Optimal. Leaf size=33 \[ -\frac {1}{13} \csc ^{13}(x)+\frac {3 \csc ^{11}(x)}{11}-\frac {\csc ^9(x)}{3}+\frac {\csc ^7(x)}{7} \]
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Rubi [A] time = 0.04, antiderivative size = 33, 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, 270} \[ -\frac {1}{13} \csc ^{13}(x)+\frac {3 \csc ^{11}(x)}{11}-\frac {\csc ^9(x)}{3}+\frac {\csc ^7(x)}{7} \]
Antiderivative was successfully verified.
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Rule 270
Rule 2606
Rule 4397
Rubi steps
\begin {align*} \int \frac {1}{(-\cos (x)+\sec (x))^7} \, dx &=\int \cot ^7(x) \csc ^7(x) \, dx\\ &=-\operatorname {Subst}\left (\int x^6 \left (-1+x^2\right )^3 \, dx,x,\csc (x)\right )\\ &=-\operatorname {Subst}\left (\int \left (-x^6+3 x^8-3 x^{10}+x^{12}\right ) \, dx,x,\csc (x)\right )\\ &=\frac {\csc ^7(x)}{7}-\frac {\csc ^9(x)}{3}+\frac {3 \csc ^{11}(x)}{11}-\frac {\csc ^{13}(x)}{13}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 33, normalized size = 1.00 \[ -\frac {1}{13} \csc ^{13}(x)+\frac {3 \csc ^{11}(x)}{11}-\frac {\csc ^9(x)}{3}+\frac {\csc ^7(x)}{7} \]
Antiderivative was successfully verified.
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fricas [B] time = 2.27, size = 64, normalized size = 1.94 \[ -\frac {429 \, \cos \relax (x)^{6} - 286 \, \cos \relax (x)^{4} + 104 \, \cos \relax (x)^{2} - 16}{3003 \, {\left (\cos \relax (x)^{12} - 6 \, \cos \relax (x)^{10} + 15 \, \cos \relax (x)^{8} - 20 \, \cos \relax (x)^{6} + 15 \, \cos \relax (x)^{4} - 6 \, \cos \relax (x)^{2} + 1\right )} \sin \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 26, normalized size = 0.79 \[ \frac {429 \, \sin \relax (x)^{6} - 1001 \, \sin \relax (x)^{4} + 819 \, \sin \relax (x)^{2} - 231}{3003 \, \sin \relax (x)^{13}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 26, normalized size = 0.79 \[ -\frac {1}{13 \sin \relax (x )^{13}}-\frac {1}{3 \sin \relax (x )^{9}}+\frac {1}{7 \sin \relax (x )^{7}}+\frac {3}{11 \sin \relax (x )^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 169, normalized size = 5.12 \[ \frac {{\left (\frac {273 \, \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + \frac {2002 \, \sin \relax (x)^{4}}{{\left (\cos \relax (x) + 1\right )}^{4}} - \frac {2574 \, \sin \relax (x)^{6}}{{\left (\cos \relax (x) + 1\right )}^{6}} - \frac {9009 \, \sin \relax (x)^{8}}{{\left (\cos \relax (x) + 1\right )}^{8}} + \frac {15015 \, \sin \relax (x)^{10}}{{\left (\cos \relax (x) + 1\right )}^{10}} + \frac {60060 \, \sin \relax (x)^{12}}{{\left (\cos \relax (x) + 1\right )}^{12}} - 231\right )} {\left (\cos \relax (x) + 1\right )}^{13}}{24600576 \, \sin \relax (x)^{13}} + \frac {5 \, \sin \relax (x)}{2048 \, {\left (\cos \relax (x) + 1\right )}} + \frac {5 \, \sin \relax (x)^{3}}{8192 \, {\left (\cos \relax (x) + 1\right )}^{3}} - \frac {3 \, \sin \relax (x)^{5}}{8192 \, {\left (\cos \relax (x) + 1\right )}^{5}} - \frac {3 \, \sin \relax (x)^{7}}{28672 \, {\left (\cos \relax (x) + 1\right )}^{7}} + \frac {\sin \relax (x)^{9}}{12288 \, {\left (\cos \relax (x) + 1\right )}^{9}} + \frac {\sin \relax (x)^{11}}{90112 \, {\left (\cos \relax (x) + 1\right )}^{11}} - \frac {\sin \relax (x)^{13}}{106496 \, {\left (\cos \relax (x) + 1\right )}^{13}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.51, size = 109, normalized size = 3.30 \[ -\frac {{\mathrm {cot}\left (\frac {x}{2}\right )}^{13}}{106496}+\frac {{\mathrm {cot}\left (\frac {x}{2}\right )}^{11}}{90112}+\frac {{\mathrm {cot}\left (\frac {x}{2}\right )}^9}{12288}-\frac {3\,{\mathrm {cot}\left (\frac {x}{2}\right )}^7}{28672}-\frac {3\,{\mathrm {cot}\left (\frac {x}{2}\right )}^5}{8192}+\frac {5\,{\mathrm {cot}\left (\frac {x}{2}\right )}^3}{8192}+\frac {5\,\mathrm {cot}\left (\frac {x}{2}\right )}{2048}-\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^{13}}{106496}+\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^{11}}{90112}+\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^9}{12288}-\frac {3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^7}{28672}-\frac {3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5}{8192}+\frac {5\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3}{8192}+\frac {5\,\mathrm {tan}\left (\frac {x}{2}\right )}{2048} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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