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

Optimal. Leaf size=185 \[ -\frac{16 a \sqrt{a+i a \sinh (e+f x)}}{3 f^2}-\frac{8 a \cosh ^2\left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (e+f x)}}{9 f^2}+\frac{8 a x \tanh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (e+f x)}}{3 f}+\frac{4 a x \sinh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \cosh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (e+f x)}}{3 f} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.133372, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {3319, 3310, 3296, 2638} \[ -\frac{16 a \sqrt{a+i a \sinh (e+f x)}}{3 f^2}-\frac{8 a \cosh ^2\left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (e+f x)}}{9 f^2}+\frac{8 a x \tanh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (e+f x)}}{3 f}+\frac{4 a x \sinh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \cosh \left (\frac{e}{2}+\frac{f x}{2}+\frac{i \pi }{4}\right ) \sqrt{a+i a \sinh (e+f x)}}{3 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3319

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])

Rule 3310

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

Rule 3296

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 2638

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

Rubi steps

\begin{align*} \int x (a+i a \sinh (e+f x))^{3/2} \, dx &=-\left (\left (2 a \text{csch}\left (\frac{e}{2}-\frac{i \pi }{4}+\frac{f x}{2}\right ) \sqrt{a+i a \sinh (e+f x)}\right ) \int x \sinh ^3\left (\frac{e}{2}-\frac{i \pi }{4}+\frac{f x}{2}\right ) \, dx\right )\\ &=-\frac{8 a \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 \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{1}{3} \left (4 a \text{csch}\left (\frac{e}{2}-\frac{i \pi }{4}+\frac{f x}{2}\right ) \sqrt{a+i a \sinh (e+f x)}\right ) \int x \sinh \left (\frac{e}{2}-\frac{i \pi }{4}+\frac{f x}{2}\right ) \, dx\\ &=-\frac{8 a \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 \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{8 a x \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{\left (8 a \text{csch}\left (\frac{e}{2}-\frac{i \pi }{4}+\frac{f x}{2}\right ) \sqrt{a+i a \sinh (e+f x)}\right ) \int \cosh \left (\frac{e}{2}-\frac{i \pi }{4}+\frac{f x}{2}\right ) \, dx}{3 f}\\ &=-\frac{16 a \sqrt{a+i a \sinh (e+f x)}}{3 f^2}-\frac{8 a \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 \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{8 a x \sqrt{a+i a \sinh (e+f x)} \tanh \left (\frac{e}{2}+\frac{i \pi }{4}+\frac{f x}{2}\right )}{3 f}\\ \end{align*}

Mathematica [A]  time = 0.776093, size = 138, normalized size = 0.75 \[ -\frac{a (\sinh (e+f x)-i) \sqrt{a+i a \sinh (e+f x)} \left (27 (f x+2 i) \cosh \left (\frac{1}{2} (e+f x)\right )+(3 f x-2 i) \cosh \left (\frac{3}{2} (e+f x)\right )+2 i \sinh \left (\frac{1}{2} (e+f x)\right ) ((3 f x+2 i) \cosh (e+f x)-12 f x+28 i)\right )}{9 f^2 \left (\cosh \left (\frac{1}{2} (e+f x)\right )+i \sinh \left (\frac{1}{2} (e+f x)\right )\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*(27*(2*I + f*x)*Cosh[(e + f*x)/2] + (-2*I + 3*f*x)*Cosh[(3*(e + f*x))/2] + (2*I)*(28*I - 12*f*x + (2*I + 3
*f*x)*Cosh[e + f*x])*Sinh[(e + f*x)/2])*(-I + Sinh[e + f*x])*Sqrt[a + I*a*Sinh[e + f*x]])/(9*f^2*(Cosh[(e + f*
x)/2] + I*Sinh[(e + f*x)/2])^3)

________________________________________________________________________________________

Maple [F]  time = 0.043, size = 0, normalized size = 0. \begin{align*} \int x \left ( a+ia\sinh \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (i \, a \sinh \left (f x + e\right ) + a\right )}^{\frac{3}{2}} x\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (i \, a \sinh \left (f x + e\right ) + a\right )}^{\frac{3}{2}} x\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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