Integrand size = 12, antiderivative size = 21 \[ \int \frac {\log \left (2+e x^n\right )}{x} \, dx=\log (2) \log (x)-\frac {\operatorname {PolyLog}\left (2,-\frac {e x^n}{2}\right )}{n} \]
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Time = 0.02 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2504, 2439, 2438} \[ \int \frac {\log \left (2+e x^n\right )}{x} \, dx=\log (2) \log (x)-\frac {\operatorname {PolyLog}\left (2,-\frac {e x^n}{2}\right )}{n} \]
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Rule 2438
Rule 2439
Rule 2504
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\log (2+e x)}{x} \, dx,x,x^n\right )}{n} \\ & = \log (2) \log (x)+\frac {\text {Subst}\left (\int \frac {\log \left (1+\frac {e x}{2}\right )}{x} \, dx,x,x^n\right )}{n} \\ & = \log (2) \log (x)-\frac {\text {Li}_2\left (-\frac {e x^n}{2}\right )}{n} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {\log \left (2+e x^n\right )}{x} \, dx=\log (2) \log (x)-\frac {\operatorname {PolyLog}\left (2,-\frac {e x^n}{2}\right )}{n} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(39\) vs. \(2(19)=38\).
Time = 0.84 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.90
method | result | size |
risch | \(\ln \left (x \right ) \ln \left (2+e \,x^{n}\right )-\frac {\operatorname {dilog}\left (\frac {e \,x^{n}}{2}+1\right )}{n}-\ln \left (x \right ) \ln \left (\frac {e \,x^{n}}{2}+1\right )\) | \(40\) |
derivativedivides | \(\frac {\left (\ln \left (2+e \,x^{n}\right )-\ln \left (\frac {e \,x^{n}}{2}+1\right )\right ) \ln \left (-\frac {e \,x^{n}}{2}\right )-\operatorname {dilog}\left (\frac {e \,x^{n}}{2}+1\right )}{n}\) | \(45\) |
default | \(\frac {\left (\ln \left (2+e \,x^{n}\right )-\ln \left (\frac {e \,x^{n}}{2}+1\right )\right ) \ln \left (-\frac {e \,x^{n}}{2}\right )-\operatorname {dilog}\left (\frac {e \,x^{n}}{2}+1\right )}{n}\) | \(45\) |
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Leaf count of result is larger than twice the leaf count of optimal. 40 vs. \(2 (18) = 36\).
Time = 0.33 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.90 \[ \int \frac {\log \left (2+e x^n\right )}{x} \, dx=\frac {n \log \left (e x^{n} + 2\right ) \log \left (x\right ) - n \log \left (\frac {1}{2} \, e x^{n} + 1\right ) \log \left (x\right ) - {\rm Li}_2\left (-\frac {1}{2} \, e x^{n}\right )}{n} \]
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Result contains complex when optimal does not.
Time = 1.65 (sec) , antiderivative size = 100, normalized size of antiderivative = 4.76 \[ \int \frac {\log \left (2+e x^n\right )}{x} \, dx=\begin {cases} - \frac {\operatorname {Li}_{2}\left (\frac {e x^{n} e^{i \pi }}{2}\right )}{n} & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\\log {\left (2 \right )} \log {\left (x \right )} - \frac {\operatorname {Li}_{2}\left (\frac {e x^{n} e^{i \pi }}{2}\right )}{n} & \text {for}\: \left |{x}\right | < 1 \\- \log {\left (2 \right )} \log {\left (\frac {1}{x} \right )} - \frac {\operatorname {Li}_{2}\left (\frac {e x^{n} e^{i \pi }}{2}\right )}{n} & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\left (2 \right )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\left (2 \right )} - \frac {\operatorname {Li}_{2}\left (\frac {e x^{n} e^{i \pi }}{2}\right )}{n} & \text {otherwise} \end {cases} \]
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\[ \int \frac {\log \left (2+e x^n\right )}{x} \, dx=\int { \frac {\log \left (e x^{n} + 2\right )}{x} \,d x } \]
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\[ \int \frac {\log \left (2+e x^n\right )}{x} \, dx=\int { \frac {\log \left (e x^{n} + 2\right )}{x} \,d x } \]
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Timed out. \[ \int \frac {\log \left (2+e x^n\right )}{x} \, dx=\int \frac {\ln \left (e\,x^n+2\right )}{x} \,d x \]
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