Integrand size = 27, antiderivative size = 60 \[ \int \cot (3 x) \left (-1+\csc ^2(3 x)\right )^3 \left (1-\sin ^2(3 x)\right )^2 \, dx=-\frac {5}{3} \csc ^2(3 x)+\frac {5}{12} \csc ^4(3 x)-\frac {1}{18} \csc ^6(3 x)-\frac {10}{3} \log (\sin (3 x))+\frac {5}{6} \sin ^2(3 x)-\frac {1}{12} \sin ^4(3 x) \]
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Time = 0.15 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {3254, 4445, 272, 45} \[ \int \cot (3 x) \left (-1+\csc ^2(3 x)\right )^3 \left (1-\sin ^2(3 x)\right )^2 \, dx=-\frac {1}{12} \sin ^4(3 x)+\frac {5}{6} \sin ^2(3 x)-\frac {1}{18} \csc ^6(3 x)+\frac {5}{12} \csc ^4(3 x)-\frac {5}{3} \csc ^2(3 x)-\frac {10}{3} \log (\sin (3 x)) \]
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Rule 45
Rule 272
Rule 3254
Rule 4445
Rubi steps \begin{align*} \text {integral}& = \int \cos ^4(3 x) \cot (3 x) \left (-1+\csc ^2(3 x)\right )^3 \, dx \\ & = \frac {1}{3} \text {Subst}\left (\int \frac {\left (1-x^2\right )^5}{x^7} \, dx,x,\sin (3 x)\right ) \\ & = \frac {1}{6} \text {Subst}\left (\int \frac {(1-x)^5}{x^4} \, dx,x,\sin ^2(3 x)\right ) \\ & = \frac {1}{6} \text {Subst}\left (\int \left (5+\frac {1}{x^4}-\frac {5}{x^3}+\frac {10}{x^2}-\frac {10}{x}-x\right ) \, dx,x,\sin ^2(3 x)\right ) \\ & = -\frac {5}{3} \csc ^2(3 x)+\frac {5}{12} \csc ^4(3 x)-\frac {1}{18} \csc ^6(3 x)-\frac {10}{3} \log (\sin (3 x))+\frac {5}{6} \sin ^2(3 x)-\frac {1}{12} \sin ^4(3 x) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00 \[ \int \cot (3 x) \left (-1+\csc ^2(3 x)\right )^3 \left (1-\sin ^2(3 x)\right )^2 \, dx=-\frac {5}{3} \csc ^2(3 x)+\frac {5}{12} \csc ^4(3 x)-\frac {1}{18} \csc ^6(3 x)-\frac {10}{3} \log (\sin (3 x))+\frac {5}{6} \sin ^2(3 x)-\frac {1}{12} \sin ^4(3 x) \]
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Time = 45.95 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.82
method | result | size |
derivativedivides | \(-\frac {\sin \left (3 x \right )^{4}}{12}-\frac {5 \cos \left (3 x \right )^{2}}{6}-\frac {10 \ln \left (\sin \left (3 x \right )\right )}{3}-\frac {5}{3 \sin \left (3 x \right )^{2}}+\frac {5}{12 \sin \left (3 x \right )^{4}}-\frac {1}{18 \sin \left (3 x \right )^{6}}\) | \(49\) |
default | \(-\frac {\sin \left (3 x \right )^{4}}{12}-\frac {5 \cos \left (3 x \right )^{2}}{6}-\frac {10 \ln \left (\sin \left (3 x \right )\right )}{3}-\frac {5}{3 \sin \left (3 x \right )^{2}}+\frac {5}{12 \sin \left (3 x \right )^{4}}-\frac {1}{18 \sin \left (3 x \right )^{6}}\) | \(49\) |
risch | \(10 i x -\frac {{\mathrm e}^{12 i x}}{192}-\frac {3 \,{\mathrm e}^{6 i x}}{16}-\frac {3 \,{\mathrm e}^{-6 i x}}{16}-\frac {{\mathrm e}^{-12 i x}}{192}+\frac {\frac {20 \,{\mathrm e}^{30 i x}}{3}-20 \,{\mathrm e}^{24 i x}+\frac {272 \,{\mathrm e}^{18 i x}}{9}-20 \,{\mathrm e}^{12 i x}+\frac {20 \,{\mathrm e}^{6 i x}}{3}}{\left ({\mathrm e}^{6 i x}-1\right )^{6}}-\frac {10 \ln \left ({\mathrm e}^{6 i x}-1\right )}{3}\) | \(91\) |
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Leaf count of result is larger than twice the leaf count of optimal. 103 vs. \(2 (48) = 96\).
