### 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]

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

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Rubi [A]  time = 0.0505343, antiderivative size = 29, normalized size of antiderivative = 1., 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 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]

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])

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*}

Mathematica [A]  time = 0.021203, 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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Maple [A]  time = 0.01, size = 22, normalized size = 0.8 \begin{align*}{\frac{1}{24} \left ( -1+4\,\ln \left ( x \right ) \right ) ^{{\frac{3}{2}}}}+{\frac{1}{8}\sqrt{-1+4\,\ln \left ( x \right ) }} \end{align*}

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 = 1.01078, size = 28, normalized size = 0.97 \begin{align*} \frac{1}{24} \,{\left (4 \, \log \left (x\right ) - 1\right )}^{\frac{3}{2}} + \frac{1}{8} \, \sqrt{4 \, \log \left (x\right ) - 1} \end{align*}

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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Fricas [A]  time = 2.01502, size = 54, normalized size = 1.86 \begin{align*} \frac{1}{12} \, \sqrt{4 \, \log \left (x\right ) - 1}{\left (2 \, \log \left (x\right ) + 1\right )} \end{align*}

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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Sympy [A]  time = 4.35269, size = 22, normalized size = 0.76 \begin{align*} \frac{\left (4 \log{\left (x \right )} - 1\right )^{\frac{3}{2}}}{24} + \frac{\sqrt{4 \log{\left (x \right )} - 1}}{8} \end{align*}

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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Giac [A]  time = 1.19953, size = 28, normalized size = 0.97 \begin{align*} \frac{1}{24} \,{\left (4 \, \log \left (x\right ) - 1\right )}^{\frac{3}{2}} + \frac{1}{8} \, \sqrt{4 \, \log \left (x\right ) - 1} \end{align*}

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)