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

Optimal. Leaf size=166 \[ -\frac{\sqrt{\pi } \text{Erf}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{6 a^3}-\frac{\sqrt{3 \pi } \text{Erf}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{2 a^3}+\frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{6 a^3}+\frac{\sqrt{3 \pi } \text{Erfi}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{2 a^3}+\frac{8 x}{3 a^2 \sqrt{\cosh ^{-1}(a x)}}-\frac{4 x^3}{\sqrt{\cosh ^{-1}(a x)}}-\frac{2 x^2 \sqrt{a x-1} \sqrt{a x+1}}{3 a \cosh ^{-1}(a x)^{3/2}} \]

[Out]

(-2*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(3*a*ArcCosh[a*x]^(3/2)) + (8*x)/(3*a^2*Sqrt[ArcCosh[a*x]]) - (4*x^3)/Sq
rt[ArcCosh[a*x]] - (Sqrt[Pi]*Erf[Sqrt[ArcCosh[a*x]]])/(6*a^3) - (Sqrt[3*Pi]*Erf[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/(
2*a^3) + (Sqrt[Pi]*Erfi[Sqrt[ArcCosh[a*x]]])/(6*a^3) + (Sqrt[3*Pi]*Erfi[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/(2*a^3)

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Rubi [A]  time = 0.626936, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 9, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, Rules used = {5668, 5775, 5670, 5448, 3308, 2180, 2204, 2205, 5658} \[ -\frac{\sqrt{\pi } \text{Erf}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{6 a^3}-\frac{\sqrt{3 \pi } \text{Erf}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{2 a^3}+\frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{6 a^3}+\frac{\sqrt{3 \pi } \text{Erfi}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{2 a^3}+\frac{8 x}{3 a^2 \sqrt{\cosh ^{-1}(a x)}}-\frac{4 x^3}{\sqrt{\cosh ^{-1}(a x)}}-\frac{2 x^2 \sqrt{a x-1} \sqrt{a x+1}}{3 a \cosh ^{-1}(a x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(3*a*ArcCosh[a*x]^(3/2)) + (8*x)/(3*a^2*Sqrt[ArcCosh[a*x]]) - (4*x^3)/Sq
rt[ArcCosh[a*x]] - (Sqrt[Pi]*Erf[Sqrt[ArcCosh[a*x]]])/(6*a^3) - (Sqrt[3*Pi]*Erf[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/(
2*a^3) + (Sqrt[Pi]*Erfi[Sqrt[ArcCosh[a*x]]])/(6*a^3) + (Sqrt[3*Pi]*Erfi[Sqrt[3]*Sqrt[ArcCosh[a*x]]])/(2*a^3)

Rule 5668

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[(c*(m + 1))/(b*(n + 1)), Int[(x^(m + 1)*(a + b*ArcCosh
[c*x])^(n + 1))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x], x] + Dist[m/(b*c*(n + 1)), Int[(x^(m - 1)*(a + b*ArcCosh[c
*x])^(n + 1))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]

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 5670

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

Rule 5448

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int
[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n,
0] && IGtQ[p, 0]

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]

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]

