\(\int x^2 (a+i a \sinh (c+d x))^{5/2} \, dx\) [131]

Optimal result

Integrand size = 21, antiderivative size = 506 \[ \int x^2 (a+i a \sinh (c+d x))^{5/2} \, dx=-\frac {256 a^2 x \sqrt {a+i a \sinh (c+d x)}}{15 d^2}-\frac {128 a^2 x \cosh ^2\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}}{45 d^2}-\frac {32 a^2 x \cosh ^4\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}}{25 d^2}+\frac {32 a^2 x^2 \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sinh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}}{15 d}+\frac {8 a^2 x^2 \cosh ^3\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sinh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}}{5 d}+\frac {9536 a^2 \sqrt {a+i a \sinh (c+d x)} \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{225 d^3}+\frac {64 a^2 x^2 \sqrt {a+i a \sinh (c+d x)} \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{15 d}+\frac {2432 a^2 \sinh ^2\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)} \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{675 d^3}+\frac {64 a^2 \sinh ^4\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)} \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{125 d^3} \]

[Out]

-256/15*a^2*x*(a+I*a*sinh(d*x+c))^(1/2)/d^2-128/45*a^2*x*cosh(1/2*c+1/4*I*Pi+1/2*d*x)^2*(a+I*a*sinh(d*x+c))^(1
/2)/d^2-32/25*a^2*x*cosh(1/2*c+1/4*I*Pi+1/2*d*x)^4*(a+I*a*sinh(d*x+c))^(1/2)/d^2+32/15*a^2*x^2*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+8/5*a^2*x^2*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+9536/225*a^2*(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^2*(a+I*a*sinh(d*x+c))^(1/2)*tanh(1/2*c+1/4*I*Pi+1/2*d*x)/d+2432/675*a^2*sin
h(1/2*c+1/4*I*Pi+1/2*d*x)^2*(a+I*a*sinh(d*x+c))^(1/2)*tanh(1/2*c+1/4*I*Pi+1/2*d*x)/d^3+64/125*a^2*sinh(1/2*c+1
/4*I*Pi+1/2*d*x)^4*(a+I*a*sinh(d*x+c))^(1/2)*tanh(1/2*c+1/4*I*Pi+1/2*d*x)/d^3

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 506, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3400, 3392, 3377, 2718, 2713} \[ \int x^2 (a+i a \sinh (c+d x))^{5/2} \, dx=\frac {64 a^2 \sinh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)}}{125 d^3}+\frac {2432 a^2 \sinh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)}}{675 d^3}+\frac {9536 a^2 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)}}{225 d^3}-\frac {256 a^2 x \sqrt {a+i a \sinh (c+d x)}}{15 d^2}-\frac {32 a^2 x \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)}}{25 d^2}-\frac {128 a^2 x \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)}}{45 d^2}+\frac {64 a^2 x^2 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)}}{15 d}+\frac {8 a^2 x^2 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)}}{5 d}+\frac {32 a^2 x^2 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)}}{15 d} \]

[In]

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

[Out]

(-256*a^2*x*Sqrt[a + I*a*Sinh[c + d*x]])/(15*d^2) - (128*a^2*x*Cosh[c/2 + (I/4)*Pi + (d*x)/2]^2*Sqrt[a + I*a*S
inh[c + d*x]])/(45*d^2) - (32*a^2*x*Cosh[c/2 + (I/4)*Pi + (d*x)/2]^4*Sqrt[a + I*a*Sinh[c + d*x]])/(25*d^2) + (
32*a^2*x^2*Cosh[c/2 + (I/4)*Pi + (d*x)/2]*Sinh[c/2 + (I/4)*Pi + (d*x)/2]*Sqrt[a + I*a*Sinh[c + d*x]])/(15*d) +
(8*a^2*x^2*Cosh[c/2 + (I/4)*Pi + (d*x)/2]^3*Sinh[c/2 + (I/4)*Pi + (d*x)/2]*Sqrt[a + I*a*Sinh[c + d*x]])/(5*d)
+ (9536*a^2*Sqrt[a + I*a*Sinh[c + d*x]]*Tanh[c/2 + (I/4)*Pi + (d*x)/2])/(225*d^3) + (64*a^2*x^2*Sqrt[a + I*a*
Sinh[c + d*x]]*Tanh[c/2 + (I/4)*Pi + (d*x)/2])/(15*d) + (2432*a^2*Sinh[c/2 + (I/4)*Pi + (d*x)/2]^2*Sqrt[a + I*
a*Sinh[c + d*x]]*Tanh[c/2 + (I/4)*Pi + (d*x)/2])/(675*d^3) + (64*a^2*Sinh[c/2 + (I/4)*Pi + (d*x)/2]^4*Sqrt[a +
I*a*Sinh[c + d*x]]*Tanh[c/2 + (I/4)*Pi + (d*x)/2])/(125*d^3)

Rule 2713

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

Rule 2718

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

Rule 3377

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

Rule 3392

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

Rule 3400

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[(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])

