Optimal. Leaf size=168 \[ -\frac{9}{16} \sqrt{-\frac{1}{n^2}} n x e^{a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{\left .-\frac{1}{3}\right /n}+\frac{9}{32} \sqrt{-\frac{1}{n^2}} n x e^{-a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{\left .\frac{1}{3}\right /n}-\frac{1}{16} \sqrt{-\frac{1}{n^2}} n x e^{-3 a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{\frac{1}{n}}+\frac{1}{8} \sqrt{-\frac{1}{n^2}} n x e^{3 a \sqrt{-\frac{1}{n^2}} n} \log (x) \left (c x^n\right )^{-1/n} \]
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Rubi [A] time = 0.104931, antiderivative size = 168, 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 = {4483, 4489} \[ -\frac{9}{16} \sqrt{-\frac{1}{n^2}} n x e^{a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{\left .-\frac{1}{3}\right /n}+\frac{9}{32} \sqrt{-\frac{1}{n^2}} n x e^{-a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{\left .\frac{1}{3}\right /n}-\frac{1}{16} \sqrt{-\frac{1}{n^2}} n x e^{-3 a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{\frac{1}{n}}+\frac{1}{8} \sqrt{-\frac{1}{n^2}} n x e^{3 a \sqrt{-\frac{1}{n^2}} n} \log (x) \left (c x^n\right )^{-1/n} \]
Antiderivative was successfully verified.
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Rule 4483
Rule 4489
Rubi steps
\begin{align*} \int \sin ^3\left (a+\frac{1}{3} \sqrt{-\frac{1}{n^2}} \log \left (c x^n\right )\right ) \, dx &=\frac{\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int x^{-1+\frac{1}{n}} \sin ^3\left (a+\frac{1}{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 \left (c x^n\right )^{-1/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}{3 n}}+3 e^{-a \sqrt{-\frac{1}{n^2}} n} x^{-1+\frac{4}{3 n}}-e^{-3 a \sqrt{-\frac{1}{n^2}} n} x^{-1+\frac{2}{n}}\right ) \, dx,x,c x^n\right )\\ &=-\frac{9}{16} e^{a \sqrt{-\frac{1}{n^2}} n} \sqrt{-\frac{1}{n^2}} n x \left (c x^n\right )^{\left .-\frac{1}{3}\right /n}+\frac{9}{32} e^{-a \sqrt{-\frac{1}{n^2}} n} \sqrt{-\frac{1}{n^2}} n x \left (c x^n\right )^{\left .\frac{1}{3}\right /n}-\frac{1}{16} e^{-3 a \sqrt{-\frac{1}{n^2}} n} \sqrt{-\frac{1}{n^2}} n x \left (c x^n\right )^{\frac{1}{n}}+\frac{1}{8} e^{3 a \sqrt{-\frac{1}{n^2}} n} \sqrt{-\frac{1}{n^2}} n x \left (c x^n\right )^{-1/n} \log (x)\\ \end{align*}
Mathematica [F] time = 0.133365, size = 0, normalized size = 0. \[ \int \sin ^3\left (a+\frac{1}{3} \sqrt{-\frac{1}{n^2}} \log \left (c x^n\right )\right ) \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.069, size = 0, normalized size = 0. \begin{align*} \int \left ( \sin \left ( a+{\frac{\ln \left ( c{x}^{n} \right ) }{3}\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 = 1.1595, size = 143, normalized size = 0.85 \begin{align*} -\frac{4 \, c^{\frac{1}{3 \, n}}{\left (x^{n}\right )}^{\frac{1}{3 \, n}} \log \left (x\right ) \sin \left (3 \, a\right ) - 9 \, c^{\frac{5}{3 \, n}} x{\left (x^{n}\right )}^{\frac{2}{3 \, n}} \sin \left (a\right ) + 2 \, c^{\frac{7}{3 \, n}} e^{\left (\frac{\log \left (x^{n}\right )}{3 \, n} + 2 \, \log \left (x\right )\right )} \sin \left (3 \, a\right ) - 18 \, c^{\left (\frac{1}{n}\right )} x \sin \left (a\right )}{32 \, c^{\frac{4}{3 \, n}}{\left (x^{n}\right )}^{\frac{1}{3 \, n}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 0.474748, size = 238, normalized size = 1.42 \begin{align*} \frac{1}{32} \,{\left (9 i \, x^{\frac{4}{3}} e^{\left (\frac{2 \,{\left (3 i \, a n - \log \left (c\right )\right )}}{3 \, n}\right )} - 2 i \, x^{2} + 12 i \, e^{\left (\frac{2 \,{\left (3 i \, a n - \log \left (c\right )\right )}}{n}\right )} \log \left (x^{\frac{1}{3}}\right ) - 18 i \, x^{\frac{2}{3}} e^{\left (\frac{4 \,{\left (3 i \, a n - \log \left (c\right )\right )}}{3 \, n}\right )}\right )} e^{\left (-\frac{3 i \, a n - \log \left (c\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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