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

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

[Out]

Li(t)

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Rubi [A]  time = 0.00, antiderivative size = 2, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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*}

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Mathematica [A]  time = 0.00, size = 2, normalized size = 1.00 \[ \operatorname {LogIntegral}(t) \]

Antiderivative was successfully verified.

[In]

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

[Out]

LogIntegral[t]

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fricas [A]  time = 0.40, size = 2, normalized size = 1.00 \[ \operatorname {log\_integral}\relax (t) \]

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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giac [A]  time = 0.01, size = 3, normalized size = 1.50 \[ {\rm Ei}\left (\log \relax (t)\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Ei(log(t))

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maple [B]  time = 0.01, size = 9, normalized size = 4.50 \[ -\Ei \left (1, -\ln \relax (t )\right ) \]

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 = 0.63, size = 3, normalized size = 1.50 \[ {\rm Ei}\left (\log \relax (t)\right ) \]

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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mupad [B]  time = 0.01, size = 2, normalized size = 1.00 \[ \mathrm {logint}\relax (t) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/log(t),t)

[Out]

logint(t)

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sympy [A]  time = 0.46, size = 2, normalized size = 1.00 \[ \operatorname {li}{\relax (t )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/ln(t),t)

[Out]

li(t)

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