\(\int x^2 (a+i a \sinh (e+f x))^{3/2} \, dx\) [126]

Optimal result

Integrand size = 21, antiderivative size = 303 \[ \int x^2 (a+i a \sinh (e+f x))^{3/2} \, dx=-\frac {32 a x \sqrt {a+i a \sinh (e+f x)}}{3 f^2}-\frac {16 a x \cosh ^2\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \sqrt {a+i a \sinh (e+f x)}}{9 f^2}+\frac {4 a x^2 \cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \sinh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \sqrt {a+i a \sinh (e+f x)}}{3 f}+\frac {224 a \sqrt {a+i a \sinh (e+f x)} \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{9 f^3}+\frac {8 a x^2 \sqrt {a+i a \sinh (e+f x)} \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{3 f}+\frac {32 a \sinh ^2\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \sqrt {a+i a \sinh (e+f x)} \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{27 f^3} \] Output:

-32/3*a*x*(a+I*a*sinh(f*x+e))^(1/2)/f^2-16/9*a*x*cosh(1/2*e+1/4*I*Pi+1/2*f 
*x)^2*(a+I*a*sinh(f*x+e))^(1/2)/f^2+4/3*a*x^2*cosh(1/2*e+1/4*I*Pi+1/2*f*x) 
*sinh(1/2*e+1/4*I*Pi+1/2*f*x)*(a+I*a*sinh(f*x+e))^(1/2)/f+224/9*a*(a+I*a*s 
inh(f*x+e))^(1/2)*tanh(1/2*e+1/4*I*Pi+1/2*f*x)/f^3+8/3*a*x^2*(a+I*a*sinh(f 
*x+e))^(1/2)*tanh(1/2*e+1/4*I*Pi+1/2*f*x)/f+32/27*a*sinh(1/2*e+1/4*I*Pi+1/ 
2*f*x)^2*(a+I*a*sinh(f*x+e))^(1/2)*tanh(1/2*e+1/4*I*Pi+1/2*f*x)/f^3
 

Mathematica [A] (verified)

Time = 7.37 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.57 \[ \int x^2 (a+i a \sinh (e+f x))^{3/2} \, dx=-\frac {a \left (81 \left (8+4 i f x+f^2 x^2\right ) \cosh \left (\frac {1}{2} (e+f x)\right )+\left (8-12 i f x+9 f^2 x^2\right ) \cosh \left (\frac {3}{2} (e+f x)\right )+2 i \left (-4 \left (80-42 i f x+9 f^2 x^2\right )+\left (8+12 i f x+9 f^2 x^2\right ) \cosh (e+f x)\right ) \sinh \left (\frac {1}{2} (e+f x)\right )\right ) (-i+\sinh (e+f x)) \sqrt {a+i a \sinh (e+f x)}}{27 f^3 \left (\cosh \left (\frac {1}{2} (e+f x)\right )+i \sinh \left (\frac {1}{2} (e+f x)\right )\right )^3} \] Input:

Integrate[x^2*(a + I*a*Sinh[e + f*x])^(3/2),x]
 

Output:

-1/27*(a*(81*(8 + (4*I)*f*x + f^2*x^2)*Cosh[(e + f*x)/2] + (8 - (12*I)*f*x 
 + 9*f^2*x^2)*Cosh[(3*(e + f*x))/2] + (2*I)*(-4*(80 - (42*I)*f*x + 9*f^2*x 
^2) + (8 + (12*I)*f*x + 9*f^2*x^2)*Cosh[e + f*x])*Sinh[(e + f*x)/2])*(-I + 
 Sinh[e + f*x])*Sqrt[a + I*a*Sinh[e + f*x]])/(f^3*(Cosh[(e + f*x)/2] + I*S 
inh[(e + f*x)/2])^3)
 

Rubi [A] (verified)

Time = 0.88 (sec) , antiderivative size = 281, normalized size of antiderivative = 0.93, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 3800, 3042, 3792, 3042, 3113, 2009, 3777, 26, 3042, 26, 3777, 3042, 3118}

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.

