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

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.00, 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 = {2209} \begin {gather*} e^2 \text {Ei}(-2 (1-t)) \end {gather*}

Antiderivative was successfully verified.

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

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Mathematica [A]
time = 0.03, size = 10, normalized size = 0.83 \begin {gather*} e^2 \text {Ei}(2 (-1+t)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(2*t)/(-1 + t),t]

[Out]

E^2*ExpIntegralEi[2*(-1 + 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[E^(2*t)/(t - 1),t]')

[Out]

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

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Maple [A]
time = 0.06, size = 12, normalized size = 1.00

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 11, normalized size = 0.92 \begin {gather*} -e^{2} E_{1}\left (-2 \, t + 2\right ) \end {gather*}

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)

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Fricas [A]
time = 0.32, size = 9, normalized size = 0.75 \begin {gather*} {\rm Ei}\left (2 \, t - 2\right ) e^{2} \end {gather*}

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{2 t}}{t - 1}\, dt \end {gather*}

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)

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Giac [A]
time = 0.00, size = 9, normalized size = 0.75 \begin {gather*} \mathrm {Ei}\left (2 t-2\right ) \mathrm {e}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Ei(2*t - 2)*e^2

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Mupad [B]
time = 0.02, size = 9, normalized size = 0.75 \begin {gather*} {\mathrm {e}}^2\,\mathrm {ei}\left (2\,t-2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(2*t)/(t - 1),t)

[Out]

exp(2)*ei(2*t - 2)

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