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

Optimal. Leaf size=179 \[ -\frac{3 \sqrt{\pi } \text{Erf}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{32 a^3}+\frac{\sqrt{\frac{\pi }{3}} \text{Erf}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{96 a^3}-\frac{3 \sqrt{\pi } \text{Erfi}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{32 a^3}+\frac{\sqrt{\frac{\pi }{3}} \text{Erfi}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{96 a^3}-\frac{x^2 \sqrt{a^2 x^2+1} \sqrt{\sinh ^{-1}(a x)}}{6 a}+\frac{\sqrt{a^2 x^2+1} \sqrt{\sinh ^{-1}(a x)}}{3 a^3}+\frac{1}{3} x^3 \sinh ^{-1}(a x)^{3/2} \]

[Out]

(Sqrt[1 + a^2*x^2]*Sqrt[ArcSinh[a*x]])/(3*a^3) - (x^2*Sqrt[1 + a^2*x^2]*Sqrt[ArcSinh[a*x]])/(6*a) + (x^3*ArcSi
nh[a*x]^(3/2))/3 - (3*Sqrt[Pi]*Erf[Sqrt[ArcSinh[a*x]]])/(32*a^3) + (Sqrt[Pi/3]*Erf[Sqrt[3]*Sqrt[ArcSinh[a*x]]]
)/(96*a^3) - (3*Sqrt[Pi]*Erfi[Sqrt[ArcSinh[a*x]]])/(32*a^3) + (Sqrt[Pi/3]*Erfi[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/(9
6*a^3)

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Rubi [A]  time = 0.371351, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 10, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.833, Rules used = {5663, 5758, 5717, 5657, 3307, 2180, 2204, 2205, 5669, 5448} \[ -\frac{3 \sqrt{\pi } \text{Erf}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{32 a^3}+\frac{\sqrt{\frac{\pi }{3}} \text{Erf}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{96 a^3}-\frac{3 \sqrt{\pi } \text{Erfi}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{32 a^3}+\frac{\sqrt{\frac{\pi }{3}} \text{Erfi}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{96 a^3}-\frac{x^2 \sqrt{a^2 x^2+1} \sqrt{\sinh ^{-1}(a x)}}{6 a}+\frac{\sqrt{a^2 x^2+1} \sqrt{\sinh ^{-1}(a x)}}{3 a^3}+\frac{1}{3} x^3 \sinh ^{-1}(a x)^{3/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 + a^2*x^2]*Sqrt[ArcSinh[a*x]])/(3*a^3) - (x^2*Sqrt[1 + a^2*x^2]*Sqrt[ArcSinh[a*x]])/(6*a) + (x^3*ArcSi
nh[a*x]^(3/2))/3 - (3*Sqrt[Pi]*Erf[Sqrt[ArcSinh[a*x]]])/(32*a^3) + (Sqrt[Pi/3]*Erf[Sqrt[3]*Sqrt[ArcSinh[a*x]]]
)/(96*a^3) - (3*Sqrt[Pi]*Erfi[Sqrt[ArcSinh[a*x]]])/(32*a^3) + (Sqrt[Pi/3]*Erfi[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/(9
6*a^3)

Rule 5663

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

Rule 5758

Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp
[(f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x])^n)/(e*m), x] + (-Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)
^(m - 2)*(a + b*ArcSinh[c*x])^n)/Sqrt[d + e*x^2], x], x] - Dist[(b*f*n*Sqrt[1 + c^2*x^2])/(c*m*Sqrt[d + e*x^2]
), Int[(f*x)^(m - 1)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] &&
GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]

Rule 5717

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x^2)
^(p + 1)*(a + b*ArcSinh[c*x])^n)/(2*e*(p + 1)), x] - Dist[(b*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p +
1)*(1 + c^2*x^2)^FracPart[p]), Int[(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] /; FreeQ[{a,
b, c, d, e, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && NeQ[p, -1]

Rule 5657

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[1/(b*c), Subst[Int[x^n*Cosh[a/b - x/b], x], x,
a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, n}, x]

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]

Rule 5669

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[(a + b*x)^n*
Sinh[x]^m*Cosh[x], x], x, ArcSinh[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]

