Optimal. Leaf size=16 \[ x \sinh ^{-1}(x)-\sqrt{x^2+1} \]
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Rubi [A] time = 0.0060857, antiderivative size = 16, 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 = {6328, 5653, 261} \[ x \sinh ^{-1}(x)-\sqrt{x^2+1} \]
Antiderivative was successfully verified.
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Rule 6328
Rule 5653
Rule 261
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*}
Mathematica [A] time = 0.0027146, 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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Maple [A] time = 0.099, size = 29, normalized size = 1.8 \begin{align*} x{\rm arccsch} \left ({x}^{-1}\right )-{ \left ( 1+{x}^{-2} \right ){x}^{2}{\frac{1}{\sqrt{ \left ( 1+{x}^{-2} \right ){x}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00029, size = 22, normalized size = 1.38 \begin{align*} x \operatorname{arcsch}\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 = 2.58506, size = 57, normalized size = 3.56 \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 [A] time = 0.147368, size = 14, normalized size = 0.88 \begin{align*} x \operatorname{acsch}{\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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{arcsch}\left (\frac{1}{x}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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