Optimal. Leaf size=5 \[ -\log (\cos (x)) \]
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Rubi [A]
time = 0.00, antiderivative size = 5, 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 (\cos (x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 3556
Rubi steps
\begin {align*} \int \tan (x) \, dx &=-\log (\cos (x))\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 5, normalized size = 1.00 \begin {gather*} -\log (\cos (x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 6, normalized size = 1.20
method | result | size |
lookup | \(-\ln \left (\cos \left (x \right )\right )\) | \(6\) |
default | \(-\ln \left (\cos \left (x \right )\right )\) | \(6\) |
derivativedivides | \(\frac {\ln \left (1+\tan ^{2}\left (x \right )\right )}{2}\) | \(10\) |
norman | \(\frac {\ln \left (1+\tan ^{2}\left (x \right )\right )}{2}\) | \(10\) |
risch | \(i x -\ln \left ({\mathrm e}^{2 i x}+1\right )\) | \(16\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.96, size = 3, normalized size = 0.60 \begin {gather*} \log \left (\sec \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. 11 vs.
\(2 (5) = 10\).
time = 0.55, size = 11, normalized size = 2.20 \begin {gather*} -\frac {1}{2} \, \log \left (\frac {1}{\tan \left (x\right )^{2} + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 5, normalized size = 1.00 \begin {gather*} - \log {\left (\cos {\left (x \right )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.89, size = 6, normalized size = 1.20 \begin {gather*} -\log \left ({\left | \cos \left (x\right ) \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 5, normalized size = 1.00 \begin {gather*} -\ln \left (\cos \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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