Optimal. Leaf size=76 \[ -\frac{1}{16} x^2 e^{-2 a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{2/n}-\frac{1}{4} x^2 e^{2 a \sqrt{-\frac{1}{n^2}} n} \log (x) \left (c x^n\right )^{-2/n}+\frac{x^2}{4} \]
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Rubi [A] time = 0.05761, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {4493, 4489} \[ -\frac{1}{16} x^2 e^{-2 a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{2/n}-\frac{1}{4} x^2 e^{2 a \sqrt{-\frac{1}{n^2}} n} \log (x) \left (c x^n\right )^{-2/n}+\frac{x^2}{4} \]
Antiderivative was successfully verified.
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Rule 4493
Rule 4489
Rubi steps
\begin{align*} \int x \sin ^2\left (a+\sqrt{-\frac{1}{n^2}} \log \left (c x^n\right )\right ) \, dx &=\frac{\left (x^2 \left (c x^n\right )^{-2/n}\right ) \operatorname{Subst}\left (\int x^{-1+\frac{2}{n}} \sin ^2\left (a+\sqrt{-\frac{1}{n^2}} \log (x)\right ) \, dx,x,c x^n\right )}{n}\\ &=-\frac{\left (x^2 \left (c x^n\right )^{-2/n}\right ) \operatorname{Subst}\left (\int \left (\frac{e^{2 a \sqrt{-\frac{1}{n^2}} n}}{x}-2 x^{-1+\frac{2}{n}}+e^{-2 a \sqrt{-\frac{1}{n^2}} n} x^{-1+\frac{4}{n}}\right ) \, dx,x,c x^n\right )}{4 n}\\ &=\frac{x^2}{4}-\frac{1}{16} e^{-2 a \sqrt{-\frac{1}{n^2}} n} x^2 \left (c x^n\right )^{2/n}-\frac{1}{4} e^{2 a \sqrt{-\frac{1}{n^2}} n} x^2 \left (c x^n\right )^{-2/n} \log (x)\\ \end{align*}
Mathematica [F] time = 0.151923, size = 0, normalized size = 0. \[ \int x \sin ^2\left (a+\sqrt{-\frac{1}{n^2}} \log \left (c x^n\right )\right ) \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.076, size = 0, normalized size = 0. \begin{align*} \int x \left ( \sin \left ( a+\ln \left ( c{x}^{n} \right ) \sqrt{-{n}^{-2}} \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11589, size = 63, normalized size = 0.83 \begin{align*} -\frac{c^{\frac{4}{n}} x^{4} \cos \left (2 \, a\right ) - 4 \, c^{\frac{2}{n}} x^{2} + 4 \, \cos \left (2 \, a\right ) \log \left (x\right )}{16 \, c^{\frac{2}{n}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 0.475053, size = 146, normalized size = 1.92 \begin{align*} -\frac{1}{16} \,{\left (x^{4} - 4 \, x^{2} e^{\left (\frac{2 \,{\left (i \, a n - \log \left (c\right )\right )}}{n}\right )} + 4 \, e^{\left (\frac{4 \,{\left (i \, a n - \log \left (c\right )\right )}}{n}\right )} \log \left (x\right )\right )} e^{\left (-\frac{2 \,{\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \sin ^{2}{\left (a + \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 = 1.82429, 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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