Optimal. Leaf size=3 \[ \tanh ^{-1}(\sin (x)) \]
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Rubi [A] time = 0.0025583, antiderivative size = 3, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 2, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3770} \[ \tanh ^{-1}(\sin (x)) \]
Antiderivative was successfully verified.
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Rule 3770
Rubi steps
\begin{align*} \int \sec (x) \, dx &=\tanh ^{-1}(\sin (x))\\ \end{align*}
Mathematica [B] time = 0.0029171, size = 33, normalized size = 11. \[ \log \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )-\log \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 7, normalized size = 2.3 \begin{align*} \ln \left ( \sec \left ( x \right ) +\tan \left ( x \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.941983, size = 20, normalized size = 6.67 \begin{align*} \frac{1}{2} \, \log \left (\sin \left (x\right ) + 1\right ) - \frac{1}{2} \, \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 = 2.09825, size = 59, normalized size = 19.67 \begin{align*} \frac{1}{2} \, \log \left (\sin \left (x\right ) + 1\right ) - \frac{1}{2} \, \log \left (-\sin \left (x\right ) + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.097763, size = 15, normalized size = 5. \begin{align*} - \frac{\log{\left (\sin{\left (x \right )} - 1 \right )}}{2} + \frac{\log{\left (\sin{\left (x \right )} + 1 \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.08769, size = 23, normalized size = 7.67 \begin{align*} \frac{1}{2} \, \log \left (\sin \left (x\right ) + 1\right ) - \frac{1}{2} \, \log \left (-\sin \left (x\right ) + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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