Optimal. Leaf size=26 \[ \frac {\text {Li}_2\left (1-c \left (e x^{-n}+d\right )\right )}{c e n} \]
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Rubi [A] time = 0.16, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {2475, 2412, 2393, 2391} \[ \frac {\text {PolyLog}\left (2,1-c \left (d+e x^{-n}\right )\right )}{c e n} \]
Antiderivative was successfully verified.
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Rule 2391
Rule 2393
Rule 2412
Rule 2475
Rubi steps
\begin {align*} \int \frac {\log \left (c \left (d+e x^{-n}\right )\right )}{x \left (c e-(1-c d) x^n\right )} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {\log (c (d+e x))}{\left (c e+\frac {-1+c d}{x}\right ) x} \, dx,x,x^{-n}\right )}{n}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {\log (c (d+e x))}{-1+c d+c e x} \, dx,x,x^{-n}\right )}{n}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,-1+c d+c e x^{-n}\right )}{c e n}\\ &=\frac {\text {Li}_2\left (1-c d-c e x^{-n}\right )}{c e n}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 34, normalized size = 1.31 \[ \frac {\text {Li}_2\left (-x^{-n} \left (c d x^n-x^n+c e\right )\right )}{c e n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 30, normalized size = 1.15 \[ \frac {{\rm Li}_2\left (-\frac {c d x^{n} + c e}{x^{n}} + 1\right )}{c e n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left (c {\left (d + \frac {e}{x^{n}}\right )}\right )}{{\left (c e + {\left (c d - 1\right )} x^{n}\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 24, normalized size = 0.92 \[ \frac {\dilog \left (c e \,x^{-n}+c d \right )}{c e n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ n \int \frac {\log \relax (x)}{c d x x^{n} + c e x}\,{d x} + \frac {\log \left (d x^{n} + e\right ) \log \relax (x) + \log \relax (c) \log \relax (x) - \log \relax (x) \log \left (x^{n}\right )}{c e} - \frac {\log \relax (c) \log \left (\frac {c e + {\left (c d - 1\right )} x^{n}}{c d - 1}\right )}{c e n} - \frac {\log \left (d x^{n} + e\right ) \log \left (\frac {c d e + {\left (c d^{2} - d\right )} x^{n} - e}{e} + 1\right ) + {\rm Li}_2\left (-\frac {c d e + {\left (c d^{2} - d\right )} x^{n} - e}{e}\right )}{c e n} + \frac {\log \left (x^{n}\right ) \log \left (\frac {{\left (c d - 1\right )} x^{n}}{c e} + 1\right ) + {\rm Li}_2\left (-\frac {{\left (c d - 1\right )} x^{n}}{c e}\right )}{c e n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\ln \left (c\,\left (d+\frac {e}{x^n}\right )\right )}{x\,\left (c\,e+x^n\,\left (c\,d-1\right )\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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