Optimal. Leaf size=9 \[ -\log (1-\sin (x)) \]
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Rubi [A] time = 0.0270939, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3159, 2667, 31} \[ -\log (1-\sin (x)) \]
Antiderivative was successfully verified.
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Rule 3159
Rule 2667
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{\sec (x)-\tan (x)} \, dx &=\int \frac{\cos (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 [A] time = 0.0195591, size = 18, normalized size = 2. \[ -2 \log \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.067, size = 8, normalized size = 0.9 \begin{align*} -\ln \left ( \sin \left ( x \right ) -1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.09662, size = 39, normalized size = 4.33 \begin{align*} -2 \, \log \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} - 1\right ) + \log \left (\frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.482665, size = 26, normalized size = 2.89 \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 = 0.233267, size = 17, normalized size = 1.89 \begin{align*} - \log{\left (- \tan{\left (x \right )} + \sec{\left (x \right )} \right )} + \frac{\log{\left (\tan ^{2}{\left (x \right )} + 1 \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.11513, size = 27, normalized size = 3. \begin{align*} \log \left (\tan \left (\frac{1}{2} \, x\right )^{2} + 1\right ) - 2 \, \log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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