Optimal. Leaf size=5 \[ \log (\sin (x)+1) \]
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Rubi [A] time = 0.013264, antiderivative size = 5, normalized size of antiderivative = 1., 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 2667
Rule 31
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*}
Mathematica [B] time = 0.0070203, size = 36, normalized size = 7.2 \[ \log (\cos (x))-\log \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )+\log \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.021, size = 6, normalized size = 1.2 \begin{align*} \ln \left ( 1+\sin \left ( x \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.935342, size = 7, normalized size = 1.4 \begin{align*} \log \left (\sin \left (x\right ) + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.07607, size = 23, normalized size = 4.6 \begin{align*} \log \left (\sin \left (x\right ) + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.317999, size = 19, normalized size = 3.8 \begin{align*} 2 \log{\left (\tan{\left (\frac{x}{2} \right )} + 1 \right )} - \log{\left (\tan ^{2}{\left (\frac{x}{2} \right )} + 1 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05386, size = 7, normalized size = 1.4 \begin{align*} \log \left (\sin \left (x\right ) + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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