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

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

[Out]

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

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Rubi [A]
time = 0.01, 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 = {2209} \begin {gather*} e^{-a-1} \text {Ei}(a-t+1) \end {gather*}

Antiderivative was successfully verified.

[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*} \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 = 0.03, size = 15, normalized size = 1.00 \begin {gather*} e^{-1-a} \text {Ei}(1+a-t) \end {gather*}

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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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded while calling a Python object} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[1/(E^t*(t - a - 1)),t]')

[Out]

cought exception: maximum recursion depth exceeded while calling a Python object

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Maple [A]
time = 0.07, size = 17, normalized size = 1.13

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.38, size = 16, normalized size = 1.07 \begin {gather*} -e^{\left (-a - 1\right )} E_{1}\left (-a + t - 1\right ) \end {gather*}

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 = 0.32, size = 14, normalized size = 0.93 \begin {gather*} {\rm Ei}\left (a - t + 1\right ) e^{\left (-a - 1\right )} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{- t}}{- a + t - 1}\, dt \end {gather*}

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 = 0.00, size = 12, normalized size = 0.80 \begin {gather*} \mathrm {Ei}\left (a-t+1\right ) \mathrm {e}^{-a-1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Ei(a - t + 1)*e^(-a - 1)

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

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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