Optimal. Leaf size=12 \[ a \tanh ^{-1}(\sin (x))-b \log (\cos (x)) \]
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Rubi [A] time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3770, 3475} \[ a \tanh ^{-1}(\sin (x))-b \log (\cos (x)) \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3770
Rubi steps
\begin {align*} \int (a \sec (x)+b \tan (x)) \, dx &=a \int \sec (x) \, dx+b \int \tan (x) \, dx\\ &=a \tanh ^{-1}(\sin (x))-b \log (\cos (x))\\ \end {align*}
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Mathematica [B] time = 0.01, size = 42, normalized size = 3.50 \[ -a \log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )+a \log \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )-b \log (\cos (x)) \]
Antiderivative was successfully verified.
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fricas [B] time = 1.49, size = 25, normalized size = 2.08 \[ \frac {1}{2} \, {\left (a - b\right )} \log \left (\sin \relax (x) + 1\right ) - \frac {1}{2} \, {\left (a + b\right )} \log \left (-\sin \relax (x) + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.13, size = 34, normalized size = 2.83 \[ \frac {1}{4} \, a {\left (\log \left ({\left | \frac {1}{\sin \relax (x)} + \sin \relax (x) + 2 \right |}\right ) - \log \left ({\left | \frac {1}{\sin \relax (x)} + \sin \relax (x) - 2 \right |}\right )\right )} - b \log \left ({\left | \cos \relax (x) \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 16, normalized size = 1.33 \[ a \ln \left (\sec \relax (x )+\tan \relax (x )\right )-b \ln \left (\cos \relax (x )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 14, normalized size = 1.17 \[ a \log \left (\sec \relax (x) + \tan \relax (x)\right ) + b \log \left (\sec \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.49, size = 37, normalized size = 3.08 \[ b\,\ln \left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )-\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )-1\right )\,\left (a+b\right )+\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )\,\left (a-b\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 24, normalized size = 2.00 \[ a \left (- \frac {\log {\left (\sin {\relax (x )} - 1 \right )}}{2} + \frac {\log {\left (\sin {\relax (x )} + 1 \right )}}{2}\right ) - b \log {\left (\cos {\relax (x )} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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