Optimal. Leaf size=67 \[ -\frac{a m \text{CannotIntegrate}\left (\frac{x^{m-1}}{\left (a x^m+b \log ^2\left (c x^n\right )\right )^2},x\right )}{2 b n}-\frac{1}{2 b n \left (a x^m+b \log ^2\left (c x^n\right )\right )} \]
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Rubi [A] time = 0.196432, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\log \left (c x^n\right )}{x \left (a x^m+b \log ^2\left (c x^n\right )\right )^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\log \left (c x^n\right )}{x \left (a x^m+b \log ^2\left (c x^n\right )\right )^2} \, dx &=-\frac{1}{2 b n \left (a x^m+b \log ^2\left (c x^n\right )\right )}-\frac{(a m) \int \frac{x^{-1+m}}{\left (a x^m+b \log ^2\left (c x^n\right )\right )^2} \, dx}{2 b n}\\ \end{align*}
Mathematica [A] time = 1.56721, size = 0, normalized size = 0. \[ \int \frac{\log \left (c x^n\right )}{x \left (a x^m+b \log ^2\left (c x^n\right )\right )^2} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 51.749, size = 0, normalized size = 0. \begin{align*} \int{\frac{\ln \left ( c{x}^{n} \right ) }{x \left ( a{x}^{m}+b \left ( \ln \left ( c{x}^{n} \right ) \right ) ^{2} \right ) ^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{m \log \left (c\right ) + m \log \left (x^{n}\right ) + 2 \, n}{4 \, b^{2} n^{2} \log \left (c\right )^{2} + a^{2} m^{2} x^{2 \, m} +{\left (m^{2} \log \left (c\right )^{2} + 4 \, n^{2}\right )} a b x^{m} +{\left (a b m^{2} x^{m} + 4 \, b^{2} n^{2}\right )} \log \left (x^{n}\right )^{2} + 2 \,{\left (a b m^{2} x^{m} \log \left (c\right ) + 4 \, b^{2} n^{2} \log \left (c\right )\right )} \log \left (x^{n}\right )} - \int \frac{a m^{4} x^{m} \log \left (x^{n}\right ) + 4 \, b m n^{3} +{\left (m^{4} \log \left (c\right ) + 3 \, m^{3} n\right )} a x^{m}}{16 \, b^{3} n^{4} x \log \left (c\right )^{2} + a^{3} m^{4} x x^{3 \, m} +{\left (m^{4} \log \left (c\right )^{2} + 8 \, m^{2} n^{2}\right )} a^{2} b x x^{2 \, m} + 8 \,{\left (m^{2} n^{2} \log \left (c\right )^{2} + 2 \, n^{4}\right )} a b^{2} x x^{m} +{\left (a^{2} b m^{4} x x^{2 \, m} + 8 \, a b^{2} m^{2} n^{2} x x^{m} + 16 \, b^{3} n^{4} x\right )} \log \left (x^{n}\right )^{2} + 2 \,{\left (a^{2} b m^{4} x x^{2 \, m} \log \left (c\right ) + 8 \, a b^{2} m^{2} n^{2} x x^{m} \log \left (c\right ) + 16 \, b^{3} n^{4} x \log \left (c\right )\right )} \log \left (x^{n}\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left (c x^{n}\right )}{b^{2} x \log \left (c x^{n}\right )^{4} + 2 \, a b x x^{m} \log \left (c x^{n}\right )^{2} + a^{2} x x^{2 \, m}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log{\left (c x^{n} \right )}}{x \left (a x^{m} + b \log{\left (c x^{n} \right )}^{2}\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (c x^{n}\right )}{{\left (b \log \left (c x^{n}\right )^{2} + a x^{m}\right )}^{2} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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