3.161 \(\int e^{\frac{1}{t}} \, dt\)

Optimal. Leaf size=14 \[ e^{\frac{1}{t}} t-\text{ExpIntegralEi}\left (\frac{1}{t}\right ) \]

[Out]

E^t^(-1)*t - ExpIntegralEi[t^(-1)]

________________________________________________________________________________________

Rubi [A]  time = 0.0120688, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 5, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {2206, 2210} \[ e^{\frac{1}{t}} t-\text{ExpIntegralEi}\left (\frac{1}{t}\right ) \]

Antiderivative was successfully verified.

[In]

Int[E^t^(-1),t]

[Out]

E^t^(-1)*t - ExpIntegralEi[t^(-1)]

Rule 2206

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_)), x_Symbol] :> Simp[((c + d*x)*F^(a + b*(c + d*x)^n))/d, x]
- Dist[b*n*Log[F], Int[(c + d*x)^n*F^(a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[2/n]
&& ILtQ[n, 0]

Rule 2210

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Simp[(F^a*ExpIntegralEi[
b*(c + d*x)^n*Log[F]])/(f*n), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[d*e - c*f, 0]

Rubi steps

\begin{align*} \int e^{\frac{1}{t}} \, dt &=e^{\frac{1}{t}} t+\int \frac{e^{\frac{1}{t}}}{t} \, dt\\ &=e^{\frac{1}{t}} t-\text{Ei}\left (\frac{1}{t}\right )\\ \end{align*}

Mathematica [A]  time = 0.0016972, size = 14, normalized size = 1. \[ e^{\frac{1}{t}} t-\text{ExpIntegralEi}\left (\frac{1}{t}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[E^t^(-1),t]

[Out]

E^t^(-1)*t - ExpIntegralEi[t^(-1)]

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 15, normalized size = 1.1 \begin{align*}{{\rm e}^{{t}^{-1}}}t+{\it Ei} \left ( 1,-{t}^{-1} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(1/t),t)

[Out]

exp(1/t)*t+Ei(1,-1/t)

________________________________________________________________________________________

Maxima [A]  time = 1.0221, size = 12, normalized size = 0.86 \begin{align*} -\Gamma \left (-1, -\frac{1}{t}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(1/t),t, algorithm="maxima")

[Out]

-gamma(-1, -1/t)

________________________________________________________________________________________

Fricas [A]  time = 0.998629, size = 28, normalized size = 2. \begin{align*} t e^{\frac{1}{t}} -{\rm Ei}\left (\frac{1}{t}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(1/t),t, algorithm="fricas")

[Out]

t*e^(1/t) - Ei(1/t)

________________________________________________________________________________________

Sympy [A]  time = 1.06457, size = 10, normalized size = 0.71 \begin{align*} t e^{\frac{1}{t}} - \operatorname{Ei}{\left (\frac{1}{t} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(1/t),t)

[Out]

t*exp(1/t) - Ei(1/t)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int e^{\frac{1}{t}}\,{d t} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(1/t),t, algorithm="giac")

[Out]

integrate(e^(1/t), t)