Optimal. Leaf size=68 \[ \frac{1}{8} x e^{-2 a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{\frac{1}{n}}+\frac{1}{4} x e^{2 a \sqrt{-\frac{1}{n^2}} n} \log (x) \left (c x^n\right )^{-1/n}+\frac{x}{2} \]
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Rubi [A] time = 0.0557935, antiderivative size = 68, 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 = {4484, 4490} \[ \frac{1}{8} x e^{-2 a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{\frac{1}{n}}+\frac{1}{4} x e^{2 a \sqrt{-\frac{1}{n^2}} n} \log (x) \left (c x^n\right )^{-1/n}+\frac{x}{2} \]
Antiderivative was successfully verified.
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Rule 4484
Rule 4490
Rubi steps
\begin{align*} \int \cos ^2\left (a+\frac{1}{2} \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}} \cos ^2\left (a+\frac{1}{2} \sqrt{-\frac{1}{n^2}} \log (x)\right ) \, dx,x,c x^n\right )}{n}\\ &=\frac{\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \left (\frac{e^{2 a \sqrt{-\frac{1}{n^2}} n}}{x}+2 x^{-1+\frac{1}{n}}+e^{-2 a \sqrt{-\frac{1}{n^2}} n} x^{-1+\frac{2}{n}}\right ) \, dx,x,c x^n\right )}{4 n}\\ &=\frac{x}{2}+\frac{1}{8} e^{-2 a \sqrt{-\frac{1}{n^2}} n} x \left (c x^n\right )^{\frac{1}{n}}+\frac{1}{4} e^{2 a \sqrt{-\frac{1}{n^2}} n} x \left (c x^n\right )^{-1/n} \log (x)\\ \end{align*}
Mathematica [F] time = 0.100558, size = 0, normalized size = 0. \[ \int \cos ^2\left (a+\frac{1}{2} \sqrt{-\frac{1}{n^2}} \log \left (c x^n\right )\right ) \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.079, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \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 = 1.29058, size = 55, normalized size = 0.81 \begin{align*} \frac{c^{\frac{2}{n}} x^{2} \cos \left (2 \, a\right ) + 4 \, c^{\left (\frac{1}{n}\right )} x + 2 \, \cos \left (2 \, a\right ) \log \left (x\right )}{8 \, c^{\left (\frac{1}{n}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 0.471724, size = 143, normalized size = 2.1 \begin{align*} \frac{1}{8} \,{\left (x^{2} + 4 \, x e^{\left (\frac{2 i \, a n - \log \left (c\right )}{n}\right )} + 2 \, e^{\left (\frac{2 \,{\left (2 i \, a n - \log \left (c\right )\right )}}{n}\right )} \log \left (x\right )\right )} e^{\left (-\frac{2 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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \cos ^{2}{\left (a + \frac{\sqrt{- \frac{1}{n^{2}}} \log{\left (c x^{n} \right )}}{2} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.61904, size = 1, normalized size = 0.01 \begin{align*} +\infty \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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