3.145 \(\int \frac{\log (x)}{x \sqrt{1+\log (x)}} \, dx\)

Optimal. Leaf size=23 \[ \frac{2}{3} (\log (x)+1)^{3/2}-2 \sqrt{\log (x)+1} \]

[Out]

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

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Rubi [A]  time = 0.043482, antiderivative size = 23, normalized size of antiderivative = 1., 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} \[ \frac{2}{3} (\log (x)+1)^{3/2}-2 \sqrt{\log (x)+1} \]

Antiderivative was successfully verified.

[In]

Int[Log[x]/(x*Sqrt[1 + Log[x]]),x]

[Out]

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

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

Mathematica [A]  time = 0.0164214, size = 16, normalized size = 0.7 \[ \frac{2}{3} (\log (x)-2) \sqrt{\log (x)+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-2 + Log[x])*Sqrt[1 + Log[x]])/3

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Maple [A]  time = 0.01, size = 18, normalized size = 0.8 \begin{align*}{\frac{2}{3} \left ( 1+\ln \left ( x \right ) \right ) ^{{\frac{3}{2}}}}-2\,\sqrt{1+\ln \left ( x \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(x)/x/(1+ln(x))^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.01181, size = 23, normalized size = 1. \begin{align*} \frac{2}{3} \,{\left (\log \left (x\right ) + 1\right )}^{\frac{3}{2}} - 2 \, \sqrt{\log \left (x\right ) + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.99506, size = 47, normalized size = 2.04 \begin{align*} \frac{2}{3} \, \sqrt{\log \left (x\right ) + 1}{\left (\log \left (x\right ) - 2\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*sqrt(log(x) + 1)*(log(x) - 2)

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Sympy [A]  time = 4.04351, size = 20, normalized size = 0.87 \begin{align*} \frac{2 \left (\log{\left (x \right )} + 1\right )^{\frac{3}{2}}}{3} - 2 \sqrt{\log{\left (x \right )} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.27333, size = 23, normalized size = 1. \begin{align*} \frac{2}{3} \,{\left (\log \left (x\right ) + 1\right )}^{\frac{3}{2}} - 2 \, \sqrt{\log \left (x\right ) + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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