Optimal. Leaf size=86 \[ \frac{\sqrt{-\frac{1}{n^2}} n e^{a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{-1/n}}{4 x}+\frac{\sqrt{-\frac{1}{n^2}} n e^{-a \sqrt{-\frac{1}{n^2}} n} \log (x) \left (c x^n\right )^{\frac{1}{n}}}{2 x} \]
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Rubi [A] time = 0.0611679, antiderivative size = 86, 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} \[ \frac{\sqrt{-\frac{1}{n^2}} n e^{a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{-1/n}}{4 x}+\frac{\sqrt{-\frac{1}{n^2}} n e^{-a \sqrt{-\frac{1}{n^2}} n} \log (x) \left (c x^n\right )^{\frac{1}{n}}}{2 x} \]
Antiderivative was successfully verified.
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Rule 4493
Rule 4489
Rubi steps
\begin{align*} \int \frac{\sin \left (a+\sqrt{-\frac{1}{n^2}} \log \left (c x^n\right )\right )}{x^2} \, dx &=\frac{\left (c x^n\right )^{\frac{1}{n}} \operatorname{Subst}\left (\int x^{-1-\frac{1}{n}} \sin \left (a+\sqrt{-\frac{1}{n^2}} \log (x)\right ) \, dx,x,c x^n\right )}{n x}\\ &=\frac{\left (\sqrt{-\frac{1}{n^2}} \left (c x^n\right )^{\frac{1}{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{2+n}{n}}\right ) \, dx,x,c x^n\right )}{2 x}\\ &=\frac{e^{a \sqrt{-\frac{1}{n^2}} n} \sqrt{-\frac{1}{n^2}} n \left (c x^n\right )^{-1/n}}{4 x}+\frac{e^{-a \sqrt{-\frac{1}{n^2}} n} \sqrt{-\frac{1}{n^2}} n \left (c x^n\right )^{\frac{1}{n}} \log (x)}{2 x}\\ \end{align*}
Mathematica [F] time = 0.0722603, size = 0, normalized size = 0. \[ \int \frac{\sin \left (a+\sqrt{-\frac{1}{n^2}} \log \left (c x^n\right )\right )}{x^2} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.038, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}}\sin \left ( a+\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 = 1.13135, size = 45, normalized size = 0.52 \begin{align*} \frac{2 \, c^{\frac{2}{n}} x^{2} \log \left (x\right ) \sin \left (a\right ) - \sin \left (a\right )}{4 \, c^{\left (\frac{1}{n}\right )} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 0.472435, size = 107, normalized size = 1.24 \begin{align*} \frac{{\left (2 i \, x^{2} \log \left (x\right ) + i \, e^{\left (\frac{2 \,{\left (i \, a n - \log \left (c\right )\right )}}{n}\right )}\right )} e^{\left (-\frac{i \, a n - \log \left (c\right )}{n}\right )}}{4 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 14.2811, size = 214, normalized size = 2.49 \begin{align*} \frac{i n \sqrt{\frac{1}{n^{2}}} \log{\left (x \right )} \cos{\left (a + i n \sqrt{\frac{1}{n^{2}}} \log{\left (x \right )} + i \sqrt{\frac{1}{n^{2}}} \log{\left (c \right )} \right )}}{2 x} + \frac{i \sqrt{\frac{1}{n^{2}}} \log{\left (c \right )} \cos{\left (a + i n \sqrt{\frac{1}{n^{2}}} \log{\left (x \right )} + i \sqrt{\frac{1}{n^{2}}} \log{\left (c \right )} \right )}}{2 x} + \frac{\log{\left (x \right )} \sin{\left (a + i n \sqrt{\frac{1}{n^{2}}} \log{\left (x \right )} + i \sqrt{\frac{1}{n^{2}}} \log{\left (c \right )} \right )}}{2 x} - \frac{\sin{\left (a + i n \sqrt{\frac{1}{n^{2}}} \log{\left (x \right )} + i \sqrt{\frac{1}{n^{2}}} \log{\left (c \right )} \right )}}{2 x} + \frac{\log{\left (c \right )} \sin{\left (a + i n \sqrt{\frac{1}{n^{2}}} \log{\left (x \right )} + i \sqrt{\frac{1}{n^{2}}} \log{\left (c \right )} \right )}}{2 n x} \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 (\sqrt{-\frac{1}{n^{2}}} \log \left (c x^{n}\right ) + a\right )}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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