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 = 0.245221, antiderivative size = 26, normalized size of antiderivative = 1., 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*}
Mathematica [A] time = 0.356638, size = 26, normalized size = 1. \[ \frac{d \log \left (c x^n\right )}{a x^m+b \log ^q\left (c x^n\right )} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.128, size = 158, normalized size = 6.1 \begin{align*}{\frac{d \left ( 2\,\ln \left ( c \right ) +2\,\ln \left ({x}^{n} \right ) -i\pi \,{\it csgn} \left ( ic \right ){\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ) +i\pi \,{\it csgn} \left ( ic \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+i\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3} \right ) }{2\,a{x}^{m}+2\,b \left ( \ln \left ( c \right ) +\ln \left ({x}^{n} \right ) -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}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.67436, size = 42, normalized size = 1.62 \begin{align*} \frac{d \log \left (c\right ) + d \log \left (x^{n}\right )}{a x^{m} + b{\left (\log \left (c\right ) + \log \left (x^{n}\right )\right )}^{q}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.93464, size = 80, normalized size = 3.08 \begin{align*} \frac{d n \log \left (x\right ) + d \log \left (c\right )}{{\left (n \log \left (x\right ) + \log \left (c\right )\right )}^{q} b + a x^{m}} \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 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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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