3.491 \(\int \sqrt{a^2+x^2} \sinh ^{-1}(\frac{x}{a})^{3/2} \, dx\)

Optimal. Leaf size=259 \[ \frac{3 \sqrt{\frac{\pi }{2}} a \sqrt{a^2+x^2} \text{Erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}\right )}{64 \sqrt{\frac{x^2}{a^2}+1}}+\frac{3 \sqrt{\frac{\pi }{2}} a \sqrt{a^2+x^2} \text{Erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}\right )}{64 \sqrt{\frac{x^2}{a^2}+1}}+\frac{a \sqrt{a^2+x^2} \sinh ^{-1}\left (\frac{x}{a}\right )^{5/2}}{5 \sqrt{\frac{x^2}{a^2}+1}}+\frac{1}{2} x \sqrt{a^2+x^2} \sinh ^{-1}\left (\frac{x}{a}\right )^{3/2}-\frac{3 x^2 \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}}{8 a \sqrt{\frac{x^2}{a^2}+1}}-\frac{3 a \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}}{16 \sqrt{\frac{x^2}{a^2}+1}} \]

[Out]

(-3*a*Sqrt[a^2 + x^2]*Sqrt[ArcSinh[x/a]])/(16*Sqrt[1 + x^2/a^2]) - (3*x^2*Sqrt[a^2 + x^2]*Sqrt[ArcSinh[x/a]])/
(8*a*Sqrt[1 + x^2/a^2]) + (x*Sqrt[a^2 + x^2]*ArcSinh[x/a]^(3/2))/2 + (a*Sqrt[a^2 + x^2]*ArcSinh[x/a]^(5/2))/(5
*Sqrt[1 + x^2/a^2]) + (3*a*Sqrt[Pi/2]*Sqrt[a^2 + x^2]*Erf[Sqrt[2]*Sqrt[ArcSinh[x/a]]])/(64*Sqrt[1 + x^2/a^2])
+ (3*a*Sqrt[Pi/2]*Sqrt[a^2 + x^2]*Erfi[Sqrt[2]*Sqrt[ArcSinh[x/a]]])/(64*Sqrt[1 + x^2/a^2])

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Rubi [A]  time = 0.275563, antiderivative size = 259, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409, Rules used = {5682, 5675, 5663, 5779, 3312, 3307, 2180, 2204, 2205} \[ \frac{3 \sqrt{\frac{\pi }{2}} a \sqrt{a^2+x^2} \text{Erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}\right )}{64 \sqrt{\frac{x^2}{a^2}+1}}+\frac{3 \sqrt{\frac{\pi }{2}} a \sqrt{a^2+x^2} \text{Erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}\right )}{64 \sqrt{\frac{x^2}{a^2}+1}}+\frac{a \sqrt{a^2+x^2} \sinh ^{-1}\left (\frac{x}{a}\right )^{5/2}}{5 \sqrt{\frac{x^2}{a^2}+1}}+\frac{1}{2} x \sqrt{a^2+x^2} \sinh ^{-1}\left (\frac{x}{a}\right )^{3/2}-\frac{3 x^2 \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}}{8 a \sqrt{\frac{x^2}{a^2}+1}}-\frac{3 a \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}}{16 \sqrt{\frac{x^2}{a^2}+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*a*Sqrt[a^2 + x^2]*Sqrt[ArcSinh[x/a]])/(16*Sqrt[1 + x^2/a^2]) - (3*x^2*Sqrt[a^2 + x^2]*Sqrt[ArcSinh[x/a]])/
(8*a*Sqrt[1 + x^2/a^2]) + (x*Sqrt[a^2 + x^2]*ArcSinh[x/a]^(3/2))/2 + (a*Sqrt[a^2 + x^2]*ArcSinh[x/a]^(5/2))/(5
*Sqrt[1 + x^2/a^2]) + (3*a*Sqrt[Pi/2]*Sqrt[a^2 + x^2]*Erf[Sqrt[2]*Sqrt[ArcSinh[x/a]]])/(64*Sqrt[1 + x^2/a^2])
+ (3*a*Sqrt[Pi/2]*Sqrt[a^2 + x^2]*Erfi[Sqrt[2]*Sqrt[ArcSinh[x/a]]])/(64*Sqrt[1 + x^2/a^2])

