Optimal. Leaf size=178 \[ -\frac{\sqrt{-\frac{1}{n^2}} n e^{3 a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{-2/n}}{32 x^2}+\frac{9 \sqrt{-\frac{1}{n^2}} n e^{a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{\left .-\frac{2}{3}\right /n}}{64 x^2}-\frac{9 \sqrt{-\frac{1}{n^2}} n e^{-a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{\left .\frac{2}{3}\right /n}}{32 x^2}-\frac{\sqrt{-\frac{1}{n^2}} n e^{-3 a \sqrt{-\frac{1}{n^2}} n} \log (x) \left (c x^n\right )^{2/n}}{8 x^2} \]
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Rubi [A] time = 0.113835, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {4493, 4489} \[ -\frac{\sqrt{-\frac{1}{n^2}} n e^{3 a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{-2/n}}{32 x^2}+\frac{9 \sqrt{-\frac{1}{n^2}} n e^{a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{\left .-\frac{2}{3}\right /n}}{64 x^2}-\frac{9 \sqrt{-\frac{1}{n^2}} n e^{-a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{\left .\frac{2}{3}\right /n}}{32 x^2}-\frac{\sqrt{-\frac{1}{n^2}} n e^{-3 a \sqrt{-\frac{1}{n^2}} n} \log (x) \left (c x^n\right )^{2/n}}{8 x^2} \]
Antiderivative was successfully verified.
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Rule 4493
Rule 4489
Rubi steps
\begin{align*} \int \frac{\sin ^3\left (a+\frac{2}{3} \sqrt{-\frac{1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx &=\frac{\left (c x^n\right )^{2/n} \operatorname{Subst}\left (\int x^{-1-\frac{2}{n}} \sin ^3\left (a+\frac{2}{3} \sqrt{-\frac{1}{n^2}} \log (x)\right ) \, dx,x,c x^n\right )}{n x^2}\\ &=-\frac{\left (\sqrt{-\frac{1}{n^2}} \left (c x^n\right )^{2/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{8}{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^{-\frac{4+n}{n}}\right ) \, dx,x,c x^n\right )}{8 x^2}\\ &=-\frac{e^{3 a \sqrt{-\frac{1}{n^2}} n} \sqrt{-\frac{1}{n^2}} n \left (c x^n\right )^{-2/n}}{32 x^2}+\frac{9 e^{a \sqrt{-\frac{1}{n^2}} n} \sqrt{-\frac{1}{n^2}} n \left (c x^n\right )^{\left .-\frac{2}{3}\right /n}}{64 x^2}-\frac{9 e^{-a \sqrt{-\frac{1}{n^2}} n} \sqrt{-\frac{1}{n^2}} n \left (c x^n\right )^{\left .\frac{2}{3}\right /n}}{32 x^2}-\frac{e^{-3 a \sqrt{-\frac{1}{n^2}} n} \sqrt{-\frac{1}{n^2}} n \left (c x^n\right )^{2/n} \log (x)}{8 x^2}\\ \end{align*}
Mathematica [F] time = 0.205036, size = 0, normalized size = 0. \[ \int \frac{\sin ^3\left (a+\frac{2}{3} \sqrt{-\frac{1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.072, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}} \left ( \sin \left ( a+{\frac{2\,\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 = 1.14344, size = 173, normalized size = 0.97 \begin{align*} -\frac{{\left (8 \, c^{\frac{14}{3 \, n}} x^{2} e^{\left (\frac{2 \, \log \left (x^{n}\right )}{3 \, n} + 4 \, \log \left (x\right )\right )} \log \left (x\right ) \sin \left (3 \, a\right ) + 9 \, c^{\frac{2}{n}} x^{4} \sin \left (a\right ) - 2 \, c^{\frac{2}{3 \, n}} x^{2}{\left (x^{n}\right )}^{\frac{2}{3 \, n}} \sin \left (3 \, a\right ) + 18 \, c^{\frac{10}{3 \, n}} e^{\left (\frac{4 \, \log \left (x^{n}\right )}{3 \, n} + 4 \, \log \left (x\right )\right )} \sin \left (a\right )\right )} e^{\left (-\frac{2 \, \log \left (x^{n}\right )}{3 \, n} - 4 \, \log \left (x\right )\right )}}{64 \, c^{\frac{8}{3 \, n}} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 0.480342, size = 255, normalized size = 1.43 \begin{align*} \frac{{\left (-24 i \, x^{4} \log \left (x^{\frac{1}{3}}\right ) - 18 i \, x^{\frac{8}{3}} e^{\left (\frac{2 \,{\left (3 i \, a n - 2 \, \log \left (c\right )\right )}}{3 \, n}\right )} + 9 i \, x^{\frac{4}{3}} e^{\left (\frac{4 \,{\left (3 i \, a n - 2 \, \log \left (c\right )\right )}}{3 \, n}\right )} - 2 i \, e^{\left (\frac{2 \,{\left (3 i \, a n - 2 \, \log \left (c\right )\right )}}{n}\right )}\right )} e^{\left (-\frac{3 i \, a n - 2 \, \log \left (c\right )}{n}\right )}}{64 \, x^{4}} \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (\frac{2}{3} \, \sqrt{-\frac{1}{n^{2}}} \log \left (c x^{n}\right ) + a\right )^{3}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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