Rubi steps \begin{align*} \text {integral}& = \left (4 a^2 \text {csch}\left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}\right ) \int x^2 \sinh ^5\left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \, dx \\ & = -\frac {32 a^2 x \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 {8 a^2 x^2 \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 {1}{5} \left (16 a^2 \text {csch}\left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}\right ) \int x^2 \sinh ^3\left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \, dx+\frac {\left (32 a^2 \text {csch}\left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}\right ) \int \sinh ^5\left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \, dx}{25 d^2} \\ & = -\frac {128 a^2 x \cosh ^2\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}}{45 d^2}-\frac {32 a^2 x \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 {32 a^2 x^2 \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 {8 a^2 x^2 \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 {1}{15} \left (32 a^2 \text {csch}\left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}\right ) \int x^2 \sinh \left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \, dx+\frac {\left (64 a^2 \text {csch}\left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}\right ) \text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\cosh \left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right )\right )}{25 d^3}-\frac {\left (128 a^2 \text {csch}\left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}\right ) \int \sinh ^3\left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \, dx}{45 d^2} \\ & = -\frac {128 a^2 x \cosh ^2\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}}{45 d^2}-\frac {32 a^2 x \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 {32 a^2 x^2 \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 {8 a^2 x^2 \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 {64 a^2 \sqrt {a+i a \sinh (c+d x)} \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{25 d^3}+\frac {64 a^2 x^2 \sqrt {a+i a \sinh (c+d x)} \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{15 d}+\frac {128 a^2 \sinh ^2\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)} \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{75 d^3}+\frac {64 a^2 \sinh ^4\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)} \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{125 d^3}+\frac {\left (256 a^2 \text {csch}\left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cosh \left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right )\right )}{45 d^3}-\frac {\left (128 a^2 \text {csch}\left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}\right ) \int x \cosh \left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \, dx}{15 d} \\ & = -\frac {256 a^2 x \sqrt {a+i a \sinh (c+d x)}}{15 d^2}-\frac {128 a^2 x \cosh ^2\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}}{45 d^2}-\frac {32 a^2 x \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 {32 a^2 x^2 \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 {8 a^2 x^2 \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 {1856 a^2 \sqrt {a+i a \sinh (c+d x)} \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{225 d^3}+\frac {64 a^2 x^2 \sqrt {a+i a \sinh (c+d x)} \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{15 d}+\frac {2432 a^2 \sinh ^2\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)} \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{675 d^3}+\frac {64 a^2 \sinh ^4\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)} \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{125 d^3}-\frac {\left (256 i a^2 \text {csch}\left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}\right ) \int \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \, dx}{15 d^2} \\ & = -\frac {256 a^2 x \sqrt {a+i a \sinh (c+d x)}}{15 d^2}-\frac {128 a^2 x \cosh ^2\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}}{45 d^2}-\frac {32 a^2 x \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 {32 a^2 x^2 \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 {8 a^2 x^2 \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 {9536 a^2 \sqrt {a+i a \sinh (c+d x)} \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{225 d^3}+\frac {64 a^2 x^2 \sqrt {a+i a \sinh (c+d x)} \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{15 d}+\frac {2432 a^2 \sinh ^2\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)} \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{675 d^3}+\frac {64 a^2 \sinh ^4\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)} \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{125 d^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 9.75 (sec) , antiderivative size = 300, normalized size of antiderivative = 0.59 \[ \int x^2 (a+i a \sinh (c+d x))^{5/2} \, dx=\frac {a^2 \sqrt {a+i a \sinh (c+d x)} \left (33750 i \left (8+4 i d x+d^2 x^2\right ) \cosh \left (\frac {1}{2} (c+d x)\right )+625 \left (8 i+12 d x+9 i d^2 x^2\right ) \cosh \left (\frac {3}{2} (c+d x)\right )-216 i \cosh \left (\frac {5}{2} (c+d x)\right )+540 d x \cosh \left (\frac {5}{2} (c+d x)\right )-675 i d^2 x^2 \cosh \left (\frac {5}{2} (c+d x)\right )+270000 \sinh \left (\frac {1}{2} (c+d x)\right )-135000 i d x \sinh \left (\frac {1}{2} (c+d x)\right )+33750 d^2 x^2 \sinh \left (\frac {1}{2} (c+d x)\right )-5000 \sinh \left (\frac {3}{2} (c+d x)\right )-7500 i d x \sinh \left (\frac {3}{2} (c+d x)\right )-5625 d^2 x^2 \sinh \left (\frac {3}{2} (c+d x)\right )-216 \sinh \left (\frac {5}{2} (c+d x)\right )+540 i d x \sinh \left (\frac {5}{2} (c+d x)\right )-675 d^2 x^2 \sinh \left (\frac {5}{2} (c+d x)\right )\right )}{6750 d^3 \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )} \]

[In]

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

[Out]

(a^2*Sqrt[a + I*a*Sinh[c + d*x]]*((33750*I)*(8 + (4*I)*d*x + d^2*x^2)*Cosh[(c + d*x)/2] + 625*(8*I + 12*d*x +
(9*I)*d^2*x^2)*Cosh[(3*(c + d*x))/2] - (216*I)*Cosh[(5*(c + d*x))/2] + 540*d*x*Cosh[(5*(c + d*x))/2] - (675*I)
*d^2*x^2*Cosh[(5*(c + d*x))/2] + 270000*Sinh[(c + d*x)/2] - (135000*I)*d*x*Sinh[(c + d*x)/2] + 33750*d^2*x^2*S
inh[(c + d*x)/2] - 5000*Sinh[(3*(c + d*x))/2] - (7500*I)*d*x*Sinh[(3*(c + d*x))/2] - 5625*d^2*x^2*Sinh[(3*(c +
d*x))/2] - 216*Sinh[(5*(c + d*x))/2] + (540*I)*d*x*Sinh[(5*(c + d*x))/2] - 675*d^2*x^2*Sinh[(5*(c + d*x))/2])
)/(6750*d^3*(Cosh[(c + d*x)/2] + I*Sinh[(c + d*x)/2]))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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