Optimal. Leaf size=11 \[ -\log (\sinh (x))-\tanh ^{-1}(\cosh (x)) \]
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Rubi [A] time = 0.01, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3475, 3770} \[ -\log (\sinh (x))-\tanh ^{-1}(\cosh (x)) \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3770
Rubi steps
\begin {align*} \int (-\coth (x)+\text {csch}(x)) \, dx &=-\int \coth (x) \, dx+\int \text {csch}(x) \, dx\\ &=-\tanh ^{-1}(\cosh (x))-\log (\sinh (x))\\ \end {align*}
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Mathematica [A] time = 0.01, size = 13, normalized size = 1.18 \[ \log \left (\tanh \left (\frac {x}{2}\right )\right )-\log (\sinh (x)) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 11, normalized size = 1.00 \[ x - 2 \, \log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.12, size = 25, normalized size = 2.27 \[ x - \log \left (e^{x} + 1\right ) - \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right ) + \log \left ({\left | e^{x} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 12, normalized size = 1.09 \[ -\ln \left (\sinh \relax (x )\right )+\ln \left (\tanh \left (\frac {x}{2}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.61, size = 11, normalized size = 1.00 \[ -\log \left (\sinh \relax (x)\right ) + \log \left (\tanh \left (\frac {1}{2} \, x\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.54, size = 9, normalized size = 0.82 \[ x-2\,\ln \left ({\mathrm {e}}^x+1\right ) \]
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 \[ \int \left (- \coth {\relax (x )} + \operatorname {csch}{\relax (x )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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