Optimal. Leaf size=5 \[ \log (\sin (x)+1) \]
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Rubi [A] time = 0.01, antiderivative size = 5, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2667, 31} \[ \log (\sin (x)+1) \]
Antiderivative was successfully verified.
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Rule 31
Rule 2667
Rubi steps
\begin {align*} \int \sec (x) (1-\sin (x)) \, dx &=-\operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,-\sin (x)\right )\\ &=\log (1+\sin (x))\\ \end {align*}
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Mathematica [B] time = 0.01, size = 36, normalized size = 7.20 \[ \log (\cos (x))-\log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )+\log \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 5, normalized size = 1.00 \[ \log \left (\sin \relax (x) + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.91, size = 5, normalized size = 1.00 \[ \log \left (\sin \relax (x) + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 6, normalized size = 1.20 \[ \ln \left (\sin \relax (x )+1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 5, normalized size = 1.00 \[ \log \left (\sin \relax (x) + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.17, size = 5, normalized size = 1.00 \[ \ln \left (\sin \relax (x)+1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.33, size = 19, normalized size = 3.80 \[ 2 \log {\left (\tan {\left (\frac {x}{2} \right )} + 1 \right )} - \log {\left (\tan ^{2}{\left (\frac {x}{2} \right )} + 1 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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