Optimal. Leaf size=9 \[ -\frac {1}{2} \log ^2(\cos (x)) \]
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Rubi [A] time = 0.01, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3475, 4339, 2301} \[ -\frac {1}{2} \log ^2(\cos (x)) \]
Antiderivative was successfully verified.
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Rule 2301
Rule 3475
Rule 4339
Rubi steps
\begin {align*} \int \log (\cos (x)) \tan (x) \, dx &=-\operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,\cos (x)\right )\\ &=-\frac {1}{2} \log ^2(\cos (x))\\ \end {align*}
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Mathematica [A] time = 0.00, size = 9, normalized size = 1.00 \[ -\frac {1}{2} \log ^2(\cos (x)) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 7, normalized size = 0.78 \[ -\frac {1}{2} \, \log \left (\cos \relax (x)\right )^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 7, normalized size = 0.78 \[ -\frac {1}{2} \, \log \left (\cos \relax (x)\right )^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.19, size = 8, normalized size = 0.89 \[ -\frac {\ln \left (\cos \relax (x )\right )^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 7, normalized size = 0.78 \[ -\frac {1}{2} \, \log \left (\cos \relax (x)\right )^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.43, size = 7, normalized size = 0.78 \[ -\frac {{\ln \left (\cos \relax (x)\right )}^2}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.53, size = 8, normalized size = 0.89 \[ - \frac {\log {\left (\cos {\relax (x )} \right )}^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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