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 = 0.24 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82
method | result | size |
derivativedivides | \(-\frac {\left (\cot ^{6}\left (x \right )\right )}{6}-\frac {\left (\cot ^{4}\left (x \right )\right )}{4}\) | \(14\) |
default | \(-\frac {\left (\cot ^{6}\left (x \right )\right )}{6}-\frac {\left (\cot ^{4}\left (x \right )\right )}{4}\) | \(14\) |
risch | \(\frac {4 \,{\mathrm e}^{8 i x}+\frac {8 \,{\mathrm e}^{6 i x}}{3}+4 \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{6}}\) | \(34\) |
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Leaf count of result is larger than twice the leaf count of optimal. 30 vs. \(2 (13) = 26\).
Time = 0.25 (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 = 0.05 (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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none
Time = 0.19 (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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none
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 = 0.22 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int \cot ^3(x) \csc ^4(x) \, dx=-\frac {{\mathrm {cot}\left (x\right )}^4\,\left (2\,{\mathrm {cot}\left (x\right )}^2+3\right )}{12} \]
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