3.173 \(\int \frac{e^{2 t}}{-1+t} \, dt\)

Optimal. Leaf size=12 \[ e^2 \text{ExpIntegralEi}(-2 (1-t)) \]

[Out]

E^2*ExpIntegralEi[-2*(1 - t)]

________________________________________________________________________________________

Rubi [A]  time = 0.0148496, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {2178} \[ e^2 \text{ExpIntegralEi}(-2 (1-t)) \]

Antiderivative was successfully verified.

[In]

Int[E^(2*t)/(-1 + t),t]

[Out]

E^2*ExpIntegralEi[-2*(1 - 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^{2 t}}{-1+t} \, dt &=e^2 \text{Ei}(-2 (1-t))\\ \end{align*}

Mathematica [A]  time = 0.0131442, size = 10, normalized size = 0.83 \[ e^2 \text{ExpIntegralEi}(2 (t-1)) \]

Antiderivative was successfully verified.

[In]

Integrate[E^(2*t)/(-1 + t),t]

[Out]

E^2*ExpIntegralEi[2*(-1 + t)]

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 12, normalized size = 1. \begin{align*} -{{\rm e}^{2}}{\it Ei} \left ( 1,-2\,t+2 \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(2*t)/(-1+t),t)

[Out]

-exp(2)*Ei(1,-2*t+2)

________________________________________________________________________________________

Maxima [A]  time = 1.03601, size = 15, normalized size = 1.25 \begin{align*} -e^{2} E_{1}\left (-2 \, t + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(2*t)/(-1+t),t, algorithm="maxima")

[Out]

-e^2*exp_integral_e(1, -2*t + 2)

________________________________________________________________________________________

Fricas [A]  time = 1.12668, size = 23, normalized size = 1.92 \begin{align*}{\rm Ei}\left (2 \, t - 2\right ) e^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(2*t)/(-1+t),t, algorithm="fricas")

[Out]

Ei(2*t - 2)*e^2

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{2 t}}{t - 1}\, dt \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(2*t)/(-1+t),t)

[Out]

Integral(exp(2*t)/(t - 1), t)

________________________________________________________________________________________

Giac [A]  time = 1.08619, size = 12, normalized size = 1. \begin{align*}{\rm Ei}\left (2 \, t - 2\right ) e^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(2*t)/(-1+t),t, algorithm="giac")

[Out]

Ei(2*t - 2)*e^2