3.162 \(\int \frac{e^{-t}}{-1-a+t} \, dt\)

Optimal. Leaf size=15 \[ e^{-a-1} \text{ExpIntegralEi}(a-t+1) \]

[Out]

E^(-1 - a)*ExpIntegralEi[1 + a - t]

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Rubi [A]  time = 0.0153825, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {2178} \[ e^{-a-1} \text{ExpIntegralEi}(a-t+1) \]

Antiderivative was successfully verified.

[In]

Int[1/(E^t*(-1 - a + t)),t]

[Out]

E^(-1 - a)*ExpIntegralEi[1 + a - t]

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rubi steps

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

Mathematica [A]  time = 0.0117154, size = 15, normalized size = 1. \[ e^{-a-1} \text{ExpIntegralEi}(a-t+1) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(E^t*(-1 - a + t)),t]

[Out]

E^(-1 - a)*ExpIntegralEi[1 + a - t]

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Maple [A]  time = 0.006, size = 17, normalized size = 1.1 \begin{align*} -{{\rm e}^{-1-a}}{\it Ei} \left ( 1,-1-a+t \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/exp(t)/(-1-a+t),t)

[Out]

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

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Maxima [A]  time = 1.03949, size = 22, normalized size = 1.47 \begin{align*} -e^{\left (-a - 1\right )} E_{1}\left (-a + t - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-e^(-a - 1)*exp_integral_e(1, -a + t - 1)

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Fricas [A]  time = 1.1479, size = 35, normalized size = 2.33 \begin{align*}{\rm Ei}\left (a - t + 1\right ) e^{\left (-a - 1\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Ei(a - t + 1)*e^(-a - 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{- t}}{- a + t - 1}\, dt \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/exp(t)/(-1-a+t),t)

[Out]

Integral(exp(-t)/(-a + t - 1), t)

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Giac [A]  time = 1.08339, size = 19, normalized size = 1.27 \begin{align*}{\rm Ei}\left (a - t + 1\right ) e^{\left (-a - 1\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Ei(a - t + 1)*e^(-a - 1)