Optimal. Leaf size=9 \[ \frac {1}{2} \log (\cosh (2 x)) \]
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Rubi [A]
time = 0.00, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3556}
\begin {gather*} \frac {1}{2} \log (\cosh (2 x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 3556
Rubi steps
\begin {align*} \int \tanh (2 x) \, dx &=\frac {1}{2} \log (\cosh (2 x))\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 9, normalized size = 1.00 \begin {gather*} \frac {1}{2} \log (\cosh (2 x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 8, normalized size = 0.89
method | result | size |
derivativedivides | \(\frac {\ln \left (\cosh \left (2 x \right )\right )}{2}\) | \(8\) |
default | \(\frac {\ln \left (\cosh \left (2 x \right )\right )}{2}\) | \(8\) |
risch | \(-x +\frac {\ln \left ({\mathrm e}^{4 x}+1\right )}{2}\) | \(14\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 5.81, size = 7, normalized size = 0.78 \begin {gather*} \frac {1}{2} \, \log \left (\cosh \left (2 \, x\right )\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. 26 vs.
\(2 (7) = 14\).
time = 0.57, size = 26, normalized size = 2.89 \begin {gather*} -x + \frac {1}{2} \, \log \left (\frac {2 \, \cosh \left (2 \, x\right )}{\cosh \left (2 \, x\right ) - \sinh \left (2 \, x\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 7, normalized size = 0.78 \begin {gather*} \frac {\log {\left (\cosh {\left (2 x \right )} \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.52, size = 13, normalized size = 1.44 \begin {gather*} -x + \frac {1}{2} \, \log \left (e^{\left (4 \, x\right )} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.19, size = 7, normalized size = 0.78 \begin {gather*} \frac {\ln \left (\mathrm {cosh}\left (2\,x\right )\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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