3.83.18 \(\int \frac {1-4 \log (x)+\log (x) \log (\log (x))}{\log (x)} \, dx\)

Optimal. Leaf size=10 \[ 11+x+x (-5+\log (\log (x))) \]

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Rubi [A]  time = 0.03, antiderivative size = 9, normalized size of antiderivative = 0.90, number of steps used = 5, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {6688, 2298, 2520} \begin {gather*} x \log (\log (x))-4 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - 4*Log[x] + Log[x]*Log[Log[x]])/Log[x],x]

[Out]

-4*x + 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 (-4+\frac {1}{\log (x)}+\log (\log (x))\right ) \, dx\\ &=-4 x+\int \frac {1}{\log (x)} \, dx+\int \log (\log (x)) \, dx\\ &=-4 x+x \log (\log (x))+\text {li}(x)-\int \frac {1}{\log (x)} \, dx\\ &=-4 x+x \log (\log (x))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 9, normalized size = 0.90 \begin {gather*} -4 x+x \log (\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - 4*Log[x] + Log[x]*Log[Log[x]])/Log[x],x]

[Out]

-4*x + x*Log[Log[x]]

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fricas [A]  time = 0.68, size = 9, normalized size = 0.90 \begin {gather*} x \log \left (\log \relax (x)\right ) - 4 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(x)*log(log(x))-4*log(x)+1)/log(x),x, algorithm="fricas")

[Out]

x*log(log(x)) - 4*x

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giac [A]  time = 0.21, size = 9, normalized size = 0.90 \begin {gather*} x \log \left (\log \relax (x)\right ) - 4 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(x)*log(log(x))-4*log(x)+1)/log(x),x, algorithm="giac")

[Out]

x*log(log(x)) - 4*x

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maple [A]  time = 0.02, size = 10, normalized size = 1.00

method	result	size
default	\(-4 x +x \ln \left (\ln \relax (x )\right )\)	\(10\)
norman	\(-4 x +x \ln \left (\ln \relax (x )\right )\)	\(10\)
risch	\(-4 x +x \ln \left (\ln \relax (x )\right )\)	\(10\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((ln(x)*ln(ln(x))-4*ln(x)+1)/ln(x),x,method=_RETURNVERBOSE)

[Out]

-4*x+x*ln(ln(x))

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maxima [A]  time = 0.39, size = 9, normalized size = 0.90 \begin {gather*} x \log \left (\log \relax (x)\right ) - 4 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(x)*log(log(x))-4*log(x)+1)/log(x),x, algorithm="maxima")

[Out]

x*log(log(x)) - 4*x

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mupad [B]  time = 5.06, size = 7, normalized size = 0.70 \begin {gather*} x\,\left (\ln \left (\ln \relax (x)\right )-4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(log(x))*log(x) - 4*log(x) + 1)/log(x),x)

[Out]

x*(log(log(x)) - 4)

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sympy [A]  time = 0.28, size = 8, normalized size = 0.80 \begin {gather*} x \log {\left (\log {\relax (x )} \right )} - 4 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((ln(x)*ln(ln(x))-4*ln(x)+1)/ln(x),x)

[Out]

x*log(log(x)) - 4*x

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