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

Optimal. Leaf size=13 \[ \text {li}(x)+\log (\log (x))-\log (x+\log (x)) \]

[Out]

Li(x)+ln(ln(x))-ln(x+ln(x))

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Rubi [A]  time = 0.14, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6742, 2353, 2298, 2302, 29, 6684} \[ \text {li}(x)+\log (\log (x))-\log (x+\log (x)) \]

Antiderivative was successfully verified.

[In]

Int[(1 + x)/(Log[x]*(x + Log[x])),x]

[Out]

Log[Log[x]] - Log[x + Log[x]] + LogIntegral[x]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 2298

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

Rule 2302

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

Rule 2353

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol]
:> With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[
{a, b, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[m] && IntegerQ[r
]))

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

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

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Mathematica [A]  time = 0.04, size = 13, normalized size = 1.00 \[ \text {li}(x)+\log (\log (x))-\log (x+\log (x)) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + x)/(Log[x]*(x + Log[x])),x]

[Out]

Log[Log[x]] - Log[x + Log[x]] + LogIntegral[x]

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fricas [A]  time = 0.40, size = 13, normalized size = 1.00 \[ -\log \left (x + \log \relax (x)\right ) + \log \left (\log \relax (x)\right ) + \operatorname {log\_integral}\relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x + log(x)) + log(log(x)) + log_integral(x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x + 1}{{\left (x + \log \relax (x)\right )} \log \relax (x)}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((x + 1)/((x + log(x))*log(x)), x)

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maple [F]  time = 0.07, size = 0, normalized size = 0.00 \[ \int \frac {x +1}{\left (x +\ln \relax (x )\right ) \ln \relax (x )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x + 1}{x \log \relax (x)}\,{d x} - \log \left (x + \log \relax (x)\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((x + 1)/(x*log(x)), x) - log(x + log(x))

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mupad [B]  time = 0.39, size = 13, normalized size = 1.00 \[ \ln \left (\ln \relax (x)\right )-\ln \left (x+\ln \relax (x)\right )+\mathrm {logint}\relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 1)/(log(x)*(x + log(x))),x)

[Out]

log(log(x)) - log(x + log(x)) + logint(x)

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sympy [A]  time = 4.30, size = 15, normalized size = 1.15 \[ - \log {\left (x + \log {\relax (x )} \right )} + \log {\left (\log {\relax (x )} \right )} + \operatorname {Ei}{\left (\log {\relax (x )} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x + log(x)) + log(log(x)) + Ei(log(x))

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