Optimal. Leaf size=88 \[ \frac{1}{12} \sqrt{-\frac{1}{n^2}} n x^3 e^{-a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{3/n}-\frac{1}{2} \sqrt{-\frac{1}{n^2}} n x^3 e^{a \sqrt{-\frac{1}{n^2}} n} \log (x) \left (c x^n\right )^{-3/n} \]
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Rubi [A] time = 0.0988647, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {4493, 4489} \[ \frac{1}{12} \sqrt{-\frac{1}{n^2}} n x^3 e^{-a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{3/n}-\frac{1}{2} \sqrt{-\frac{1}{n^2}} n x^3 e^{a \sqrt{-\frac{1}{n^2}} n} \log (x) \left (c x^n\right )^{-3/n} \]
Antiderivative was successfully verified.
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Rule 4493
Rule 4489
Rubi steps
\begin{align*} \int x^2 \sin \left (a+3 \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 \left (a+3 \sqrt{-\frac{1}{n^2}} \log (x)\right ) \, dx,x,c x^n\right )}{n}\\ &=-\left (\frac{1}{2} \left (\sqrt{-\frac{1}{n^2}} x^3 \left (c x^n\right )^{-3/n}\right ) \operatorname{Subst}\left (\int \left (\frac{e^{a \sqrt{-\frac{1}{n^2}} n}}{x}-e^{-a \sqrt{-\frac{1}{n^2}} n} x^{-1+\frac{6}{n}}\right ) \, dx,x,c x^n\right )\right )\\ &=\frac{1}{12} e^{-a \sqrt{-\frac{1}{n^2}} n} \sqrt{-\frac{1}{n^2}} n x^3 \left (c x^n\right )^{3/n}-\frac{1}{2} e^{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 = 0.145668, size = 0, normalized size = 0. \[ \int x^2 \sin \left (a+3 \sqrt{-\frac{1}{n^2}} \log \left (c x^n\right )\right ) \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.037, size = 0, normalized size = 0. \begin{align*} \int{x}^{2}\sin \left ( a+3\,\ln \left ( c{x}^{n} \right ) \sqrt{-{n}^{-2}} \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09538, size = 42, normalized size = 0.48 \begin{align*} \frac{c^{\frac{6}{n}} x^{6} \sin \left (a\right ) + 6 \, \log \left (x\right ) \sin \left (a\right )}{12 \, c^{\frac{3}{n}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 0.468447, size = 108, normalized size = 1.23 \begin{align*} \frac{1}{12} \,{\left (i \, x^{6} - 6 i \, e^{\left (\frac{2 \,{\left (i \, a n - 3 \, \log \left (c\right )\right )}}{n}\right )} \log \left (x\right )\right )} e^{\left (-\frac{i \, a n - 3 \, \log \left (c\right )}{n}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \sin{\left (a + 3 \sqrt{- \frac{1}{n^{2}}} \log{\left (c x^{n} \right )} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.46388, size = 1, normalized size = 0.01 \begin{align*} +\infty \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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