Optimal. Leaf size=14 \[ \frac {\cos ^2(x)}{2}-\log (\cos (x)) \]
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Rubi [A] time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2590, 14} \[ \frac {\cos ^2(x)}{2}-\log (\cos (x)) \]
Antiderivative was successfully verified.
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Rule 14
Rule 2590
Rubi steps
\begin {align*} \int \sin ^2(x) \tan (x) \, dx &=-\operatorname {Subst}\left (\int \frac {1-x^2}{x} \, dx,x,\cos (x)\right )\\ &=-\operatorname {Subst}\left (\int \left (\frac {1}{x}-x\right ) \, dx,x,\cos (x)\right )\\ &=\frac {\cos ^2(x)}{2}-\log (\cos (x))\\ \end {align*}
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Mathematica [A] time = 0.01, size = 14, normalized size = 1.00 \[ \frac {\cos ^2(x)}{2}-\log (\cos (x)) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 14, normalized size = 1.00 \[ \frac {1}{2} \, \cos \relax (x)^{2} - \log \left (-\cos \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.08, size = 13, normalized size = 0.93 \[ \frac {1}{2} \, \cos \relax (x)^{2} - \log \left ({\left | \cos \relax (x) \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 13, normalized size = 0.93 \[ -\frac {\left (\sin ^{2}\relax (x )\right )}{2}-\ln \left (\cos \relax (x )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 16, normalized size = 1.14 \[ -\frac {1}{2} \, \sin \relax (x)^{2} - \frac {1}{2} \, \log \left (\sin \relax (x)^{2} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.20, size = 16, normalized size = 1.14 \[ \frac {{\cos \relax (x)}^2}{2}+\frac {\ln \left ({\mathrm {tan}\relax (x)}^2+1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 10, normalized size = 0.71 \[ - \log {\left (\cos {\relax (x )} \right )} + \frac {\cos ^{2}{\relax (x )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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