3.13 \(\int \frac{-x^2-\log (x)+2 \log ^2(x)}{-x^2 \log (x)+\log ^3(x)} \, dx\)

Optimal. Leaf size=23 \[ \text{LogIntegral}(x)-\frac{1}{2} \log (x-\log (x))+\frac{1}{2} \log (x+\log (x)) \]

[Out]

-Log[x - Log[x]]/2 + Log[x + Log[x]]/2 + LogIntegral[x]

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Rubi [A]  time = 0.247992, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129, Rules used = {6741, 6742, 6684, 2298} \[ \text{LogIntegral}(x)-\frac{1}{2} \log (x-\log (x))+\frac{1}{2} \log (x+\log (x)) \]

Antiderivative was successfully verified.

[In]

Int[(-x^2 - Log[x] + 2*Log[x]^2)/(-(x^2*Log[x]) + Log[x]^3),x]

[Out]

-Log[x - Log[x]]/2 + Log[x + Log[x]]/2 + LogIntegral[x]

Rule 6741

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

Rule 6742

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

Rule 6684

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

Rule 2298

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

Rubi steps

\begin{align*} \int \frac{-x^2-\log (x)+2 \log ^2(x)}{-x^2 \log (x)+\log ^3(x)} \, dx &=\int \frac{x^2+\log (x)-2 \log ^2(x)}{\log (x) \left (x^2-\log ^2(x)\right )} \, dx\\ &=\int \left (\frac{1-x}{2 x (x-\log (x))}+\frac{1}{\log (x)}+\frac{1+x}{2 x (x+\log (x))}\right ) \, dx\\ &=\frac{1}{2} \int \frac{1-x}{x (x-\log (x))} \, dx+\frac{1}{2} \int \frac{1+x}{x (x+\log (x))} \, dx+\int \frac{1}{\log (x)} \, dx\\ &=-\frac{1}{2} \log (x-\log (x))+\frac{1}{2} \log (x+\log (x))+\text{li}(x)\\ \end{align*}

Mathematica [A]  time = 0.0901356, size = 23, normalized size = 1. \[ \text{LogIntegral}(x)-\frac{1}{2} \log (x-\log (x))+\frac{1}{2} \log (x+\log (x)) \]

Antiderivative was successfully verified.

[In]

Integrate[(-x^2 - Log[x] + 2*Log[x]^2)/(-(x^2*Log[x]) + Log[x]^3),x]

[Out]

-Log[x - Log[x]]/2 + Log[x + Log[x]]/2 + LogIntegral[x]

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Maple [F]  time = 0.017, size = 0, normalized size = 0. \begin{align*} \int{\frac{-{x}^{2}-\ln \left ( x \right ) +2\, \left ( \ln \left ( x \right ) \right ) ^{2}}{-{x}^{2}\ln \left ( x \right ) + \left ( \ln \left ( x \right ) \right ) ^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x^2-ln(x)+2*ln(x)^2)/(-x^2*ln(x)+ln(x)^3),x)

[Out]

int((-x^2-ln(x)+2*ln(x)^2)/(-x^2*ln(x)+ln(x)^3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\log \left (x\right )}\,{d x} + \frac{1}{2} \, \log \left (x + \log \left (x\right )\right ) - \frac{1}{2} \, \log \left (-x + \log \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2-log(x)+2*log(x)^2)/(-x^2*log(x)+log(x)^3),x, algorithm="maxima")

[Out]

integrate(1/log(x), x) + 1/2*log(x + log(x)) - 1/2*log(-x + log(x))

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Fricas [A]  time = 2.35184, size = 84, normalized size = 3.65 \begin{align*} \frac{1}{2} \, \log \left (x + \log \left (x\right )\right ) - \frac{1}{2} \, \log \left (-x + \log \left (x\right )\right ) + \logintegral \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2-log(x)+2*log(x)^2)/(-x^2*log(x)+log(x)^3),x, algorithm="fricas")

[Out]

1/2*log(x + log(x)) - 1/2*log(-x + log(x)) + log_integral(x)

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Sympy [A]  time = 0.549804, size = 19, normalized size = 0.83 \begin{align*} - \frac{\log{\left (- x + \log{\left (x \right )} \right )}}{2} + \frac{\log{\left (x + \log{\left (x \right )} \right )}}{2} + \operatorname{li}{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x**2-ln(x)+2*ln(x)**2)/(-x**2*ln(x)+ln(x)**3),x)

[Out]

-log(-x + log(x))/2 + log(x + log(x))/2 + li(x)

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Giac [A]  time = 1.09277, size = 32, normalized size = 1.39 \begin{align*}{\rm Ei}\left (\log \left (x\right )\right ) - \frac{1}{2} \, \log \left (x - \log \left (x\right )\right ) + \frac{1}{2} \, \log \left (-x - \log \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2-log(x)+2*log(x)^2)/(-x^2*log(x)+log(x)^3),x, algorithm="giac")

[Out]

Ei(log(x)) - 1/2*log(x - log(x)) + 1/2*log(-x - log(x))