3.159 \(\int \frac{e^{a t}}{t} \, dt\)

Optimal. Leaf size=4 \[ \text{ExpIntegralEi}(a t) \]

[Out]

ExpIntegralEi[a*t]

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Rubi [A]  time = 0.0120414, antiderivative size = 4, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2178} \[ \text{ExpIntegralEi}(a t) \]

Antiderivative was successfully verified.

[In]

Int[E^(a*t)/t,t]

[Out]

ExpIntegralEi[a*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^{a t}}{t} \, dt &=\text{Ei}(a t)\\ \end{align*}

Mathematica [A]  time = 0.005839, size = 4, normalized size = 1. \[ \text{ExpIntegralEi}(a t) \]

Antiderivative was successfully verified.

[In]

Integrate[E^(a*t)/t,t]

[Out]

ExpIntegralEi[a*t]

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Maple [A]  time = 0.003, size = 9, normalized size = 2.3 \begin{align*} -{\it Ei} \left ( 1,-at \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(a*t)/t,t)

[Out]

-Ei(1,-a*t)

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Maxima [A]  time = 1.03016, size = 5, normalized size = 1.25 \begin{align*}{\rm Ei}\left (a t\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(a*t)/t,t, algorithm="maxima")

[Out]

Ei(a*t)

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Fricas [A]  time = 0.997875, size = 12, normalized size = 3. \begin{align*}{\rm Ei}\left (a t\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(a*t)/t,t, algorithm="fricas")

[Out]

Ei(a*t)

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Sympy [A]  time = 0.78138, size = 3, normalized size = 0.75 \begin{align*} \operatorname{Ei}{\left (a t \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(a*t)/t,t)

[Out]

Ei(a*t)

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Giac [A]  time = 1.07686, size = 5, normalized size = 1.25 \begin{align*}{\rm Ei}\left (a t\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(a*t)/t,t, algorithm="giac")

[Out]

Ei(a*t)