Optimal. Leaf size=16 \[ \frac {1}{2} \tan (x) \sec (x)-\frac {1}{2} \tanh ^{-1}(\sin (x)) \]
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Rubi [A] time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2611, 3770} \[ \frac {1}{2} \tan (x) \sec (x)-\frac {1}{2} \tanh ^{-1}(\sin (x)) \]
Antiderivative was successfully verified.
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Rule 2611
Rule 3770
Rubi steps
\begin {align*} \int \sec (x) \tan ^2(x) \, dx &=\frac {1}{2} \sec (x) \tan (x)-\frac {1}{2} \int \sec (x) \, dx\\ &=-\frac {1}{2} \tanh ^{-1}(\sin (x))+\frac {1}{2} \sec (x) \tan (x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 16, normalized size = 1.00 \[ \frac {1}{2} \tan (x) \sec (x)-\frac {1}{2} \tanh ^{-1}(\sin (x)) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 34, normalized size = 2.12 \[ -\frac {\cos \relax (x)^{2} \log \left (\sin \relax (x) + 1\right ) - \cos \relax (x)^{2} \log \left (-\sin \relax (x) + 1\right ) - 2 \, \sin \relax (x)}{4 \, \cos \relax (x)^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.13, size = 29, normalized size = 1.81 \[ -\frac {\sin \relax (x)}{2 \, {\left (\sin \relax (x)^{2} - 1\right )}} - \frac {1}{4} \, \log \left (\sin \relax (x) + 1\right ) + \frac {1}{4} \, \log \left (-\sin \relax (x) + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 24, normalized size = 1.50 \[ \frac {\sin ^{3}\relax (x )}{2 \cos \relax (x )^{2}}-\frac {\ln \left (\sec \relax (x )+\tan \relax (x )\right )}{2}+\frac {\sin \relax (x )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.58, size = 27, normalized size = 1.69 \[ -\frac {\sin \relax (x)}{2 \, {\left (\sin \relax (x)^{2} - 1\right )}} - \frac {1}{4} \, \log \left (\sin \relax (x) + 1\right ) + \frac {1}{4} \, \log \left (\sin \relax (x) - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.00, size = 30, normalized size = 1.88 \[ \frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^3+\mathrm {tan}\left (\frac {x}{2}\right )}{{\left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2-1\right )}^2}-\mathrm {atanh}\left (\mathrm {tan}\left (\frac {x}{2}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 27, normalized size = 1.69 \[ \frac {\log {\left (\sin {\relax (x )} - 1 \right )}}{4} - \frac {\log {\left (\sin {\relax (x )} + 1 \right )}}{4} - \frac {\sin {\relax (x )}}{2 \sin ^{2}{\relax (x )} - 2} \]
Verification of antiderivative is not currently implemented for this CAS.
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