\(\int \frac {e^{a x}}{x} \, dx\) [153]

Optimal result

Integrand size = 9, antiderivative size = 4 \[ \int \frac {e^{a x}}{x} \, dx=\operatorname {ExpIntegralEi}(a x) \]

[Out]

Ei(a*x)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 4, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2209} \[ \int \frac {e^{a x}}{x} \, dx=\operatorname {ExpIntegralEi}(a x) \]

[In]

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

[Out]

ExpIntegralEi[a*x]

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}(a x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 4, normalized size of antiderivative = 1.00 \[ \int \frac {e^{a x}}{x} \, dx=\operatorname {ExpIntegralEi}(a x) \]

[In]

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

[Out]

ExpIntegralEi[a*x]

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 9, normalized size of antiderivative = 2.25

method	result	size
derivativedivides	\(-\operatorname {Ei}_{1}\left (-a x \right )\)	\(9\)
default	\(-\operatorname {Ei}_{1}\left (-a x \right )\)	\(9\)
risch	\(-\operatorname {Ei}_{1}\left (-a x \right )\)	\(9\)
meijerg	\(\ln \left (x \right )+\ln \left (-a \right )-\ln \left (-a x \right )-\operatorname {Ei}_{1}\left (-a x \right )\)	\(23\)

[In]

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

[Out]

-Ei(1,-a*x)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 4, normalized size of antiderivative = 1.00 \[ \int \frac {e^{a x}}{x} \, dx={\rm Ei}\left (a x\right ) \]

[In]

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

[Out]

Ei(a*x)

Sympy [A] (verification not implemented)

Time = 0.47 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.75 \[ \int \frac {e^{a x}}{x} \, dx=\operatorname {Ei}{\left (a x \right )} \]

[In]

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

[Out]

Ei(a*x)

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 4, normalized size of antiderivative = 1.00 \[ \int \frac {e^{a x}}{x} \, dx={\rm Ei}\left (a x\right ) \]

[In]

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

[Out]

Ei(a*x)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 4, normalized size of antiderivative = 1.00 \[ \int \frac {e^{a x}}{x} \, dx={\rm Ei}\left (a x\right ) \]

[In]

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

[Out]

Ei(a*x)

Mupad [B] (verification not implemented)

Time = 0.01 (sec) , antiderivative size = 4, normalized size of antiderivative = 1.00 \[ \int \frac {e^{a x}}{x} \, dx=\mathrm {ei}\left (a\,x\right ) \]

[In]

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

[Out]

ei(a*x)