Rule 5682

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

Rule 5675

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

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 5779

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

Rule 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 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 \sqrt{a^2+x^2} \sinh ^{-1}\left (\frac{x}{a}\right )^{3/2} \, dx &=\frac{1}{2} x \sqrt{a^2+x^2} \sinh ^{-1}\left (\frac{x}{a}\right )^{3/2}+\frac{\sqrt{a^2+x^2} \int \frac{\sinh ^{-1}\left (\frac{x}{a}\right )^{3/2}}{\sqrt{1+\frac{x^2}{a^2}}} \, dx}{2 \sqrt{1+\frac{x^2}{a^2}}}-\frac{\left (3 \sqrt{a^2+x^2}\right ) \int x \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )} \, dx}{4 a \sqrt{1+\frac{x^2}{a^2}}}\\ &=-\frac{3 x^2 \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}}{8 a \sqrt{1+\frac{x^2}{a^2}}}+\frac{1}{2} x \sqrt{a^2+x^2} \sinh ^{-1}\left (\frac{x}{a}\right )^{3/2}+\frac{a \sqrt{a^2+x^2} \sinh ^{-1}\left (\frac{x}{a}\right )^{5/2}}{5 \sqrt{1+\frac{x^2}{a^2}}}+\frac{\left (3 \sqrt{a^2+x^2}\right ) \int \frac{x^2}{\sqrt{1+\frac{x^2}{a^2}} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}} \, dx}{16 a^2 \sqrt{1+\frac{x^2}{a^2}}}\\ &=-\frac{3 x^2 \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}}{8 a \sqrt{1+\frac{x^2}{a^2}}}+\frac{1}{2} x \sqrt{a^2+x^2} \sinh ^{-1}\left (\frac{x}{a}\right )^{3/2}+\frac{a \sqrt{a^2+x^2} \sinh ^{-1}\left (\frac{x}{a}\right )^{5/2}}{5 \sqrt{1+\frac{x^2}{a^2}}}+\frac{\left (3 a \sqrt{a^2+x^2}\right ) \operatorname{Subst}\left (\int \frac{\sinh ^2(x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}\left (\frac{x}{a}\right )\right )}{16 \sqrt{1+\frac{x^2}{a^2}}}\\ &=-\frac{3 x^2 \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}}{8 a \sqrt{1+\frac{x^2}{a^2}}}+\frac{1}{2} x \sqrt{a^2+x^2} \sinh ^{-1}\left (\frac{x}{a}\right )^{3/2}+\frac{a \sqrt{a^2+x^2} \sinh ^{-1}\left (\frac{x}{a}\right )^{5/2}}{5 \sqrt{1+\frac{x^2}{a^2}}}-\frac{\left (3 a \sqrt{a^2+x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{2 \sqrt{x}}-\frac{\cosh (2 x)}{2 \sqrt{x}}\right ) \, dx,x,\sinh ^{-1}\left (\frac{x}{a}\right )\right )}{16 \sqrt{1+\frac{x^2}{a^2}}}\\ &=-\frac{3 a \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}}{16 \sqrt{1+\frac{x^2}{a^2}}}-\frac{3 x^2 \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}}{8 a \sqrt{1+\frac{x^2}{a^2}}}+\frac{1}{2} x \sqrt{a^2+x^2} \sinh ^{-1}\left (\frac{x}{a}\right )^{3/2}+\frac{a \sqrt{a^2+x^2} \sinh ^{-1}\left (\frac{x}{a}\right )^{5/2}}{5 \sqrt{1+\frac{x^2}{a^2}}}+\frac{\left (3 a \sqrt{a^2+x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh (2 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}\left (\frac{x}{a}\right )\right )}{32 \sqrt{1+\frac{x^2}{a^2}}}\\ &=-\frac{3 a \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}}{16 \sqrt{1+\frac{x^2}{a^2}}}-\frac{3 x^2 \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}}{8 a \sqrt{1+\frac{x^2}{a^2}}}+\frac{1}{2} x \sqrt{a^2+x^2} \sinh ^{-1}\left (\frac{x}{a}\right )^{3/2}+\frac{a \sqrt{a^2+x^2} \sinh ^{-1}\left (\frac{x}{a}\right )^{5/2}}{5 \sqrt{1+\frac{x^2}{a^2}}}+\frac{\left (3 a \sqrt{a^2+x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}\left (\frac{x}{a}\right )\right )}{64 \sqrt{1+\frac{x^2}{a^2}}}+\frac{\left (3 a \sqrt{a^2+x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}\left (\frac{x}{a}\right )\right )}{64 \sqrt{1+\frac{x^2}{a^2}}}\\ &=-\frac{3 a \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}}{16 \sqrt{1+\frac{x^2}{a^2}}}-\frac{3 x^2 \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}}{8 a \sqrt{1+\frac{x^2}{a^2}}}+\frac{1}{2} x \sqrt{a^2+x^2} \sinh ^{-1}\left (\frac{x}{a}\right )^{3/2}+\frac{a \sqrt{a^2+x^2} \sinh ^{-1}\left (\frac{x}{a}\right )^{5/2}}{5 \sqrt{1+\frac{x^2}{a^2}}}+\frac{\left (3 a \sqrt{a^2+x^2}\right ) \operatorname{Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}\right )}{32 \sqrt{1+\frac{x^2}{a^2}}}+\frac{\left (3 a \sqrt{a^2+x^2}\right ) \operatorname{Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}\right )}{32 \sqrt{1+\frac{x^2}{a^2}}}\\ &=-\frac{3 a \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}}{16 \sqrt{1+\frac{x^2}{a^2}}}-\frac{3 x^2 \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}}{8 a \sqrt{1+\frac{x^2}{a^2}}}+\frac{1}{2} x \sqrt{a^2+x^2} \sinh ^{-1}\left (\frac{x}{a}\right )^{3/2}+\frac{a \sqrt{a^2+x^2} \sinh ^{-1}\left (\frac{x}{a}\right )^{5/2}}{5 \sqrt{1+\frac{x^2}{a^2}}}+\frac{3 a \sqrt{\frac{\pi }{2}} \sqrt{a^2+x^2} \text{erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}\right )}{64 \sqrt{1+\frac{x^2}{a^2}}}+\frac{3 a \sqrt{\frac{\pi }{2}} \sqrt{a^2+x^2} \text{erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}\right )}{64 \sqrt{1+\frac{x^2}{a^2}}}\\ \end{align*}

