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

Optimal. Leaf size=127 \[ -\frac{3 \sqrt{\frac{\pi }{2}} \text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{64 a^2}+\frac{3 \sqrt{\frac{\pi }{2}} \text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{64 a^2}-\frac{\cosh ^{-1}(a x)^{3/2}}{4 a^2}+\frac{1}{2} x^2 \cosh ^{-1}(a x)^{3/2}-\frac{3 x \sqrt{a x-1} \sqrt{a x+1} \sqrt{\cosh ^{-1}(a x)}}{8 a} \]

[Out]

(-3*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Sqrt[ArcCosh[a*x]])/(8*a) - ArcCosh[a*x]^(3/2)/(4*a^2) + (x^2*ArcCosh[a*x]^
(3/2))/2 - (3*Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(64*a^2) + (3*Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a
*x]]])/(64*a^2)

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Rubi [A]  time = 0.437723, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {5664, 5759, 5676, 5670, 5448, 12, 3308, 2180, 2204, 2205} \[ -\frac{3 \sqrt{\frac{\pi }{2}} \text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{64 a^2}+\frac{3 \sqrt{\frac{\pi }{2}} \text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{64 a^2}-\frac{\cosh ^{-1}(a x)^{3/2}}{4 a^2}+\frac{1}{2} x^2 \cosh ^{-1}(a x)^{3/2}-\frac{3 x \sqrt{a x-1} \sqrt{a x+1} \sqrt{\cosh ^{-1}(a x)}}{8 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Sqrt[ArcCosh[a*x]])/(8*a) - ArcCosh[a*x]^(3/2)/(4*a^2) + (x^2*ArcCosh[a*x]^
(3/2))/2 - (3*Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(64*a^2) + (3*Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a
*x]]])/(64*a^2)

Rule 5664

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[(x^(m + 1)*(a + b*ArcCosh[c*x])^n)/
(m + 1), x] - Dist[(b*c*n)/(m + 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] && GtQ[n, 0]

Rule 5759

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_
.)*(x_)]), x_Symbol] :> Simp[(f*(f*x)^(m - 1)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n)/(e1*e2*m
), x] + (Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m - 2)*(a + b*ArcCosh[c*x])^n)/(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*
x]), x], x] + Dist[(b*f*n*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(c*d1*d2*m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[(f*x)
^(m - 1)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f}, x] && EqQ[e1 - c*d1, 0]
&& EqQ[e2 + c*d2, 0] && GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]