\(\displaystyle \int x^2 (a+i a \sinh (e+f x))^{3/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int x^2 (a+a \sin (i e+i f x))^{3/2}dx\)

\(\Big \downarrow \) 3800

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3792

\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {8 \int \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )dx}{9 f^2}+\frac {2}{3} \int x^2 \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )dx-\frac {8 x \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{9 f^2}+\frac {2 x^2 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {8 \int \sin \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )^3dx}{9 f^2}+\frac {2}{3} \int x^2 \sin \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )dx-\frac {8 x \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{9 f^2}+\frac {2 x^2 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 3113

\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {16 i \int \left (\sinh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )+1\right )d\left (-i \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )\right )}{9 f^3}+\frac {2}{3} \int x^2 \sin \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )dx-\frac {8 x \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{9 f^2}+\frac {2 x^2 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {2}{3} \int x^2 \sin \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )dx+\frac {16 i \left (-\frac {1}{3} i \sinh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )-i \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )\right )}{9 f^3}-\frac {8 x \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{9 f^2}+\frac {2 x^2 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 3777

\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {2}{3} \left (\frac {2 x^2 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 i \int -i x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )dx}{f}\right )+\frac {16 i \left (-\frac {1}{3} i \sinh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )-i \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )\right )}{9 f^3}-\frac {8 x \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{9 f^2}+\frac {2 x^2 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {2}{3} \left (\frac {2 x^2 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 \int x \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )dx}{f}\right )+\frac {16 i \left (-\frac {1}{3} i \sinh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )-i \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )\right )}{9 f^3}-\frac {8 x \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{9 f^2}+\frac {2 x^2 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {2}{3} \left (\frac {2 x^2 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 \int -i x \sin \left (\frac {i e}{2}+\frac {i f x}{2}-\frac {\pi }{4}\right )dx}{f}\right )+\frac {16 i \left (-\frac {1}{3} i \sinh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )-i \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )\right )}{9 f^3}-\frac {8 x \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{9 f^2}+\frac {2 x^2 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {2}{3} \left (\frac {4 i \int x \sin \left (\frac {i e}{2}+\frac {i f x}{2}-\frac {\pi }{4}\right )dx}{f}+\frac {2 x^2 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )+\frac {16 i \left (-\frac {1}{3} i \sinh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )-i \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )\right )}{9 f^3}-\frac {8 x \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{9 f^2}+\frac {2 x^2 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 3777

\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {2}{3} \left (\frac {4 i \left (\frac {2 i x \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {2 i \int \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )dx}{f}\right )}{f}+\frac {2 x^2 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )+\frac {16 i \left (-\frac {1}{3} i \sinh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )-i \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )\right )}{9 f^3}-\frac {8 x \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{9 f^2}+\frac {2 x^2 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {2}{3} \left (\frac {4 i \left (\frac {2 i x \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {2 i \int \sin \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )dx}{f}\right )}{f}+\frac {2 x^2 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )+\frac {16 i \left (-\frac {1}{3} i \sinh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )-i \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )\right )}{9 f^3}-\frac {8 x \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{9 f^2}+\frac {2 x^2 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\)

\(\Big \downarrow \) 3118

\(\displaystyle 2 a \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)} \left (\frac {16 i \left (-\frac {1}{3} i \sinh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )-i \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )\right )}{9 f^3}+\frac {2}{3} \left (\frac {4 i \left (\frac {2 i x \cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 i \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f^2}\right )}{f}+\frac {2 x^2 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )-\frac {8 x \cosh ^3\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{9 f^2}+\frac {2 x^2 \sinh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \cosh ^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}\right )\)

Input:

Int[x^2*(a + I*a*Sinh[e + f*x])^(3/2),x]
 

Output:

2*a*Sech[e/2 + (I/4)*Pi + (f*x)/2]*((-8*x*Cosh[e/2 + (I/4)*Pi + (f*x)/2]^3 
)/(9*f^2) + (2*x^2*Cosh[e/2 + (I/4)*Pi + (f*x)/2]^2*Sinh[e/2 + (I/4)*Pi + 
(f*x)/2])/(3*f) + (((16*I)/9)*((-I)*Sinh[e/2 + (I/4)*Pi + (f*x)/2] - (I/3) 
*Sinh[e/2 + (I/4)*Pi + (f*x)/2]^3))/f^3 + (2*((2*x^2*Sinh[e/2 + (I/4)*Pi + 
 (f*x)/2])/f + ((4*I)*(((2*I)*x*Cosh[e/2 + (I/4)*Pi + (f*x)/2])/f - ((4*I) 
*Sinh[e/2 + (I/4)*Pi + (f*x)/2])/f^2))/f))/3)*Sqrt[a + I*a*Sinh[e + f*x]]
 

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3113

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1)   Subst[Int[Exp 
and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] 
 && IGtQ[(n - 1)/2, 0]
 

rule 3118

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ 
[{c, d}, x]
 

rule 3777

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( 
-(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1)*C 
os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
 

rule 3792

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

rule 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])
 

