3.1.61 \(\int \frac {1}{\log (1+x)} \, dx\) [61]

Optimal. Leaf size=4 \[ \text {li}(1+x) \]

[Out]

Li(1+x)

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Rubi [A]
time = 0.00, antiderivative size = 4, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2436, 2335} \begin {gather*} \text {LogIntegral}(x+1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Log[1 + x]^(-1),x]

[Out]

LogIntegral[1 + x]

Rule 2335

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

Rule 2436

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 4, normalized size = 1.00 \begin {gather*} \text {li}(1+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Log[1 + x]^(-1),x]

[Out]

LogIntegral[1 + x]

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(10\) vs. \(2(4)=8\).
time = 0.01, size = 11, normalized size = 2.75

method	result	size
derivativedivides	\(-\expIntegral \left (1, -\ln \left (1+x \right )\right )\)	\(11\)
default	\(-\expIntegral \left (1, -\ln \left (1+x \right )\right )\)	\(11\)
risch	\(-\expIntegral \left (1, -\ln \left (1+x \right )\right )\)	\(11\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Ei(1,-ln(1+x))

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Maxima [A]
time = 1.42, size = 5, normalized size = 1.25 \begin {gather*} {\rm Ei}\left (\log \left (x + 1\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Ei(log(x + 1))

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Fricas [A]
time = 1.13, size = 4, normalized size = 1.00 \begin {gather*} \operatorname {log\_integral}\left (x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log_integral(x + 1)

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Sympy [A]
time = 0.23, size = 3, normalized size = 0.75 \begin {gather*} \operatorname {li}{\left (x + 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/ln(1+x),x)

[Out]

li(x + 1)

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Giac [A]
time = 0.86, size = 5, normalized size = 1.25 \begin {gather*} {\rm Ei}\left (\log \left (x + 1\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Ei(log(x + 1))

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Mupad [B]
time = 0.02, size = 4, normalized size = 1.00 \begin {gather*} \mathrm {logint}\left (x+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/log(x + 1),x)

[Out]

logint(x + 1)

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