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

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

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 15, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, size = 15, normalized size = 1.00 \[ e^{-a-1} \text {Ei}(a-t+1) \]

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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fricas [A]  time = 0.40, size = 14, normalized size = 0.93 \[ {\rm Ei}\left (a - t + 1\right ) e^{\left (-a - 1\right )} \]

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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giac [A]  time = 0.01, size = 14, normalized size = 0.93 \[ {\rm Ei}\left (a - t + 1\right ) e^{\left (-a - 1\right )} \]

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)

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maple [A]  time = 0.01, size = 17, normalized size = 1.13 \[ -\Ei \left (1, -a +t -1\right ) {\mathrm e}^{-a -1} \]

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 = 0.87, size = 16, normalized size = 1.07 \[ -e^{\left (-a - 1\right )} E_{1}\left (-a + t - 1\right ) \]

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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mupad [B]  time = 0.03, size = 14, normalized size = 0.93 \[ {\mathrm {e}}^{-a-1}\,\mathrm {ei}\left (a-t+1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(-t)/(a - t + 1),t)

[Out]

exp(- a - 1)*ei(a - t + 1)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{- t}}{- a + t - 1}\, dt \]

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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