Integrand size = 12, antiderivative size = 13 \[ \int \frac {\log \left (1+e x^n\right )}{x} \, dx=-\frac {\operatorname {PolyLog}\left (2,-e x^n\right )}{n} \]
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Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2438} \[ \int \frac {\log \left (1+e x^n\right )}{x} \, dx=-\frac {\operatorname {PolyLog}\left (2,-e x^n\right )}{n} \]
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Rule 2438
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Li}_2\left (-e x^n\right )}{n} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {\log \left (1+e x^n\right )}{x} \, dx=-\frac {\operatorname {PolyLog}\left (2,-e x^n\right )}{n} \]
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Time = 0.85 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08
method | result | size |
derivativedivides | \(-\frac {\operatorname {dilog}\left (1+e \,x^{n}\right )}{n}\) | \(14\) |
default | \(-\frac {\operatorname {dilog}\left (1+e \,x^{n}\right )}{n}\) | \(14\) |
meijerg | \(-\frac {\operatorname {Li}_{2}\left (-e \,x^{n}\right )}{n}\) | \(14\) |
risch | \(-\frac {\operatorname {dilog}\left (1+e \,x^{n}\right )}{n}\) | \(14\) |
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none
Time = 0.29 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92 \[ \int \frac {\log \left (1+e x^n\right )}{x} \, dx=-\frac {{\rm Li}_2\left (-e x^{n}\right )}{n} \]
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Result contains complex when optimal does not.
Time = 1.46 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08 \[ \int \frac {\log \left (1+e x^n\right )}{x} \, dx=- \frac {\operatorname {Li}_{2}\left (e x^{n} e^{i \pi }\right )}{n} \]
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\[ \int \frac {\log \left (1+e x^n\right )}{x} \, dx=\int { \frac {\log \left (e x^{n} + 1\right )}{x} \,d x } \]
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\[ \int \frac {\log \left (1+e x^n\right )}{x} \, dx=\int { \frac {\log \left (e x^{n} + 1\right )}{x} \,d x } \]
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Timed out. \[ \int \frac {\log \left (1+e x^n\right )}{x} \, dx=\int \frac {\ln \left (e\,x^n+1\right )}{x} \,d x \]
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