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

Optimal. Leaf size=68 \[ \frac{\sqrt{\pi } \text{Erf}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{a}+\frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{a}-\frac{2 \sqrt{a x-1} \sqrt{a x+1}}{a \sqrt{\cosh ^{-1}(a x)}} \]

[Out]

(-2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*Sqrt[ArcCosh[a*x]]) + (Sqrt[Pi]*Erf[Sqrt[ArcCosh[a*x]]])/a + (Sqrt[Pi]*Er
fi[Sqrt[ArcCosh[a*x]]])/a

________________________________________________________________________________________

Rubi [A]  time = 0.217745, antiderivative size = 68, 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 = {5656, 5781, 3307, 2180, 2204, 2205} \[ \frac{\sqrt{\pi } \text{Erf}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{a}+\frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{a}-\frac{2 \sqrt{a x-1} \sqrt{a x+1}}{a \sqrt{\cosh ^{-1}(a x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*Sqrt[ArcCosh[a*x]]) + (Sqrt[Pi]*Erf[Sqrt[ArcCosh[a*x]]])/a + (Sqrt[Pi]*Er
fi[Sqrt[ArcCosh[a*x]]])/a

Rule 5656

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

Rule 5781

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_
.), x_Symbol] :> Dist[(-(d1*d2))^p/c^(m + 1), Subst[Int[(a + b*x)^n*Cosh[x]^m*Sinh[x]^(2*p + 1), x], x, ArcCos
h[c*x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[p
+ 1/2] && GtQ[p, -1] && IGtQ[m, 0] && (GtQ[d1, 0] && LtQ[d2, 0])

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

Mathematica [A]  time = 0.0604272, size = 76, normalized size = 1.12 \[ \frac{\sqrt{-\cosh ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-\cosh ^{-1}(a x)\right )-\sqrt{\cosh ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},\cosh ^{-1}(a x)\right )-2 \sqrt{\frac{a x-1}{a x+1}} (a x+1)}{a \sqrt{\cosh ^{-1}(a x)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.095, size = 66, normalized size = 1. \begin{align*}{\frac{1}{\sqrt{\pi }a{\rm arccosh} \left (ax\right )} \left ( -2\,\sqrt{{\rm arccosh} \left (ax\right )}\sqrt{\pi }\sqrt{ax+1}\sqrt{ax-1}+{\rm arccosh} \left (ax\right )\pi \,{\it Erf} \left ( \sqrt{{\rm arccosh} \left (ax\right )} \right ) +{\rm arccosh} \left (ax\right )\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(1/arccosh(a*x)^(3/2),x)

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\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(1/arccosh(a*x)^(3/2),x, algorithm="maxima")

[Out]

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

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acosh(a*x)**(-3/2), x)

________________________________________________________________________________________

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)^(3/2),x, algorithm="giac")

[Out]

sage0*x