Optimal. Leaf size=13 \[ 2 x-\coth (x)+2 \log (\sinh (x)) \]
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Rubi [A] time = 0.0133208, antiderivative size = 13, normalized size of antiderivative = 1., 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 3477
Rule 3475
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*}
Mathematica [A] time = 0.0043895, size = 13, normalized size = 1. \[ 2 x-\coth (x)+2 \log (\sinh (x)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 13, normalized size = 1. \begin{align*} -{\rm coth} \left (x\right )-2\,\ln \left ({\rm coth} \left (x\right )-1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.13003, size = 26, normalized size = 2. \begin{align*} 2 \, x + \frac{2}{e^{\left (-2 \, x\right )} - 1} + 2 \, \log \left (\sinh \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.95735, size = 189, normalized size = 14.54 \begin{align*} \frac{2 \,{\left ({\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1\right )} \log \left (\frac{2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) - 1\right )}}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.745568, size = 22, normalized size = 1.69 \begin{align*} 4 x - 2 \log{\left (\tanh{\left (x \right )} + 1 \right )} + 2 \log{\left (\tanh{\left (x \right )} \right )} - \frac{1}{\tanh{\left (x \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18667, size = 28, normalized size = 2.15 \begin{align*} -\frac{2}{e^{\left (2 \, x\right )} - 1} + 2 \, \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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