3.2.59 \(\int \frac {e^{a t}}{t} \, dt\) [159]

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

[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 = {2209} \begin {gather*} \text {ExpIntegralEi}(a t) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

ExpIntegralEi[a*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^{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 \begin {gather*} \text {Ei}(a t) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

ExpIntegralEi[a*t]

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Maple [A]
time = 0.02, size = 9, normalized size = 2.25

method	result	size
derivativedivides	\(-\expIntegral \left (1, -a t \right )\)	\(9\)
default	\(-\expIntegral \left (1, -a t \right )\)	\(9\)
risch	\(-\expIntegral \left (1, -a t \right )\)	\(9\)
meijerg	\(\ln \left (t \right )+\ln \left (-a \right )-\ln \left (-a t \right )-\expIntegral \left (1, -a t \right )\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(a*t)/t,t,method=_RETURNVERBOSE)

[Out]

-Ei(1,-a*t)

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Maxima [A]
time = 6.97, size = 4, normalized size = 1.00 \begin {gather*} {\rm Ei}\left (a t\right ) \end {gather*}

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 = 0.86, size = 4, normalized size = 1.00 \begin {gather*} {\rm Ei}\left (a t\right ) \end {gather*}

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 = 0.34, size = 3, normalized size = 0.75 \begin {gather*} \operatorname {Ei}{\left (a t \right )} \end {gather*}

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 = 0.45, size = 4, normalized size = 1.00 \begin {gather*} {\rm Ei}\left (a t\right ) \end {gather*}

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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Mupad [B]
time = 0.01, size = 4, normalized size = 1.00 \begin {gather*} \mathrm {ei}\left (a\,t\right ) \end {gather*}

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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