Optimal. Leaf size=17 \[ -\frac{1}{7} \cot ^7(x)-\frac{\cot ^5(x)}{5} \]
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Rubi [A] time = 0.0171948, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 0, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ -\frac{1}{7} \cot ^7(x)-\frac{\cot ^5(x)}{5} \]
Antiderivative was successfully verified.
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Rubi steps
\begin{align*} \int \frac{1}{(-\cos (x)+\sec (x))^4} \, dx &=\operatorname{Subst}\left (\int \left (\frac{1}{x^8}+\frac{1}{x^6}\right ) \, dx,x,\tan (x)\right )\\ &=-\frac{1}{5} \cot ^5(x)-\frac{\cot ^7(x)}{7}\\ \end{align*}
Mathematica [B] time = 0.0224526, size = 37, normalized size = 2.18 \[ -\frac{2 \cot (x)}{35}-\frac{1}{7} \cot (x) \csc ^6(x)+\frac{8}{35} \cot (x) \csc ^4(x)-\frac{1}{35} \cot (x) \csc ^2(x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.037, size = 14, normalized size = 0.8 \begin{align*} -{\frac{1}{7\, \left ( \tan \left ( x \right ) \right ) ^{7}}}-{\frac{1}{5\, \left ( \tan \left ( x \right ) \right ) ^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.997813, size = 19, normalized size = 1.12 \begin{align*} -\frac{7 \, \tan \left (x\right )^{2} + 5}{35 \, \tan \left (x\right )^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.30063, size = 112, normalized size = 6.59 \begin{align*} -\frac{2 \, \cos \left (x\right )^{7} - 7 \, \cos \left (x\right )^{5}}{35 \,{\left (\cos \left (x\right )^{6} - 3 \, \cos \left (x\right )^{4} + 3 \, \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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1702, size = 19, normalized size = 1.12 \begin{align*} -\frac{7 \, \tan \left (x\right )^{2} + 5}{35 \, \tan \left (x\right )^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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