Optimal. Leaf size=8 \[ \tanh ^{-1}(\sin (x))-\sin (x) \]
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Rubi [A] time = 0.00, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2637, 3770} \[ \tanh ^{-1}(\sin (x))-\sin (x) \]
Antiderivative was successfully verified.
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Rule 2637
Rule 3770
Rubi steps
\begin {align*} \int (-\cos (x)+\sec (x)) \, dx &=-\int \cos (x) \, dx+\int \sec (x) \, dx\\ &=\tanh ^{-1}(\sin (x))-\sin (x)\\ \end {align*}
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Mathematica [B] time = 0.00, size = 37, normalized size = 4.62 \[ -\sin (x)-\log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )+\log \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.95, size = 21, normalized size = 2.62 \[ \frac {1}{2} \, \log \left (\sin \relax (x) + 1\right ) - \frac {1}{2} \, \log \left (-\sin \relax (x) + 1\right ) - \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 29, normalized size = 3.62 \[ \frac {1}{4} \, \log \left ({\left | \frac {1}{\sin \relax (x)} + \sin \relax (x) + 2 \right |}\right ) - \frac {1}{4} \, \log \left ({\left | \frac {1}{\sin \relax (x)} + \sin \relax (x) - 2 \right |}\right ) - \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 12, normalized size = 1.50 \[ -\sin \relax (x )+\ln \left (\sec \relax (x )+\tan \relax (x )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 11, normalized size = 1.38 \[ \log \left (\sec \relax (x) + \tan \relax (x)\right ) - \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.30, size = 14, normalized size = 1.75 \[ \ln \left (\mathrm {tan}\left (\frac {x}{2}+\frac {\pi }{4}\right )\right )-\sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.09, size = 19, normalized size = 2.38 \[ - \frac {\log {\left (\sin {\relax (x )} - 1 \right )}}{2} + \frac {\log {\left (\sin {\relax (x )} + 1 \right )}}{2} - \sin {\relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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