Integrand size = 5, antiderivative size = 14 \[ \int e^{\frac {1}{t}} \, dt=e^{\frac {1}{t}} t-\operatorname {ExpIntegralEi}\left (\frac {1}{t}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2237, 2241} \[ \int e^{\frac {1}{t}} \, dt=e^{\frac {1}{t}} t-\operatorname {ExpIntegralEi}\left (\frac {1}{t}\right ) \]
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Rule 2237
Rule 2241
Rubi steps \begin{align*} \text {integral}& = e^{\frac {1}{t}} t+\int \frac {e^{\frac {1}{t}}}{t} \, dt \\ & = e^{\frac {1}{t}} t-\operatorname {ExpIntegralEi}\left (\frac {1}{t}\right ) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int e^{\frac {1}{t}} \, dt=e^{\frac {1}{t}} t-\operatorname {ExpIntegralEi}\left (\frac {1}{t}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.07
method | result | size |
derivativedivides | \({\mathrm e}^{\frac {1}{t}} t +\operatorname {Ei}_{1}\left (-\frac {1}{t}\right )\) | \(15\) |
default | \({\mathrm e}^{\frac {1}{t}} t +\operatorname {Ei}_{1}\left (-\frac {1}{t}\right )\) | \(15\) |
risch | \({\mathrm e}^{\frac {1}{t}} t +\operatorname {Ei}_{1}\left (-\frac {1}{t}\right )\) | \(15\) |
meijerg | \(t +1+\ln \left (t \right )-i \pi -\frac {t \left (2+\frac {2}{t}\right )}{2}+{\mathrm e}^{\frac {1}{t}} t +\ln \left (-\frac {1}{t}\right )+\operatorname {Ei}_{1}\left (-\frac {1}{t}\right )\) | \(39\) |
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none
Time = 0.23 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93 \[ \int e^{\frac {1}{t}} \, dt=t e^{\frac {1}{t}} - {\rm Ei}\left (\frac {1}{t}\right ) \]
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Time = 0.62 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71 \[ \int e^{\frac {1}{t}} \, dt=t e^{\frac {1}{t}} - \operatorname {Ei}{\left (\frac {1}{t} \right )} \]
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none
Time = 0.21 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.64 \[ \int e^{\frac {1}{t}} \, dt=-\Gamma \left (-1, -\frac {1}{t}\right ) \]
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none
Time = 0.25 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.29 \[ \int e^{\frac {1}{t}} \, dt=-t {\left (\frac {{\rm Ei}\left (\frac {1}{t}\right )}{t} - e^{\frac {1}{t}}\right )} \]
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Time = 0.02 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.64 \[ \int e^{\frac {1}{t}} \, dt=t\,\mathrm {expint}\left (2,-\frac {1}{t}\right ) \]
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