Integrand size = 28, antiderivative size = 76 \[ \int x^2 \sin ^2\left (a+\frac {3}{2} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\frac {x^3}{6}-\frac {1}{24} e^{-2 a \sqrt {-\frac {1}{n^2}} n} x^3 \left (c x^n\right )^{3/n}-\frac {1}{4} e^{2 a \sqrt {-\frac {1}{n^2}} n} x^3 \left (c x^n\right )^{-3/n} \log (x) \]
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Time = 0.10 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {4581, 4577} \[ \int x^2 \sin ^2\left (a+\frac {3}{2} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=-\frac {1}{24} x^3 e^{-2 a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{3/n}-\frac {1}{4} x^3 e^{2 a \sqrt {-\frac {1}{n^2}} n} \log (x) \left (c x^n\right )^{-3/n}+\frac {x^3}{6} \]
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Rule 4577
Rule 4581
Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^3 \left (c x^n\right )^{-3/n}\right ) \text {Subst}\left (\int x^{-1+\frac {3}{n}} \sin ^2\left (a+\frac {3}{2} \sqrt {-\frac {1}{n^2}} \log (x)\right ) \, dx,x,c x^n\right )}{n} \\ & = -\frac {\left (x^3 \left (c x^n\right )^{-3/n}\right ) \text {Subst}\left (\int \left (\frac {e^{2 a \sqrt {-\frac {1}{n^2}} n}}{x}-2 x^{-1+\frac {3}{n}}+e^{-2 a \sqrt {-\frac {1}{n^2}} n} x^{-1+\frac {6}{n}}\right ) \, dx,x,c x^n\right )}{4 n} \\ & = \frac {x^3}{6}-\frac {1}{24} e^{-2 a \sqrt {-\frac {1}{n^2}} n} x^3 \left (c x^n\right )^{3/n}-\frac {1}{4} e^{2 a \sqrt {-\frac {1}{n^2}} n} x^3 \left (c x^n\right )^{-3/n} \log (x) \\ \end{align*}
\[ \int x^2 \sin ^2\left (a+\frac {3}{2} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\int x^2 \sin ^2\left (a+\frac {3}{2} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx \]
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\[\int x^{2} {\sin \left (a +\frac {3 \ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}}{2}\right )}^{2}d x\]
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Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.78 \[ \int x^2 \sin ^2\left (a+\frac {3}{2} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=-\frac {1}{24} \, {\left (x^{6} - 4 \, x^{3} e^{\left (\frac {2 i \, a n - 3 \, \log \left (c\right )}{n}\right )} + 6 \, e^{\left (\frac {2 \, {\left (2 i \, a n - 3 \, \log \left (c\right )\right )}}{n}\right )} \log \left (x\right )\right )} e^{\left (-\frac {2 i \, a n - 3 \, \log \left (c\right )}{n}\right )} \]
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\[ \int x^2 \sin ^2\left (a+\frac {3}{2} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\int x^{2} \sin ^{2}{\left (a + \frac {3 \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )}}{2} \right )}\, dx \]
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Time = 0.23 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.62 \[ \int x^2 \sin ^2\left (a+\frac {3}{2} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=-\frac {c^{\frac {6}{n}} x^{6} \cos \left (2 \, a\right ) - 4 \, c^{\frac {3}{n}} x^{3} + 6 \, \cos \left (2 \, a\right ) \log \left (x\right )}{24 \, c^{\frac {3}{n}}} \]
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Time = 0.77 (sec) , antiderivative size = 1, normalized size of antiderivative = 0.01 \[ \int x^2 \sin ^2\left (a+\frac {3}{2} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=+\infty \]
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Time = 26.81 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.21 \[ \int x^2 \sin ^2\left (a+\frac {3}{2} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\frac {x^3}{6}-\frac {x^3\,{\mathrm {e}}^{-a\,2{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{\sqrt {-\frac {1}{n^2}}\,3{}\mathrm {i}}}\,1{}\mathrm {i}}{12\,n\,\sqrt {-\frac {1}{n^2}}+12{}\mathrm {i}}+\frac {x^3\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{\sqrt {-\frac {1}{n^2}}\,3{}\mathrm {i}}\,1{}\mathrm {i}}{12\,n\,\sqrt {-\frac {1}{n^2}}-12{}\mathrm {i}} \]
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