Optimal. Leaf size=15 \[ -\frac{1}{2} \tanh ^{-1}\left (\sin \left (2 x+\frac{\pi }{4}\right )\right ) \]
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Rubi [A] time = 0.0034344, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {3770} \[ -\frac{1}{2} \tanh ^{-1}\left (\sin \left (2 x+\frac{\pi }{4}\right )\right ) \]
Antiderivative was successfully verified.
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Rule 3770
Rubi steps
\begin{align*} \int -\sec \left (\frac{\pi }{4}+2 x\right ) \, dx &=-\frac{1}{2} \tanh ^{-1}\left (\sin \left (\frac{\pi }{4}+2 x\right )\right )\\ \end{align*}
Mathematica [A] time = 0.0058907, size = 15, normalized size = 1. \[ -\frac{1}{2} \tanh ^{-1}\left (\sin \left (2 x+\frac{\pi }{4}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 21, normalized size = 1.4 \begin{align*} -{\frac{1}{2}\ln \left ( \sec \left ({\frac{\pi }{4}}+2\,x \right ) +\tan \left ({\frac{\pi }{4}}+2\,x \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.929331, size = 36, normalized size = 2.4 \begin{align*} -\frac{1}{4} \, \log \left (\sin \left (\frac{1}{4} \, \pi + 2 \, x\right ) + 1\right ) + \frac{1}{4} \, \log \left (\sin \left (\frac{1}{4} \, \pi + 2 \, x\right ) - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.07409, size = 90, normalized size = 6. \begin{align*} -\frac{1}{4} \, \log \left (\sin \left (\frac{1}{4} \, \pi + 2 \, x\right ) + 1\right ) + \frac{1}{4} \, \log \left (-\sin \left (\frac{1}{4} \, \pi + 2 \, x\right ) + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.198141, size = 22, normalized size = 1.47 \begin{align*} \frac{\log{\left (\tan{\left (x + \frac{\pi }{8} \right )} - 1 \right )}}{2} - \frac{\log{\left (\tan{\left (x + \frac{\pi }{8} \right )} + 1 \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.05812, size = 39, normalized size = 2.6 \begin{align*} -\frac{1}{4} \, \log \left (\sin \left (\frac{1}{4} \, \pi + 2 \, x\right ) + 1\right ) + \frac{1}{4} \, \log \left (-\sin \left (\frac{1}{4} \, \pi + 2 \, x\right ) + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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