Optimal. Leaf size=7 \[ -\log (\cos (x)+1) \]
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Rubi [A] time = 0.031622, antiderivative size = 7, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {4339, 31} \[ -\log (\cos (x)+1) \]
Antiderivative was successfully verified.
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Rule 4339
Rule 31
Rubi steps
\begin{align*} \int \frac{\sec (x) \tan (x)}{\sec (x)+\sec ^2(x)} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,\cos (x)\right )\\ &=-\log (1+\cos (x))\\ \end{align*}
Mathematica [A] time = 0.0055143, size = 9, normalized size = 1.29 \[ -2 \log \left (\cos \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.028, size = 12, normalized size = 1.7 \begin{align*} \ln \left ( \sec \left ( x \right ) \right ) -\ln \left ( 1+\sec \left ( x \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.954864, size = 9, normalized size = 1.29 \begin{align*} -\log \left (\cos \left (x\right ) + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.41407, size = 32, normalized size = 4.57 \begin{align*} -\log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.256751, size = 15, normalized size = 2.14 \begin{align*} \frac{\log{\left (\tan ^{2}{\left (x \right )} + 1 \right )}}{2} - \log{\left (\sec{\left (x \right )} + 1 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.0896, size = 9, normalized size = 1.29 \begin{align*} -\log \left (\cos \left (x\right ) + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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