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 = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, 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 2298
Rule 2520
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*}
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Mathematica [A] time = 0.02, size = 22, normalized size = 1.00 \[ x \log \left (c \log ^p(d x)\right )-\frac {p \text {li}(d x)}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 26, normalized size = 1.18 \[ \frac {d p x \log \left (\log \left (d x\right )\right ) + d x \log \relax (c) - p \operatorname {log\_integral}\left (d x\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 26, normalized size = 1.18 \[ p x \log \left (\log \relax (d) + \log \relax (x)\right ) + x \log \relax (c) - \frac {p {\rm Ei}\left (\log \relax (d) + \log \relax (x)\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 26, normalized size = 1.18 \[ x \ln \left (c \ln \left (d x \right )^{p}\right )+\frac {p \Ei \left (1, -\ln \left (d x \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.62, size = 23, normalized size = 1.05 \[ x \log \left (c \log \left (d x\right )^{p}\right ) - \frac {p {\rm Ei}\left (\log \left (d x\right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.31, size = 22, normalized size = 1.00 \[ x\,\ln \left (c\,{\ln \left (d\,x\right )}^p\right )-\frac {p\,\mathrm {logint}\left (d\,x\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.19, size = 19, normalized size = 0.86 \[ x \log {\left (c \log {\left (d x \right )}^{p} \right )} - \frac {p \operatorname {li}{\left (d x \right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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