3.90.96 \(\int \frac {2+4 \log (x)}{x \log (x)} \, dx\)

Optimal. Leaf size=12 \[ \log \left (\frac {390625 x^4 \log ^2(x)}{276922881}\right ) \]

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Rubi [A]  time = 0.03, antiderivative size = 10, normalized size of antiderivative = 0.83, number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2365, 43} \begin {gather*} 4 \log (x)+2 \log (\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + 4*Log[x])/(x*Log[x]),x]

[Out]

4*Log[x] + 2*Log[Log[x]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2365

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\operatorname {Subst}\left (\int \frac {2+4 x}{x} \, dx,x,\log (x)\right )\\ &=\operatorname {Subst}\left (\int \left (4+\frac {2}{x}\right ) \, dx,x,\log (x)\right )\\ &=4 \log (x)+2 \log (\log (x))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 10, normalized size = 0.83 \begin {gather*} 4 \log (x)+2 \log (\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + 4*Log[x])/(x*Log[x]),x]

[Out]

4*Log[x] + 2*Log[Log[x]]

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fricas [A]  time = 0.45, size = 10, normalized size = 0.83 \begin {gather*} 4 \, \log \relax (x) + 2 \, \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*log(x)+2)/x/log(x),x, algorithm="fricas")

[Out]

4*log(x) + 2*log(log(x))

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giac [A]  time = 0.13, size = 11, normalized size = 0.92 \begin {gather*} 4 \, \log \relax (x) + 2 \, \log \left ({\left | \log \relax (x) \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*log(x)+2)/x/log(x),x, algorithm="giac")

[Out]

4*log(x) + 2*log(abs(log(x)))

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maple [A]  time = 0.03, size = 11, normalized size = 0.92

method	result	size
derivativedivides	\(4 \ln \relax (x )+2 \ln \left (\ln \relax (x )\right )\)	\(11\)
default	\(4 \ln \relax (x )+2 \ln \left (\ln \relax (x )\right )\)	\(11\)
norman	\(4 \ln \relax (x )+2 \ln \left (\ln \relax (x )\right )\)	\(11\)
risch	\(4 \ln \relax (x )+2 \ln \left (\ln \relax (x )\right )\)	\(11\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*ln(x)+2)/x/ln(x),x,method=_RETURNVERBOSE)

[Out]

4*ln(x)+2*ln(ln(x))

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maxima [A]  time = 0.35, size = 10, normalized size = 0.83 \begin {gather*} 4 \, \log \relax (x) + 2 \, \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*log(x)+2)/x/log(x),x, algorithm="maxima")

[Out]

4*log(x) + 2*log(log(x))

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mupad [B]  time = 6.87, size = 10, normalized size = 0.83 \begin {gather*} 2\,\ln \left (\ln \relax (x)\right )+4\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*log(x) + 2)/(x*log(x)),x)

[Out]

2*log(log(x)) + 4*log(x)

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sympy [A]  time = 0.12, size = 10, normalized size = 0.83 \begin {gather*} 4 \log {\relax (x )} + 2 \log {\left (\log {\relax (x )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*ln(x)+2)/x/ln(x),x)

[Out]

4*log(x) + 2*log(log(x))

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