Optimal. Leaf size=19 \[ -\frac {x}{2}+\log (\cosh (x))-\frac {1}{2 (1+\tanh (x))} \]
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Rubi [A]
time = 0.03, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3621, 3556}
\begin {gather*} -\frac {x}{2}-\frac {1}{2 (\tanh (x)+1)}+\log (\cosh (x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 3556
Rule 3621
Rubi steps
\begin {align*} \int \frac {\tanh ^2(x)}{1+\tanh (x)} \, dx &=-\frac {1}{2 (1+\tanh (x))}-\frac {1}{2} \int (1-2 \tanh (x)) \, dx\\ &=-\frac {x}{2}-\frac {1}{2 (1+\tanh (x))}+\int \tanh (x) \, dx\\ &=-\frac {x}{2}+\log (\cosh (x))-\frac {1}{2 (1+\tanh (x))}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 23, normalized size = 1.21 \begin {gather*} \frac {1}{4} (-2 x-\cosh (2 x)+4 \log (\cosh (x))+\sinh (2 x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.29, size = 24, normalized size = 1.26
method | result | size |
risch | \(-\frac {3 x}{2}-\frac {{\mathrm e}^{-2 x}}{4}+\ln \left (1+{\mathrm e}^{2 x}\right )\) | \(18\) |
derivativedivides | \(-\frac {1}{2 \left (1+\tanh \left (x \right )\right )}-\frac {3 \ln \left (1+\tanh \left (x \right )\right )}{4}-\frac {\ln \left (\tanh \left (x \right )-1\right )}{4}\) | \(24\) |
default | \(-\frac {1}{2 \left (1+\tanh \left (x \right )\right )}-\frac {3 \ln \left (1+\tanh \left (x \right )\right )}{4}-\frac {\ln \left (\tanh \left (x \right )-1\right )}{4}\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 17, normalized size = 0.89 \begin {gather*} \frac {1}{2} \, x - \frac {1}{4} \, e^{\left (-2 \, x\right )} + \log \left (e^{\left (-2 \, x\right )} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 73 vs.
\(2 (15) = 30\).
time = 0.33, size = 73, normalized size = 3.84 \begin {gather*} -\frac {6 \, x \cosh \left (x\right )^{2} + 12 \, x \cosh \left (x\right ) \sinh \left (x\right ) + 6 \, x \sinh \left (x\right )^{2} - 4 \, {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}\right )} \log \left (\frac {2 \, \cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 1}{4 \, {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 61 vs.
\(2 (15) = 30\).
time = 0.17, size = 61, normalized size = 3.21 \begin {gather*} \frac {x \tanh {\left (x \right )}}{2 \tanh {\left (x \right )} + 2} + \frac {x}{2 \tanh {\left (x \right )} + 2} - \frac {2 \log {\left (\tanh {\left (x \right )} + 1 \right )} \tanh {\left (x \right )}}{2 \tanh {\left (x \right )} + 2} - \frac {2 \log {\left (\tanh {\left (x \right )} + 1 \right )}}{2 \tanh {\left (x \right )} + 2} - \frac {1}{2 \tanh {\left (x \right )} + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 17, normalized size = 0.89 \begin {gather*} -\frac {3}{2} \, x - \frac {1}{4} \, e^{\left (-2 \, x\right )} + \log \left (e^{\left (2 \, x\right )} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.06, size = 21, normalized size = 1.11 \begin {gather*} \frac {x}{2}-\ln \left (\mathrm {tanh}\left (x\right )+1\right )-\frac {1}{2\,\left (\mathrm {tanh}\left (x\right )+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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