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 = 0.00, 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 = {2600, 2335}
\begin {gather*} x \log \left (c \log ^p(d x)\right )-\frac {p \text {li}(d x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2335
Rule 2600
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 = 0.01, size = 22, normalized size = 1.00 \begin {gather*} x \log \left (c \log ^p(d x)\right )-\frac {p \text {li}(d x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 26, normalized size = 1.18
method | result | size |
default | \(x \ln \left (c \ln \left (d x \right )^{p}\right )+\frac {p \expIntegral \left (1, -\ln \left (d x \right )\right )}{d}\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.34, size = 23, normalized size = 1.05 \begin {gather*} x \log \left (c \log \left (d x\right )^{p}\right ) - \frac {p {\rm Ei}\left (\log \left (d x\right )\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 26, normalized size = 1.18 \begin {gather*} \frac {d p x \log \left (\log \left (d x\right )\right ) + d x \log \left (c\right ) - p \operatorname {log\_integral}\left (d x\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.57, size = 19, normalized size = 0.86 \begin {gather*} x \log {\left (c \log {\left (d x \right )}^{p} \right )} - \frac {p \operatorname {li}{\left (d x \right )}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.95, size = 26, normalized size = 1.18 \begin {gather*} 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 {gather*}
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 \begin {gather*} x\,\ln \left (c\,{\ln \left (d\,x\right )}^p\right )-\frac {p\,\mathrm {logint}\left (d\,x\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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