Integrand size = 26, antiderivative size = 178 \[ \int x \sin ^3\left (a+\frac {2}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=-\frac {9}{32} e^{a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x^2 \left (c x^n\right )^{\left .-\frac {2}{3}\right /n}+\frac {9}{64} e^{-a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x^2 \left (c x^n\right )^{\left .\frac {2}{3}\right /n}-\frac {1}{32} e^{-3 a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x^2 \left (c x^n\right )^{2/n}+\frac {1}{8} e^{3 a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x^2 \left (c x^n\right )^{-2/n} \log (x) \]
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Time = 0.13 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {4581, 4577} \[ \int x \sin ^3\left (a+\frac {2}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=-\frac {9}{32} \sqrt {-\frac {1}{n^2}} n x^2 e^{a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{\left .-\frac {2}{3}\right /n}+\frac {9}{64} \sqrt {-\frac {1}{n^2}} n x^2 e^{-a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{\left .\frac {2}{3}\right /n}-\frac {1}{32} \sqrt {-\frac {1}{n^2}} n x^2 e^{-3 a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{2/n}+\frac {1}{8} \sqrt {-\frac {1}{n^2}} n x^2 e^{3 a \sqrt {-\frac {1}{n^2}} n} \log (x) \left (c x^n\right )^{-2/n} \]
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Rule 4577
Rule 4581
Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^2 \left (c x^n\right )^{-2/n}\right ) \text {Subst}\left (\int x^{-1+\frac {2}{n}} \sin ^3\left (a+\frac {2}{3} \sqrt {-\frac {1}{n^2}} \log (x)\right ) \, dx,x,c x^n\right )}{n} \\ & = \frac {1}{8} \left (\sqrt {-\frac {1}{n^2}} x^2 \left (c x^n\right )^{-2/n}\right ) \text {Subst}\left (\int \left (\frac {e^{3 a \sqrt {-\frac {1}{n^2}} n}}{x}-3 e^{a \sqrt {-\frac {1}{n^2}} n} x^{-1+\frac {4}{3 n}}+3 e^{-a \sqrt {-\frac {1}{n^2}} n} x^{-1+\frac {8}{3 n}}-e^{-3 a \sqrt {-\frac {1}{n^2}} n} x^{-1+\frac {4}{n}}\right ) \, dx,x,c x^n\right ) \\ & = -\frac {9}{32} e^{a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x^2 \left (c x^n\right )^{\left .-\frac {2}{3}\right /n}+\frac {9}{64} e^{-a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x^2 \left (c x^n\right )^{\left .\frac {2}{3}\right /n}-\frac {1}{32} e^{-3 a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x^2 \left (c x^n\right )^{2/n}+\frac {1}{8} e^{3 a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x^2 \left (c x^n\right )^{-2/n} \log (x) \\ \end{align*}
\[ \int x \sin ^3\left (a+\frac {2}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\int x \sin ^3\left (a+\frac {2}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx \]
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\[\int x {\sin \left (a +\frac {2 \ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}}{3}\right )}^{3}d x\]
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Result contains complex when optimal does not.
Time = 0.24 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.47 \[ \int x \sin ^3\left (a+\frac {2}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\frac {1}{64} \, {\left (-2 i \, x^{4} + 9 i \, x^{\frac {8}{3}} e^{\left (\frac {2 \, {\left (3 i \, a n - 2 \, \log \left (c\right )\right )}}{3 \, n}\right )} - 18 i \, x^{\frac {4}{3}} e^{\left (\frac {4 \, {\left (3 i \, a n - 2 \, \log \left (c\right )\right )}}{3 \, n}\right )} + 24 i \, e^{\left (\frac {2 \, {\left (3 i \, a n - 2 \, \log \left (c\right )\right )}}{n}\right )} \log \left (x^{\frac {1}{3}}\right )\right )} e^{\left (-\frac {3 i \, a n - 2 \, \log \left (c\right )}{n}\right )} \]
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\[ \int x \sin ^3\left (a+\frac {2}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\int x \sin ^{3}{\left (a + \frac {2 \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )}}{3} \right )}\, dx \]
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none
Time = 0.24 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.63 \[ \int x \sin ^3\left (a+\frac {2}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\frac {9 \, c^{\frac {10}{3 \, n}} x^{2} {\left (x^{n}\right )}^{\frac {4}{3 \, n}} \sin \left (a\right ) - 8 \, c^{\frac {2}{3 \, n}} {\left (x^{n}\right )}^{\frac {2}{3 \, n}} \log \left (x\right ) \sin \left (3 \, a\right ) + 18 \, c^{\frac {2}{n}} x^{2} \sin \left (a\right ) - 2 \, c^{\frac {14}{3 \, n}} e^{\left (\frac {2 \, \log \left (x^{n}\right )}{3 \, n} + 4 \, \log \left (x\right )\right )} \sin \left (3 \, a\right )}{64 \, c^{\frac {8}{3 \, n}} {\left (x^{n}\right )}^{\frac {2}{3 \, n}}} \]
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Exception generated. \[ \int x \sin ^3\left (a+\frac {2}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\text {Exception raised: NotImplementedError} \]
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Time = 29.19 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.92 \[ \int x \sin ^3\left (a+\frac {2}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=-x^2\,{\mathrm {e}}^{-a\,1{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{\frac {\sqrt {-\frac {1}{n^2}}\,2{}\mathrm {i}}{3}}}\,\left (\frac {9\,n\,\sqrt {-\frac {1}{n^2}}}{128}-\frac {27}{128}{}\mathrm {i}\right )-x^2\,{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (c\,x^n\right )}^{\frac {\sqrt {-\frac {1}{n^2}}\,2{}\mathrm {i}}{3}}\,\left (\frac {9\,n\,\sqrt {-\frac {1}{n^2}}}{128}+\frac {27}{128}{}\mathrm {i}\right )+\frac {x^2\,{\mathrm {e}}^{-a\,3{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{\sqrt {-\frac {1}{n^2}}\,2{}\mathrm {i}}}}{16\,n\,\sqrt {-\frac {1}{n^2}}+16{}\mathrm {i}}+\frac {x^2\,{\mathrm {e}}^{a\,3{}\mathrm {i}}\,{\left (c\,x^n\right )}^{\sqrt {-\frac {1}{n^2}}\,2{}\mathrm {i}}}{16\,n\,\sqrt {-\frac {1}{n^2}}-16{}\mathrm {i}} \]
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