3.103 \(\int \frac{1}{\sinh ^{-1}(a x)^{3/2}} \, dx\)

Optimal. Leaf size=64 \[ -\frac{2 \sqrt{a^2 x^2+1}}{a \sqrt{\sinh ^{-1}(a x)}}-\frac{\sqrt{\pi } \text{Erf}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{a}+\frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{a} \]

[Out]

(-2*Sqrt[1 + a^2*x^2])/(a*Sqrt[ArcSinh[a*x]]) - (Sqrt[Pi]*Erf[Sqrt[ArcSinh[a*x]]])/a + (Sqrt[Pi]*Erfi[Sqrt[Arc
Sinh[a*x]]])/a

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Rubi [A]  time = 0.1103, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, Rules used = {5655, 5779, 3308, 2180, 2204, 2205} \[ -\frac{2 \sqrt{a^2 x^2+1}}{a \sqrt{\sinh ^{-1}(a x)}}-\frac{\sqrt{\pi } \text{Erf}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{a}+\frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[1 + a^2*x^2])/(a*Sqrt[ArcSinh[a*x]]) - (Sqrt[Pi]*Erf[Sqrt[ArcSinh[a*x]]])/a + (Sqrt[Pi]*Erfi[Sqrt[Arc
Sinh[a*x]]])/a

Rule 5655

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^(n + 1
))/(b*c*(n + 1)), x] - Dist[c/(b*(n + 1)), Int[(x*(a + b*ArcSinh[c*x])^(n + 1))/Sqrt[1 + c^2*x^2], x], x] /; F
reeQ[{a, b, c}, x] && LtQ[n, -1]

Rule 5779

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[d^p/c^
(m + 1), Subst[Int[(a + b*x)^n*Sinh[x]^m*Cosh[x]^(2*p + 1), x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e,
n}, x] && EqQ[e, c^2*d] && IntegerQ[2*p] && GtQ[p, -1] && IGtQ[m, 0] && (IntegerQ[p] || GtQ[d, 0])

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

Mathematica [A]  time = 0.0680336, size = 69, normalized size = 1.08 \[ \frac{\sqrt{-\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-\sinh ^{-1}(a x)\right )+\sqrt{\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},\sinh ^{-1}(a x)\right )-e^{-\sinh ^{-1}(a x)}-e^{\sinh ^{-1}(a x)}}{a \sqrt{\sinh ^{-1}(a x)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-E^(-ArcSinh[a*x]) - E^ArcSinh[a*x] + Sqrt[-ArcSinh[a*x]]*Gamma[1/2, -ArcSinh[a*x]] + Sqrt[ArcSinh[a*x]]*Gamm
a[1/2, ArcSinh[a*x]])/(a*Sqrt[ArcSinh[a*x]])

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Maple [A]  time = 0.073, size = 65, normalized size = 1. \begin{align*} -{\frac{1}{\sqrt{\pi }a{\it Arcsinh} \left ( ax \right ) } \left ({\it Arcsinh} \left ( ax \right ) \pi \,{\it Erf} \left ( \sqrt{{\it Arcsinh} \left ( ax \right ) } \right ) -{\it Arcsinh} \left ( ax \right ) \pi \,{\it erfi} \left ( \sqrt{{\it Arcsinh} \left ( ax \right ) } \right ) +2\,\sqrt{{\it Arcsinh} \left ( ax \right ) }\sqrt{\pi }\sqrt{{a}^{2}{x}^{2}+1} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/arcsinh(a*x)^(3/2),x)

[Out]

-(arcsinh(a*x)*Pi*erf(arcsinh(a*x)^(1/2))-arcsinh(a*x)*Pi*erfi(arcsinh(a*x)^(1/2))+2*arcsinh(a*x)^(1/2)*Pi^(1/
2)*(a^2*x^2+1)^(1/2))/Pi^(1/2)/a/arcsinh(a*x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\operatorname{arsinh}\left (a x\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arcsinh(a*x)^(3/2),x, algorithm="maxima")

[Out]

integrate(arcsinh(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(1/arcsinh(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 \frac{1}{\operatorname{asinh}^{\frac{3}{2}}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/asinh(a*x)**(3/2),x)

[Out]

Integral(asinh(a*x)**(-3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\operatorname{arsinh}\left (a x\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arcsinh(a*x)^(3/2),x, algorithm="giac")

[Out]

integrate(arcsinh(a*x)^(-3/2), x)