Integrand size = 9, antiderivative size = 33 \[ \int \frac {1}{(-\cos (x)+\sec (x))^7} \, dx=\frac {\csc ^7(x)}{7}-\frac {\csc ^9(x)}{3}+\frac {3 \csc ^{11}(x)}{11}-\frac {\csc ^{13}(x)}{13} \]
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Time = 0.05 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4482, 2686, 276} \[ \int \frac {1}{(-\cos (x)+\sec (x))^7} \, dx=-\frac {1}{13} \csc ^{13}(x)+\frac {3 \csc ^{11}(x)}{11}-\frac {\csc ^9(x)}{3}+\frac {\csc ^7(x)}{7} \]
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Rule 276
Rule 2686
Rule 4482
Rubi steps \begin{align*} \text {integral}& = \int \cot ^7(x) \csc ^7(x) \, dx \\ & = -\text {Subst}\left (\int x^6 \left (-1+x^2\right )^3 \, dx,x,\csc (x)\right ) \\ & = -\text {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*}
Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(-\cos (x)+\sec (x))^7} \, dx=\frac {\csc ^7(x)}{7}-\frac {\csc ^9(x)}{3}+\frac {3 \csc ^{11}(x)}{11}-\frac {\csc ^{13}(x)}{13} \]
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Time = 0.59 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79
method | result | size |
default | \(-\frac {1}{13 \sin \left (x \right )^{13}}+\frac {1}{7 \sin \left (x \right )^{7}}+\frac {3}{11 \sin \left (x \right )^{11}}-\frac {1}{3 \sin \left (x \right )^{9}}\) | \(26\) |
parallelrisch | \(-\frac {\csc \left (x \right )^{13} \left (2010+429 \cos \left (6 x \right )+1430 \cos \left (4 x \right )+3523 \cos \left (2 x \right )\right )}{96096}\) | \(27\) |
risch | \(-\frac {128 i \left (429 \,{\mathrm e}^{19 i x}+1430 \,{\mathrm e}^{17 i x}+3523 \,{\mathrm e}^{15 i x}+4020 \,{\mathrm e}^{13 i x}+3523 \,{\mathrm e}^{11 i x}+1430 \,{\mathrm e}^{9 i x}+429 \,{\mathrm e}^{7 i x}\right )}{3003 \left ({\mathrm e}^{2 i x}-1\right )^{13}}\) | \(63\) |
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Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (25) = 50\).
Time = 0.23 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.94 \[ \int \frac {1}{(-\cos (x)+\sec (x))^7} \, dx=-\frac {429 \, \cos \left (x\right )^{6} - 286 \, \cos \left (x\right )^{4} + 104 \, \cos \left (x\right )^{2} - 16}{3003 \, {\left (\cos \left (x\right )^{12} - 6 \, \cos \left (x\right )^{10} + 15 \, \cos \left (x\right )^{8} - 20 \, \cos \left (x\right )^{6} + 15 \, \cos \left (x\right )^{4} - 6 \, \cos \left (x\right )^{2} + 1\right )} \sin \left (x\right )} \]
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Timed out. \[ \int \frac {1}{(-\cos (x)+\sec (x))^7} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 169 vs. \(2 (25) = 50\).
Time = 0.22 (sec) , antiderivative size = 169, normalized size of antiderivative = 5.12 \[ \int \frac {1}{(-\cos (x)+\sec (x))^7} \, dx=\frac {{\left (\frac {273 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {2002 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} - \frac {2574 \, \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} - \frac {9009 \, \sin \left (x\right )^{8}}{{\left (\cos \left (x\right ) + 1\right )}^{8}} + \frac {15015 \, \sin \left (x\right )^{10}}{{\left (\cos \left (x\right ) + 1\right )}^{10}} + \frac {60060 \, \sin \left (x\right )^{12}}{{\left (\cos \left (x\right ) + 1\right )}^{12}} - 231\right )} {\left (\cos \left (x\right ) + 1\right )}^{13}}{24600576 \, \sin \left (x\right )^{13}} + \frac {5 \, \sin \left (x\right )}{2048 \, {\left (\cos \left (x\right ) + 1\right )}} + \frac {5 \, \sin \left (x\right )^{3}}{8192 \, {\left (\cos \left (x\right ) + 1\right )}^{3}} - \frac {3 \, \sin \left (x\right )^{5}}{8192 \, {\left (\cos \left (x\right ) + 1\right )}^{5}} - \frac {3 \, \sin \left (x\right )^{7}}{28672 \, {\left (\cos \left (x\right ) + 1\right )}^{7}} + \frac {\sin \left (x\right )^{9}}{12288 \, {\left (\cos \left (x\right ) + 1\right )}^{9}} + \frac {\sin \left (x\right )^{11}}{90112 \, {\left (\cos \left (x\right ) + 1\right )}^{11}} - \frac {\sin \left (x\right )^{13}}{106496 \, {\left (\cos \left (x\right ) + 1\right )}^{13}} \]
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Time = 0.28 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79 \[ \int \frac {1}{(-\cos (x)+\sec (x))^7} \, dx=\frac {429 \, \sin \left (x\right )^{6} - 1001 \, \sin \left (x\right )^{4} + 819 \, \sin \left (x\right )^{2} - 231}{3003 \, \sin \left (x\right )^{13}} \]
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Time = 27.34 (sec) , antiderivative size = 109, normalized size of antiderivative = 3.30 \[ \int \frac {1}{(-\cos (x)+\sec (x))^7} \, dx=-\frac {{\mathrm {cot}\left (\frac {x}{2}\right )}^{13}}{106496}+\frac {{\mathrm {cot}\left (\frac {x}{2}\right )}^{11}}{90112}+\frac {{\mathrm {cot}\left (\frac {x}{2}\right )}^9}{12288}-\frac {3\,{\mathrm {cot}\left (\frac {x}{2}\right )}^7}{28672}-\frac {3\,{\mathrm {cot}\left (\frac {x}{2}\right )}^5}{8192}+\frac {5\,{\mathrm {cot}\left (\frac {x}{2}\right )}^3}{8192}+\frac {5\,\mathrm {cot}\left (\frac {x}{2}\right )}{2048}-\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^{13}}{106496}+\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^{11}}{90112}+\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^9}{12288}-\frac {3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^7}{28672}-\frac {3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5}{8192}+\frac {5\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3}{8192}+\frac {5\,\mathrm {tan}\left (\frac {x}{2}\right )}{2048} \]
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