Optimal. Leaf size=88 \[ \frac{\sqrt{-\frac{1}{n^2}} n e^{a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{-2/n}}{8 x^2}+\frac{\sqrt{-\frac{1}{n^2}} n e^{-a \sqrt{-\frac{1}{n^2}} n} \log (x) \left (c x^n\right )^{2/n}}{2 x^2} \]
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Rubi [A] time = 0.0531861, 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} \[ \frac{\sqrt{-\frac{1}{n^2}} n e^{a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{-2/n}}{8 x^2}+\frac{\sqrt{-\frac{1}{n^2}} n e^{-a \sqrt{-\frac{1}{n^2}} n} \log (x) \left (c x^n\right )^{2/n}}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 4493
Rule 4489
Rubi steps
\begin{align*} \int \frac{\sin \left (a+2 \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 \left (a+2 \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^{-a \sqrt{-\frac{1}{n^2}} n}}{x}-e^{a \sqrt{-\frac{1}{n^2}} n} x^{-\frac{4+n}{n}}\right ) \, dx,x,c x^n\right )}{2 x^2}\\ &=\frac{e^{a \sqrt{-\frac{1}{n^2}} n} \sqrt{-\frac{1}{n^2}} n \left (c x^n\right )^{-2/n}}{8 x^2}+\frac{e^{-a \sqrt{-\frac{1}{n^2}} n} \sqrt{-\frac{1}{n^2}} n \left (c x^n\right )^{2/n} \log (x)}{2 x^2}\\ \end{align*}
Mathematica [F] time = 0.0895486, size = 0, normalized size = 0. \[ \int \frac{\sin \left (a+2 \sqrt{-\frac{1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.035, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}}\sin \left ( a+2\,\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 = 1.24162, size = 47, normalized size = 0.53 \begin{align*} \frac{4 \, c^{\frac{4}{n}} x^{4} \log \left (x\right ) \sin \left (a\right ) - \sin \left (a\right )}{8 \, c^{\frac{2}{n}} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 0.471725, size = 112, normalized size = 1.27 \begin{align*} \frac{{\left (4 i \, x^{4} \log \left (x\right ) + i \, e^{\left (\frac{2 \,{\left (i \, a n - 2 \, \log \left (c\right )\right )}}{n}\right )}\right )} e^{\left (-\frac{i \, a n - 2 \, \log \left (c\right )}{n}\right )}}{8 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 71.6194, size = 240, normalized size = 2.73 \begin{align*} \frac{i n \sqrt{\frac{1}{n^{2}}} \log{\left (x \right )} \cos{\left (a + 2 i n \sqrt{\frac{1}{n^{2}}} \log{\left (x \right )} + 2 i \sqrt{\frac{1}{n^{2}}} \log{\left (c \right )} \right )}}{2 x^{2}} + \frac{i \sqrt{\frac{1}{n^{2}}} \log{\left (c \right )} \cos{\left (a + 2 i n \sqrt{\frac{1}{n^{2}}} \log{\left (x \right )} + 2 i \sqrt{\frac{1}{n^{2}}} \log{\left (c \right )} \right )}}{2 x^{2}} + \frac{\log{\left (x \right )} \sin{\left (a + 2 i n \sqrt{\frac{1}{n^{2}}} \log{\left (x \right )} + 2 i \sqrt{\frac{1}{n^{2}}} \log{\left (c \right )} \right )}}{2 x^{2}} - \frac{\sin{\left (a + 2 i n \sqrt{\frac{1}{n^{2}}} \log{\left (x \right )} + 2 i \sqrt{\frac{1}{n^{2}}} \log{\left (c \right )} \right )}}{4 x^{2}} + \frac{\log{\left (c \right )} \sin{\left (a + 2 i n \sqrt{\frac{1}{n^{2}}} \log{\left (x \right )} + 2 i \sqrt{\frac{1}{n^{2}}} \log{\left (c \right )} \right )}}{2 n x^{2}} \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 (2 \, \sqrt{-\frac{1}{n^{2}}} \log \left (c x^{n}\right ) + a\right )}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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