3.82 \(\int \cosh ^{-1}(a x)^{3/2} \, dx\)

Optimal. Leaf size=86 \[ -\frac{3 \sqrt{\pi } \text{Erf}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{8 a}+\frac{3 \sqrt{\pi } \text{Erfi}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{8 a}+x \cosh ^{-1}(a x)^{3/2}-\frac{3 \sqrt{a x-1} \sqrt{a x+1} \sqrt{\cosh ^{-1}(a x)}}{2 a} \]

[Out]

(-3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Sqrt[ArcCosh[a*x]])/(2*a) + x*ArcCosh[a*x]^(3/2) - (3*Sqrt[Pi]*Erf[Sqrt[ArcCo
sh[a*x]]])/(8*a) + (3*Sqrt[Pi]*Erfi[Sqrt[ArcCosh[a*x]]])/(8*a)

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Rubi [A]  time = 0.220478, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.875, Rules used = {5654, 5718, 5658, 3308, 2180, 2204, 2205} \[ -\frac{3 \sqrt{\pi } \text{Erf}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{8 a}+\frac{3 \sqrt{\pi } \text{Erfi}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{8 a}+x \cosh ^{-1}(a x)^{3/2}-\frac{3 \sqrt{a x-1} \sqrt{a x+1} \sqrt{\cosh ^{-1}(a x)}}{2 a} \]

Antiderivative was successfully verified.

[In]

Int[ArcCosh[a*x]^(3/2),x]

[Out]

(-3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Sqrt[ArcCosh[a*x]])/(2*a) + x*ArcCosh[a*x]^(3/2) - (3*Sqrt[Pi]*Erf[Sqrt[ArcCo
sh[a*x]]])/(8*a) + (3*Sqrt[Pi]*Erfi[Sqrt[ArcCosh[a*x]]])/(8*a)

Rule 5654

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcCosh[c*x])^n, x] - Dist[b*c*n, In
t[(x*(a + b*ArcCosh[c*x])^(n - 1))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 5718

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_
Symbol] :> Simp[((d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(2*e1*e2*(p + 1)), x] - Dist[
(b*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/(2*c*(p + 1)*(1 + c*x)^FracPart[p]
*(-1 + c*x)^FracPart[p]), Int[(-1 + c^2*x^2)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c,
d1, e1, d2, e2, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && NeQ[p, -1] && IntegerQ[p + 1
/2]

Rule 5658

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Dist[(b*c)^(-1), Subst[Int[x^n*Sinh[a/b - x/b], x]
, x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, n}, x]

Rule 3308

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Dist[I/2, Int[(c + d*x)^m/E^(I*(e + f*x))
, x], x] - Dist[I/2, Int[(c + d*x)^m*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]

Rule 2180

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[F^(g*(e - (c*
f)/d) + (f*g*x^2)/d), x], x, Sqrt[c + d*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rule 2204

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2
]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2205

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erf[(c + d*x)*Rt[-(b*Log[F]),
2]])/(2*d*Rt[-(b*Log[F]), 2]), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rubi steps

\begin{align*} \int \cosh ^{-1}(a x)^{3/2} \, dx &=x \cosh ^{-1}(a x)^{3/2}-\frac{1}{2} (3 a) \int \frac{x \sqrt{\cosh ^{-1}(a x)}}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=-\frac{3 \sqrt{-1+a x} \sqrt{1+a x} \sqrt{\cosh ^{-1}(a x)}}{2 a}+x \cosh ^{-1}(a x)^{3/2}+\frac{3}{4} \int \frac{1}{\sqrt{\cosh ^{-1}(a x)}} \, dx\\ &=-\frac{3 \sqrt{-1+a x} \sqrt{1+a x} \sqrt{\cosh ^{-1}(a x)}}{2 a}+x \cosh ^{-1}(a x)^{3/2}+\frac{3 \operatorname{Subst}\left (\int \frac{\sinh (x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{4 a}\\ &=-\frac{3 \sqrt{-1+a x} \sqrt{1+a x} \sqrt{\cosh ^{-1}(a x)}}{2 a}+x \cosh ^{-1}(a x)^{3/2}-\frac{3 \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{8 a}+\frac{3 \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{8 a}\\ &=-\frac{3 \sqrt{-1+a x} \sqrt{1+a x} \sqrt{\cosh ^{-1}(a x)}}{2 a}+x \cosh ^{-1}(a x)^{3/2}-\frac{3 \operatorname{Subst}\left (\int e^{-x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{4 a}+\frac{3 \operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{4 a}\\ &=-\frac{3 \sqrt{-1+a x} \sqrt{1+a x} \sqrt{\cosh ^{-1}(a x)}}{2 a}+x \cosh ^{-1}(a x)^{3/2}-\frac{3 \sqrt{\pi } \text{erf}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{8 a}+\frac{3 \sqrt{\pi } \text{erfi}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{8 a}\\ \end{align*}

Mathematica [A]  time = 0.0287667, size = 45, normalized size = 0.52 \[ \frac{\frac{\sqrt{-\cosh ^{-1}(a x)} \text{Gamma}\left (\frac{5}{2},-\cosh ^{-1}(a x)\right )}{\sqrt{\cosh ^{-1}(a x)}}+\text{Gamma}\left (\frac{5}{2},\cosh ^{-1}(a x)\right )}{2 a} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcCosh[a*x]^(3/2),x]

[Out]

((Sqrt[-ArcCosh[a*x]]*Gamma[5/2, -ArcCosh[a*x]])/Sqrt[ArcCosh[a*x]] + Gamma[5/2, ArcCosh[a*x]])/(2*a)

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Maple [A]  time = 0.095, size = 68, normalized size = 0.8 \begin{align*} -{\frac{1}{8\,\sqrt{\pi }a} \left ( -8\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3/2}\sqrt{\pi }xa+12\,\sqrt{{\rm arccosh} \left (ax\right )}\sqrt{\pi }\sqrt{ax+1}\sqrt{ax-1}+3\,\pi \,{\it Erf} \left ( \sqrt{{\rm arccosh} \left (ax\right )} \right ) -3\,\pi \,{\it erfi} \left ( \sqrt{{\rm arccosh} \left (ax\right )} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccosh(a*x)^(3/2),x)

[Out]

-1/8*(-8*arccosh(a*x)^(3/2)*Pi^(1/2)*x*a+12*arccosh(a*x)^(1/2)*Pi^(1/2)*(a*x+1)^(1/2)*(a*x-1)^(1/2)+3*Pi*erf(a
rccosh(a*x)^(1/2))-3*Pi*erfi(arccosh(a*x)^(1/2)))/Pi^(1/2)/a

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(a*x)^(3/2),x, algorithm="maxima")

[Out]

integrate(arccosh(a*x)^(3/2), x)

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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(arccosh(a*x)^(3/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{acosh}^{\frac{3}{2}}{\left (a x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acosh(a*x)**(3/2),x)

[Out]

Integral(acosh(a*x)**(3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(a*x)^(3/2),x, algorithm="giac")

[Out]

sage0*x