Rule 5676

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

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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 x \cosh ^{-1}(a x)^{3/2} \, dx &=\frac{1}{2} x^2 \cosh ^{-1}(a x)^{3/2}-\frac{1}{4} (3 a) \int \frac{x^2 \sqrt{\cosh ^{-1}(a x)}}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=-\frac{3 x \sqrt{-1+a x} \sqrt{1+a x} \sqrt{\cosh ^{-1}(a x)}}{8 a}+\frac{1}{2} x^2 \cosh ^{-1}(a x)^{3/2}+\frac{3}{16} \int \frac{x}{\sqrt{\cosh ^{-1}(a x)}} \, dx-\frac{3 \int \frac{\sqrt{\cosh ^{-1}(a x)}}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{8 a}\\ &=-\frac{3 x \sqrt{-1+a x} \sqrt{1+a x} \sqrt{\cosh ^{-1}(a x)}}{8 a}-\frac{\cosh ^{-1}(a x)^{3/2}}{4 a^2}+\frac{1}{2} x^2 \cosh ^{-1}(a x)^{3/2}+\frac{3 \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh (x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{16 a^2}\\ &=-\frac{3 x \sqrt{-1+a x} \sqrt{1+a x} \sqrt{\cosh ^{-1}(a x)}}{8 a}-\frac{\cosh ^{-1}(a x)^{3/2}}{4 a^2}+\frac{1}{2} x^2 \cosh ^{-1}(a x)^{3/2}+\frac{3 \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{2 \sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{16 a^2}\\ &=-\frac{3 x \sqrt{-1+a x} \sqrt{1+a x} \sqrt{\cosh ^{-1}(a x)}}{8 a}-\frac{\cosh ^{-1}(a x)^{3/2}}{4 a^2}+\frac{1}{2} x^2 \cosh ^{-1}(a x)^{3/2}+\frac{3 \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{32 a^2}\\ &=-\frac{3 x \sqrt{-1+a x} \sqrt{1+a x} \sqrt{\cosh ^{-1}(a x)}}{8 a}-\frac{\cosh ^{-1}(a x)^{3/2}}{4 a^2}+\frac{1}{2} x^2 \cosh ^{-1}(a x)^{3/2}-\frac{3 \operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{64 a^2}+\frac{3 \operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{64 a^2}\\ &=-\frac{3 x \sqrt{-1+a x} \sqrt{1+a x} \sqrt{\cosh ^{-1}(a x)}}{8 a}-\frac{\cosh ^{-1}(a x)^{3/2}}{4 a^2}+\frac{1}{2} x^2 \cosh ^{-1}(a x)^{3/2}-\frac{3 \operatorname{Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{32 a^2}+\frac{3 \operatorname{Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{32 a^2}\\ &=-\frac{3 x \sqrt{-1+a x} \sqrt{1+a x} \sqrt{\cosh ^{-1}(a x)}}{8 a}-\frac{\cosh ^{-1}(a x)^{3/2}}{4 a^2}+\frac{1}{2} x^2 \cosh ^{-1}(a x)^{3/2}-\frac{3 \sqrt{\frac{\pi }{2}} \text{erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{64 a^2}+\frac{3 \sqrt{\frac{\pi }{2}} \text{erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{64 a^2}\\ \end{align*}

Mathematica [A]  time = 0.125301, size = 84, normalized size = 0.66 \[ \frac{3 \sqrt{2 \pi } \left (\text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )-\text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )\right )+32 \cosh \left (2 \cosh ^{-1}(a x)\right ) \cosh ^{-1}(a x)^{3/2}-24 \sqrt{\cosh ^{-1}(a x)} \sinh \left (2 \cosh ^{-1}(a x)\right )}{128 a^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(32*ArcCosh[a*x]^(3/2)*Cosh[2*ArcCosh[a*x]] + 3*Sqrt[2*Pi]*(-Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]] + Erfi[Sqrt[2]*Sq
rt[ArcCosh[a*x]]]) - 24*Sqrt[ArcCosh[a*x]]*Sinh[2*ArcCosh[a*x]])/(128*a^2)

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Maple [A]  time = 0.112, size = 105, normalized size = 0.8 \begin{align*} -{\frac{\sqrt{2}}{128\,\sqrt{\pi }{a}^{2}} \left ( -32\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3/2}\sqrt{2}\sqrt{\pi }{x}^{2}{a}^{2}+24\,\sqrt{2}\sqrt{{\rm arccosh} \left (ax\right )}\sqrt{\pi }\sqrt{ax+1}\sqrt{ax-1}xa+16\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3/2}\sqrt{2}\sqrt{\pi }+3\,\pi \,{\it Erf} \left ( \sqrt{2}\sqrt{{\rm arccosh} \left (ax\right )} \right ) -3\,\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]

-1/128*2^(1/2)*(-32*arccosh(a*x)^(3/2)*2^(1/2)*Pi^(1/2)*x^2*a^2+24*2^(1/2)*arccosh(a*x)^(1/2)*Pi^(1/2)*(a*x+1)
^(1/2)*(a*x-1)^(1/2)*x*a+16*arccosh(a*x)^(3/2)*2^(1/2)*Pi^(1/2)+3*Pi*erf(2^(1/2)*arccosh(a*x)^(1/2))-3*Pi*erfi
(2^(1/2)*arccosh(a*x)^(1/2)))/Pi^(1/2)/a^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int 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 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