3.3.38 \(\int \cosh ^{-1}(\frac {1}{x}) \, dx\) [238]

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

[Out]

x*arcsech(x)+arcsin(x)*(1/(1+x))^(1/2)*(1+x)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6012, 6412, 222} \begin {gather*} \sqrt {\frac {1}{x+1}} \sqrt {x+1} \text {ArcSin}(x)+x \text {sech}^{-1}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 222

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]

Rule 6012

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 6412

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]

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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(59\) vs. \(2(24)=48\).
time = 0.13, size = 59, normalized size = 2.46 \begin {gather*} x \cosh ^{-1}\left (\frac {1}{x}\right )-\frac {2 \sqrt {1-x^2} \text {ArcTan}\left (\frac {\sqrt {1-x}}{\sqrt {1+x}}\right )}{\sqrt {-1+\frac {1}{x}} \sqrt {1+\frac {1}{x}} x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.04, size = 38, normalized size = 1.58

method	result	size
derivativedivides	\(x \,\mathrm {arccosh}\left (\frac {1}{x}\right )+\frac {\sqrt {\frac {1}{x}-1}\, \sqrt {\frac {1}{x}+1}\, \arctan \left (\frac {1}{\sqrt {\frac {1}{x^{2}}-1}}\right )}{\sqrt {\frac {1}{x^{2}}-1}}\)	\(38\)
default	\(x \,\mathrm {arccosh}\left (\frac {1}{x}\right )+\frac {\sqrt {\frac {1}{x}-1}\, \sqrt {\frac {1}{x}+1}\, \arctan \left (\frac {1}{\sqrt {\frac {1}{x^{2}}-1}}\right )}{\sqrt {\frac {1}{x^{2}}-1}}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccosh(1/x),x,method=_RETURNVERBOSE)

[Out]

x*arccosh(1/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] Leaf count of result is larger than twice the leaf count of optimal. 17 vs. \(2 (7) = 14\).
time = 0.51, size = 17, normalized size = 0.71 \begin {gather*} x \operatorname {arcosh}\left (\frac {1}{x}\right ) - \arctan \left (\sqrt {\frac {1}{x^{2}} - 1}\right ) \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (7) = 14\).
time = 0.38, size = 72, normalized size = 3.00 \begin {gather*} {\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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \operatorname {acosh}{\left (\frac {1}{x} \right )}\, dx \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 22 vs. \(2 (7) = 14\).
time = 0.40, size = 22, normalized size = 0.92 \begin {gather*} x \log \left (\sqrt {\frac {1}{x^{2}} - 1} + \frac {1}{x}\right ) + \frac {\arcsin \left (x\right )}{\mathrm {sgn}\left (x\right )} \end {gather*}

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)

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Mupad [B]
time = 0.37, size = 23, normalized size = 0.96 \begin {gather*} \mathrm {atan}\left (\frac {1}{\sqrt {\frac {1}{x}-1}\,\sqrt {\frac {1}{x}+1}}\right )+x\,\mathrm {acosh}\left (\frac {1}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acosh(1/x),x)

[Out]

atan(1/((1/x - 1)^(1/2)*(1/x + 1)^(1/2))) + x*acosh(1/x)

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