Integrand size = 4, antiderivative size = 13 \[ \int \csc ^4(x) \, dx=-\cot (x)-\frac {\cot ^3(x)}{3} \]
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Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3852} \[ \int \csc ^4(x) \, dx=-\frac {1}{3} \cot ^3(x)-\cot (x) \]
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Rule 3852
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (x)\right ) \\ & = -\cot (x)-\frac {\cot ^3(x)}{3} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.31 \[ \int \csc ^4(x) \, dx=-\frac {2 \cot (x)}{3}-\frac {1}{3} \cot (x) \csc ^2(x) \]
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Time = 0.24 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92
method | result | size |
default | \(\left (-\frac {2}{3}-\frac {\left (\csc ^{2}\left (x \right )\right )}{3}\right ) \cot \left (x \right )\) | \(12\) |
parallelrisch | \(\frac {2 \left (\cot ^{3}\left (x \right )\right )}{3}-\cot \left (x \right ) \left (\csc ^{2}\left (x \right )\right )\) | \(16\) |
risch | \(\frac {4 i \left (3 \,{\mathrm e}^{2 i x}-1\right )}{3 \left ({\mathrm e}^{2 i x}-1\right )^{3}}\) | \(22\) |
norman | \(\frac {-\frac {1}{24}-\frac {3 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{8}+\frac {3 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{8}+\frac {\left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{24}}{\tan \left (\frac {x}{2}\right )^{3}}\) | \(34\) |
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Leaf count of result is larger than twice the leaf count of optimal. 25 vs. \(2 (11) = 22\).
Time = 0.24 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.92 \[ \int \csc ^4(x) \, dx=-\frac {2 \, \cos \left (x\right )^{3} - 3 \, \cos \left (x\right )}{3 \, {\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )} \]
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Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.54 \[ \int \csc ^4(x) \, dx=- \frac {2 \cos {\left (x \right )}}{3 \sin {\left (x \right )}} - \frac {\cos {\left (x \right )}}{3 \sin ^{3}{\left (x \right )}} \]
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none
Time = 0.19 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08 \[ \int \csc ^4(x) \, dx=-\frac {3 \, \tan \left (x\right )^{2} + 1}{3 \, \tan \left (x\right )^{3}} \]
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none
Time = 0.28 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08 \[ \int \csc ^4(x) \, dx=-\frac {3 \, \tan \left (x\right )^{2} + 1}{3 \, \tan \left (x\right )^{3}} \]
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Time = 0.03 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.31 \[ \int \csc ^4(x) \, dx=-\frac {2\,\cos \left (x\right )\,{\sin \left (x\right )}^2+\cos \left (x\right )}{3\,{\sin \left (x\right )}^3} \]
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