Rubi steps

\begin{align*} \int x^2 \sinh ^{-1}(a x)^{3/2} \, dx &=\frac{1}{3} x^3 \sinh ^{-1}(a x)^{3/2}-\frac{1}{2} a \int \frac{x^3 \sqrt{\sinh ^{-1}(a x)}}{\sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{x^2 \sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}}{6 a}+\frac{1}{3} x^3 \sinh ^{-1}(a x)^{3/2}+\frac{1}{12} \int \frac{x^2}{\sqrt{\sinh ^{-1}(a x)}} \, dx+\frac{\int \frac{x \sqrt{\sinh ^{-1}(a x)}}{\sqrt{1+a^2 x^2}} \, dx}{3 a}\\ &=\frac{\sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}}{3 a^3}-\frac{x^2 \sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}}{6 a}+\frac{1}{3} x^3 \sinh ^{-1}(a x)^{3/2}+\frac{\operatorname{Subst}\left (\int \frac{\cosh (x) \sinh ^2(x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{12 a^3}-\frac{\int \frac{1}{\sqrt{\sinh ^{-1}(a x)}} \, dx}{6 a^2}\\ &=\frac{\sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}}{3 a^3}-\frac{x^2 \sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}}{6 a}+\frac{1}{3} x^3 \sinh ^{-1}(a x)^{3/2}+\frac{\operatorname{Subst}\left (\int \left (-\frac{\cosh (x)}{4 \sqrt{x}}+\frac{\cosh (3 x)}{4 \sqrt{x}}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{12 a^3}-\frac{\operatorname{Subst}\left (\int \frac{\cosh (x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{6 a^3}\\ &=\frac{\sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}}{3 a^3}-\frac{x^2 \sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}}{6 a}+\frac{1}{3} x^3 \sinh ^{-1}(a x)^{3/2}-\frac{\operatorname{Subst}\left (\int \frac{\cosh (x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{48 a^3}+\frac{\operatorname{Subst}\left (\int \frac{\cosh (3 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{48 a^3}-\frac{\operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{12 a^3}-\frac{\operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{12 a^3}\\ &=\frac{\sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}}{3 a^3}-\frac{x^2 \sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}}{6 a}+\frac{1}{3} x^3 \sinh ^{-1}(a x)^{3/2}+\frac{\operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{96 a^3}-\frac{\operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{96 a^3}-\frac{\operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{96 a^3}+\frac{\operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{96 a^3}-\frac{\operatorname{Subst}\left (\int e^{-x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{6 a^3}-\frac{\operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{6 a^3}\\ &=\frac{\sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}}{3 a^3}-\frac{x^2 \sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}}{6 a}+\frac{1}{3} x^3 \sinh ^{-1}(a x)^{3/2}-\frac{\sqrt{\pi } \text{erf}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{12 a^3}-\frac{\sqrt{\pi } \text{erfi}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{12 a^3}+\frac{\operatorname{Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{48 a^3}-\frac{\operatorname{Subst}\left (\int e^{-x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{48 a^3}-\frac{\operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{48 a^3}+\frac{\operatorname{Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{48 a^3}\\ &=\frac{\sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}}{3 a^3}-\frac{x^2 \sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)}}{6 a}+\frac{1}{3} x^3 \sinh ^{-1}(a x)^{3/2}-\frac{3 \sqrt{\pi } \text{erf}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{32 a^3}+\frac{\sqrt{\frac{\pi }{3}} \text{erf}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{96 a^3}-\frac{3 \sqrt{\pi } \text{erfi}\left (\sqrt{\sinh ^{-1}(a x)}\right )}{32 a^3}+\frac{\sqrt{\frac{\pi }{3}} \text{erfi}\left (\sqrt{3} \sqrt{\sinh ^{-1}(a x)}\right )}{96 a^3}\\ \end{align*}

Mathematica [A]  time = 0.0339384, size = 102, normalized size = 0.57 \[ \frac{-\sqrt{3} \sqrt{\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{5}{2},-3 \sinh ^{-1}(a x)\right )+27 \sqrt{\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{5}{2},-\sinh ^{-1}(a x)\right )+\sqrt{-\sinh ^{-1}(a x)} \left (27 \text{Gamma}\left (\frac{5}{2},\sinh ^{-1}(a x)\right )-\sqrt{3} \text{Gamma}\left (\frac{5}{2},3 \sinh ^{-1}(a x)\right )\right )}{216 a^3 \sqrt{-\sinh ^{-1}(a x)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-(Sqrt[3]*Sqrt[ArcSinh[a*x]]*Gamma[5/2, -3*ArcSinh[a*x]]) + 27*Sqrt[ArcSinh[a*x]]*Gamma[5/2, -ArcSinh[a*x]] +
Sqrt[-ArcSinh[a*x]]*(27*Gamma[5/2, ArcSinh[a*x]] - Sqrt[3]*Gamma[5/2, 3*ArcSinh[a*x]]))/(216*a^3*Sqrt[-ArcSin
h[a*x]])

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Maple [F]  time = 0.099, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arcsinh(a*x)^(3/2),x, algorithm="maxima")

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2*asinh(a*x)**(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arcsinh(a*x)^(3/2),x, algorithm="giac")

[Out]

integrate(x^2*arcsinh(a*x)^(3/2), x)