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

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

[Out]

Ei(a*t)

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Rubi [A]  time = 0.01, antiderivative size = 4, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, size = 4, normalized size = 1.00 \[ \operatorname {ExpIntegralEi}(a t) \]

Antiderivative was successfully verified.

[In]

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

[Out]

ExpIntegralEi[a*t]

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fricas [A]  time = 0.38, size = 4, normalized size = 1.00 \[ {\rm Ei}\left (a t\right ) \]

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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giac [A]  time = 0.00, size = 4, normalized size = 1.00 \[ {\rm Ei}\left (a t\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Ei(a*t)

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maple [A]  time = 0.00, size = 9, normalized size = 2.25 \[ -\Ei \left (1, -a t \right ) \]

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 = 0.82, size = 4, normalized size = 1.00 \[ {\rm Ei}\left (a t\right ) \]

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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mupad [B]  time = 0.01, size = 4, normalized size = 1.00 \[ \mathrm {ei}\left (a\,t\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ei(a*t)

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sympy [A]  time = 0.78, size = 3, normalized size = 0.75 \[ \operatorname {Ei}{\left (a t \right )} \]

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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