Integrand size = 9, antiderivative size = 17 \[ \int \frac {1}{(\csc (x)-\sin (x))^3} \, dx=-\frac {1}{3} \sec ^3(x)+\frac {\sec ^5(x)}{5} \]
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Time = 0.04 (sec) , antiderivative size = 17, 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, 14} \[ \int \frac {1}{(\csc (x)-\sin (x))^3} \, dx=\frac {\sec ^5(x)}{5}-\frac {\sec ^3(x)}{3} \]
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Rule 14
Rule 2686
Rule 4482
Rubi steps \begin{align*} \text {integral}& = \int \sec ^3(x) \tan ^3(x) \, dx \\ & = \text {Subst}\left (\int x^2 \left (-1+x^2\right ) \, dx,x,\sec (x)\right ) \\ & = \text {Subst}\left (\int \left (-x^2+x^4\right ) \, dx,x,\sec (x)\right ) \\ & = -\frac {1}{3} \sec ^3(x)+\frac {\sec ^5(x)}{5} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(\csc (x)-\sin (x))^3} \, dx=-\frac {1}{3} \sec ^3(x)+\frac {\sec ^5(x)}{5} \]
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Time = 0.61 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82
method | result | size |
default | \(-\frac {1}{3 \cos \left (x \right )^{3}}+\frac {1}{5 \cos \left (x \right )^{5}}\) | \(14\) |
parallelrisch | \(\frac {4}{15}-\frac {\sec \left (x \right )^{3}}{3}+\frac {\sec \left (x \right )^{5}}{5}\) | \(15\) |
risch | \(-\frac {8 \left (5 \,{\mathrm e}^{7 i x}-2 \,{\mathrm e}^{5 i x}+5 \,{\mathrm e}^{3 i x}\right )}{15 \left ({\mathrm e}^{2 i x}+1\right )^{5}}\) | \(34\) |
norman | \(\frac {-4 \tan \left (\frac {x}{2}\right )^{6}-\frac {4 \tan \left (\frac {x}{2}\right )^{4}}{3}-\frac {4 \tan \left (\frac {x}{2}\right )^{2}}{3}+\frac {4}{15}}{\left (\tan \left (\frac {x}{2}\right )^{2}-1\right )^{5}}\) | \(38\) |
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none
Time = 0.23 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int \frac {1}{(\csc (x)-\sin (x))^3} \, dx=-\frac {5 \, \cos \left (x\right )^{2} - 3}{15 \, \cos \left (x\right )^{5}} \]
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\[ \int \frac {1}{(\csc (x)-\sin (x))^3} \, dx=\int \frac {1}{\left (- \sin {\left (x \right )} + \csc {\left (x \right )}\right )^{3}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 103 vs. \(2 (13) = 26\).
Time = 0.23 (sec) , antiderivative size = 103, normalized size of antiderivative = 6.06 \[ \int \frac {1}{(\csc (x)-\sin (x))^3} \, dx=-\frac {4 \, {\left (\frac {5 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {5 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {15 \, \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} - 1\right )}}{15 \, {\left (\frac {5 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac {10 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {10 \, \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} - \frac {5 \, \sin \left (x\right )^{8}}{{\left (\cos \left (x\right ) + 1\right )}^{8}} + \frac {\sin \left (x\right )^{10}}{{\left (\cos \left (x\right ) + 1\right )}^{10}} - 1\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (13) = 26\).
Time = 0.27 (sec) , antiderivative size = 59, normalized size of antiderivative = 3.47 \[ \int \frac {1}{(\csc (x)-\sin (x))^3} \, dx=-\frac {4 \, {\left (\frac {5 \, {\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} - \frac {5 \, {\left (\cos \left (x\right ) - 1\right )}^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {15 \, {\left (\cos \left (x\right ) - 1\right )}^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + 1\right )}}{15 \, {\left (\frac {\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1} + 1\right )}^{5}} \]
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Time = 28.04 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \frac {1}{(\csc (x)-\sin (x))^3} \, dx=\frac {1}{5\,{\cos \left (x\right )}^5}-\frac {1}{3\,{\cos \left (x\right )}^3} \]
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