Optimal. Leaf size=19 \[ \log \left (a x^{m-1}+b \log ^q\left (c x^n\right )\right ) \]
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Rubi [A] time = 0.32, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.044, Rules used = {2561, 2541} \[ \log \left (a x^{m-1}+b \log ^q\left (c x^n\right )\right ) \]
Antiderivative was successfully verified.
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Rule 2541
Rule 2561
Rubi steps
\begin {align*} \int \frac {a (-1+m) x^{-1+m}+b n q \log ^{-1+q}\left (c x^n\right )}{a x^m+b x \log ^q\left (c x^n\right )} \, dx &=\int \frac {a (-1+m) x^{-1+m}+b n q \log ^{-1+q}\left (c x^n\right )}{x \left (a x^{-1+m}+b \log ^q\left (c x^n\right )\right )} \, dx\\ &=\log \left (a x^{-1+m}+b \log ^q\left (c x^n\right )\right )\\ \end {align*}
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Mathematica [A] time = 0.48, size = 23, normalized size = 1.21 \[ \log \left (a x^m+b x \log ^q\left (c x^n\right )\right )-\log (x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 23, normalized size = 1.21 \[ \log \left (\frac {{\left (n \log \relax (x) + \log \relax (c)\right )}^{q} b x + a x^{m}}{x}\right ) \]
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 \[ \int \frac {b n q \log \left (c x^{n}\right )^{q - 1} + a {\left (m - 1\right )} x^{m - 1}}{b x \log \left (c x^{n}\right )^{q} + a x^{m}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.60, size = 215, normalized size = 11.32 \[ -q \ln \left (-\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (\mathrm {csgn}\left (i x^{n}\right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2}+\ln \relax (c )+\ln \left (x^{n}\right )\right )+q \ln \left (-\frac {i \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{2}+\ln \relax (c )+\ln \left (x^{n}\right )\right )+\ln \left (\left (-\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (\mathrm {csgn}\left (i x^{n}\right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2}+\ln \relax (c )+\ln \left (x^{n}\right )\right )^{q}+\frac {a \,x^{m}}{b x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.03, size = 26, normalized size = 1.37 \[ \log \left (\frac {b x {\left (\log \relax (c) + \log \left (x^{n}\right )\right )}^{q} + a x^{m}}{b x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {a\,x^{m-1}\,\left (m-1\right )+b\,n\,q\,{\ln \left (c\,x^n\right )}^{q-1}}{a\,x^m+b\,x\,{\ln \left (c\,x^n\right )}^q} \,d x \]
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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