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

Optimal. Leaf size=89 \[ -\frac{2 \sqrt{\pi } \text{Erf}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{3 a}+\frac{2 \sqrt{\pi } \text{Erfi}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{3 a}-\frac{4 x}{3 \sqrt{\cosh ^{-1}(a x)}}-\frac{2 \sqrt{a x-1} \sqrt{a x+1}}{3 a \cosh ^{-1}(a x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.234149, antiderivative size = 89, 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 = {5656, 5775, 5658, 3308, 2180, 2204, 2205} \[ -\frac{2 \sqrt{\pi } \text{Erf}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{3 a}+\frac{2 \sqrt{\pi } \text{Erfi}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{3 a}-\frac{4 x}{3 \sqrt{\cosh ^{-1}(a x)}}-\frac{2 \sqrt{a x-1} \sqrt{a x+1}}{3 a \cosh ^{-1}(a x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(3*a*ArcCosh[a*x]^(3/2)) - (4*x)/(3*Sqrt[ArcCosh[a*x]]) - (2*Sqrt[Pi]*Erf[Sq
rt[ArcCosh[a*x]]])/(3*a) + (2*Sqrt[Pi]*Erfi[Sqrt[ArcCosh[a*x]]])/(3*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 5775

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_
.)*(x_)]), x_Symbol] :> Simp[((f*x)^m*(a + b*ArcCosh[c*x])^(n + 1))/(b*c*Sqrt[-(d1*d2)]*(n + 1)), x] - Dist[(f
*m)/(b*c*Sqrt[-(d1*d2)]*(n + 1)), Int[(f*x)^(m - 1)*(a + b*ArcCosh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d1
, e1, d2, e2, f, m}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && LtQ[n, -1] && GtQ[d1, 0] && LtQ[d2, 0]

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

Mathematica [A]  time = 0.150791, size = 121, normalized size = 1.36 \[ -\frac{2 e^{-\cosh ^{-1}(a x)} \left (e^{\cosh ^{-1}(a x)} \left (-\cosh ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-\cosh ^{-1}(a x)\right )-e^{\cosh ^{-1}(a x)} \cosh ^{-1}(a x)^{3/2} \text{Gamma}\left (\frac{1}{2},\cosh ^{-1}(a x)\right )+\sqrt{\frac{a x-1}{a x+1}} (a x+1) e^{\cosh ^{-1}(a x)}+e^{2 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)+\cosh ^{-1}(a x)\right )}{3 a \cosh ^{-1}(a x)^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-2*(E^ArcCosh[a*x]*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x) + ArcCosh[a*x] + E^(2*ArcCosh[a*x])*ArcCosh[a*x] + E^
ArcCosh[a*x]*(-ArcCosh[a*x])^(3/2)*Gamma[1/2, -ArcCosh[a*x]] - E^ArcCosh[a*x]*ArcCosh[a*x]^(3/2)*Gamma[1/2, Ar
cCosh[a*x]]))/(3*a*E^ArcCosh[a*x]*ArcCosh[a*x]^(3/2))

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Maple [A]  time = 0.106, size = 84, normalized size = 0.9 \begin{align*} -{\frac{2}{3\,\sqrt{\pi }a \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}} \left ( 2\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3/2}\sqrt{\pi }xa+ \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}\pi \,{\it Erf} \left ( \sqrt{{\rm arccosh} \left (ax\right )} \right ) - \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}\pi \,{\it erfi} \left ( \sqrt{{\rm arccosh} \left (ax\right )} \right ) +\sqrt{{\rm arccosh} \left (ax\right )}\sqrt{\pi }\sqrt{ax+1}\sqrt{ax-1} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/arccosh(a*x)^(5/2),x)

[Out]

-2/3*(2*arccosh(a*x)^(3/2)*Pi^(1/2)*x*a+arccosh(a*x)^2*Pi*erf(arccosh(a*x)^(1/2))-arccosh(a*x)^2*Pi*erfi(arcco
sh(a*x)^(1/2))+arccosh(a*x)^(1/2)*Pi^(1/2)*(a*x+1)^(1/2)*(a*x-1)^(1/2))/Pi^(1/2)/a/arccosh(a*x)^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\operatorname{arcosh}\left (a x\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arccosh(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/arccosh(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{acosh}^{\frac{5}{2}}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acosh(a*x)**(-5/2), 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(1/arccosh(a*x)^(5/2),x, algorithm="giac")

[Out]

sage0*x