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

Optimal. Leaf size=89 \[ \frac{\sqrt{\frac{\pi }{2}} \text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{a^2}+\frac{\sqrt{\frac{\pi }{2}} \text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{a^2}-\frac{2 x \sqrt{a x-1} \sqrt{a x+1}}{a \sqrt{\cosh ^{-1}(a x)}} \]

[Out]

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

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Rubi [A]  time = 0.0658254, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5666, 3307, 2180, 2204, 2205} \[ \frac{\sqrt{\frac{\pi }{2}} \text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{a^2}+\frac{\sqrt{\frac{\pi }{2}} \text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{a^2}-\frac{2 x \sqrt{a x-1} \sqrt{a x+1}}{a \sqrt{\cosh ^{-1}(a x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5666

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[(x^m*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(
a + b*ArcCosh[c*x])^(n + 1))/(b*c*(n + 1)), x] + Dist[1/(b*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[(a +
b*x)^(n + 1)*Cosh[x]^(m - 1)*(m - (m + 1)*Cosh[x]^2), x], x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]

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

Mathematica [A]  time = 0.10018, size = 63, normalized size = 0.71 \[ \frac{\sqrt{\frac{\pi }{2}} \left (\text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )+\text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )\right )-\frac{\sinh \left (2 \cosh ^{-1}(a x)\right )}{\sqrt{\cosh ^{-1}(a x)}}}{a^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]  time = 0.111, size = 83, normalized size = 0.9 \begin{align*}{\frac{\sqrt{2}}{2\,\sqrt{\pi }{a}^{2}{\rm arccosh} \left (ax\right )} \left ( -2\,\sqrt{2}\sqrt{{\rm arccosh} \left (ax\right )}\sqrt{\pi }\sqrt{ax+1}\sqrt{ax-1}xa+{\rm arccosh} \left (ax\right )\pi \,{\it Erf} \left ( \sqrt{2}\sqrt{{\rm arccosh} \left (ax\right )} \right ) +{\rm arccosh} \left (ax\right )\pi \,{\it erfi} \left ( \sqrt{2}\sqrt{{\rm arccosh} \left (ax\right )} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/arccosh(a*x)^(3/2),x)

[Out]

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

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x/acosh(a*x)**(3/2), 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(x/arccosh(a*x)^(3/2),x, algorithm="giac")

[Out]

sage0*x