Optimal. Leaf size=11 \[ \frac{\cos (x)}{1-\sin (x)} \]
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Rubi [A] time = 0.0287537, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3165, 2648} \[ \frac{\cos (x)}{1-\sin (x)} \]
Antiderivative was successfully verified.
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Rule 3165
Rule 2648
Rubi steps
\begin{align*} \int \frac{\sec (x)}{\sec (x)-\tan (x)} \, dx &=\int \frac{1}{1-\sin (x)} \, dx\\ &=\frac{\cos (x)}{1-\sin (x)}\\ \end{align*}
Mathematica [B] time = 0.0163485, size = 25, normalized size = 2.27 \[ \frac{2 \sin \left (\frac{x}{2}\right )}{\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.059, size = 11, normalized size = 1. \begin{align*} -2\, \left ( \tan \left ( x/2 \right ) -1 \right ) ^{-1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.982344, size = 20, normalized size = 1.82 \begin{align*} -\frac{2}{\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.457423, size = 61, normalized size = 5.55 \begin{align*} \frac{\cos \left (x\right ) + \sin \left (x\right ) + 1}{\cos \left (x\right ) - \sin \left (x\right ) + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec{\left (x \right )}}{- \tan{\left (x \right )} + \sec{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15903, size = 14, normalized size = 1.27 \begin{align*} -\frac{2}{\tan \left (\frac{1}{2} \, x\right ) - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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