Optimal. Leaf size=17 \[ \log \left (a x^m+b \log ^q\left (c x^n\right )\right ) \]
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Rubi [A] time = 0.19, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {2541} \[ \log \left (a x^m+b \log ^q\left (c x^n\right )\right ) \]
Antiderivative was successfully verified.
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Rule 2541
Rubi steps
\begin {align*} \int \frac {a m x^m+b n q \log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )} \, dx &=\log \left (a x^m+b \log ^q\left (c x^n\right )\right )\\ \end {align*}
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Mathematica [A] time = 0.24, size = 17, normalized size = 1.00 \[ \log \left (a x^m+b \log ^q\left (c x^n\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 18, normalized size = 1.06 \[ \log \left ({\left (n \log \relax (x) + \log \relax (c)\right )}^{q} b + a x^{m}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b n q \log \left (c x^{n}\right )^{q - 1} + a m x^{m}}{{\left (a x^{m} + b \log \left (c x^{n}\right )^{q}\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.58, size = 212, normalized size = 12.47 \[ -q \ln \left (-\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (\mathrm {csgn}\left (i x^{n}\right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2}+\ln \relax (c )+\ln \left (x^{n}\right )\right )+q \ln \left (-\frac {i \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{2}+\ln \relax (c )+\ln \left (x^{n}\right )\right )+\ln \left (\frac {a \,x^{m}}{b}+\left (-\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (\mathrm {csgn}\left (i x^{n}\right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2}+\ln \relax (c )+\ln \left (x^{n}\right )\right )^{q}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.03, size = 22, normalized size = 1.29 \[ \log \left (\frac {a x^{m} + b {\left (\log \relax (c) + \log \left (x^{n}\right )\right )}^{q}}{b}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.06 \[ \int \frac {a\,m\,x^m+b\,n\,q\,{\ln \left (c\,x^n\right )}^{q-1}}{x\,\left (a\,x^m+b\,{\ln \left (c\,x^n\right )}^q\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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