Optimal. Leaf size=26 \[ \frac {\text {Li}_2\left (1-c \left (d+e x^{-n}\right )\right )}{c e n} \]
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Rubi [A]
time = 0.10, 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 = {2525, 2459,
2440, 2438} \begin {gather*} \frac {\text {PolyLog}\left (2,1-c \left (d+e x^{-n}\right )\right )}{c e n} \end {gather*}
Antiderivative was successfully verified.
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Rule 2438
Rule 2440
Rule 2459
Rule 2525
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 {\text {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 {\text {Subst}\left (\int \frac {\log (c (d+e x))}{-1+c d+c e x} \, dx,x,x^{-n}\right )}{n}\\ &=-\frac {\text {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 = 0.05, size = 34, normalized size = 1.31 \begin {gather*} \frac {\text {Li}_2\left (-x^{-n} \left (c e-x^n+c d x^n\right )\right )}{c e n} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.57, size = 24, normalized size = 0.92
method | result | size |
derivativedivides | \(\frac {\dilog \left (c d +c e \,x^{-n}\right )}{n c e}\) | \(24\) |
default | \(\frac {\dilog \left (c d +c e \,x^{-n}\right )}{n c e}\) | \(24\) |
risch | \(\text {Expression too large to display}\) | \(1900\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.45, size = 30, normalized size = 1.15 \begin {gather*} \frac {{\rm Li}_2\left (-\frac {c d x^{n} + c e}{x^{n}} + 1\right ) e^{\left (-1\right )}}{c n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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