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 = 0.0338097, 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 \[ e^{-a-1} \text{ExpIntegralEi}(a-t+1) \]

Antiderivative was successfully verified.

[In]  Int[1/(E^t*(-1 - a + t)),t]

[Out]

E^(-1 - a)*ExpIntegralEi[1 + a - t]

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Rubi in Sympy [A]  time = 2.02819, size = 12, normalized size = 0.8 \[ e^{- a - 1} \operatorname{Ei}{\left (a - t + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/exp(t)/(-1-a+t),t)

[Out]

exp(-a - 1)*Ei(a - t + 1)

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Mathematica [A]  time = 0.00724314, 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 = 0.023, size = 17, normalized size = 1.1 \[ -{{\rm e}^{-1-a}}{\it Ei} \left ( 1,-1-a+t \right ) \]

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 = 1.44247, size = 22, normalized size = 1.47 \[ -e^{\left (-a - 1\right )} exp_integral_e\left (1, -a + t - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-e^(-t)/(a - t + 1),t, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.199377, size = 19, normalized size = 1.27 \[{\rm Ei}\left (a - t + 1\right ) e^{\left (-a - 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-e^(-t)/(a - t + 1),t, algorithm="fricas")

[Out]

Ei(a - t + 1)*e^(-a - 1)

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

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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GIAC/XCAS [A]  time = 0.232198, size = 19, normalized size = 1.27 \[{\rm Ei}\left (a - t + 1\right ) e^{\left (-a - 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-e^(-t)/(a - t + 1),t, algorithm="giac")

[Out]

Ei(a - t + 1)*e^(-a - 1)