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 = 0.0152064, antiderivative size = 16, normalized size of antiderivative = 1., 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*}
Mathematica [A] time = 0.005882, size = 16, normalized size = 1. \[ \frac{1}{2} \tan (x) \sec (x)-\frac{1}{2} \tanh ^{-1}(\sin (x)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 24, normalized size = 1.5 \begin{align*}{\frac{ \left ( \sin \left ( x \right ) \right ) ^{3}}{2\, \left ( \cos \left ( x \right ) \right ) ^{2}}}+{\frac{\sin \left ( x \right ) }{2}}-{\frac{\ln \left ( \sec \left ( x \right ) +\tan \left ( x \right ) \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.928819, size = 36, normalized size = 2.25 \begin{align*} -\frac{\sin \left (x\right )}{2 \,{\left (\sin \left (x\right )^{2} - 1\right )}} - \frac{1}{4} \, \log \left (\sin \left (x\right ) + 1\right ) + \frac{1}{4} \, \log \left (\sin \left (x\right ) - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.17193, size = 109, normalized size = 6.81 \begin{align*} -\frac{\cos \left (x\right )^{2} \log \left (\sin \left (x\right ) + 1\right ) - \cos \left (x\right )^{2} \log \left (-\sin \left (x\right ) + 1\right ) - 2 \, \sin \left (x\right )}{4 \, \cos \left (x\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.109246, size = 27, normalized size = 1.69 \begin{align*} \frac{\log{\left (\sin{\left (x \right )} - 1 \right )}}{4} - \frac{\log{\left (\sin{\left (x \right )} + 1 \right )}}{4} - \frac{\sin{\left (x \right )}}{2 \sin ^{2}{\left (x \right )} - 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.05921, size = 39, normalized size = 2.44 \begin{align*} -\frac{\sin \left (x\right )}{2 \,{\left (\sin \left (x\right )^{2} - 1\right )}} - \frac{1}{4} \, \log \left (\sin \left (x\right ) + 1\right ) + \frac{1}{4} \, \log \left (-\sin \left (x\right ) + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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