Optimal. Leaf size=20 \[ -p \log (x)+\log (d x) \log \left (c \log ^p(d x)\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.00, 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 = {2601}
\begin {gather*} \log (d x) \log \left (c \log ^p(d x)\right )-p \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 2601
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*}
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Mathematica [A]
time = 0.01, size = 22, normalized size = 1.10 \begin {gather*} -p \log (d x)+\log (d x) \log \left (c \log ^p(d x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 23, normalized size = 1.15
method | result | size |
derivativedivides | \(\ln \left (d x \right ) \ln \left (c \ln \left (d x \right )^{p}\right )-\ln \left (d x \right ) p\) | \(23\) |
default | \(\ln \left (d x \right ) \ln \left (c \ln \left (d x \right )^{p}\right )-\ln \left (d x \right ) p\) | \(23\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 22, normalized size = 1.10 \begin {gather*} -p \log \left (d x\right ) + \log \left (d x\right ) \log \left (c \log \left (d x\right )^{p}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 24, normalized size = 1.20 \begin {gather*} 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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\log {\left (c \log {\left (d x \right )}^{p} \right )}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.18, size = 32, normalized size = 1.60 \begin {gather*} {\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.28, size = 20, normalized size = 1.00 \begin {gather*} \ln \left (c\,{\ln \left (d\,x\right )}^p\right )\,\ln \left (d\,x\right )-p\,\ln \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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