Mathematica [A]  time = 0.24483, size = 133, normalized size = 0.51 \[ \frac{a \sqrt{a^2+x^2} \left (15 \sqrt{2 \pi } \text{Erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}\right )+15 \sqrt{2 \pi } \text{Erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}\right )+8 \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )} \left (16 \sinh ^{-1}\left (\frac{x}{a}\right )^2+20 \sinh \left (2 \sinh ^{-1}\left (\frac{x}{a}\right )\right ) \sinh ^{-1}\left (\frac{x}{a}\right )-15 \cosh \left (2 \sinh ^{-1}\left (\frac{x}{a}\right )\right )\right )\right )}{640 \sqrt{\frac{x^2}{a^2}+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*Sqrt[a^2 + x^2]*(15*Sqrt[2*Pi]*Erf[Sqrt[2]*Sqrt[ArcSinh[x/a]]] + 15*Sqrt[2*Pi]*Erfi[Sqrt[2]*Sqrt[ArcSinh[x/
a]]] + 8*Sqrt[ArcSinh[x/a]]*(16*ArcSinh[x/a]^2 - 15*Cosh[2*ArcSinh[x/a]] + 20*ArcSinh[x/a]*Sinh[2*ArcSinh[x/a]
])))/(640*Sqrt[1 + x^2/a^2])

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Maple [F]  time = 0.214, size = 0, normalized size = 0. \begin{align*} \int \left ({\it Arcsinh} \left ({\frac{x}{a}} \right ) \right ) ^{{\frac{3}{2}}}\sqrt{{a}^{2}+{x}^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsinh(x/a)^(3/2)*(a^2+x^2)^(1/2),x)

[Out]

int(arcsinh(x/a)^(3/2)*(a^2+x^2)^(1/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(x/a)^(3/2)*(a^2+x^2)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(a^2 + x^2)*arcsinh(x/a)^(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(arcsinh(x/a)^(3/2)*(a^2+x^2)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asinh(x/a)**(3/2)*(a**2+x**2)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(x/a)^(3/2)*(a^2+x^2)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(a^2 + x^2)*arcsinh(x/a)^(3/2), x)