Optimal. Leaf size=26 \[ \frac {d \log \left (c x^n\right )}{a x^m+b \log ^q\left (c x^n\right )} \]
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Rubi [A] time = 0.25, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 60, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.017, Rules used = {2546} \[ \frac {d \log \left (c x^n\right )}{a x^m+b \log ^q\left (c x^n\right )} \]
Antiderivative was successfully verified.
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Rule 2546
Rubi steps
\begin {align*} \int \frac {a d n x^m-a d m x^m \log \left (c x^n\right )-b d n (-1+q) \log ^q\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )^2} \, dx &=\frac {d \log \left (c x^n\right )}{a x^m+b \log ^q\left (c x^n\right )}\\ \end {align*}
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Mathematica [A] time = 0.37, size = 26, normalized size = 1.00 \[ \frac {d \log \left (c x^n\right )}{a x^m+b \log ^q\left (c x^n\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 30, normalized size = 1.15 \[ \frac {d n \log \relax (x) + d \log \relax (c)}{{\left (n \log \relax (x) + \log \relax (c)\right )}^{q} b + a x^{m}} \]
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 \[ \int -\frac {b d n {\left (q - 1\right )} \log \left (c x^{n}\right )^{q} + a d m x^{m} \log \left (c x^{n}\right ) - a d n x^{m}}{{\left (a x^{m} + b \log \left (c x^{n}\right )^{q}\right )}^{2} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.51, size = 158, normalized size = 6.08 \[ \frac {\left (-i \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+i \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+2 \ln \relax (c )+2 \ln \left (x^{n}\right )\right ) d}{2 a \,x^{m}+2 b \left (-\frac {i \pi \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \left (\mathrm {csgn}\left (i x^{n}\right )-\mathrm {csgn}\left (i c \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2}+\ln \relax (c )+\ln \left (x^{n}\right )\right )^{q}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.25, size = 31, normalized size = 1.19 \[ \frac {d \log \relax (c) + d \log \left (x^{n}\right )}{a x^{m} + b {\left (\log \relax (c) + \log \left (x^{n}\right )\right )}^{q}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.33, size = 26, normalized size = 1.00 \[ \frac {d\,\ln \left (c\,x^n\right )}{a\,x^m+b\,{\ln \left (c\,x^n\right )}^q} \]
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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