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

Optimal. Leaf size=176 \[ \frac{\sqrt{\frac{\pi }{2}} a \sqrt{a^2+x^2} \text{Erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}\right )}{16 \sqrt{\frac{x^2}{a^2}+1}}-\frac{\sqrt{\frac{\pi }{2}} a \sqrt{a^2+x^2} \text{Erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}\right )}{16 \sqrt{\frac{x^2}{a^2}+1}}+\frac{a \sqrt{a^2+x^2} \sinh ^{-1}\left (\frac{x}{a}\right )^{3/2}}{3 \sqrt{\frac{x^2}{a^2}+1}}+\frac{1}{2} x \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )} \]

[Out]

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

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Rubi [A]  time = 0.148164, antiderivative size = 176, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409, Rules used = {5682, 5675, 5669, 5448, 12, 3308, 2180, 2204, 2205} \[ \frac{\sqrt{\frac{\pi }{2}} a \sqrt{a^2+x^2} \text{Erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}\right )}{16 \sqrt{\frac{x^2}{a^2}+1}}-\frac{\sqrt{\frac{\pi }{2}} a \sqrt{a^2+x^2} \text{Erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}\right )}{16 \sqrt{\frac{x^2}{a^2}+1}}+\frac{a \sqrt{a^2+x^2} \sinh ^{-1}\left (\frac{x}{a}\right )^{3/2}}{3 \sqrt{\frac{x^2}{a^2}+1}}+\frac{1}{2} x \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )} \, dx &=\frac{1}{2} x \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}+\frac{\sqrt{a^2+x^2} \int \frac{\sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}}{\sqrt{1+\frac{x^2}{a^2}}} \, dx}{2 \sqrt{1+\frac{x^2}{a^2}}}-\frac{\sqrt{a^2+x^2} \int \frac{x}{\sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}} \, dx}{4 a \sqrt{1+\frac{x^2}{a^2}}}\\ &=\frac{1}{2} x \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}+\frac{a \sqrt{a^2+x^2} \sinh ^{-1}\left (\frac{x}{a}\right )^{3/2}}{3 \sqrt{1+\frac{x^2}{a^2}}}-\frac{\left (a \sqrt{a^2+x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh (x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}\left (\frac{x}{a}\right )\right )}{4 \sqrt{1+\frac{x^2}{a^2}}}\\ &=\frac{1}{2} x \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}+\frac{a \sqrt{a^2+x^2} \sinh ^{-1}\left (\frac{x}{a}\right )^{3/2}}{3 \sqrt{1+\frac{x^2}{a^2}}}-\frac{\left (a \sqrt{a^2+x^2}\right ) \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{2 \sqrt{x}} \, dx,x,\sinh ^{-1}\left (\frac{x}{a}\right )\right )}{4 \sqrt{1+\frac{x^2}{a^2}}}\\ &=\frac{1}{2} x \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}+\frac{a \sqrt{a^2+x^2} \sinh ^{-1}\left (\frac{x}{a}\right )^{3/2}}{3 \sqrt{1+\frac{x^2}{a^2}}}-\frac{\left (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 )}{8 \sqrt{1+\frac{x^2}{a^2}}}\\ &=\frac{1}{2} x \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}+\frac{a \sqrt{a^2+x^2} \sinh ^{-1}\left (\frac{x}{a}\right )^{3/2}}{3 \sqrt{1+\frac{x^2}{a^2}}}+\frac{\left (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 )}{16 \sqrt{1+\frac{x^2}{a^2}}}-\frac{\left (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 )}{16 \sqrt{1+\frac{x^2}{a^2}}}\\ &=\frac{1}{2} x \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}+\frac{a \sqrt{a^2+x^2} \sinh ^{-1}\left (\frac{x}{a}\right )^{3/2}}{3 \sqrt{1+\frac{x^2}{a^2}}}+\frac{\left (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 )}{8 \sqrt{1+\frac{x^2}{a^2}}}-\frac{\left (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 )}{8 \sqrt{1+\frac{x^2}{a^2}}}\\ &=\frac{1}{2} x \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}+\frac{a \sqrt{a^2+x^2} \sinh ^{-1}\left (\frac{x}{a}\right )^{3/2}}{3 \sqrt{1+\frac{x^2}{a^2}}}+\frac{a \sqrt{\frac{\pi }{2}} \sqrt{a^2+x^2} \text{erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}\right )}{16 \sqrt{1+\frac{x^2}{a^2}}}-\frac{a \sqrt{\frac{\pi }{2}} \sqrt{a^2+x^2} \text{erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}\right )}{16 \sqrt{1+\frac{x^2}{a^2}}}\\ \end{align*}

Mathematica [A]  time = 0.0872034, size = 110, normalized size = 0.62 \[ \frac{a \sqrt{a^2+x^2} \left (-3 \sqrt{2} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )} \text{Gamma}\left (\frac{3}{2},2 \sinh ^{-1}\left (\frac{x}{a}\right )\right )-3 \sqrt{2} \sqrt{-\sinh ^{-1}\left (\frac{x}{a}\right )} \text{Gamma}\left (\frac{3}{2},-2 \sinh ^{-1}\left (\frac{x}{a}\right )\right )+16 \sinh ^{-1}\left (\frac{x}{a}\right )^2\right )}{48 \sqrt{\frac{x^2}{a^2}+1} \sqrt{\sinh ^{-1}\left (\frac{x}{a}\right )}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a*Sqrt[a^2 + x^2]*(16*ArcSinh[x/a]^2 - 3*Sqrt[2]*Sqrt[-ArcSinh[x/a]]*Gamma[3/2, -2*ArcSinh[x/a]] - 3*Sqrt[2]*
Sqrt[ArcSinh[x/a]]*Gamma[3/2, 2*ArcSinh[x/a]]))/(48*Sqrt[1 + x^2/a^2]*Sqrt[ArcSinh[x/a]])

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Maple [F]  time = 0.224, size = 0, normalized size = 0. \begin{align*} \int \sqrt{{a}^{2}+{x}^{2}}\sqrt{{\it Arcsinh} \left ({\frac{x}{a}} \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: UnboundLocalError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a**2 + x**2)*sqrt(asinh(x/a)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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