Optimal. Leaf size=17 \[ \frac{\csc ^3(x)}{3}-\frac{\csc ^5(x)}{5} \]
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Rubi [A] time = 0.0379673, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4397, 2606, 14} \[ \frac{\csc ^3(x)}{3}-\frac{\csc ^5(x)}{5} \]
Antiderivative was successfully verified.
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Rule 4397
Rule 2606
Rule 14
Rubi steps
\begin{align*} \int \frac{1}{(-\cos (x)+\sec (x))^3} \, dx &=\int \cot ^3(x) \csc ^3(x) \, dx\\ &=-\operatorname{Subst}\left (\int x^2 \left (-1+x^2\right ) \, dx,x,\csc (x)\right )\\ &=-\operatorname{Subst}\left (\int \left (-x^2+x^4\right ) \, dx,x,\csc (x)\right )\\ &=\frac{\csc ^3(x)}{3}-\frac{\csc ^5(x)}{5}\\ \end{align*}
Mathematica [A] time = 0.0091952, size = 17, normalized size = 1. \[ \frac{\csc ^3(x)}{3}-\frac{\csc ^5(x)}{5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 14, normalized size = 0.8 \begin{align*}{\frac{1}{3\, \left ( \sin \left ( x \right ) \right ) ^{3}}}-{\frac{1}{5\, \left ( \sin \left ( x \right ) \right ) ^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.988393, size = 99, normalized size = 5.82 \begin{align*} \frac{{\left (\frac{5 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac{30 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} - 3\right )}{\left (\cos \left (x\right ) + 1\right )}^{5}}{480 \, \sin \left (x\right )^{5}} + \frac{\sin \left (x\right )}{16 \,{\left (\cos \left (x\right ) + 1\right )}} + \frac{\sin \left (x\right )^{3}}{96 \,{\left (\cos \left (x\right ) + 1\right )}^{3}} - \frac{\sin \left (x\right )^{5}}{160 \,{\left (\cos \left (x\right ) + 1\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.095, size = 82, normalized size = 4.82 \begin{align*} -\frac{5 \, \cos \left (x\right )^{2} - 2}{15 \,{\left (\cos \left (x\right )^{4} - 2 \, \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}{\cos ^{3}{\left (x \right )} - 3 \cos ^{2}{\left (x \right )} \sec{\left (x \right )} + 3 \cos{\left (x \right )} \sec ^{2}{\left (x \right )} - \sec ^{3}{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11208, size = 19, normalized size = 1.12 \begin{align*} \frac{5 \, \sin \left (x\right )^{2} - 3}{15 \, \sin \left (x\right )^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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