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

Optimal. Leaf size=33 \[ \frac{1}{2} \sqrt{\pi } \text{Erfi}\left (\sqrt{\sinh ^{-1}(x)}\right )-\frac{1}{2} \sqrt{\pi } \text{Erf}\left (\sqrt{\sinh ^{-1}(x)}\right ) \]

[Out]

-(Sqrt[Pi]*Erf[Sqrt[ArcSinh[x]]])/2 + (Sqrt[Pi]*Erfi[Sqrt[ArcSinh[x]]])/2

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Rubi [A]  time = 0.0799689, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {5779, 3308, 2180, 2204, 2205} \[ \frac{1}{2} \sqrt{\pi } \text{Erfi}\left (\sqrt{\sinh ^{-1}(x)}\right )-\frac{1}{2} \sqrt{\pi } \text{Erf}\left (\sqrt{\sinh ^{-1}(x)}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x/(Sqrt[1 + x^2]*Sqrt[ArcSinh[x]]),x]

[Out]

-(Sqrt[Pi]*Erf[Sqrt[ArcSinh[x]]])/2 + (Sqrt[Pi]*Erfi[Sqrt[ArcSinh[x]]])/2

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 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 \frac{x}{\sqrt{1+x^2} \sqrt{\sinh ^{-1}(x)}} \, dx &=\operatorname{Subst}\left (\int \frac{\sinh (x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(x)\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(x)\right )\right )+\frac{1}{2} \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,\sinh ^{-1}(x)\right )\\ &=-\operatorname{Subst}\left (\int e^{-x^2} \, dx,x,\sqrt{\sinh ^{-1}(x)}\right )+\operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{\sinh ^{-1}(x)}\right )\\ &=-\frac{1}{2} \sqrt{\pi } \text{erf}\left (\sqrt{\sinh ^{-1}(x)}\right )+\frac{1}{2} \sqrt{\pi } \text{erfi}\left (\sqrt{\sinh ^{-1}(x)}\right )\\ \end{align*}

Mathematica [A]  time = 0.068194, size = 34, normalized size = 1.03 \[ \frac{1}{2} \left (\frac{\sqrt{-\sinh ^{-1}(x)} \text{Gamma}\left (\frac{1}{2},-\sinh ^{-1}(x)\right )}{\sqrt{\sinh ^{-1}(x)}}+\text{Gamma}\left (\frac{1}{2},\sinh ^{-1}(x)\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x/(Sqrt[1 + x^2]*Sqrt[ArcSinh[x]]),x]

[Out]

((Sqrt[-ArcSinh[x]]*Gamma[1/2, -ArcSinh[x]])/Sqrt[ArcSinh[x]] + Gamma[1/2, ArcSinh[x]])/2

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Maple [F]  time = 0.105, size = 0, normalized size = 0. \begin{align*} \int{x{\frac{1}{\sqrt{{x}^{2}+1}}}{\frac{1}{\sqrt{{\it Arcsinh} \left ( x \right ) }}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{x^{2} + 1} \sqrt{\operatorname{arsinh}\left (x\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x/(sqrt(x^2 + 1)*sqrt(arcsinh(x))), 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/(x^2+1)^(1/2)/arcsinh(x)^(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 \frac{x}{\sqrt{x^{2} + 1} \sqrt{\operatorname{asinh}{\left (x \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x/(sqrt(x**2 + 1)*sqrt(asinh(x))), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{x^{2} + 1} \sqrt{\operatorname{arsinh}\left (x\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x/(sqrt(x^2 + 1)*sqrt(arcsinh(x))), x)