Optimal. Leaf size=27 \[ \sinh ^{-1}(\sinh (x))+\log \left (\sinh ^{-1}(\sinh (x))\right ) \left (x \sqrt {\cosh ^2(x)} \text {sech}(x)-\sinh ^{-1}(\sinh (x))\right ) \]
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Rubi [F] time = 0.04, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {x}{\sinh ^{-1}(\sinh (x))} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {x}{\sinh ^{-1}(\sinh (x))} \, dx &=\int \frac {x}{\sinh ^{-1}(\sinh (x))} \, dx\\ \end {align*}
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Mathematica [A] time = 0.56, size = 28, normalized size = 1.04 \[ x \sqrt {\cosh ^2(x)} \text {sech}(x) \log \left (\sinh ^{-1}(\sinh (x))\right )-\sinh ^{-1}(\sinh (x)) \left (\log \left (\sinh ^{-1}(\sinh (x))\right )-1\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 1, normalized size = 0.04 \[ x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 1, normalized size = 0.04 \[ x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {x}{\arcsinh \left (\sinh \relax (x )\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.78, size = 1, normalized size = 0.04 \[ x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.22, size = 1, normalized size = 0.04 \[ x \]
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 \[ \int \frac {x}{\operatorname {asinh}{\left (\sinh {\relax (x )} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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