\(\int \frac {e^{-t}}{-1-a+t} \, dt\) [162]

Optimal result

Integrand size = 14, antiderivative size = 15 \[ \int \frac {e^{-t}}{-1-a+t} \, dt=e^{-1-a} \operatorname {ExpIntegralEi}(1+a-t) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2209} \[ \int \frac {e^{-t}}{-1-a+t} \, dt=e^{-a-1} \operatorname {ExpIntegralEi}(a-t+1) \]

[In]

Int[1/(E^t*(-1 - a + t)),t]

[Out]

E^(-1 - a)*ExpIntegralEi[1 + a - t]

Rule 2209

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

Rubi steps \begin{align*} \text {integral}& = e^{-1-a} \operatorname {ExpIntegralEi}(1+a-t) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-t}}{-1-a+t} \, dt=e^{-1-a} \operatorname {ExpIntegralEi}(1+a-t) \]

[In]

Integrate[1/(E^t*(-1 - a + t)),t]

[Out]

E^(-1 - a)*ExpIntegralEi[1 + a - t]

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13

method	result	size
default	\(-{\mathrm e}^{-1-a} \operatorname {Ei}_{1}\left (-1-a +t \right )\)	\(17\)
risch	\(-{\mathrm e}^{-1-a} \operatorname {Ei}_{1}\left (-1-a +t \right )\)	\(17\)

[In]

int(1/exp(t)/(-1-a+t),t,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93 \[ \int \frac {e^{-t}}{-1-a+t} \, dt={\rm Ei}\left (a - t + 1\right ) e^{\left (-a - 1\right )} \]

[In]

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

[Out]

Ei(a - t + 1)*e^(-a - 1)

Sympy [F]

\[ \int \frac {e^{-t}}{-1-a+t} \, dt=\int \frac {e^{- t}}{- a + t - 1}\, dt \]

[In]

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

[Out]

Integral(exp(-t)/(-a + t - 1), t)

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.07 \[ \int \frac {e^{-t}}{-1-a+t} \, dt=-e^{\left (-a - 1\right )} E_{1}\left (-a + t - 1\right ) \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93 \[ \int \frac {e^{-t}}{-1-a+t} \, dt={\rm Ei}\left (a - t + 1\right ) e^{\left (-a - 1\right )} \]

[In]

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

[Out]

Ei(a - t + 1)*e^(-a - 1)

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93 \[ \int \frac {e^{-t}}{-1-a+t} \, dt={\mathrm {e}}^{-a-1}\,\mathrm {ei}\left (a-t+1\right ) \]

[In]

int(-exp(-t)/(a - t + 1),t)

[Out]

exp(- a - 1)*ei(a - t + 1)