3.94 \(\int \frac{1}{\sqrt{\cosh ^{-1}(a x)}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0466218, 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 = {5658, 3308, 2180, 2204, 2205} \[ \frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{2 a}-\frac{\sqrt{\pi } \text{Erf}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{2 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0282835, size = 45, normalized size = 1.05 \[ \frac{\frac{\sqrt{-\cosh ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-\cosh ^{-1}(a x)\right )}{\sqrt{\cosh ^{-1}(a x)}}+\text{Gamma}\left (\frac{1}{2},\cosh ^{-1}(a x)\right )}{2 a} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]  time = 0.043, size = 26, normalized size = 0.6 \begin{align*} -{\frac{\sqrt{\pi }}{2\,a} \left ({\it Erf} \left ( \sqrt{{\rm arccosh} \left (ax\right )} \right ) -{\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(1/arccosh(a*x)^(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(arccosh(a*x)), 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/arccosh(a*x)^(1/2),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}{\sqrt{\operatorname{acosh}{\left (a x \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(acosh(a*x)), 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(1/arccosh(a*x)^(1/2),x, algorithm="giac")

[Out]

sage0*x