Optimal. Leaf size=22 \[ x \log \left (c \log ^p(d x)\right )-\frac{p \text{li}(d x)}{d} \]
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Rubi [A] time = 0.0067795, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2520, 2298} \[ x \log \left (c \log ^p(d x)\right )-\frac{p \text{li}(d x)}{d} \]
Antiderivative was successfully verified.
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Rule 2520
Rule 2298
Rubi steps
\begin{align*} \int \log \left (c \log ^p(d x)\right ) \, dx &=x \log \left (c \log ^p(d x)\right )-p \int \frac{1}{\log (d x)} \, dx\\ &=x \log \left (c \log ^p(d x)\right )-\frac{p \text{li}(d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0140017, size = 22, normalized size = 1. \[ x \log \left (c \log ^p(d x)\right )-\frac{p \text{li}(d x)}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 26, normalized size = 1.2 \begin{align*} x\ln \left ( c \left ( \ln \left ( dx \right ) \right ) ^{p} \right ) +{\frac{p{\it Ei} \left ( 1,-\ln \left ( dx \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14172, size = 31, normalized size = 1.41 \begin{align*} x \log \left (c \log \left (d x\right )^{p}\right ) - \frac{p{\rm Ei}\left (\log \left (d x\right )\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64578, size = 81, normalized size = 3.68 \begin{align*} \frac{d p x \log \left (\log \left (d x\right )\right ) + d x \log \left (c\right ) - p \logintegral \left (d x\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.31437, size = 19, normalized size = 0.86 \begin{align*} x \log{\left (c \log{\left (d x \right )}^{p} \right )} - \frac{p \operatorname{li}{\left (d x \right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20494, size = 35, normalized size = 1.59 \begin{align*} p x \log \left (\log \left (d\right ) + \log \left (x\right )\right ) + x \log \left (c\right ) - \frac{p{\rm Ei}\left (\log \left (d\right ) + \log \left (x\right )\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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