Optimal. Leaf size=27 \[ \frac{\log \left (d x^n\right ) \log \left (c \log ^p\left (d x^n\right )\right )}{n}-p \log (x) \]
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Rubi [A] time = 0.0211121, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {2521} \[ \frac{\log \left (d x^n\right ) \log \left (c \log ^p\left (d x^n\right )\right )}{n}-p \log (x) \]
Antiderivative was successfully verified.
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Rule 2521
Rubi steps
\begin{align*} \int \frac{\log \left (c \log ^p\left (d x^n\right )\right )}{x} \, dx &=-p \log (x)+\frac{\log \left (d x^n\right ) \log \left (c \log ^p\left (d x^n\right )\right )}{n}\\ \end{align*}
Mathematica [A] time = 0.0101253, size = 34, normalized size = 1.26 \[ \frac{\log \left (d x^n\right ) \log \left (c \log ^p\left (d x^n\right )\right )}{n}-\frac{p \log \left (d x^n\right )}{n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 35, normalized size = 1.3 \begin{align*}{\frac{\ln \left ( c \left ( \ln \left ( d{x}^{n} \right ) \right ) ^{p} \right ) \ln \left ( d{x}^{n} \right ) }{n}}-{\frac{p\ln \left ( d{x}^{n} \right ) }{n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.01719, size = 74, normalized size = 2.74 \begin{align*} -p \log \left (x\right ) \log \left (\log \left (d x^{n}\right )\right ) + \log \left (c \log \left (d x^{n}\right )^{p}\right ) \log \left (x\right ) + \frac{{\left (\log \left (d x^{n}\right ) \log \left (\log \left (d x^{n}\right )\right ) - \log \left (d x^{n}\right )\right )} p}{n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52428, size = 105, normalized size = 3.89 \begin{align*} \frac{{\left (n p \log \left (x\right ) + p \log \left (d\right )\right )} \log \left (n \log \left (x\right ) + \log \left (d\right )\right ) -{\left (n p - n \log \left (c\right )\right )} \log \left (x\right )}{n} \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^{n} \right )}^{p} \right )}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.36748, size = 58, normalized size = 2.15 \begin{align*} \frac{{\left ({\left (n \log \left (x\right ) + \log \left (d\right )\right )} \log \left (n \log \left (x\right ) + \log \left (d\right )\right ) - n \log \left (x\right ) - \log \left (d\right )\right )} p +{\left (n \log \left (x\right ) + \log \left (d\right )\right )} \log \left (c\right )}{n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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