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 = 0.18538, antiderivative size = 17, normalized size of antiderivative = 1., 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*}
Mathematica [A] time = 0.233164, size = 17, normalized size = 1. \[ \log \left (a x^m+b \log ^q\left (c x^n\right )\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.164, size = 213, normalized size = 12.5 \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}}{b}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.55993, size = 30, normalized size = 1.76 \begin{align*} \log \left (\frac{a x^{m} + b{\left (\log \left (c\right ) + \log \left (x^{n}\right )\right )}^{q}}{b}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.82366, size = 51, normalized size = 3. \begin{align*} \log \left ({\left (n \log \left (x\right ) + \log \left (c\right )\right )}^{q} b + a x^{m}\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 m x^{m}}{{\left (a x^{m} + b \log \left (c x^{n}\right )^{q}\right )} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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