Optimal. Leaf size=32 \[ \frac {\log \left (d x^n\right ) \left (a+b \log \left (c \log ^p\left (d x^n\right )\right )\right )}{n}-b p \log (x) \]
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Rubi [A] time = 0.02, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2521} \[ \frac {\log \left (d x^n\right ) \left (a+b \log \left (c \log ^p\left (d x^n\right )\right )\right )}{n}-b p \log (x) \]
Antiderivative was successfully verified.
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Rule 2521
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c \log ^p\left (d x^n\right )\right )}{x} \, dx &=-b p \log (x)+\frac {\log \left (d x^n\right ) \left (a+b \log \left (c \log ^p\left (d x^n\right )\right )\right )}{n}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 40, normalized size = 1.25 \[ a \log (x)+\frac {b \log \left (d x^n\right ) \log \left (c \log ^p\left (d x^n\right )\right )}{n}-\frac {b p \log \left (d x^n\right )}{n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 45, normalized size = 1.41 \[ \frac {{\left (b n p \log \relax (x) + b p \log \relax (d)\right )} \log \left (n \log \relax (x) + \log \relax (d)\right ) - {\left (b n p - b n \log \relax (c) - a n\right )} \log \relax (x)}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 54, normalized size = 1.69 \[ \frac {{\left ({\left (n \log \relax (x) + \log \relax (d)\right )} \log \left (n \log \relax (x) + \log \relax (d)\right ) - n \log \relax (x) - \log \relax (d)\right )} b p + {\left (n \log \relax (x) + \log \relax (d)\right )} b \log \relax (c) + {\left (n \log \relax (x) + \log \relax (d)\right )} a}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 48, normalized size = 1.50 \[ -\frac {b p \ln \left (d \,x^{n}\right )}{n}+\frac {b \ln \left (c \ln \left (d \,x^{n}\right )^{p}\right ) \ln \left (d \,x^{n}\right )}{n}+\frac {a \ln \left (d \,x^{n}\right )}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 64, normalized size = 2.00 \[ b \log \left (c \log \left (d x^{n}\right )^{p}\right ) \log \relax (x) - {\left (p \log \relax (x) \log \left (\log \left (d x^{n}\right )\right ) - \frac {{\left (\log \left (d x^{n}\right ) \log \left (\log \left (d x^{n}\right )\right ) - \log \left (d x^{n}\right )\right )} p}{n}\right )} b + a \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.32, size = 32, normalized size = 1.00 \[ \ln \relax (x)\,\left (a-b\,p\right )+\frac {b\,\ln \left (c\,{\ln \left (d\,x^n\right )}^p\right )\,\ln \left (d\,x^n\right )}{n} \]
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 \[ \int \frac {a + b \log {\left (c \log {\left (d x^{n} \right )}^{p} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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