3.238 \(\int \cosh ^{-1}(\frac{1}{x}) \, dx\)

Optimal. Leaf size=24 \[ \sqrt{\frac{1}{x+1}} \sqrt{x+1} \sin ^{-1}(x)+x \text{sech}^{-1}(x) \]

[Out]

x*ArcSech[x] + Sqrt[(1 + x)^(-1)]*Sqrt[1 + x]*ArcSin[x]

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Rubi [A]  time = 0.008589, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, Rules used = {5893, 6277, 216} \[ \sqrt{\frac{1}{x+1}} \sqrt{x+1} \sin ^{-1}(x)+x \text{sech}^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[ArcCosh[x^(-1)],x]

[Out]

x*ArcSech[x] + Sqrt[(1 + x)^(-1)]*Sqrt[1 + x]*ArcSin[x]

Rule 5893

Int[ArcCosh[(c_.)/((a_.) + (b_.)*(x_)^(n_.))]^(m_.)*(u_.), x_Symbol] :> Int[u*ArcSech[a/c + (b*x^n)/c]^m, x] /
; FreeQ[{a, b, c, n, m}, x]

Rule 6277

Int[ArcSech[(c_.)*(x_)], x_Symbol] :> Simp[x*ArcSech[c*x], x] + Dist[Sqrt[1 + c*x]*Sqrt[1/(1 + c*x)], Int[1/Sq
rt[1 - c^2*x^2], x], x] /; FreeQ[c, x]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin{align*} \int \cosh ^{-1}\left (\frac{1}{x}\right ) \, dx &=\int \text{sech}^{-1}(x) \, dx\\ &=x \text{sech}^{-1}(x)+\left (\sqrt{\frac{1}{1+x}} \sqrt{1+x}\right ) \int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=x \text{sech}^{-1}(x)+\sqrt{\frac{1}{1+x}} \sqrt{1+x} \sin ^{-1}(x)\\ \end{align*}

Mathematica [A]  time = 0.0458795, size = 46, normalized size = 1.92 \[ x \cosh ^{-1}\left (\frac{1}{x}\right )-\frac{\sqrt{\frac{1}{x^2}-1} \tan ^{-1}\left (\sqrt{\frac{1}{x^2}-1}\right )}{\sqrt{\frac{1}{x}-1} \sqrt{\frac{1}{x}+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCosh[x^(-1)],x]

[Out]

x*ArcCosh[x^(-1)] - (Sqrt[-1 + x^(-2)]*ArcTan[Sqrt[-1 + x^(-2)]])/(Sqrt[-1 + x^(-1)]*Sqrt[1 + x^(-1)])

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Maple [A]  time = 0.025, size = 38, normalized size = 1.6 \begin{align*}{\rm arccosh} \left ({x}^{-1}\right )x+{\sqrt{{x}^{-1}-1}\sqrt{{x}^{-1}+1}\arctan \left ({\frac{1}{\sqrt{{x}^{-2}-1}}} \right ){\frac{1}{\sqrt{{x}^{-2}-1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccosh(1/x),x)

[Out]

arccosh(1/x)*x+(1/x-1)^(1/2)*(1/x+1)^(1/2)/(1/x^2-1)^(1/2)*arctan(1/(1/x^2-1)^(1/2))

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Maxima [B]  time = 1.53753, size = 23, normalized size = 0.96 \begin{align*} x \operatorname{arcosh}\left (\frac{1}{x}\right ) - \arctan \left (\sqrt{\frac{1}{x^{2}} - 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(1/x),x, algorithm="maxima")

[Out]

x*arccosh(1/x) - arctan(sqrt(1/x^2 - 1))

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Fricas [B]  time = 2.15316, size = 173, normalized size = 7.21 \begin{align*}{\left (x - 2\right )} \log \left (\frac{x \sqrt{-\frac{x^{2} - 1}{x^{2}}} + 1}{x}\right ) - 2 \, \arctan \left (\frac{x \sqrt{-\frac{x^{2} - 1}{x^{2}}} - 1}{x}\right ) - 2 \, \log \left (\frac{x \sqrt{-\frac{x^{2} - 1}{x^{2}}} - 1}{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(1/x),x, algorithm="fricas")

[Out]

(x - 2)*log((x*sqrt(-(x^2 - 1)/x^2) + 1)/x) - 2*arctan((x*sqrt(-(x^2 - 1)/x^2) - 1)/x) - 2*log((x*sqrt(-(x^2 -
1)/x^2) - 1)/x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{acosh}{\left (\frac{1}{x} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acosh(1/x),x)

[Out]

Integral(acosh(1/x), x)

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Giac [B]  time = 1.10403, size = 30, normalized size = 1.25 \begin{align*} x \log \left (\sqrt{\frac{1}{x^{2}} - 1} + \frac{1}{x}\right ) + \frac{\arcsin \left (x\right )}{\mathrm{sgn}\left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(1/x),x, algorithm="giac")

[Out]

x*log(sqrt(1/x^2 - 1) + 1/x) + arcsin(x)/sgn(x)