Optimal. Leaf size=17 \[ -\frac {1}{7} \cot ^7(x)-\frac {\cot ^5(x)}{5} \]
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Rubi [A] time = 0.02, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 0, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}
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Mathematica [B] time = 0.02, 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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fricas [B] time = 0.65, size = 39, normalized size = 2.29 \[ -\frac {2 \, \cos \relax (x)^{7} - 7 \, \cos \relax (x)^{5}}{35 \, {\left (\cos \relax (x)^{6} - 3 \, \cos \relax (x)^{4} + 3 \, \cos \relax (x)^{2} - 1\right )} \sin \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 14, normalized size = 0.82 \[ -\frac {7 \, \tan \relax (x)^{2} + 5}{35 \, \tan \relax (x)^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 14, normalized size = 0.82 \[ -\frac {1}{7 \tan \relax (x )^{7}}-\frac {1}{5 \tan \relax (x )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 14, normalized size = 0.82 \[ -\frac {7 \, \tan \relax (x)^{2} + 5}{35 \, \tan \relax (x)^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.43, size = 16, normalized size = 0.94 \[ \frac {{\cos \relax (x)}^5\,\left (\cos \left (2\,x\right )-6\right )}{35\,{\sin \relax (x)}^7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (- \cos {\relax (x )} + \sec {\relax (x )}\right )^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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