3.2.61 \(\int e^{\frac {1}{t}} \, dt\) [161]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2237, 2241} \begin {gather*} e^{\frac {1}{t}} t-\text {ExpIntegralEi}\left (\frac {1}{t}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^t^(-1)*t - ExpIntegralEi[t^(-1)]

Rule 2237

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 2241

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*}

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Mathematica [A]
time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} e^{\frac {1}{t}} t-\text {Ei}\left (\frac {1}{t}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^t^(-1)*t - ExpIntegralEi[t^(-1)]

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Maple [A]
time = 0.02, size = 15, normalized size = 1.07

method	result	size
derivativedivides	\({\mathrm e}^{\frac {1}{t}} t +\expIntegral \left (1, -\frac {1}{t}\right )\)	\(15\)
default	\({\mathrm e}^{\frac {1}{t}} t +\expIntegral \left (1, -\frac {1}{t}\right )\)	\(15\)
risch	\({\mathrm e}^{\frac {1}{t}} t +\expIntegral \left (1, -\frac {1}{t}\right )\)	\(15\)
meijerg	\(t +1+\ln \left (t \right )-i \pi -\frac {t \left (\frac {2}{t}+2\right )}{2}+{\mathrm e}^{\frac {1}{t}} t +\ln \left (-\frac {1}{t}\right )+\expIntegral \left (1, -\frac {1}{t}\right )\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(1/t),t,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 1.30, size = 9, normalized size = 0.64 \begin {gather*} -\Gamma \left (-1, -\frac {1}{t}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-gamma(-1, -1/t)

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Fricas [A]
time = 0.81, size = 13, normalized size = 0.93 \begin {gather*} t e^{\frac {1}{t}} - {\rm Ei}\left (\frac {1}{t}\right ) \end {gather*}

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)

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Sympy [A]
time = 0.47, size = 10, normalized size = 0.71 \begin {gather*} t e^{\frac {1}{t}} - \operatorname {Ei}{\left (\frac {1}{t} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(1/t),t)

[Out]

t*exp(1/t) - Ei(1/t)

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Giac [A]
time = 0.46, size = 18, normalized size = 1.29 \begin {gather*} -t {\left (\frac {{\rm Ei}\left (\frac {1}{t}\right )}{t} - e^{\frac {1}{t}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-t*(Ei(1/t)/t - e^(1/t))

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Mupad [B]
time = 0.02, size = 9, normalized size = 0.64 \begin {gather*} t\,\mathrm {expint}\left (2,-\frac {1}{t}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(1/t),t)

[Out]

t*expint(2, -1/t)

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