3.1.60 \(\int \frac {1}{\log (x)} \, dx\) [60]

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

[Out]

Li(x)

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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 = {2335} \begin {gather*} \text {LogIntegral}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

LogIntegral[x]

Rule 2335

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*}

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Mathematica [A]
time = 0.00, size = 2, normalized size = 1.00 \begin {gather*} \text {li}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

LogIntegral[x]

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(8\) vs. \(2(2)=4\).
time = 0.00, size = 9, normalized size = 4.50

method	result	size
default	\(-\expIntegral \left (1, -\ln \left (x \right )\right )\)	\(9\)
risch	\(-\expIntegral \left (1, -\ln \left (x \right )\right )\)	\(9\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/ln(x),x,method=_RETURNVERBOSE)

[Out]

-Ei(1,-ln(x))

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Maxima [A]
time = 1.18, size = 3, normalized size = 1.50 \begin {gather*} {\rm Ei}\left (\log \left (x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Ei(log(x))

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Fricas [A]
time = 0.82, size = 2, normalized size = 1.00 \begin {gather*} \operatorname {log\_integral}\left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log_integral(x)

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Sympy [A]
time = 0.24, size = 2, normalized size = 1.00 \begin {gather*} \operatorname {li}{\left (x \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/ln(x),x)

[Out]

li(x)

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Giac [A]
time = 0.77, size = 3, normalized size = 1.50 \begin {gather*} {\rm Ei}\left (\log \left (x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Ei(log(x))

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Mupad [B]
time = 0.00, size = 2, normalized size = 1.00 \begin {gather*} \mathrm {logint}\left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/log(x),x)

[Out]

logint(x)

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