3.57 \(\int \frac{1}{\log (x)} \, dx\)

Optimal. Leaf size=2 \[ \text{LogIntegral}(x) \]

[Out]

LogIntegral[x]

________________________________________________________________________________________

Rubi [A]  time = 0.001793, antiderivative size = 2, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2298} \[ \text{LogIntegral}(x) \]

Antiderivative was successfully verified.

[In]

Int[Log[x]^(-1),x]

[Out]

LogIntegral[x]

Rule 2298

Int[Log[(c_.)*(x_)]^(-1), x_Symbol] :> Simp[LogIntegral[c*x]/c, x] /; FreeQ[c, x]

Rubi steps

\begin{align*} \int \frac{1}{\log (x)} \, dx &=\text{li}(x)\\ \end{align*}

Mathematica [F]  time = 0.0021106, size = 0, normalized size = 0. \[ \int \frac{1}{\log (x)} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Log[x]^(-1),x]

[Out]

Integrate[Log[x]^(-1), x]

________________________________________________________________________________________

Maple [B]  time = 0., size = 9, normalized size = 4.5 \begin{align*} -{\it Ei} \left ( 1,-\ln \left ( x \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/ln(x),x)

[Out]

-Ei(1,-ln(x))

________________________________________________________________________________________

Maxima [A]  time = 1.0224, size = 4, normalized size = 2. \begin{align*}{\rm Ei}\left (\log \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/log(x),x, algorithm="maxima")

[Out]

Ei(log(x))

________________________________________________________________________________________

Fricas [A]  time = 1.80083, size = 23, normalized size = 11.5 \begin{align*} \logintegral \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/log(x),x, algorithm="fricas")

[Out]

log_integral(x)

________________________________________________________________________________________

Sympy [A]  time = 0.439018, size = 2, normalized size = 1. \begin{align*} \operatorname{li}{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/ln(x),x)

[Out]

li(x)

________________________________________________________________________________________

Giac [A]  time = 1.12029, size = 4, normalized size = 2. \begin{align*}{\rm Ei}\left (\log \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/log(x),x, algorithm="giac")

[Out]

Ei(log(x))