\(\int \frac {\log (x)}{x \sqrt {-1+4 \log (x)}} \, dx\) [146]

Optimal result

Integrand size = 16, antiderivative size = 29 \[ \int \frac {\log (x)}{x \sqrt {-1+4 \log (x)}} \, dx=\frac {1}{8} \sqrt {-1+4 \log (x)}+\frac {1}{24} (-1+4 \log (x))^{3/2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2412, 45} \[ \int \frac {\log (x)}{x \sqrt {-1+4 \log (x)}} \, dx=\frac {1}{24} (4 \log (x)-1)^{3/2}+\frac {1}{8} \sqrt {4 \log (x)-1} \]

[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 45

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 2412

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*} \text {integral}& = \text {Subst}\left (\int \frac {x}{\sqrt {-1+4 x}} \, dx,x,\log (x)\right ) \\ & = \text {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*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.69 \[ \int \frac {\log (x)}{x \sqrt {-1+4 \log (x)}} \, dx=\frac {1}{12} (1+2 \log (x)) \sqrt {-1+4 \log (x)} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.76

method	result	size
derivativedivides	\(\frac {\left (-1+4 \ln \left (x \right )\right )^{\frac {3}{2}}}{24}+\frac {\sqrt {-1+4 \ln \left (x \right )}}{8}\)	\(22\)
default	\(\frac {\left (-1+4 \ln \left (x \right )\right )^{\frac {3}{2}}}{24}+\frac {\sqrt {-1+4 \ln \left (x \right )}}{8}\)	\(22\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.55 \[ \int \frac {\log (x)}{x \sqrt {-1+4 \log (x)}} \, dx=\frac {1}{12} \, \sqrt {4 \, \log \left (x\right ) - 1} {\left (2 \, \log \left (x\right ) + 1\right )} \]

[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)

Sympy [A] (verification not implemented)

Time = 1.11 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.76 \[ \int \frac {\log (x)}{x \sqrt {-1+4 \log (x)}} \, dx=\frac {\left (4 \log {\left (x \right )} - 1\right )^{\frac {3}{2}}}{24} + \frac {\sqrt {4 \log {\left (x \right )} - 1}}{8} \]

[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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.72 \[ \int \frac {\log (x)}{x \sqrt {-1+4 \log (x)}} \, dx=\frac {1}{24} \, {\left (4 \, \log \left (x\right ) - 1\right )}^{\frac {3}{2}} + \frac {1}{8} \, \sqrt {4 \, \log \left (x\right ) - 1} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.72 \[ \int \frac {\log (x)}{x \sqrt {-1+4 \log (x)}} \, dx=\frac {1}{24} \, {\left (4 \, \log \left (x\right ) - 1\right )}^{\frac {3}{2}} + \frac {1}{8} \, \sqrt {4 \, \log \left (x\right ) - 1} \]

[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)

Mupad [B] (verification not implemented)

Time = 1.60 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.52 \[ \int \frac {\log (x)}{x \sqrt {-1+4 \log (x)}} \, dx=\sqrt {4\,\ln \left (x\right )-1}\,\left (\frac {\ln \left (x\right )}{6}+\frac {1}{12}\right ) \]

[In]

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

[Out]

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