Optimal. Leaf size=21 \[ x \cosh ^{-1}(x)-\sqrt {x-1} \sqrt {x+1} \]
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Rubi [A] time = 0.01, antiderivative size = 21, 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 = {6327, 5654, 74} \[ x \cosh ^{-1}(x)-\sqrt {x-1} \sqrt {x+1} \]
Antiderivative was successfully verified.
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Rule 74
Rule 5654
Rule 6327
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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 0.63, size = 22, normalized size = 1.05 \[ x \log \left (x + \sqrt {x^{2} - 1}\right ) - \sqrt {x^{2} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {arsech}\left (\frac {1}{x}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 29, normalized size = 1.38 \[ x \,\mathrm {arcsech}\left (\frac {1}{x}\right )-\sqrt {-\left (\frac {1}{x}-1\right ) x}\, \sqrt {\left (\frac {1}{x}+1\right ) x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 16, normalized size = 0.76 \[ x \operatorname {arsech}\left (\frac {1}{x}\right ) - \sqrt {x^{2} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.35, size = 17, normalized size = 0.81 \[ x\,\mathrm {acosh}\relax (x)-\sqrt {x-1}\,\sqrt {x+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {asech}{\left (\frac {1}{x} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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