Integrand size = 20, antiderivative size = 6 \[ \int \frac {e^{e^x+x} \left (1+e^x\right )}{e^x+x} \, dx=\operatorname {ExpIntegralEi}\left (e^x+x\right ) \]
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Time = 0.09 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6839, 2209} \[ \int \frac {e^{e^x+x} \left (1+e^x\right )}{e^x+x} \, dx=\operatorname {ExpIntegralEi}\left (x+e^x\right ) \]
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Rule 2209
Rule 6839
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {e^x}{x} \, dx,x,e^x+x\right ) \\ & = \operatorname {ExpIntegralEi}\left (e^x+x\right ) \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00 \[ \int \frac {e^{e^x+x} \left (1+e^x\right )}{e^x+x} \, dx=\operatorname {ExpIntegralEi}\left (e^x+x\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(12\) vs. \(2(5)=10\).
Time = 0.05 (sec) , antiderivative size = 13, normalized size of antiderivative = 2.17
method | result | size |
derivativedivides | \(-\operatorname {Ei}_{1}\left (-{\mathrm e}^{x}-x \right )\) | \(13\) |
default | \(-\operatorname {Ei}_{1}\left (-{\mathrm e}^{x}-x \right )\) | \(13\) |
risch | \(-\operatorname {Ei}_{1}\left (-{\mathrm e}^{x}-x \right )\) | \(13\) |
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none
Time = 0.27 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.83 \[ \int \frac {e^{e^x+x} \left (1+e^x\right )}{e^x+x} \, dx={\rm Ei}\left (x + e^{x}\right ) \]
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Time = 0.97 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.83 \[ \int \frac {e^{e^x+x} \left (1+e^x\right )}{e^x+x} \, dx=\operatorname {Ei}{\left (x + e^{x} \right )} \]
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\[ \int \frac {e^{e^x+x} \left (1+e^x\right )}{e^x+x} \, dx=\int { \frac {{\left (e^{x} + 1\right )} e^{\left (x + e^{x}\right )}}{x + e^{x}} \,d x } \]
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none
Time = 0.30 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.83 \[ \int \frac {e^{e^x+x} \left (1+e^x\right )}{e^x+x} \, dx={\rm Ei}\left (x + e^{x}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.83 \[ \int \frac {e^{e^x+x} \left (1+e^x\right )}{e^x+x} \, dx=\mathrm {ei}\left (x+{\mathrm {e}}^x\right ) \]
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