Optimal. Leaf size=27 \[ \frac {\text {Li}_2\left (\frac {(1-c) \left (b+a x^{-m}\right )}{b}\right )}{a m} \]
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Rubi [A]
time = 0.12, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.139, Rules used = {2530, 2525,
2459, 2440, 2438} \begin {gather*} \frac {\text {PolyLog}\left (2,\frac {(1-c) \left (a x^{-m}+b\right )}{b}\right )}{a m} \end {gather*}
Antiderivative was successfully verified.
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Rule 2438
Rule 2440
Rule 2459
Rule 2525
Rule 2530
Rubi steps
\begin {align*} \int \frac {\log \left (\frac {x^{-m} \left (-a+a c+b c x^m\right )}{b}\right )}{x \left (a+b x^m\right )} \, dx &=\int \frac {\log \left (c+\frac {(-a+a c) x^{-m}}{b}\right )}{x \left (a+b x^m\right )} \, dx\\ &=-\frac {\text {Subst}\left (\int \frac {\log \left (c+\frac {(-a+a c) x}{b}\right )}{\left (a+\frac {b}{x}\right ) x} \, dx,x,x^{-m}\right )}{m}\\ &=-\frac {\text {Subst}\left (\int \frac {\log \left (c+\frac {(-a+a c) x}{b}\right )}{b+a x} \, dx,x,x^{-m}\right )}{m}\\ &=-\frac {\text {Subst}\left (\int \frac {\log \left (1+\frac {(-a+a c) x}{a b}\right )}{x} \, dx,x,b+a x^{-m}\right )}{a m}\\ &=\frac {\text {Li}_2\left (\frac {(1-c) \left (b+a x^{-m}\right )}{b}\right )}{a m}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 29, normalized size = 1.07 \begin {gather*} \frac {\text {Li}_2\left (-\frac {(-1+c) x^{-m} \left (a+b x^m\right )}{b}\right )}{a m} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.93, size = 24, normalized size = 0.89
method | result | size |
derivativedivides | \(\frac {\dilog \left (c +\frac {a \left (-1+c \right ) x^{-m}}{b}\right )}{m a}\) | \(24\) |
default | \(\frac {\dilog \left (c +\frac {a \left (-1+c \right ) x^{-m}}{b}\right )}{m a}\) | \(24\) |
risch | \(\text {Expression too large to display}\) | \(1267\) |
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 = 0.36, size = 33, normalized size = 1.22 \begin {gather*} \frac {{\rm Li}_2\left (-\frac {b c x^{m} + a c - a}{b x^{m}} + 1\right )}{a m} \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 (\frac {a\,c-a+b\,c\,x^m}{b\,x^m}\right )}{x\,\left (a+b\,x^m\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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