Optimal. Leaf size=68 \[ \frac {\log \left (a x^m+b \log ^q\left (c x^n\right )\right )}{b n q}-\frac {a m \text {Int}\left (\frac {x^{m-1}}{a x^m+b \log ^q\left (c x^n\right )},x\right )}{b n q} \]
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Rubi [A] time = 0.24, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )} \, dx &=\frac {\log \left (a x^m+b \log ^q\left (c x^n\right )\right )}{b n q}-\frac {(a m) \int \frac {x^{-1+m}}{a x^m+b \log ^q\left (c x^n\right )} \, dx}{b n q}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 0, normalized size = 0.00 \[ \int \frac {\log ^{-1+q}\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left (c x^{n}\right )^{q - 1}}{a x x^{m} + b x \log \left (c x^{n}\right )^{q}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left (c x^{n}\right )^{q - 1}}{{\left (a x^{m} + b \log \left (c x^{n}\right )^{q}\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 29.52, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (c \,x^{n}\right )^{q -1}}{\left (a \,x^{m}+b \ln \left (c \,x^{n}\right )^{q}\right ) x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ -a \int \frac {x^{m}}{a b x x^{m} \log \relax (c) + a b x x^{m} \log \left (x^{n}\right ) + {\left (b^{2} x \log \relax (c) + b^{2} x \log \left (x^{n}\right )\right )} {\left (\log \relax (c) + \log \left (x^{n}\right )\right )}^{q}}\,{d x} + \frac {\log \left (\log \relax (c) + \log \left (x^{n}\right )\right )}{b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\ln \left (c\,x^n\right )}^{q-1}}{x\,\left (a\,x^m+b\,{\ln \left (c\,x^n\right )}^q\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log {\left (c x^{n} \right )}^{q - 1}}{x \left (a x^{m} + b \log {\left (c x^{n} \right )}^{q}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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