\(\int \frac {e^t}{t} \, dt\) [158]

Optimal result

Integrand size = 7, antiderivative size = 2 \[ \int \frac {e^t}{t} \, dt=\operatorname {ExpIntegralEi}(t) \]

[Out]

Ei(t)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 2, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2209} \[ \int \frac {e^t}{t} \, dt=\operatorname {ExpIntegralEi}(t) \]

[In]

Int[E^t/t,t]

[Out]

ExpIntegralEi[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}& = \operatorname {ExpIntegralEi}(t) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 2, normalized size of antiderivative = 1.00 \[ \int \frac {e^t}{t} \, dt=\operatorname {ExpIntegralEi}(t) \]

[In]

Integrate[E^t/t,t]

[Out]

ExpIntegralEi[t]

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(7\) vs. \(2(2)=4\).

Time = 0.04 (sec) , antiderivative size = 8, normalized size of antiderivative = 4.00

method	result	size
default	\(-\operatorname {Ei}_{1}\left (-t \right )\)	\(8\)
risch	\(-\operatorname {Ei}_{1}\left (-t \right )\)	\(8\)
meijerg	\(\ln \left (t \right )+i \pi -\ln \left (-t \right )-\operatorname {Ei}_{1}\left (-t \right )\)	\(21\)

[In]

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

[Out]

-Ei(1,-t)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 2, normalized size of antiderivative = 1.00 \[ \int \frac {e^t}{t} \, dt={\rm Ei}\left (t\right ) \]

[In]

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

[Out]

Ei(t)

Sympy [A] (verification not implemented)

Time = 0.39 (sec) , antiderivative size = 2, normalized size of antiderivative = 1.00 \[ \int \frac {e^t}{t} \, dt=\operatorname {Ei}{\left (t \right )} \]

[In]

integrate(exp(t)/t,t)

[Out]

Ei(t)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 2, normalized size of antiderivative = 1.00 \[ \int \frac {e^t}{t} \, dt={\rm Ei}\left (t\right ) \]

[In]

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

[Out]

Ei(t)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 2, normalized size of antiderivative = 1.00 \[ \int \frac {e^t}{t} \, dt={\rm Ei}\left (t\right ) \]

[In]

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

[Out]

Ei(t)

Mupad [B] (verification not implemented)

Time = 0.01 (sec) , antiderivative size = 2, normalized size of antiderivative = 1.00 \[ \int \frac {e^t}{t} \, dt=\mathrm {ei}\left (t\right ) \]

[In]

int(exp(t)/t,t)

[Out]

ei(t)