3.109 \(\int \frac{1}{\sinh ^{-1}(a x)^{5/2}} \, dx\)

Optimal. Leaf size=84 \[ -\frac{2 \sqrt{a^2 x^2+1}}{3 a \sinh ^{-1}(a x)^{3/2}}+\frac{2 \sqrt{\pi } \text{Erf}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{3 a}+\frac{2 \sqrt{\pi } \text{Erfi}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{3 a}-\frac{4 x}{3 \sqrt{\sinh ^{-1}(a x)}} \]

[Out]

(-2*Sqrt[1 + a^2*x^2])/(3*a*ArcSinh[a*x]^(3/2)) - (4*x)/(3*Sqrt[ArcSinh[a*x]]) + (2*Sqrt[Pi]*Erf[Sqrt[ArcSinh[
a*x]]])/(3*a) + (2*Sqrt[Pi]*Erfi[Sqrt[ArcSinh[a*x]]])/(3*a)

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Rubi [A]  time = 0.117399, antiderivative size = 84, 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 = {5655, 5774, 5657, 3307, 2180, 2204, 2205} \[ -\frac{2 \sqrt{a^2 x^2+1}}{3 a \sinh ^{-1}(a x)^{3/2}}+\frac{2 \sqrt{\pi } \text{Erf}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{3 a}+\frac{2 \sqrt{\pi } \text{Erfi}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{3 a}-\frac{4 x}{3 \sqrt{\sinh ^{-1}(a x)}} \]

Antiderivative was successfully verified.

[In]

Int[ArcSinh[a*x]^(-5/2),x]

[Out]

(-2*Sqrt[1 + a^2*x^2])/(3*a*ArcSinh[a*x]^(3/2)) - (4*x)/(3*Sqrt[ArcSinh[a*x]]) + (2*Sqrt[Pi]*Erf[Sqrt[ArcSinh[
a*x]]])/(3*a) + (2*Sqrt[Pi]*Erfi[Sqrt[ArcSinh[a*x]]])/(3*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 5774

Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp
[((f*x)^m*(a + b*ArcSinh[c*x])^(n + 1))/(b*c*Sqrt[d]*(n + 1)), x] - Dist[(f*m)/(b*c*Sqrt[d]*(n + 1)), Int[(f*x
)^(m - 1)*(a + b*ArcSinh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && LtQ[n, -
1] && GtQ[d, 0]

Rule 5657

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

Rule 3307

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

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)^{5/2}} \, dx &=-\frac{2 \sqrt{1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^{3/2}}+\frac{1}{3} (2 a) \int \frac{x}{\sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}} \, dx\\ &=-\frac{2 \sqrt{1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac{4 x}{3 \sqrt{\sinh ^{-1}(a x)}}+\frac{4}{3} \int \frac{1}{\sqrt{\sinh ^{-1}(a x)}} \, dx\\ &=-\frac{2 \sqrt{1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac{4 x}{3 \sqrt{\sinh ^{-1}(a x)}}+\frac{4 \operatorname{Subst}\left (\int \frac{\cosh (x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{3 a}\\ &=-\frac{2 \sqrt{1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac{4 x}{3 \sqrt{\sinh ^{-1}(a x)}}+\frac{2 \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{3 a}+\frac{2 \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{3 a}\\ &=-\frac{2 \sqrt{1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac{4 x}{3 \sqrt{\sinh ^{-1}(a x)}}+\frac{4 \operatorname{Subst}\left (\int e^{-x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{3 a}+\frac{4 \operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{3 a}\\ &=-\frac{2 \sqrt{1+a^2 x^2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac{4 x}{3 \sqrt{\sinh ^{-1}(a x)}}+\frac{2 \sqrt{\pi } \text{erf}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{3 a}+\frac{2 \sqrt{\pi } \text{erfi}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{3 a}\\ \end{align*}

Mathematica [A]  time = 0.0983834, size = 105, normalized size = 1.25 \[ -\frac{e^{-\sinh ^{-1}(a x)} \left (2 e^{\sinh ^{-1}(a x)} \left (-\sinh ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-\sinh ^{-1}(a x)\right )+2 e^{\sinh ^{-1}(a x)} \sinh ^{-1}(a x)^{3/2} \text{Gamma}\left (\frac{1}{2},\sinh ^{-1}(a x)\right )+e^{2 \sinh ^{-1}(a x)}+2 e^{2 \sinh ^{-1}(a x)} \sinh ^{-1}(a x)-2 \sinh ^{-1}(a x)+1\right )}{3 a \sinh ^{-1}(a x)^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcSinh[a*x]^(-5/2),x]

[Out]

-(1 + E^(2*ArcSinh[a*x]) - 2*ArcSinh[a*x] + 2*E^(2*ArcSinh[a*x])*ArcSinh[a*x] + 2*E^ArcSinh[a*x]*(-ArcSinh[a*x
])^(3/2)*Gamma[1/2, -ArcSinh[a*x]] + 2*E^ArcSinh[a*x]*ArcSinh[a*x]^(3/2)*Gamma[1/2, ArcSinh[a*x]])/(3*a*E^ArcS
inh[a*x]*ArcSinh[a*x]^(3/2))

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Maple [A]  time = 0.076, size = 81, normalized size = 1. \begin{align*}{\frac{2}{3\,\sqrt{\pi }a \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}} \left ( -2\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3/2}\sqrt{\pi }xa+ \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}\pi \,{\it Erf} \left ( \sqrt{{\it Arcsinh} \left ( ax \right ) } \right ) + \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}\pi \,{\it erfi} \left ( \sqrt{{\it Arcsinh} \left ( ax \right ) } \right ) -\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)^(5/2),x)

[Out]

2/3*(-2*arcsinh(a*x)^(3/2)*Pi^(1/2)*x*a+arcsinh(a*x)^2*Pi*erf(arcsinh(a*x)^(1/2))+arcsinh(a*x)^2*Pi*erfi(arcsi
nh(a*x)^(1/2))-arcsinh(a*x)^(1/2)*Pi^(1/2)*(a^2*x^2+1)^(1/2))/Pi^(1/2)/a/arcsinh(a*x)^2

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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{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arcsinh(a*x)^(5/2),x, algorithm="maxima")

[Out]

integrate(arcsinh(a*x)^(-5/2), 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)^(5/2),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{5}{2}}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/asinh(a*x)**(5/2),x)

[Out]

Integral(asinh(a*x)**(-5/2), 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{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arcsinh(a*x)^(5/2),x, algorithm="giac")

[Out]

integrate(arcsinh(a*x)^(-5/2), x)