\(\int \frac {e^{2 t}}{-1+t} \, dt\) [173]

Optimal result

Integrand size = 11, antiderivative size = 12 \[ \int \frac {e^{2 t}}{-1+t} \, dt=e^2 \operatorname {ExpIntegralEi}(-2 (1-t)) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2209} \[ \int \frac {e^{2 t}}{-1+t} \, dt=e^2 \operatorname {ExpIntegralEi}(-2 (1-t)) \]

[In]

Int[E^(2*t)/(-1 + t),t]

[Out]

E^2*ExpIntegralEi[-2*(1 - 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^2 \operatorname {ExpIntegralEi}(-2 (1-t)) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {e^{2 t}}{-1+t} \, dt=e^2 \operatorname {ExpIntegralEi}(2 (-1+t)) \]

[In]

Integrate[E^(2*t)/(-1 + t),t]

[Out]

E^2*ExpIntegralEi[2*(-1 + t)]

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00

method	result	size
derivativedivides	\(-{\mathrm e}^{2} \operatorname {Ei}_{1}\left (-2 t +2\right )\)	\(12\)
default	\(-{\mathrm e}^{2} \operatorname {Ei}_{1}\left (-2 t +2\right )\)	\(12\)
risch	\(-{\mathrm e}^{2} \operatorname {Ei}_{1}\left (-2 t +2\right )\)	\(12\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.75 \[ \int \frac {e^{2 t}}{-1+t} \, dt={\rm Ei}\left (2 \, t - 2\right ) e^{2} \]

[In]

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

[Out]

Ei(2*t - 2)*e^2

Sympy [F]

\[ \int \frac {e^{2 t}}{-1+t} \, dt=\int \frac {e^{2 t}}{t - 1}\, dt \]

[In]

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

[Out]

Integral(exp(2*t)/(t - 1), t)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.92 \[ \int \frac {e^{2 t}}{-1+t} \, dt=-e^{2} E_{1}\left (-2 \, t + 2\right ) \]

[In]

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

[Out]

-e^2*exp_integral_e(1, -2*t + 2)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.75 \[ \int \frac {e^{2 t}}{-1+t} \, dt={\rm Ei}\left (2 \, t - 2\right ) e^{2} \]

[In]

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

[Out]

Ei(2*t - 2)*e^2

Mupad [B] (verification not implemented)

Time = 0.01 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.75 \[ \int \frac {e^{2 t}}{-1+t} \, dt={\mathrm {e}}^2\,\mathrm {ei}\left (2\,t-2\right ) \]

[In]

int(exp(2*t)/(t - 1),t)

[Out]

exp(2)*ei(2*t - 2)