Integrand size = 9, antiderivative size = 17 \[ \int \cot ^3(x) \csc ^4(x) \, dx=\frac {\csc ^4(x)}{4}-\frac {\csc ^6(x)}{6} \]
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Time = 0.02 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2686, 14} \[ \int \cot ^3(x) \csc ^4(x) \, dx=\frac {\csc ^4(x)}{4}-\frac {\csc ^6(x)}{6} \]
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Rule 14
Rule 2686
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int x^3 \left (-1+x^2\right ) \, dx,x,\csc (x)\right ) \\ & = -\text {Subst}\left (\int \left (-x^3+x^5\right ) \, dx,x,\csc (x)\right ) \\ & = \frac {\csc ^4(x)}{4}-\frac {\csc ^6(x)}{6} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \cot ^3(x) \csc ^4(x) \, dx=\frac {\csc ^4(x)}{4}-\frac {\csc ^6(x)}{6} \]
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Time = 0.07 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.29
method | result | size |
default | \(-\frac {\cos ^{4}\left (x \right )}{6 \sin \left (x \right )^{6}}-\frac {\cos ^{4}\left (x \right )}{12 \sin \left (x \right )^{4}}\) | \(22\) |
norman | \(\frac {-\frac {1}{384}+\frac {3 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{128}+\frac {3 \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{128}-\frac {\left (\tan ^{12}\left (\frac {x}{2}\right )\right )}{384}}{\tan \left (\frac {x}{2}\right )^{6}}\) | \(34\) |
risch | \(\frac {4 \,{\mathrm e}^{8 i x}+\frac {8 \,{\mathrm e}^{6 i x}}{3}+4 \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{6}}\) | \(34\) |
parallelrisch | \(\frac {-\left (\tan ^{12}\left (\frac {x}{2}\right )\right )+9 \left (\tan ^{8}\left (\frac {x}{2}\right )\right )+9 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )-1}{384 \tan \left (\frac {x}{2}\right )^{6}}\) | \(35\) |
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Leaf count of result is larger than twice the leaf count of optimal. 30 vs. \(2 (13) = 26\).
Time = 0.30 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.76 \[ \int \cot ^3(x) \csc ^4(x) \, dx=\frac {3 \, \cos \left (x\right )^{2} - 1}{12 \, {\left (\cos \left (x\right )^{6} - 3 \, \cos \left (x\right )^{4} + 3 \, \cos \left (x\right )^{2} - 1\right )}} \]
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Time = 0.04 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \cot ^3(x) \csc ^4(x) \, dx=- \frac {2 - 3 \sin ^{2}{\left (x \right )}}{12 \sin ^{6}{\left (x \right )}} \]
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Time = 0.18 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int \cot ^3(x) \csc ^4(x) \, dx=\frac {3 \, \sin \left (x\right )^{2} - 2}{12 \, \sin \left (x\right )^{6}} \]
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Time = 0.30 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int \cot ^3(x) \csc ^4(x) \, dx=\frac {3 \, \sin \left (x\right )^{2} - 2}{12 \, \sin \left (x\right )^{6}} \]
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Time = 0.15 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \cot ^3(x) \csc ^4(x) \, dx=\frac {\frac {{\sin \left (x\right )}^2}{4}-\frac {1}{6}}{{\sin \left (x\right )}^6} \]
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