Optimal. Leaf size=8 \[ -\frac{1}{3} \cot ^3(x) \]
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Rubi [A] time = 0.0140351, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {30} \[ -\frac{1}{3} \cot ^3(x) \]
Antiderivative was successfully verified.
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Rule 30
Rubi steps
\begin{align*} \int \frac{1}{(-\cos (x)+\sec (x))^2} \, dx &=\operatorname{Subst}\left (\int \frac{1}{x^4} \, dx,x,\tan (x)\right )\\ &=-\frac{1}{3} \cot ^3(x)\\ \end{align*}
Mathematica [A] time = 0.0025405, size = 8, normalized size = 1. \[ -\frac{1}{3} \cot ^3(x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 7, normalized size = 0.9 \begin{align*} -{\frac{1}{3\, \left ( \tan \left ( x \right ) \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.980036, size = 8, normalized size = 1. \begin{align*} -\frac{1}{3 \, \tan \left (x\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.03443, size = 51, normalized size = 6.38 \begin{align*} \frac{\cos \left (x\right )^{3}}{3 \,{\left (\cos \left (x\right )^{2} - 1\right )} \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}{\left (- \cos{\left (x \right )} + \sec{\left (x \right )}\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14738, size = 8, normalized size = 1. \begin{align*} -\frac{1}{3 \, \tan \left (x\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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