Optimal. Leaf size=43 \[ -\frac {x}{b}-\frac {(1-a-b x) \log (1-a-b x)}{b^2}+\frac {a \text {Li}_2(a+b x)}{b^2} \]
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Rubi [A]
time = 0.05, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {45, 2463, 2436,
2332, 2440, 2438} \begin {gather*} \frac {a \text {PolyLog}(2,a+b x)}{b^2}-\frac {(-a-b x+1) \log (-a-b x+1)}{b^2}-\frac {x}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2332
Rule 2436
Rule 2438
Rule 2440
Rule 2463
Rubi steps
\begin {align*} \int \frac {x \log (1-a-b x)}{a+b x} \, dx &=\int \left (\frac {\log (1-a-b x)}{b}-\frac {a \log (1-a-b x)}{b (a+b x)}\right ) \, dx\\ &=\frac {\int \log (1-a-b x) \, dx}{b}-\frac {a \int \frac {\log (1-a-b x)}{a+b x} \, dx}{b}\\ &=-\frac {\text {Subst}(\int \log (x) \, dx,x,1-a-b x)}{b^2}-\frac {a \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,a+b x\right )}{b^2}\\ &=-\frac {x}{b}-\frac {(1-a-b x) \log (1-a-b x)}{b^2}+\frac {a \text {Li}_2(a+b x)}{b^2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 35, normalized size = 0.81 \begin {gather*} \frac {-b x+(-1+a+b x) \log (1-a-b x)+a \text {Li}_2(a+b x)}{b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 47, normalized size = 1.09
method | result | size |
derivativedivides | \(\frac {-\left (-b x -a +1\right ) \ln \left (-b x -a +1\right )-b x -a +1+\dilog \left (-b x -a +1\right ) a}{b^{2}}\) | \(47\) |
default | \(\frac {-\left (-b x -a +1\right ) \ln \left (-b x -a +1\right )-b x -a +1+\dilog \left (-b x -a +1\right ) a}{b^{2}}\) | \(47\) |
risch | \(\frac {x \ln \left (-b x -a +1\right )}{b}+\frac {\dilog \left (-b x -a +1\right ) a}{b^{2}}+\frac {\ln \left (-b x -a +1\right ) a}{b^{2}}-\frac {x}{b}-\frac {\ln \left (-b x -a +1\right )}{b^{2}}-\frac {a}{b^{2}}+\frac {1}{b^{2}}\) | \(77\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 82 vs.
\(2 (38) = 76\).
time = 0.31, size = 82, normalized size = 1.91 \begin {gather*} b {\left (\frac {{\left (\log \left (b x + a\right ) \log \left (-b x - a + 1\right ) + {\rm Li}_2\left (b x + a\right )\right )} a}{b^{3}} - \frac {x}{b^{2}} + \frac {{\left (a - 1\right )} \log \left (b x + a - 1\right )}{b^{3}}\right )} + {\left (\frac {x}{b} - \frac {a \log \left (b x + a\right )}{b^{2}}\right )} \log \left (-b x - a + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \log {\left (- a - b x + 1 \right )}}{a + b x}\, dx \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 [B]
time = 0.32, size = 59, normalized size = 1.37 \begin {gather*} -\frac {\ln \left (1-b\,x-a\right )+b\,\left (x-x\,\ln \left (1-b\,x-a\right )\right )-a\,{\mathrm {Li}}_{\mathrm {2}}\left (1-b\,x-a\right )-a\,\ln \left (1-b\,x-a\right )}{b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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