Optimal. Leaf size=74 \[ \frac{e^{2 a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{-1/n}}{8 x}-\frac{e^{-2 a \sqrt{-\frac{1}{n^2}} n} \log (x) \left (c x^n\right )^{\frac{1}{n}}}{4 x}-\frac{1}{2 x} \]
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Rubi [A] time = 0.068497, antiderivative size = 74, 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} \[ \frac{e^{2 a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{-1/n}}{8 x}-\frac{e^{-2 a \sqrt{-\frac{1}{n^2}} n} \log (x) \left (c x^n\right )^{\frac{1}{n}}}{4 x}-\frac{1}{2 x} \]
Antiderivative was successfully verified.
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Rule 4493
Rule 4489
Rubi steps
\begin{align*} \int \frac{\sin ^2\left (a+\frac{1}{2} \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 ^2\left (a+\frac{1}{2} \sqrt{-\frac{1}{n^2}} \log (x)\right ) \, dx,x,c x^n\right )}{n x}\\ &=-\frac{\left (c x^n\right )^{\frac{1}{n}} \operatorname{Subst}\left (\int \left (\frac{e^{-2 a \sqrt{-\frac{1}{n^2}} n}}{x}-2 x^{-\frac{1+n}{n}}+e^{2 a \sqrt{-\frac{1}{n^2}} n} x^{-\frac{2+n}{n}}\right ) \, dx,x,c x^n\right )}{4 n x}\\ &=-\frac{1}{2 x}+\frac{e^{2 a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{-1/n}}{8 x}-\frac{e^{-2 a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{\frac{1}{n}} \log (x)}{4 x}\\ \end{align*}
Mathematica [F] time = 0.149265, size = 0, normalized size = 0. \[ \int \frac{\sin ^2\left (a+\frac{1}{2} \sqrt{-\frac{1}{n^2}} \log \left (c x^n\right )\right )}{x^2} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.065, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \left ( \sin \left ( a+{\frac{\ln \left ( c{x}^{n} \right ) }{2}\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 = 1.1636, size = 65, normalized size = 0.88 \begin{align*} -\frac{2 \, c^{\frac{2}{n}} x^{3} \cos \left (2 \, a\right ) \log \left (x\right ) + 4 \, c^{\left (\frac{1}{n}\right )} x^{2} - x \cos \left (2 \, a\right )}{8 \, c^{\left (\frac{1}{n}\right )} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 0.471367, size = 150, normalized size = 2.03 \begin{align*} -\frac{{\left (2 \, x^{2} \log \left (x\right ) + 4 \, x e^{\left (\frac{2 i \, a n - \log \left (c\right )}{n}\right )} - e^{\left (\frac{2 \,{\left (2 i \, a n - \log \left (c\right )\right )}}{n}\right )}\right )} e^{\left (-\frac{2 i \, a n - \log \left (c\right )}{n}\right )}}{8 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 75.7666, size = 240, normalized size = 3.24 \begin{align*} \frac{i n \sqrt{\frac{1}{n^{2}}} \log{\left (x \right )} \sin{\left (2 a + i n \sqrt{\frac{1}{n^{2}}} \log{\left (x \right )} + i \sqrt{\frac{1}{n^{2}}} \log{\left (c \right )} \right )}}{4 x} + \frac{i n \sqrt{\frac{1}{n^{2}}} \sin{\left (2 a + i n \sqrt{\frac{1}{n^{2}}} \log{\left (x \right )} + i \sqrt{\frac{1}{n^{2}}} \log{\left (c \right )} \right )}}{4 x} + \frac{i \sqrt{\frac{1}{n^{2}}} \log{\left (c \right )} \sin{\left (2 a + i n \sqrt{\frac{1}{n^{2}}} \log{\left (x \right )} + i \sqrt{\frac{1}{n^{2}}} \log{\left (c \right )} \right )}}{4 x} - \frac{\log{\left (x \right )} \cos{\left (2 a + i n \sqrt{\frac{1}{n^{2}}} \log{\left (x \right )} + i \sqrt{\frac{1}{n^{2}}} \log{\left (c \right )} \right )}}{4 x} - \frac{1}{2 x} - \frac{\log{\left (c \right )} \cos{\left (2 a + i n \sqrt{\frac{1}{n^{2}}} \log{\left (x \right )} + i \sqrt{\frac{1}{n^{2}}} \log{\left (c \right )} \right )}}{4 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 (\frac{1}{2} \, \sqrt{-\frac{1}{n^{2}}} \log \left (c x^{n}\right ) + a\right )^{2}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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