Optimal. Leaf size=3 \[ \log (\cosh (x)) \]
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Rubi [A]
time = 0.00, antiderivative size = 3, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3556}
\begin {gather*} \log (\cosh (x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 3556
Rubi steps
\begin {gather*} \begin {aligned} \text {Integral} &=\log (\cosh (x))\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.00, size = 3, normalized size = 1.00 \begin {gather*} \log (\cosh (x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 4, normalized size = 1.33
method | result | size |
lookup | \(\ln \left (\cosh \left (x \right )\right )\) | \(4\) |
derivativedivides | \(\ln \left (\cosh \left (x \right )\right )\) | \(4\) |
default | \(\ln \left (\cosh \left (x \right )\right )\) | \(4\) |
risch | \(-x +\ln \left ({\mathrm e}^{2 x}+1\right )\) | \(12\) |
parallelrisch | \(-\ln \left (1-\tanh \left (x \right )\right )-x\) | \(14\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.35, size = 3, normalized size = 1.00 \begin {gather*} \log \left (\cosh \left (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. 18 vs.
\(2 (3) = 6\).
time = 0.58, size = 18, normalized size = 6.00 \begin {gather*} -x + \log \left (\frac {2 \, \cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\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. 7 vs.
\(2 (3) = 6\).
time = 0.04, size = 7, normalized size = 2.33 \begin {gather*} x - \log {\left (\tanh {\left (x \right )} + 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 11 vs.
\(2 (3) = 6\).
time = 0.43, size = 11, normalized size = 3.67 \begin {gather*} -x + \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 = 0.01, size = 3, normalized size = 1.00 \begin {gather*} \ln \left (\mathrm {cosh}\left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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