3.28 \(\int \text{sech}^{-1}(\frac{1}{x}) \, dx\)

Optimal. Leaf size=21 \[ x \cosh ^{-1}(x)-\sqrt{x-1} \sqrt{x+1} \]

[Out]

-(Sqrt[-1 + x]*Sqrt[1 + x]) + x*ArcCosh[x]

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Rubi [A]  time = 0.0069606, antiderivative size = 21, 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 = {6327, 5654, 74} \[ x \cosh ^{-1}(x)-\sqrt{x-1} \sqrt{x+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[-1 + x]*Sqrt[1 + x]) + x*ArcCosh[x]

Rule 6327

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

Rule 5654

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcCosh[c*x])^n, x] - Dist[b*c*n, In
t[(x*(a + b*ArcCosh[c*x])^(n - 1))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 74

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

Rubi steps

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

Mathematica [A]  time = 0.0235574, size = 25, normalized size = 1.19 \[ x \text{sech}^{-1}\left (\frac{1}{x}\right )-\frac{x-1}{\sqrt{\frac{x-1}{x+1}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.101, size = 29, normalized size = 1.4 \begin{align*} x{\rm arcsech} \left ({x}^{-1}\right )-\sqrt{- \left ({x}^{-1}-1 \right ) x}\sqrt{ \left ({x}^{-1}+1 \right ) x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsech(1/x),x)

[Out]

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

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Maxima [A]  time = 1.02406, size = 22, normalized size = 1.05 \begin{align*} x \operatorname{arsech}\left (\frac{1}{x}\right ) - \sqrt{x^{2} - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*arcsech(1/x) - sqrt(x^2 - 1)

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Fricas [A]  time = 1.81089, size = 57, normalized size = 2.71 \begin{align*} x \log \left (x + \sqrt{x^{2} - 1}\right ) - \sqrt{x^{2} - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asech(1/x),x)

[Out]

Integral(asech(1/x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arcsech(1/x), x)