Rubi steps

\begin{align*} \int \frac{x^2}{\cosh ^{-1}(a x)^{5/2}} \, dx &=-\frac{2 x^2 \sqrt{-1+a x} \sqrt{1+a x}}{3 a \cosh ^{-1}(a x)^{3/2}}-\frac{4 \int \frac{x}{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}} \, dx}{3 a}+(2 a) \int \frac{x^3}{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}} \, dx\\ &=-\frac{2 x^2 \sqrt{-1+a x} \sqrt{1+a x}}{3 a \cosh ^{-1}(a x)^{3/2}}+\frac{8 x}{3 a^2 \sqrt{\cosh ^{-1}(a x)}}-\frac{4 x^3}{\sqrt{\cosh ^{-1}(a x)}}+12 \int \frac{x^2}{\sqrt{\cosh ^{-1}(a x)}} \, dx-\frac{8 \int \frac{1}{\sqrt{\cosh ^{-1}(a x)}} \, dx}{3 a^2}\\ &=-\frac{2 x^2 \sqrt{-1+a x} \sqrt{1+a x}}{3 a \cosh ^{-1}(a x)^{3/2}}+\frac{8 x}{3 a^2 \sqrt{\cosh ^{-1}(a x)}}-\frac{4 x^3}{\sqrt{\cosh ^{-1}(a x)}}-\frac{8 \operatorname{Subst}\left (\int \frac{\sinh (x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{3 a^3}+\frac{12 \operatorname{Subst}\left (\int \frac{\cosh ^2(x) \sinh (x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{a^3}\\ &=-\frac{2 x^2 \sqrt{-1+a x} \sqrt{1+a x}}{3 a \cosh ^{-1}(a x)^{3/2}}+\frac{8 x}{3 a^2 \sqrt{\cosh ^{-1}(a x)}}-\frac{4 x^3}{\sqrt{\cosh ^{-1}(a x)}}+\frac{4 \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{3 a^3}-\frac{4 \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{3 a^3}+\frac{12 \operatorname{Subst}\left (\int \left (\frac{\sinh (x)}{4 \sqrt{x}}+\frac{\sinh (3 x)}{4 \sqrt{x}}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a^3}\\ &=-\frac{2 x^2 \sqrt{-1+a x} \sqrt{1+a x}}{3 a \cosh ^{-1}(a x)^{3/2}}+\frac{8 x}{3 a^2 \sqrt{\cosh ^{-1}(a x)}}-\frac{4 x^3}{\sqrt{\cosh ^{-1}(a x)}}+\frac{8 \operatorname{Subst}\left (\int e^{-x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{3 a^3}-\frac{8 \operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{3 a^3}+\frac{3 \operatorname{Subst}\left (\int \frac{\sinh (x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{a^3}+\frac{3 \operatorname{Subst}\left (\int \frac{\sinh (3 x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{a^3}\\ &=-\frac{2 x^2 \sqrt{-1+a x} \sqrt{1+a x}}{3 a \cosh ^{-1}(a x)^{3/2}}+\frac{8 x}{3 a^2 \sqrt{\cosh ^{-1}(a x)}}-\frac{4 x^3}{\sqrt{\cosh ^{-1}(a x)}}+\frac{4 \sqrt{\pi } \text{erf}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{3 a^3}-\frac{4 \sqrt{\pi } \text{erfi}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{3 a^3}-\frac{3 \operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{2 a^3}-\frac{3 \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{2 a^3}+\frac{3 \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{2 a^3}+\frac{3 \operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{2 a^3}\\ &=-\frac{2 x^2 \sqrt{-1+a x} \sqrt{1+a x}}{3 a \cosh ^{-1}(a x)^{3/2}}+\frac{8 x}{3 a^2 \sqrt{\cosh ^{-1}(a x)}}-\frac{4 x^3}{\sqrt{\cosh ^{-1}(a x)}}+\frac{4 \sqrt{\pi } \text{erf}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{3 a^3}-\frac{4 \sqrt{\pi } \text{erfi}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{3 a^3}-\frac{3 \operatorname{Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{a^3}-\frac{3 \operatorname{Subst}\left (\int e^{-x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{a^3}+\frac{3 \operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{a^3}+\frac{3 \operatorname{Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{a^3}\\ &=-\frac{2 x^2 \sqrt{-1+a x} \sqrt{1+a x}}{3 a \cosh ^{-1}(a x)^{3/2}}+\frac{8 x}{3 a^2 \sqrt{\cosh ^{-1}(a x)}}-\frac{4 x^3}{\sqrt{\cosh ^{-1}(a x)}}-\frac{\sqrt{\pi } \text{erf}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{6 a^3}-\frac{\sqrt{3 \pi } \text{erf}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{2 a^3}+\frac{\sqrt{\pi } \text{erfi}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{6 a^3}+\frac{\sqrt{3 \pi } \text{erfi}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{2 a^3}\\ \end{align*}

Mathematica [A]  time = 0.659596, size = 194, normalized size = 1.17 \[ \frac{-3 \sqrt{3} \left (-\cosh ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-3 \cosh ^{-1}(a x)\right )-\left (-\cosh ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-\cosh ^{-1}(a x)\right )+\cosh ^{-1}(a x)^{3/2} \text{Gamma}\left (\frac{1}{2},\cosh ^{-1}(a x)\right )+3 \sqrt{3} \cosh ^{-1}(a x)^{3/2} \text{Gamma}\left (\frac{1}{2},3 \cosh ^{-1}(a x)\right )-\sqrt{\frac{a x-1}{a x+1}} (a x+1)-3 e^{-3 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)-e^{-\cosh ^{-1}(a x)} \cosh ^{-1}(a x)-e^{\cosh ^{-1}(a x)} \cosh ^{-1}(a x)-3 e^{3 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)-\sinh \left (3 \cosh ^{-1}(a x)\right )}{6 a^3 \cosh ^{-1}(a x)^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-(Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)) - (3*ArcCosh[a*x])/E^(3*ArcCosh[a*x]) - ArcCosh[a*x]/E^ArcCosh[a*x] -
E^ArcCosh[a*x]*ArcCosh[a*x] - 3*E^(3*ArcCosh[a*x])*ArcCosh[a*x] - 3*Sqrt[3]*(-ArcCosh[a*x])^(3/2)*Gamma[1/2,
-3*ArcCosh[a*x]] - (-ArcCosh[a*x])^(3/2)*Gamma[1/2, -ArcCosh[a*x]] + ArcCosh[a*x]^(3/2)*Gamma[1/2, ArcCosh[a*x
]] + 3*Sqrt[3]*ArcCosh[a*x]^(3/2)*Gamma[1/2, 3*ArcCosh[a*x]] - Sinh[3*ArcCosh[a*x]])/(6*a^3*ArcCosh[a*x]^(3/2)
)

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Maple [F]  time = 0.086, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2} \left ({\rm arccosh} \left (ax\right ) \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\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(x^2/arccosh(a*x)^(5/2),x, algorithm="maxima")

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

sage0*x