Optimal. Leaf size=18 \[ \frac{\sin \left (a+b \log \left (c x^n\right )\right )}{b n} \]
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Rubi [A] time = 0.0147802, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {2637} \[ \frac{\sin \left (a+b \log \left (c x^n\right )\right )}{b n} \]
Antiderivative was successfully verified.
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Rule 2637
Rubi steps
\begin{align*} \int \frac{\cos \left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \cos (a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac{\sin \left (a+b \log \left (c x^n\right )\right )}{b n}\\ \end{align*}
Mathematica [B] time = 0.0269324, size = 37, normalized size = 2.06 \[ \frac{\sin (a) \cos \left (b \log \left (c x^n\right )\right )}{b n}+\frac{\cos (a) \sin \left (b \log \left (c x^n\right )\right )}{b n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.021, size = 19, normalized size = 1.1 \begin{align*}{\frac{\sin \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }{bn}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.989078, size = 24, normalized size = 1.33 \begin{align*} \frac{\sin \left (b \log \left (c x^{n}\right ) + a\right )}{b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.48318, size = 51, normalized size = 2.83 \begin{align*} \frac{\sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.39403, size = 37, normalized size = 2.06 \begin{align*} \begin{cases} \log{\left (x \right )} \cos{\left (a \right )} & \text{for}\: b = 0 \wedge \left (b = 0 \vee n = 0\right ) \\\log{\left (x \right )} \cos{\left (a + b \log{\left (c \right )} \right )} & \text{for}\: n = 0 \\\frac{\sin{\left (a + b n \log{\left (x \right )} + b \log{\left (c \right )} \right )}}{b n} & \text{otherwise} \end{cases} \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 )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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