Optimal. Leaf size=172 \[ -\frac{3}{16} \sqrt{-\frac{1}{n^2}} n x^3 e^{a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{-1/n}-\frac{1}{48} \sqrt{-\frac{1}{n^2}} n x^3 e^{-3 a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{3/n}+\frac{3}{32} \sqrt{-\frac{1}{n^2}} n x^3 e^{-a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{\frac{1}{n}}+\frac{1}{8} \sqrt{-\frac{1}{n^2}} n x^3 e^{3 a \sqrt{-\frac{1}{n^2}} n} \log (x) \left (c x^n\right )^{-3/n} \]
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Rubi [A] time = 0.160594, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {4493, 4489} \[ -\frac{3}{16} \sqrt{-\frac{1}{n^2}} n x^3 e^{a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{-1/n}-\frac{1}{48} \sqrt{-\frac{1}{n^2}} n x^3 e^{-3 a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{3/n}+\frac{3}{32} \sqrt{-\frac{1}{n^2}} n x^3 e^{-a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{\frac{1}{n}}+\frac{1}{8} \sqrt{-\frac{1}{n^2}} n x^3 e^{3 a \sqrt{-\frac{1}{n^2}} n} \log (x) \left (c x^n\right )^{-3/n} \]
Antiderivative was successfully verified.
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Rule 4493
Rule 4489
Rubi steps
\begin{align*} \int x^2 \sin ^3\left (a+\sqrt{-\frac{1}{n^2}} \log \left (c x^n\right )\right ) \, dx &=\frac{\left (x^3 \left (c x^n\right )^{-3/n}\right ) \operatorname{Subst}\left (\int x^{-1+\frac{3}{n}} \sin ^3\left (a+\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^3 \left (c x^n\right )^{-3/n}\right ) \operatorname{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{2}{n}}+3 e^{-a \sqrt{-\frac{1}{n^2}} n} x^{-1+\frac{4}{n}}-e^{-3 a \sqrt{-\frac{1}{n^2}} n} x^{-1+\frac{6}{n}}\right ) \, dx,x,c x^n\right )\\ &=-\frac{3}{16} e^{a \sqrt{-\frac{1}{n^2}} n} \sqrt{-\frac{1}{n^2}} n x^3 \left (c x^n\right )^{-1/n}+\frac{3}{32} e^{-a \sqrt{-\frac{1}{n^2}} n} \sqrt{-\frac{1}{n^2}} n x^3 \left (c x^n\right )^{\frac{1}{n}}-\frac{1}{48} e^{-3 a \sqrt{-\frac{1}{n^2}} n} \sqrt{-\frac{1}{n^2}} n x^3 \left (c x^n\right )^{3/n}+\frac{1}{8} e^{3 a \sqrt{-\frac{1}{n^2}} n} \sqrt{-\frac{1}{n^2}} n x^3 \left (c x^n\right )^{-3/n} \log (x)\\ \end{align*}
Mathematica [F] time = 0.215729, size = 0, normalized size = 0. \[ \int x^2 \sin ^3\left (a+\sqrt{-\frac{1}{n^2}} \log \left (c x^n\right )\right ) \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.072, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( \sin \left ( a+\ln \left ( c{x}^{n} \right ) \sqrt{-{n}^{-2}} \right ) \right ) ^{3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.19871, size = 122, normalized size = 0.71 \begin{align*} \frac{18 \, c^{\frac{2}{n}} x^{3} \sin \left (a\right ) - 12 \,{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )} \log \left (x\right ) \sin \left (3 \, a\right ) -{\left (2 \, c^{\frac{6}{n}} x^{6} \sin \left (3 \, a\right ) - 9 \, c^{\frac{4}{n}} x^{4} \sin \left (a\right )\right )}{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )}}{96 \, c^{\frac{3}{n}}{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 0.470377, size = 207, normalized size = 1.2 \begin{align*} \frac{1}{96} \,{\left (-2 i \, x^{6} + 9 i \, x^{4} e^{\left (\frac{2 \,{\left (i \, a n - \log \left (c\right )\right )}}{n}\right )} - 18 i \, x^{2} e^{\left (\frac{4 \,{\left (i \, a n - \log \left (c\right )\right )}}{n}\right )} + 12 i \, e^{\left (\frac{6 \,{\left (i \, a n - \log \left (c\right )\right )}}{n}\right )} \log \left (x\right )\right )} e^{\left (-\frac{3 \,{\left (i \, a n - \log \left (c\right )\right )}}{n}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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