Optimal. Leaf size=31 \[ \log \left (\frac {e x}{2}\right ) (a+b \log (-2+e x))+b \text {Li}_2\left (1-\frac {e x}{2}\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2441, 2352}
\begin {gather*} b \text {PolyLog}\left (2,1-\frac {e x}{2}\right )+\log \left (\frac {e x}{2}\right ) (a+b \log (e x-2)) \end {gather*}
Antiderivative was successfully verified.
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Rule 2352
Rule 2441
Rubi steps
\begin {align*} \int \frac {a+b \log (-2+e x)}{x} \, dx &=\log \left (\frac {e x}{2}\right ) (a+b \log (-2+e x))-(b e) \int \frac {\log \left (\frac {e x}{2}\right )}{-2+e x} \, dx\\ &=\log \left (\frac {e x}{2}\right ) (a+b \log (-2+e x))+b \text {Li}_2\left (1-\frac {e x}{2}\right )\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 34, normalized size = 1.10 \begin {gather*} a \log (x)+b \log \left (\frac {e x}{2}\right ) \log (-2+e x)+b \text {Li}_2\left (\frac {1}{2} (2-e x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.19, size = 28, normalized size = 0.90
method | result | size |
risch | \(\ln \left (x \right ) a +\ln \left (e x -2\right ) \ln \left (\frac {e x}{2}\right ) b +\dilog \left (\frac {e x}{2}\right ) b\) | \(26\) |
derivativedivides | \(a \ln \left (e x \right )+\ln \left (e x -2\right ) \ln \left (\frac {e x}{2}\right ) b +\dilog \left (\frac {e x}{2}\right ) b\) | \(28\) |
default | \(a \ln \left (e x \right )+\ln \left (e x -2\right ) \ln \left (\frac {e x}{2}\right ) b +\dilog \left (\frac {e x}{2}\right ) b\) | \(28\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 30, normalized size = 0.97 \begin {gather*} {\left (\log \left (x e - 2\right ) \log \left (\frac {1}{2} \, x e\right ) + {\rm Li}_2\left (-\frac {1}{2} \, x e + 1\right )\right )} b + a \log \left (x\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 [A]
time = 2.79, size = 109, normalized size = 3.52 \begin {gather*} a \log {\left (x \right )} + b \left (\begin {cases} - \operatorname {Li}_{2}\left (\frac {e x}{2}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\\log {\left (2 \right )} \log {\left (x \right )} + 3 i \pi \log {\left (x \right )} - \operatorname {Li}_{2}\left (\frac {e x}{2}\right ) & \text {for}\: \left |{x}\right | < 1 \\- \log {\left (2 \right )} \log {\left (\frac {1}{x} \right )} - 3 i \pi \log {\left (\frac {1}{x} \right )} - \operatorname {Li}_{2}\left (\frac {e x}{2}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\left (2 \right )} - 3 i \pi {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\left (2 \right )} + 3 i \pi {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} - \operatorname {Li}_{2}\left (\frac {e x}{2}\right ) & \text {otherwise} \end {cases}\right ) \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 = 0.16, size = 25, normalized size = 0.81 \begin {gather*} b\,{\mathrm {Li}}_{\mathrm {2}}\left (\frac {e\,x}{2}\right )+a\,\ln \left (x\right )+b\,\ln \left (e\,x-2\right )\,\ln \left (\frac {e\,x}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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