3.96 \(\int \frac{1}{\sqrt{\sinh ^{-1}(a x)}} \, dx\)

Optimal. Leaf size=43 \[ \frac{\sqrt{\pi } \text{Erf}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{2 a}+\frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{2 a} \]

[Out]

(Sqrt[Pi]*Erf[Sqrt[ArcSinh[a*x]]])/(2*a) + (Sqrt[Pi]*Erfi[Sqrt[ArcSinh[a*x]]])/(2*a)

________________________________________________________________________________________

Rubi [A]  time = 0.0471502, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {5657, 3307, 2180, 2204, 2205} \[ \frac{\sqrt{\pi } \text{Erf}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{2 a}+\frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{2 a} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[ArcSinh[a*x]],x]

[Out]

(Sqrt[Pi]*Erf[Sqrt[ArcSinh[a*x]]])/(2*a) + (Sqrt[Pi]*Erfi[Sqrt[ArcSinh[a*x]]])/(2*a)

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

Mathematica [A]  time = 0.0272014, size = 47, normalized size = 1.09 \[ \frac{\frac{\sqrt{-\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-\sinh ^{-1}(a x)\right )}{\sqrt{\sinh ^{-1}(a x)}}-\text{Gamma}\left (\frac{1}{2},\sinh ^{-1}(a x)\right )}{2 a} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/Sqrt[ArcSinh[a*x]],x]

[Out]

((Sqrt[-ArcSinh[a*x]]*Gamma[1/2, -ArcSinh[a*x]])/Sqrt[ArcSinh[a*x]] - Gamma[1/2, ArcSinh[a*x]])/(2*a)

________________________________________________________________________________________

Maple [A]  time = 0.036, size = 24, normalized size = 0.6 \begin{align*}{\frac{\sqrt{\pi }}{2\,a} \left ({\it Erf} \left ( \sqrt{{\it Arcsinh} \left ( ax \right ) } \right ) +{\it erfi} \left ( \sqrt{{\it Arcsinh} \left ( ax \right ) } \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(arcsinh(a*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(1/arcsinh(a*x)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{\operatorname{asinh}{\left (a x \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(asinh(a*x)), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(arcsinh(a*x)), x)