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 = 0.320846, antiderivative size = 19, normalized size of antiderivative = 1., 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 2561
Rule 2541
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*}
Mathematica [A] time = 0.460488, 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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Maple [C] time = 0.191, size = 216, normalized size = 11.4 \begin{align*} q\ln \left ( \ln \left ({x}^{n} \right ) -{\frac{i}{2}} \left ( \pi \,{\it csgn} \left ( ic \right ){\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ) -\pi \,{\it csgn} \left ( ic \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+2\,i\ln \left ( c \right ) \right ) \right ) -q\ln \left ( \ln \left ( c \right ) +\ln \left ({x}^{n} \right ) -{\frac{i}{2}}\pi \,{\it csgn} \left ( ic{x}^{n} \right ) \left ( -{\it csgn} \left ( ic{x}^{n} \right ) +{\it csgn} \left ( ic \right ) \right ) \left ( -{\it csgn} \left ( ic{x}^{n} \right ) +{\it csgn} \left ( i{x}^{n} \right ) \right ) \right ) +\ln \left ( \left ( \ln \left ( c \right ) +\ln \left ({x}^{n} \right ) -{\frac{i}{2}}\pi \,{\it csgn} \left ( ic{x}^{n} \right ) \left ( -{\it csgn} \left ( ic{x}^{n} \right ) +{\it csgn} \left ( ic \right ) \right ) \left ( -{\it csgn} \left ( ic{x}^{n} \right ) +{\it csgn} \left ( i{x}^{n} \right ) \right ) \right ) ^{q}+{\frac{a{x}^{m}}{bx}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.54302, size = 35, normalized size = 1.84 \begin{align*} \log \left (\frac{b x{\left (\log \left (c\right ) + \log \left (x^{n}\right )\right )}^{q} + a x^{m}}{b x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.92569, size = 59, normalized size = 3.11 \begin{align*} \log \left (\frac{{\left (n \log \left (x\right ) + \log \left (c\right )\right )}^{q} b x + a x^{m}}{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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