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

Optimal. Leaf size=10 \[ 23+4 x-\log (\log (x)) \]

________________________________________________________________________________________

Rubi [A]  time = 0.08, antiderivative size = 9, normalized size of antiderivative = 0.90, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6688, 2302, 29} \begin {gather*} 4 x-\log (\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + 4*x*Log[x])/(x*Log[x]),x]

[Out]

4*x - Log[Log[x]]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], 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 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}{x \log (x)}\right ) \, dx\\ &=4 x-\int \frac {1}{x \log (x)} \, dx\\ &=4 x-\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log (x)\right )\\ &=4 x-\log (\log (x))\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 9, normalized size = 0.90 \begin {gather*} 4 x-\log (\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

4*x - Log[Log[x]]

________________________________________________________________________________________

fricas [A]  time = 0.75, size = 9, normalized size = 0.90 \begin {gather*} 4 \, x - \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*x - log(log(x))

________________________________________________________________________________________

giac [A]  time = 0.23, size = 9, normalized size = 0.90 \begin {gather*} 4 \, x - \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*x - log(log(x))

________________________________________________________________________________________

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*x-ln(ln(x))

________________________________________________________________________________________

maxima [A]  time = 0.35, size = 9, normalized size = 0.90 \begin {gather*} 4 \, x - \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*x - log(log(x))

________________________________________________________________________________________

mupad [B]  time = 5.23, size = 9, normalized size = 0.90 \begin {gather*} 4\,x-\ln \left (\ln \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x*log(x) - 1)/(x*log(x)),x)

[Out]

4*x - log(log(x))

________________________________________________________________________________________

sympy [A]  time = 0.09, size = 7, normalized size = 0.70 \begin {gather*} 4 x - \log {\left (\log {\relax (x )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*x - log(log(x))

________________________________________________________________________________________