Optimal. Leaf size=95 \[ -\frac{\cos ^2\left (a+b \log \left (c x^n\right )\right )}{x \left (4 b^2 n^2+1\right )}+\frac{2 b n \sin \left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{x \left (4 b^2 n^2+1\right )}-\frac{2 b^2 n^2}{x \left (4 b^2 n^2+1\right )} \]
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Rubi [A] time = 0.026947, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {4488, 30} \[ -\frac{\cos ^2\left (a+b \log \left (c x^n\right )\right )}{x \left (4 b^2 n^2+1\right )}+\frac{2 b n \sin \left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{x \left (4 b^2 n^2+1\right )}-\frac{2 b^2 n^2}{x \left (4 b^2 n^2+1\right )} \]
Antiderivative was successfully verified.
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Rule 4488
Rule 30
Rubi steps
\begin{align*} \int \frac{\cos ^2\left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx &=-\frac{\cos ^2\left (a+b \log \left (c x^n\right )\right )}{\left (1+4 b^2 n^2\right ) x}+\frac{2 b n \cos \left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{\left (1+4 b^2 n^2\right ) x}+\frac{\left (2 b^2 n^2\right ) \int \frac{1}{x^2} \, dx}{1+4 b^2 n^2}\\ &=-\frac{2 b^2 n^2}{\left (1+4 b^2 n^2\right ) x}-\frac{\cos ^2\left (a+b \log \left (c x^n\right )\right )}{\left (1+4 b^2 n^2\right ) x}+\frac{2 b n \cos \left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{\left (1+4 b^2 n^2\right ) x}\\ \end{align*}
Mathematica [A] time = 0.117084, size = 57, normalized size = 0.6 \[ -\frac{-2 b n \sin \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+\cos \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+4 b^2 n^2+1}{2 \left (4 b^2 n^2 x+x\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.057, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \cos \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}}{{x}^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.37307, size = 385, normalized size = 4.05 \begin{align*} -\frac{8 \,{\left (b^{2} \cos \left (2 \, b \log \left (c\right )\right )^{2} + b^{2} \sin \left (2 \, b \log \left (c\right )\right )^{2}\right )} n^{2} + 2 \, \cos \left (2 \, b \log \left (c\right )\right )^{2} -{\left (2 \,{\left (b \cos \left (2 \, b \log \left (c\right )\right ) \sin \left (4 \, b \log \left (c\right )\right ) - b \cos \left (4 \, b \log \left (c\right )\right ) \sin \left (2 \, b \log \left (c\right )\right ) + b \sin \left (2 \, b \log \left (c\right )\right )\right )} n - \cos \left (4 \, b \log \left (c\right )\right ) \cos \left (2 \, b \log \left (c\right )\right ) - \sin \left (4 \, b \log \left (c\right )\right ) \sin \left (2 \, b \log \left (c\right )\right ) - \cos \left (2 \, b \log \left (c\right )\right )\right )} \cos \left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right ) + 2 \, \sin \left (2 \, b \log \left (c\right )\right )^{2} -{\left (2 \,{\left (b \cos \left (4 \, b \log \left (c\right )\right ) \cos \left (2 \, b \log \left (c\right )\right ) + b \sin \left (4 \, b \log \left (c\right )\right ) \sin \left (2 \, b \log \left (c\right )\right ) + b \cos \left (2 \, b \log \left (c\right )\right )\right )} n + \cos \left (2 \, b \log \left (c\right )\right ) \sin \left (4 \, b \log \left (c\right )\right ) - \cos \left (4 \, b \log \left (c\right )\right ) \sin \left (2 \, b \log \left (c\right )\right ) + \sin \left (2 \, b \log \left (c\right )\right )\right )} \sin \left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )}{4 \,{\left (4 \,{\left (b^{2} \cos \left (2 \, b \log \left (c\right )\right )^{2} + b^{2} \sin \left (2 \, b \log \left (c\right )\right )^{2}\right )} n^{2} + \cos \left (2 \, b \log \left (c\right )\right )^{2} + \sin \left (2 \, b \log \left (c\right )\right )^{2}\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.504295, size = 188, normalized size = 1.98 \begin{align*} -\frac{2 \, b^{2} n^{2} - 2 \, b n \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2}}{{\left (4 \, b^{2} n^{2} + 1\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \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{\cos \left (b \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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