Optimal. Leaf size=51 \[ \frac {b n x \sin \left (a+b \log \left (c x^n\right )\right )}{b^2 n^2+1}+\frac {x \cos \left (a+b \log \left (c x^n\right )\right )}{b^2 n^2+1} \]
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Rubi [A] time = 0.01, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {4476} \[ \frac {b n x \sin \left (a+b \log \left (c x^n\right )\right )}{b^2 n^2+1}+\frac {x \cos \left (a+b \log \left (c x^n\right )\right )}{b^2 n^2+1} \]
Antiderivative was successfully verified.
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Rule 4476
Rubi steps
\begin {align*} \int \cos \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {x \cos \left (a+b \log \left (c x^n\right )\right )}{1+b^2 n^2}+\frac {b n x \sin \left (a+b \log \left (c x^n\right )\right )}{1+b^2 n^2}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 39, normalized size = 0.76 \[ \frac {x \left (b n \sin \left (a+b \log \left (c x^n\right )\right )+\cos \left (a+b \log \left (c x^n\right )\right )\right )}{b^2 n^2+1} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 43, normalized size = 0.84 \[ \frac {b n x \sin \left (b n \log \relax (x) + b \log \relax (c) + a\right ) + x \cos \left (b n \log \relax (x) + b \log \relax (c) + a\right )}{b^{2} n^{2} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.28, size = 878, normalized size = 17.22 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int \cos \left (a +b \ln \left (c \,x^{n}\right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.36, size = 205, normalized size = 4.02 \[ \frac {{\left ({\left (b \cos \left (b \log \relax (c)\right ) \sin \left (2 \, b \log \relax (c)\right ) - b \cos \left (2 \, b \log \relax (c)\right ) \sin \left (b \log \relax (c)\right ) + b \sin \left (b \log \relax (c)\right )\right )} n + \cos \left (2 \, b \log \relax (c)\right ) \cos \left (b \log \relax (c)\right ) + \sin \left (2 \, b \log \relax (c)\right ) \sin \left (b \log \relax (c)\right ) + \cos \left (b \log \relax (c)\right )\right )} x \cos \left (b \log \left (x^{n}\right ) + a\right ) + {\left ({\left (b \cos \left (2 \, b \log \relax (c)\right ) \cos \left (b \log \relax (c)\right ) + b \sin \left (2 \, b \log \relax (c)\right ) \sin \left (b \log \relax (c)\right ) + b \cos \left (b \log \relax (c)\right )\right )} n - \cos \left (b \log \relax (c)\right ) \sin \left (2 \, b \log \relax (c)\right ) + \cos \left (2 \, b \log \relax (c)\right ) \sin \left (b \log \relax (c)\right ) - \sin \left (b \log \relax (c)\right )\right )} x \sin \left (b \log \left (x^{n}\right ) + a\right )}{2 \, {\left ({\left (b^{2} \cos \left (b \log \relax (c)\right )^{2} + b^{2} \sin \left (b \log \relax (c)\right )^{2}\right )} n^{2} + \cos \left (b \log \relax (c)\right )^{2} + \sin \left (b \log \relax (c)\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.35, size = 39, normalized size = 0.76 \[ \frac {x\,\left (\cos \left (a+b\,\ln \left (c\,x^n\right )\right )+b\,n\,\sin \left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{b^2\,n^2+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \begin {cases} \int \cos {\left (a - \frac {i \log {\left (c x^{n} \right )}}{n} \right )}\, dx & \text {for}\: b = - \frac {i}{n} \\\int \cos {\left (a + \frac {i \log {\left (c x^{n} \right )}}{n} \right )}\, dx & \text {for}\: b = \frac {i}{n} \\\frac {b n x \sin {\left (a + b n \log {\relax (x )} + b \log {\relax (c )} \right )}}{b^{2} n^{2} + 1} + \frac {x \cos {\left (a + b n \log {\relax (x )} + b \log {\relax (c )} \right )}}{b^{2} n^{2} + 1} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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