Optimal. Leaf size=13 \[ 2 x-\coth (x)+2 \log (\sinh (x)) \]
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Rubi [A] time = 0.01, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3477, 3475} \[ 2 x-\coth (x)+2 \log (\sinh (x)) \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3477
Rubi steps
\begin {align*} \int (1+\coth (x))^2 \, dx &=2 x-\coth (x)+2 \int \coth (x) \, dx\\ &=2 x-\coth (x)+2 \log (\sinh (x))\\ \end {align*}
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Mathematica [A] time = 0.00, size = 13, normalized size = 1.00 \[ 2 x-\coth (x)+2 \log (\sinh (x)) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 53, normalized size = 4.08 \[ \frac {2 \, {\left ({\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} - 1\right )} \log \left (\frac {2 \, \sinh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}\right ) - 1\right )}}{\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 21, normalized size = 1.62 \[ -\frac {2}{e^{\left (2 \, x\right )} - 1} + 2 \, \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 13, normalized size = 1.00 \[ -\coth \relax (x )-2 \ln \left (\coth \relax (x )-1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 19, normalized size = 1.46 \[ 2 \, x + \frac {2}{e^{\left (-2 \, x\right )} - 1} + 2 \, \log \left (\sinh \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.14, size = 20, normalized size = 1.54 \[ 2\,\ln \left ({\mathrm {e}}^{2\,x}-1\right )-\frac {2}{{\mathrm {e}}^{2\,x}-1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.37, size = 22, normalized size = 1.69 \[ 4 x - 2 \log {\left (\tanh {\relax (x )} + 1 \right )} + 2 \log {\left (\tanh {\relax (x )} \right )} - \frac {1}{\tanh {\relax (x )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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