3.61.62 \(\int \frac {1-2 x \log (x)+\log (x) \log (\log (x))}{\log (x)} \, dx\)

Optimal. Leaf size=11 \[ 3+x (-x+\log (\log (x))) \]

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Rubi [A]  time = 0.03, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6688, 2298, 2520} \begin {gather*} x \log (\log (x))-x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x*Log[x] + Log[x]*Log[Log[x]])/Log[x],x]

[Out]

-x^2 + x*Log[Log[x]]

Rule 2298

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

Rule 2520

Int[Log[Log[(d_.)*(x_)^(n_.)]^(p_.)*(c_.)], x_Symbol] :> Simp[x*Log[c*Log[d*x^n]^p], x] - Dist[n*p, Int[1/Log[
d*x^n], x], x] /; FreeQ[{c, d, n, p}, x]

Rule 6688

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

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 11, normalized size = 1.00 \begin {gather*} -x^2+x \log (\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x*Log[x] + Log[x]*Log[Log[x]])/Log[x],x]

[Out]

-x^2 + x*Log[Log[x]]

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fricas [A]  time = 1.18, size = 11, normalized size = 1.00 \begin {gather*} -x^{2} + x \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(x)*log(log(x))-2*x*log(x)+1)/log(x),x, algorithm="fricas")

[Out]

-x^2 + x*log(log(x))

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giac [A]  time = 0.22, size = 11, normalized size = 1.00 \begin {gather*} -x^{2} + x \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(x)*log(log(x))-2*x*log(x)+1)/log(x),x, algorithm="giac")

[Out]

-x^2 + x*log(log(x))

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maple [A]  time = 0.03, size = 12, normalized size = 1.09

method	result	size
default	\(-x^{2}+x \ln \left (\ln \relax (x )\right )\)	\(12\)
norman	\(-x^{2}+x \ln \left (\ln \relax (x )\right )\)	\(12\)
risch	\(-x^{2}+x \ln \left (\ln \relax (x )\right )\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((ln(x)*ln(ln(x))-2*x*ln(x)+1)/ln(x),x,method=_RETURNVERBOSE)

[Out]

-x^2+x*ln(ln(x))

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maxima [A]  time = 0.37, size = 11, normalized size = 1.00 \begin {gather*} -x^{2} + x \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(x)*log(log(x))-2*x*log(x)+1)/log(x),x, algorithm="maxima")

[Out]

-x^2 + x*log(log(x))

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mupad [B]  time = 4.30, size = 10, normalized size = 0.91 \begin {gather*} -x\,\left (x-\ln \left (\ln \relax (x)\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x*(x - log(log(x)))

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sympy [A]  time = 0.28, size = 8, normalized size = 0.73 \begin {gather*} - x^{2} + x \log {\left (\log {\relax (x )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x**2 + x*log(log(x))

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