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

Optimal. Leaf size=29 \[ \frac {1}{24} (4 \log (x)-1)^{3/2}+\frac {1}{8} \sqrt {4 \log (x)-1} \]

[Out]

1/24*(-1+4*ln(x))^(3/2)+1/8*(-1+4*ln(x))^(1/2)

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Rubi [A]  time = 0.05, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2365, 43} \[ \frac {1}{24} (4 \log (x)-1)^{3/2}+\frac {1}{8} \sqrt {4 \log (x)-1} \]

Antiderivative was successfully verified.

[In]

Int[Log[x]/(x*Sqrt[-1 + 4*Log[x]]),x]

[Out]

Sqrt[-1 + 4*Log[x]]/8 + (-1 + 4*Log[x])^(3/2)/24

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 {align*} \int \frac {\log (x)}{x \sqrt {-1+4 \log (x)}} \, dx &=\operatorname {Subst}\left (\int \frac {x}{\sqrt {-1+4 x}} \, dx,x,\log (x)\right )\\ &=\operatorname {Subst}\left (\int \left (\frac {1}{4 \sqrt {-1+4 x}}+\frac {1}{4} \sqrt {-1+4 x}\right ) \, dx,x,\log (x)\right )\\ &=\frac {1}{8} \sqrt {-1+4 \log (x)}+\frac {1}{24} (-1+4 \log (x))^{3/2}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 20, normalized size = 0.69 \[ \frac {1}{12} (2 \log (x)+1) \sqrt {4 \log (x)-1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((1 + 2*Log[x])*Sqrt[-1 + 4*Log[x]])/12

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fricas [A]  time = 1.64, size = 16, normalized size = 0.55 \[ \frac {1}{12} \, \sqrt {4 \, \log \relax (x) - 1} {\left (2 \, \log \relax (x) + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(x)/x/(-1+4*log(x))^(1/2),x, algorithm="fricas")

[Out]

1/12*sqrt(4*log(x) - 1)*(2*log(x) + 1)

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giac [A]  time = 0.15, size = 21, normalized size = 0.72 \[ \frac {1}{24} \, {\left (4 \, \log \relax (x) - 1\right )}^{\frac {3}{2}} + \frac {1}{8} \, \sqrt {4 \, \log \relax (x) - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(x)/x/(-1+4*log(x))^(1/2),x, algorithm="giac")

[Out]

1/24*(4*log(x) - 1)^(3/2) + 1/8*sqrt(4*log(x) - 1)

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maple [A]  time = 0.07, size = 22, normalized size = 0.76 \[ \frac {\left (4 \ln \relax (x )-1\right )^{\frac {3}{2}}}{24}+\frac {\sqrt {4 \ln \relax (x )-1}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(x)/x/(-1+4*ln(x))^(1/2),x)

[Out]

1/24*(-1+4*ln(x))^(3/2)+1/8*(-1+4*ln(x))^(1/2)

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maxima [A]  time = 0.75, size = 21, normalized size = 0.72 \[ \frac {1}{24} \, {\left (4 \, \log \relax (x) - 1\right )}^{\frac {3}{2}} + \frac {1}{8} \, \sqrt {4 \, \log \relax (x) - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(x)/x/(-1+4*log(x))^(1/2),x, algorithm="maxima")

[Out]

1/24*(4*log(x) - 1)^(3/2) + 1/8*sqrt(4*log(x) - 1)

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mupad [B]  time = 0.44, size = 15, normalized size = 0.52 \[ \sqrt {4\,\ln \relax (x)-1}\,\left (\frac {\ln \relax (x)}{6}+\frac {1}{12}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(x)/(x*(4*log(x) - 1)^(1/2)),x)

[Out]

(4*log(x) - 1)^(1/2)*(log(x)/6 + 1/12)

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sympy [A]  time = 5.43, size = 22, normalized size = 0.76 \[ \frac {\left (4 \log {\relax (x )} - 1\right )^{\frac {3}{2}}}{24} + \frac {\sqrt {4 \log {\relax (x )} - 1}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(x)/x/(-1+4*ln(x))**(1/2),x)

[Out]

(4*log(x) - 1)**(3/2)/24 + sqrt(4*log(x) - 1)/8

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