Optimal. Leaf size=13 \[ -2 \log \left (\cos \left (\frac {1}{4} (2 x+\pi )\right )\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 9, normalized size of antiderivative = 0.69, number of steps used = 3, number of rules used = 2, integrand size = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3770, 3475} \[ \tanh ^{-1}(\sin (x))-\log (\cos (x)) \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3770
Rubi steps
\begin {align*} \int (\sec (x)+\tan (x)) \, dx &=\int \sec (x) \, dx+\int \tan (x) \, dx\\ &=\tanh ^{-1}(\sin (x))-\log (\cos (x))\\ \end {align*}
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Mathematica [B] time = 0.00, size = 38, normalized size = 2.92 \[ -\log (\cos (x))-\log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )+\log \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 3.01, size = 9, normalized size = 0.69 \[ -\log \left (-\sin \relax (x) + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 31, normalized size = 2.38 \[ \frac {1}{4} \, \log \left ({\left | \frac {1}{\sin \relax (x)} + \sin \relax (x) + 2 \right |}\right ) - \frac {1}{4} \, \log \left ({\left | \frac {1}{\sin \relax (x)} + \sin \relax (x) - 2 \right |}\right ) - \log \left ({\left | \cos \relax (x) \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 13, normalized size = 1.00 \[ \ln \left (\sec \relax (x )+\tan \relax (x )\right )-\ln \left (\cos \relax (x )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 10, normalized size = 0.77 \[ \log \left (\sec \relax (x) + \tan \relax (x)\right ) + \log \left (\sec \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.40, size = 19, normalized size = 1.46 \[ \ln \left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )-2\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )-1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 20, normalized size = 1.54 \[ - \frac {\log {\left (\sin {\relax (x )} - 1 \right )}}{2} + \frac {\log {\left (\sin {\relax (x )} + 1 \right )}}{2} - \log {\left (\cos {\relax (x )} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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