Optimal. Leaf size=21 \[ x \cosh ^{-1}(x)-\sqrt{x-1} \sqrt{x+1} \]
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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.
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Rule 6327
Rule 5654
Rule 74
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.
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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.
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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.
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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.
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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.
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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.
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