Integrand size = 10, antiderivative size = 16 \[ \int \frac {\log (3+e x)}{x} \, dx=\log (3) \log (x)-\operatorname {PolyLog}\left (2,-\frac {e x}{3}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2439, 2438} \[ \int \frac {\log (3+e x)}{x} \, dx=\log (3) \log (x)-\operatorname {PolyLog}\left (2,-\frac {e x}{3}\right ) \]
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Rule 2438
Rule 2439
Rubi steps \begin{align*} \text {integral}& = \log (3) \log (x)+\int \frac {\log \left (1+\frac {e x}{3}\right )}{x} \, dx \\ & = \log (3) \log (x)-\text {Li}_2\left (-\frac {e x}{3}\right ) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {\log (3+e x)}{x} \, dx=\log (3) \log (x)-\operatorname {PolyLog}\left (2,-\frac {e x}{3}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(32\) vs. \(2(14)=28\).
Time = 0.10 (sec) , antiderivative size = 33, normalized size of antiderivative = 2.06
method | result | size |
derivativedivides | \(\left (\ln \left (e x +3\right )-\ln \left (\frac {e x}{3}+1\right )\right ) \ln \left (-\frac {e x}{3}\right )-\operatorname {dilog}\left (\frac {e x}{3}+1\right )\) | \(33\) |
default | \(\left (\ln \left (e x +3\right )-\ln \left (\frac {e x}{3}+1\right )\right ) \ln \left (-\frac {e x}{3}\right )-\operatorname {dilog}\left (\frac {e x}{3}+1\right )\) | \(33\) |
risch | \(\left (\ln \left (e x +3\right )-\ln \left (\frac {e x}{3}+1\right )\right ) \ln \left (-\frac {e x}{3}\right )-\operatorname {dilog}\left (\frac {e x}{3}+1\right )\) | \(33\) |
parts | \(\ln \left (e x +3\right ) \ln \left (x \right )-e \left (\frac {\operatorname {dilog}\left (\frac {e x}{3}+1\right )}{e}+\frac {\ln \left (x \right ) \ln \left (\frac {e x}{3}+1\right )}{e}\right )\) | \(39\) |
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\[ \int \frac {\log (3+e x)}{x} \, dx=\int { \frac {\log \left (e x + 3\right )}{x} \,d x } \]
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Result contains complex when optimal does not.
Time = 1.48 (sec) , antiderivative size = 87, normalized size of antiderivative = 5.44 \[ \int \frac {\log (3+e x)}{x} \, dx=\begin {cases} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{3}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\\log {\left (3 \right )} \log {\left (x \right )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{3}\right ) & \text {for}\: \left |{x}\right | < 1 \\- \log {\left (3 \right )} \log {\left (\frac {1}{x} \right )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{3}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\left (3 \right )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\left (3 \right )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{3}\right ) & \text {otherwise} \end {cases} \]
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none
Time = 0.21 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.25 \[ \int \frac {\log (3+e x)}{x} \, dx=\log \left (e x + 3\right ) \log \left (-\frac {1}{3} \, e x\right ) + {\rm Li}_2\left (\frac {1}{3} \, e x + 1\right ) \]
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\[ \int \frac {\log (3+e x)}{x} \, dx=\int { \frac {\log \left (e x + 3\right )}{x} \,d x } \]
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Time = 0.03 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {\log (3+e x)}{x} \, dx={\mathrm {Li}}_{\mathrm {2}}\left (-\frac {e\,x}{3}\right )+\ln \left (e\,x+3\right )\,\ln \left (-\frac {e\,x}{3}\right ) \]
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