Integrand size = 9, antiderivative size = 25 \[ \int \frac {1}{(\csc (x)-\sin (x))^5} \, dx=\frac {\sec ^5(x)}{5}-\frac {2 \sec ^7(x)}{7}+\frac {\sec ^9(x)}{9} \]
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Time = 0.05 (sec) , antiderivative size = 25, 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} \[ \int \frac {1}{(\csc (x)-\sin (x))^5} \, dx=\frac {\sec ^9(x)}{9}-\frac {2 \sec ^7(x)}{7}+\frac {\sec ^5(x)}{5} \]
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Rule 276
Rule 2686
Rule 4482
Rubi steps \begin{align*} \text {integral}& = \int \sec ^5(x) \tan ^5(x) \, dx \\ & = \text {Subst}\left (\int x^4 \left (-1+x^2\right )^2 \, dx,x,\sec (x)\right ) \\ & = \text {Subst}\left (\int \left (x^4-2 x^6+x^8\right ) \, dx,x,\sec (x)\right ) \\ & = \frac {\sec ^5(x)}{5}-\frac {2 \sec ^7(x)}{7}+\frac {\sec ^9(x)}{9} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(\csc (x)-\sin (x))^5} \, dx=\frac {\sec ^5(x)}{5}-\frac {2 \sec ^7(x)}{7}+\frac {\sec ^9(x)}{9} \]
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Time = 0.94 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80
method | result | size |
default | \(\frac {1}{9 \cos \left (x \right )^{9}}-\frac {2}{7 \cos \left (x \right )^{7}}+\frac {1}{5 \cos \left (x \right )^{5}}\) | \(20\) |
parallelrisch | \(\frac {32}{315}+\frac {\sec \left (x \right )^{5}}{5}-\frac {2 \sec \left (x \right )^{7}}{7}+\frac {\sec \left (x \right )^{9}}{9}\) | \(21\) |
risch | \(\frac {\frac {32 \,{\mathrm e}^{13 i x}}{5}-\frac {384 \,{\mathrm e}^{11 i x}}{35}+\frac {6976 \,{\mathrm e}^{9 i x}}{315}-\frac {384 \,{\mathrm e}^{7 i x}}{35}+\frac {32 \,{\mathrm e}^{5 i x}}{5}}{\left ({\mathrm e}^{2 i x}+1\right )^{9}}\) | \(48\) |
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Time = 0.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {1}{(\csc (x)-\sin (x))^5} \, dx=\frac {63 \, \cos \left (x\right )^{4} - 90 \, \cos \left (x\right )^{2} + 35}{315 \, \cos \left (x\right )^{9}} \]
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\[ \int \frac {1}{(\csc (x)-\sin (x))^5} \, dx=\int \frac {1}{\left (- \sin {\left (x \right )} + \csc {\left (x \right )}\right )^{5}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 187 vs. \(2 (19) = 38\).
Time = 0.23 (sec) , antiderivative size = 187, normalized size of antiderivative = 7.48 \[ \int \frac {1}{(\csc (x)-\sin (x))^5} \, dx=\frac {16 \, {\left (\frac {9 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac {36 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} - \frac {126 \, \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} - \frac {441 \, \sin \left (x\right )^{8}}{{\left (\cos \left (x\right ) + 1\right )}^{8}} - \frac {315 \, \sin \left (x\right )^{10}}{{\left (\cos \left (x\right ) + 1\right )}^{10}} - \frac {210 \, \sin \left (x\right )^{12}}{{\left (\cos \left (x\right ) + 1\right )}^{12}} - 1\right )}}{315 \, {\left (\frac {9 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac {36 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {84 \, \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} - \frac {126 \, \sin \left (x\right )^{8}}{{\left (\cos \left (x\right ) + 1\right )}^{8}} + \frac {126 \, \sin \left (x\right )^{10}}{{\left (\cos \left (x\right ) + 1\right )}^{10}} - \frac {84 \, \sin \left (x\right )^{12}}{{\left (\cos \left (x\right ) + 1\right )}^{12}} + \frac {36 \, \sin \left (x\right )^{14}}{{\left (\cos \left (x\right ) + 1\right )}^{14}} - \frac {9 \, \sin \left (x\right )^{16}}{{\left (\cos \left (x\right ) + 1\right )}^{16}} + \frac {\sin \left (x\right )^{18}}{{\left (\cos \left (x\right ) + 1\right )}^{18}} - 1\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 101 vs. \(2 (19) = 38\).
Time = 0.29 (sec) , antiderivative size = 101, normalized size of antiderivative = 4.04 \[ \int \frac {1}{(\csc (x)-\sin (x))^5} \, dx=\frac {16 \, {\left (\frac {9 \, {\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} + \frac {36 \, {\left (\cos \left (x\right ) - 1\right )}^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac {126 \, {\left (\cos \left (x\right ) - 1\right )}^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {441 \, {\left (\cos \left (x\right ) - 1\right )}^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} - \frac {315 \, {\left (\cos \left (x\right ) - 1\right )}^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac {210 \, {\left (\cos \left (x\right ) - 1\right )}^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + 1\right )}}{315 \, {\left (\frac {\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1} + 1\right )}^{9}} \]
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Time = 29.59 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \frac {1}{(\csc (x)-\sin (x))^5} \, dx=\frac {1}{5\,{\cos \left (x\right )}^5}-\frac {2}{7\,{\cos \left (x\right )}^7}+\frac {1}{9\,{\cos \left (x\right )}^9} \]
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