Maple [F]

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

Input:

int(x^2*(a+I*a*sinh(f*x+e))^(3/2),x)
 

Output:

int(x^2*(a+I*a*sinh(f*x+e))^(3/2),x)
 

Fricas [F(-2)]

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

integrate(x^2*(a+I*a*sinh(f*x+e))^(3/2),x, algorithm="fricas")
 

Output:

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

Sympy [F]

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

integrate(x**2*(a+I*a*sinh(f*x+e))**(3/2),x)
 

Output:

Integral(x**2*(I*a*(sinh(e + f*x) - I))**(3/2), x)
 

Maxima [F]

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

integrate(x^2*(a+I*a*sinh(f*x+e))^(3/2),x, algorithm="maxima")
 

Output:

integrate((I*a*sinh(f*x + e) + a)^(3/2)*x^2, x)
                                                                                    
                                                                                    
 

Giac [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.59 \[ \int x^2 (a+i a \sinh (e+f x))^{3/2} \, dx=-\frac {{\left (-\left (9 i - 9\right ) \, a^{\frac {3}{2}} f^{2} x^{2} e^{\left (3 \, f x + 3 \, e\right )} - \left (81 i + 81\right ) \, a^{\frac {3}{2}} f^{2} x^{2} e^{\left (2 \, f x + 2 \, e\right )} - \left (81 i - 81\right ) \, a^{\frac {3}{2}} f^{2} x^{2} e^{\left (f x + e\right )} - \left (9 i + 9\right ) \, a^{\frac {3}{2}} f^{2} x^{2} + \left (12 i - 12\right ) \, a^{\frac {3}{2}} f x e^{\left (3 \, f x + 3 \, e\right )} + \left (324 i + 324\right ) \, a^{\frac {3}{2}} f x e^{\left (2 \, f x + 2 \, e\right )} - \left (324 i - 324\right ) \, a^{\frac {3}{2}} f x e^{\left (f x + e\right )} - \left (12 i + 12\right ) \, a^{\frac {3}{2}} f x - \left (8 i - 8\right ) \, a^{\frac {3}{2}} e^{\left (3 \, f x + 3 \, e\right )} - \left (648 i + 648\right ) \, a^{\frac {3}{2}} e^{\left (2 \, f x + 2 \, e\right )} - \left (648 i - 648\right ) \, a^{\frac {3}{2}} e^{\left (f x + e\right )} - \left (8 i + 8\right ) \, a^{\frac {3}{2}}\right )} e^{\left (-\frac {3}{2} \, f x - \frac {3}{2} \, e\right )}}{54 \, f^{3}} \] Input:

integrate(x^2*(a+I*a*sinh(f*x+e))^(3/2),x, algorithm="giac")
 

Output:

-1/54*(-(9*I - 9)*a^(3/2)*f^2*x^2*e^(3*f*x + 3*e) - (81*I + 81)*a^(3/2)*f^ 
2*x^2*e^(2*f*x + 2*e) - (81*I - 81)*a^(3/2)*f^2*x^2*e^(f*x + e) - (9*I + 9 
)*a^(3/2)*f^2*x^2 + (12*I - 12)*a^(3/2)*f*x*e^(3*f*x + 3*e) + (324*I + 324 
)*a^(3/2)*f*x*e^(2*f*x + 2*e) - (324*I - 324)*a^(3/2)*f*x*e^(f*x + e) - (1 
2*I + 12)*a^(3/2)*f*x - (8*I - 8)*a^(3/2)*e^(3*f*x + 3*e) - (648*I + 648)* 
a^(3/2)*e^(2*f*x + 2*e) - (648*I - 648)*a^(3/2)*e^(f*x + e) - (8*I + 8)*a^ 
(3/2))*e^(-3/2*f*x - 3/2*e)/f^3
 

Mupad [F(-1)]

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

int(x^2*(a + a*sinh(e + f*x)*1i)^(3/2),x)
 

Output:

int(x^2*(a + a*sinh(e + f*x)*1i)^(3/2), x)
 

Reduce [F]

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

int(x^2*(a+I*a*sinh(f*x+e))^(3/2),x)
 

Output:

sqrt(a)*a*(int(sqrt(sinh(e + f*x)*i + 1)*x**2,x) + int(sqrt(sinh(e + f*x)* 
i + 1)*sinh(e + f*x)*x**2,x)*i)