
 

Fricas [F(-2)]

Exception generated. \[ \int x^3 (a+i a \sinh (c+d x))^{5/2} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (has polynomial part)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

Reduce [F]

\[ \int x^3 (a+i a \sinh (c+d x))^{5/2} \, dx=\sqrt {a}\, a^{2} \left (-\left (\int \sqrt {\sinh \left (d x +c \right ) i +1}\, \sinh \left (d x +c \right )^{2} x^{3}d x \right )+\int \sqrt {\sinh \left (d x +c \right ) i +1}\, x^{3}d x +2 \left (\int \sqrt {\sinh \left (d x +c \right ) i +1}\, \sinh \left (d x +c \right ) x^{3}d x \right ) i \right ) \] Input:

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

Output:

sqrt(a)*a**2*( - int(sqrt(sinh(c + d*x)*i + 1)*sinh(c + d*x)**2*x**3,x) + 
int(sqrt(sinh(c + d*x)*i + 1)*x**3,x) + 2*int(sqrt(sinh(c + d*x)*i + 1)*si 
nh(c + d*x)*x**3,x)*i)