Optimal. Leaf size=9 \[ \frac {1}{2} \log ^2(\tan (x)) \]
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Rubi [A] time = 0.02, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2620, 29, 6686} \[ \frac {1}{2} \log ^2(\tan (x)) \]
Antiderivative was successfully verified.
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Rule 29
Rule 2620
Rule 6686
Rubi steps
\begin {align*} \int \csc (x) \log (\tan (x)) \sec (x) \, dx &=\frac {1}{2} \log ^2(\tan (x))\\ \end {align*}
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Mathematica [A] time = 0.01, size = 9, normalized size = 1.00 \[ \frac {1}{2} \log ^2(\tan (x)) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 12, normalized size = 1.33 \[ \frac {1}{2} \, \log \left (\frac {\sin \relax (x)}{\cos \relax (x)}\right )^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.08, size = 7, normalized size = 0.78 \[ \frac {1}{2} \, \log \left (\tan \relax (x)\right )^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 8, normalized size = 0.89 \[ \frac {\ln \left (\tan \relax (x )\right )^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 7, normalized size = 0.78 \[ \frac {1}{2} \, \log \left (\tan \relax (x)\right )^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.59, size = 27, normalized size = 3.00 \[ \frac {{\ln \left (-\frac {{\mathrm {e}}^{x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}}{{\mathrm {e}}^{x\,2{}\mathrm {i}}+1}\right )}^2}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log {\left (\tan {\relax (x )} \right )}}{\sin {\relax (x )} \cos {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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