Optimal. Leaf size=60 \[ -\frac {1}{12} \sin ^4(3 x)+\frac {5}{6} \sin ^2(3 x)-\frac {1}{18} \csc ^6(3 x)+\frac {5}{12} \csc ^4(3 x)-\frac {5}{3} \csc ^2(3 x)-\frac {10}{3} \log (\sin (3 x)) \]
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Rubi [A] time = 0.13, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {3175, 4360, 266, 43} \[ -\frac {1}{12} \sin ^4(3 x)+\frac {5}{6} \sin ^2(3 x)-\frac {1}{18} \csc ^6(3 x)+\frac {5}{12} \csc ^4(3 x)-\frac {5}{3} \csc ^2(3 x)-\frac {10}{3} \log (\sin (3 x)) \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 3175
Rule 4360
Rubi steps
\begin {align*} \int \cot (3 x) \left (-1+\csc ^2(3 x)\right )^3 \left (1-\sin ^2(3 x)\right )^2 \, dx &=\int \cos ^4(3 x) \cot (3 x) \left (-1+\csc ^2(3 x)\right )^3 \, dx\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^5}{x^7} \, dx,x,\sin (3 x)\right )\\ &=\frac {1}{6} \operatorname {Subst}\left (\int \frac {(1-x)^5}{x^4} \, dx,x,\sin ^2(3 x)\right )\\ &=\frac {1}{6} \operatorname {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*}
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Mathematica [A] time = 0.13, size = 52, normalized size = 0.87 \[ \frac {1}{36} \left (-3 \sin ^4(3 x)+30 \sin ^2(3 x)-2 \csc ^6(3 x)+15 \csc ^4(3 x)-60 \csc ^2(3 x)-120 \log (\sin (3 x))\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.84, size = 103, normalized size = 1.72 \[ -\frac {24 \, \cos \left (3 \, x\right )^{10} + 120 \, \cos \left (3 \, x\right )^{8} - 609 \, \cos \left (3 \, x\right )^{6} + 387 \, \cos \left (3 \, x\right )^{4} + 333 \, \cos \left (3 \, x\right )^{2} + 960 \, {\left (\cos \left (3 \, x\right )^{6} - 3 \, \cos \left (3 \, x\right )^{4} + 3 \, \cos \left (3 \, x\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \sin \left (3 \, x\right )\right ) - 271}{288 \, {\left (\cos \left (3 \, x\right )^{6} - 3 \, \cos \left (3 \, x\right )^{4} + 3 \, \cos \left (3 \, x\right )^{2} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 60, normalized size = 1.00 \[ -\frac {1}{12} \, \sin \left (3 \, x\right )^{4} + \frac {5}{6} \, \sin \left (3 \, x\right )^{2} + \frac {110 \, \sin \left (3 \, x\right )^{6} - 60 \, \sin \left (3 \, x\right )^{4} + 15 \, \sin \left (3 \, x\right )^{2} - 2}{36 \, \sin \left (3 \, x\right )^{6}} - \frac {5}{3} \, \log \left (\sin \left (3 \, x\right )^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 49, normalized size = 0.82 \[ -\frac {\left (\sin ^{4}\left (3 x \right )\right )}{12}-\frac {5 \left (\cos ^{2}\left (3 x \right )\right )}{6}-\frac {10 \ln \left (\sin \left (3 x \right )\right )}{3}-\frac {5}{3 \sin \left (3 x \right )^{2}}+\frac {5}{12 \sin \left (3 x \right )^{4}}-\frac {1}{18 \sin \left (3 x \right )^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 52, normalized size = 0.87 \[ -\frac {1}{12} \, \sin \left (3 \, x\right )^{4} + \frac {5}{6} \, \sin \left (3 \, x\right )^{2} - \frac {60 \, \sin \left (3 \, x\right )^{4} - 15 \, \sin \left (3 \, x\right )^{2} + 2}{36 \, \sin \left (3 \, x\right )^{6}} - \frac {5}{3} \, \log \left (\sin \left (3 \, x\right )^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.67, size = 84, normalized size = 1.40 \[ \frac {\ln \left ({\left ({\mathrm {tan}\left (3\,x\right )}^2+1\right )}^5\right )}{3}-\frac {10\,\ln \left (\mathrm {tan}\left (3\,x\right )\right )}{3}-\frac {5\,{\mathrm {tan}\left (3\,x\right )}^8+\frac {15\,{\mathrm {tan}\left (3\,x\right )}^6}{2}+\frac {5\,{\mathrm {tan}\left (3\,x\right )}^4}{3}-\frac {5\,{\mathrm {tan}\left (3\,x\right )}^2}{12}+\frac {1}{6}}{3\,\left ({\mathrm {tan}\left (3\,x\right )}^{10}+2\,{\mathrm {tan}\left (3\,x\right )}^8+{\mathrm {tan}\left (3\,x\right )}^6\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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