Optimal. Leaf size=17 \[ \frac {\csc ^3(x)}{3}-\frac {\csc ^5(x)}{5} \]
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Rubi [A] time = 0.04, antiderivative size = 17, normalized size of antiderivative = 1.00, 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 14
Rule 2606
Rule 4397
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*}
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Mathematica [A] time = 0.01, size = 17, normalized size = 1.00 \[ \frac {\csc ^3(x)}{3}-\frac {\csc ^5(x)}{5} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.02, size = 28, normalized size = 1.65 \[ -\frac {5 \, \cos \relax (x)^{2} - 2}{15 \, {\left (\cos \relax (x)^{4} - 2 \, \cos \relax (x)^{2} + 1\right )} \sin \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 14, normalized size = 0.82 \[ \frac {5 \, \sin \relax (x)^{2} - 3}{15 \, \sin \relax (x)^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 14, normalized size = 0.82 \[ \frac {1}{3 \sin \relax (x )^{3}}-\frac {1}{5 \sin \relax (x )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 73, normalized size = 4.29 \[ \frac {{\left (\frac {5 \, \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + \frac {30 \, \sin \relax (x)^{4}}{{\left (\cos \relax (x) + 1\right )}^{4}} - 3\right )} {\left (\cos \relax (x) + 1\right )}^{5}}{480 \, \sin \relax (x)^{5}} + \frac {\sin \relax (x)}{16 \, {\left (\cos \relax (x) + 1\right )}} + \frac {\sin \relax (x)^{3}}{96 \, {\left (\cos \relax (x) + 1\right )}^{3}} - \frac {\sin \relax (x)^{5}}{160 \, {\left (\cos \relax (x) + 1\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.38, size = 14, normalized size = 0.82 \[ \frac {5\,{\sin \relax (x)}^2-3}{15\,{\sin \relax (x)}^5} \]
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 \[ - \int \frac {1}{\cos ^{3}{\relax (x )} - 3 \cos ^{2}{\relax (x )} \sec {\relax (x )} + 3 \cos {\relax (x )} \sec ^{2}{\relax (x )} - \sec ^{3}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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