Optimal. Leaf size=20 \[ \log (d x) \log \left (c \log ^p(d x)\right )-p \log (x) \]
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Rubi [A] time = 0.0198224, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {2521} \[ \log (d x) \log \left (c \log ^p(d x)\right )-p \log (x) \]
Antiderivative was successfully verified.
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Rule 2521
Rubi steps
\begin{align*} \int \frac{\log \left (c \log ^p(d x)\right )}{x} \, dx &=-p \log (x)+\log (d x) \log \left (c \log ^p(d x)\right )\\ \end{align*}
Mathematica [A] time = 0.0065151, size = 22, normalized size = 1.1 \[ \log (d x) \log \left (c \log ^p(d x)\right )-p \log (d x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 23, normalized size = 1.2 \begin{align*} \ln \left ( dx \right ) \ln \left ( c \left ( \ln \left ( dx \right ) \right ) ^{p} \right ) -p\ln \left ( dx \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.993782, size = 30, normalized size = 1.5 \begin{align*} -p \log \left (d x\right ) + \log \left (d x\right ) \log \left (c \log \left (d x\right )^{p}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63894, size = 68, normalized size = 3.4 \begin{align*} p \log \left (d x\right ) \log \left (\log \left (d x\right )\right ) -{\left (p - \log \left (c\right )\right )} \log \left (d x\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 \frac{\log{\left (c \log{\left (d x \right )}^{p} \right )}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.30266, size = 43, normalized size = 2.15 \begin{align*}{\left ({\left (\log \left (d\right ) + \log \left (x\right )\right )} \log \left (\log \left (d\right ) + \log \left (x\right )\right ) - \log \left (d\right ) - \log \left (x\right )\right )} p +{\left (\log \left (d\right ) + \log \left (x\right )\right )} \log \left (c\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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