3.170 \(\int \frac{1}{\log (t)} \, dt\)

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

[Out]

LogIntegral[t]

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Rubi [A]  time = 0.001934, 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} \[ \text{LogIntegral}(t) \]

Antiderivative was successfully verified.

[In]

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

[Out]

LogIntegral[t]

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 (t)} \, dt &=\text{li}(t)\\ \end{align*}

Mathematica [A]  time = 0.0168389, size = 2, normalized size = 1. \[ \text{LogIntegral}(t) \]

Antiderivative was successfully verified.

[In]

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

[Out]

LogIntegral[t]

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Maple [B]  time = 0.003, size = 9, normalized size = 4.5 \begin{align*} -{\it Ei} \left ( 1,-\ln \left ( t \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/ln(t),t)

[Out]

-Ei(1,-ln(t))

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Maxima [A]  time = 1.03128, size = 4, normalized size = 2. \begin{align*}{\rm Ei}\left (\log \left (t\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Ei(log(t))

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Fricas [A]  time = 1.03717, size = 23, normalized size = 11.5 \begin{align*} \logintegral \left (t\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log_integral(t)

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Sympy [A]  time = 0.461404, size = 2, normalized size = 1. \begin{align*} \operatorname{li}{\left (t \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/ln(t),t)

[Out]

li(t)

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Giac [A]  time = 1.08247, size = 4, normalized size = 2. \begin{align*}{\rm Ei}\left (\log \left (t\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Ei(log(t))