Optimal. Leaf size=4 \[ -\csc (x) \]
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Rubi [A] time = 0.0175251, antiderivative size = 4, 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 = {4397, 2606, 8} \[ -\csc (x) \]
Antiderivative was successfully verified.
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Rule 4397
Rule 2606
Rule 8
Rubi steps
\begin{align*} \int \frac{1}{-\cos (x)+\sec (x)} \, dx &=\int \cot (x) \csc (x) \, dx\\ &=-\operatorname{Subst}(\int 1 \, dx,x,\csc (x))\\ &=-\csc (x)\\ \end{align*}
Mathematica [A] time = 0.0026784, size = 4, normalized size = 1. \[ -\csc (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 7, normalized size = 1.8 \begin{align*} - \left ( \sin \left ( x \right ) \right ) ^{-1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.996598, size = 28, normalized size = 7. \begin{align*} -\frac{\cos \left (x\right ) + 1}{2 \, \sin \left (x\right )} - \frac{\sin \left (x\right )}{2 \,{\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 = 1.95337, size = 15, normalized size = 3.75 \begin{align*} -\frac{1}{\sin \left (x\right )} \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{1}{\cos{\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 = 1.10473, size = 8, normalized size = 2. \begin{align*} -\frac{1}{\sin \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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