Optimal. Leaf size=16 \[ x \sinh ^{-1}(x)-\sqrt {x^2+1} \]
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Rubi [A] time = 0.01, antiderivative size = 16, 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 = {6328, 5653, 261} \[ x \sinh ^{-1}(x)-\sqrt {x^2+1} \]
Antiderivative was successfully verified.
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Rule 261
Rule 5653
Rule 6328
Rubi steps
\begin {align*} \int \text {csch}^{-1}\left (\frac {1}{x}\right ) \, dx &=\int \sinh ^{-1}(x) \, dx\\ &=x \sinh ^{-1}(x)-\int \frac {x}{\sqrt {1+x^2}} \, dx\\ &=-\sqrt {1+x^2}+x \sinh ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 18, normalized size = 1.12 \[ x \text {csch}^{-1}\left (\frac {1}{x}\right )-\sqrt {x^2+1} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 22, normalized size = 1.38 \[ x \log \left (x + \sqrt {x^{2} + 1}\right ) - \sqrt {x^{2} + 1} \]
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 \[ \int \operatorname {arcsch}\left (\frac {1}{x}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 29, normalized size = 1.81 \[ x \,\mathrm {arccsch}\left (\frac {1}{x}\right )-\frac {x^{2} \left (1+\frac {1}{x^{2}}\right )}{\sqrt {\left (1+\frac {1}{x^{2}}\right ) x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 16, normalized size = 1.00 \[ x \operatorname {arcsch}\left (\frac {1}{x}\right ) - \sqrt {x^{2} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 14, normalized size = 0.88 \[ x\,\mathrm {asinh}\relax (x)-\sqrt {x^2+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 14, normalized size = 0.88 \[ x \operatorname {acsch}{\left (\frac {1}{x} \right )} - \sqrt {x^{2} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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