3.401 \(\int \frac{x}{\sqrt{1-x^2} \sqrt{\cosh ^{-1}(x)}} \, dx\)

Optimal. Leaf size=65 \[ \frac{\sqrt{\pi } \sqrt{x-1} \text{Erf}\left (\sqrt{\cosh ^{-1}(x)}\right )}{2 \sqrt{1-x}}+\frac{\sqrt{\pi } \sqrt{x-1} \text{Erfi}\left (\sqrt{\cosh ^{-1}(x)}\right )}{2 \sqrt{1-x}} \]

[Out]

(Sqrt[Pi]*Sqrt[-1 + x]*Erf[Sqrt[ArcCosh[x]]])/(2*Sqrt[1 - x]) + (Sqrt[Pi]*Sqrt[-1 + x]*Erfi[Sqrt[ArcCosh[x]]])
/(2*Sqrt[1 - x])

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Rubi [A]  time = 0.191704, antiderivative size = 83, normalized size of antiderivative = 1.28, number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {5798, 5781, 3307, 2180, 2204, 2205} \[ \frac{\sqrt{\pi } \sqrt{x-1} \sqrt{x+1} \text{Erf}\left (\sqrt{\cosh ^{-1}(x)}\right )}{2 \sqrt{1-x^2}}+\frac{\sqrt{\pi } \sqrt{x-1} \sqrt{x+1} \text{Erfi}\left (\sqrt{\cosh ^{-1}(x)}\right )}{2 \sqrt{1-x^2}} \]

Antiderivative was successfully verified.

[In]

Int[x/(Sqrt[1 - x^2]*Sqrt[ArcCosh[x]]),x]

[Out]

(Sqrt[Pi]*Sqrt[-1 + x]*Sqrt[1 + x]*Erf[Sqrt[ArcCosh[x]]])/(2*Sqrt[1 - x^2]) + (Sqrt[Pi]*Sqrt[-1 + x]*Sqrt[1 +
x]*Erfi[Sqrt[ArcCosh[x]]])/(2*Sqrt[1 - x^2])

Rule 5798

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

Rule 5781

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_
.), x_Symbol] :> Dist[(-(d1*d2))^p/c^(m + 1), Subst[Int[(a + b*x)^n*Cosh[x]^m*Sinh[x]^(2*p + 1), x], x, ArcCos
h[c*x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[p
+ 1/2] && GtQ[p, -1] && IGtQ[m, 0] && (GtQ[d1, 0] && LtQ[d2, 0])

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

Mathematica [A]  time = 0.0987592, size = 72, normalized size = 1.11 \[ -\frac{\sqrt{-(x-1) (x+1)} \left (\sqrt{-\cosh ^{-1}(x)} \text{Gamma}\left (\frac{1}{2},-\cosh ^{-1}(x)\right )-\sqrt{\cosh ^{-1}(x)} \text{Gamma}\left (\frac{1}{2},\cosh ^{-1}(x)\right )\right )}{2 \sqrt{\frac{x-1}{x+1}} (x+1) \sqrt{\cosh ^{-1}(x)}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x/(Sqrt[1 - x^2]*Sqrt[ArcCosh[x]]),x]

[Out]

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

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Maple [F]  time = 0.312, size = 0, normalized size = 0. \begin{align*} \int{x{\frac{1}{\sqrt{-{x}^{2}+1}}}{\frac{1}{\sqrt{{\rm arccosh} \left (x\right )}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(-x^2+1)^(1/2)/arccosh(x)^(1/2),x)

[Out]

int(x/(-x^2+1)^(1/2)/arccosh(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{arcosh}\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)/arccosh(x)^(1/2),x, algorithm="maxima")

[Out]

integrate(x/(sqrt(-x^2 + 1)*sqrt(arccosh(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)/arccosh(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{- \left (x - 1\right ) \left (x + 1\right )} \sqrt{\operatorname{acosh}{\left (x \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-x**2+1)**(1/2)/acosh(x)**(1/2),x)

[Out]

Integral(x/(sqrt(-(x - 1)*(x + 1))*sqrt(acosh(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{arcosh}\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)/arccosh(x)^(1/2),x, algorithm="giac")

[Out]

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