Optimal. Leaf size=28 \[ \frac {\text {Li}_2\left (\frac {(1-a c) \left (d x^{-m}+e\right )}{e}\right )}{d m} \]
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Rubi [A] time = 0.13, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2475, 2412, 2393, 2391} \[ \frac {\text {PolyLog}\left (2,\frac {(1-a c) \left (d x^{-m}+e\right )}{e}\right )}{d m} \]
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 (a-\frac {(d-a c d) x^{-m}}{c e}\right )\right )}{x \left (d+e x^m\right )} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {\log \left (c \left (a-\frac {(d-a c d) x}{c e}\right )\right )}{\left (d+\frac {e}{x}\right ) x} \, dx,x,x^{-m}\right )}{m}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {\log \left (c \left (a-\frac {(d-a c d) x}{c e}\right )\right )}{e+d x} \, dx,x,x^{-m}\right )}{m}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {\log \left (1-\frac {(d-a c d) x}{d e}\right )}{x} \, dx,x,e+d x^{-m}\right )}{d m}\\ &=\frac {\text {Li}_2\left (\frac {(1-a c) \left (e+d x^{-m}\right )}{e}\right )}{d m}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 31, normalized size = 1.11 \[ \frac {\text {Li}_2\left (-\frac {(a c-1) x^{-m} \left (e x^m+d\right )}{e}\right )}{d m} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 35, normalized size = 1.25 \[ \frac {{\rm Li}_2\left (-\frac {a c e x^{m} + {\left (a c - 1\right )} d}{e x^{m}} + 1\right )}{d m} \]
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 \[ \int \frac {\log \left ({\left (a + \frac {a c d - d}{c e x^{m}}\right )} c\right )}{{\left (e x^{m} + d\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 28, normalized size = 1.00 \[ \frac {\dilog \left (a c +\frac {\left (a c -1\right ) d \,x^{-m}}{e}\right )}{d m} \]
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 \[ {\left (a c m - m\right )} \int \frac {\log \relax (x)}{a c e x x^{m} + {\left (a c d - d\right )} x}\,{d x} + \frac {\log \left (a c e x^{m} + {\left (a c - 1\right )} d\right ) \log \relax (x) - \log \relax (e) \log \relax (x) - \log \relax (x) \log \left (x^{m}\right )}{d} + \frac {\log \relax (e) \log \left (\frac {e x^{m} + d}{e}\right )}{d m} + \frac {\log \left (x^{m}\right ) \log \left (\frac {e x^{m}}{d} + 1\right ) + {\rm Li}_2\left (-\frac {e x^{m}}{d}\right )}{d m} - \frac {\log \left (a c e x^{m} + {\left (a c - 1\right )} d\right ) \log \left (\frac {a c e x^{m} + a c d - d}{d} + 1\right ) + {\rm Li}_2\left (-\frac {a c e x^{m} + a c d - d}{d}\right )}{d m} \]
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 \[ \int \frac {\ln \left (c\,\left (a-\frac {d-a\,c\,d}{c\,e\,x^m}\right )\right )}{x\,\left (d+e\,x^m\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: HeuristicGCDFailed} \]
Verification of antiderivative is not currently implemented for this CAS.
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