Time = 0.26 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.72 \[ \int \cot (3 x) \left (-1+\csc ^2(3 x)\right )^3 \left (1-\sin ^2(3 x)\right )^2 \, dx=-\frac {24 \, \cos \left (3 \, x\right )^{10} + 120 \, \cos \left (3 \, x\right )^{8} - 609 \, \cos \left (3 \, x\right )^{6} + 387 \, \cos \left (3 \, x\right )^{4} + 333 \, \cos \left (3 \, x\right )^{2} + 960 \, {\left (\cos \left (3 \, x\right )^{6} - 3 \, \cos \left (3 \, x\right )^{4} + 3 \, \cos \left (3 \, x\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \sin \left (3 \, x\right )\right ) - 271}{288 \, {\left (\cos \left (3 \, x\right )^{6} - 3 \, \cos \left (3 \, x\right )^{4} + 3 \, \cos \left (3 \, x\right )^{2} - 1\right )}} \]
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Timed out. \[ \int \cot (3 x) \left (-1+\csc ^2(3 x)\right )^3 \left (1-\sin ^2(3 x)\right )^2 \, dx=\text {Timed out} \]
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Time = 0.20 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.87 \[ \int \cot (3 x) \left (-1+\csc ^2(3 x)\right )^3 \left (1-\sin ^2(3 x)\right )^2 \, dx=-\frac {1}{12} \, \sin \left (3 \, x\right )^{4} + \frac {5}{6} \, \sin \left (3 \, x\right )^{2} - \frac {60 \, \sin \left (3 \, x\right )^{4} - 15 \, \sin \left (3 \, x\right )^{2} + 2}{36 \, \sin \left (3 \, x\right )^{6}} - \frac {5}{3} \, \log \left (\sin \left (3 \, x\right )^{2}\right ) \]
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Time = 0.34 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00 \[ \int \cot (3 x) \left (-1+\csc ^2(3 x)\right )^3 \left (1-\sin ^2(3 x)\right )^2 \, dx=-\frac {1}{12} \, \sin \left (3 \, x\right )^{4} + \frac {5}{6} \, \sin \left (3 \, x\right )^{2} + \frac {110 \, \sin \left (3 \, x\right )^{6} - 60 \, \sin \left (3 \, x\right )^{4} + 15 \, \sin \left (3 \, x\right )^{2} - 2}{36 \, \sin \left (3 \, x\right )^{6}} - \frac {5}{3} \, \log \left (\sin \left (3 \, x\right )^{2}\right ) \]
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Time = 28.17 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.40 \[ \int \cot (3 x) \left (-1+\csc ^2(3 x)\right )^3 \left (1-\sin ^2(3 x)\right )^2 \, dx=\frac {\ln \left ({\left ({\mathrm {tan}\left (3\,x\right )}^2+1\right )}^5\right )}{3}-\frac {10\,\ln \left (\mathrm {tan}\left (3\,x\right )\right )}{3}-\frac {5\,{\mathrm {tan}\left (3\,x\right )}^8+\frac {15\,{\mathrm {tan}\left (3\,x\right )}^6}{2}+\frac {5\,{\mathrm {tan}\left (3\,x\right )}^4}{3}-\frac {5\,{\mathrm {tan}\left (3\,x\right )}^2}{12}+\frac {1}{6}}{3\,\left ({\mathrm {tan}\left (3\,x\right )}^{10}+2\,{\mathrm {tan}\left (3\,x\right )}^8+{\mathrm {tan}\left (3\,x\right )}^6\right )} \]
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