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

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

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 23, normalized size of antiderivative = 1.00, 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 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+\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*}

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Mathematica [A]  time = 0.02, size = 16, normalized size = 0.70 \[ \frac {2}{3} (\log (x)-2) \sqrt {\log (x)+1} \]

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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fricas [A]  time = 0.40, size = 12, normalized size = 0.52 \[ \frac {2}{3} \, \sqrt {\log \relax (x) + 1} {\left (\log \relax (x) - 2\right )} \]

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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giac [A]  time = 0.01, size = 17, normalized size = 0.74 \[ \frac {2}{3} \, {\left (\log \relax (x) + 1\right )}^{\frac {3}{2}} - 2 \, \sqrt {\log \relax (x) + 1} \]

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)

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maple [A]  time = 0.01, size = 18, normalized size = 0.78 \[ \frac {2 \left (\ln \relax (x )+1\right )^{\frac {3}{2}}}{3}-2 \sqrt {\ln \relax (x )+1} \]

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 = 0.60, size = 17, normalized size = 0.74 \[ \frac {2}{3} \, {\left (\log \relax (x) + 1\right )}^{\frac {3}{2}} - 2 \, \sqrt {\log \relax (x) + 1} \]

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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mupad [B]  time = 0.17, size = 13, normalized size = 0.57 \[ \sqrt {\ln \relax (x)+1}\,\left (\frac {2\,\ln \relax (x)}{3}-\frac {4}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(x)/(x*(log(x) + 1)^(1/2)),x)

[Out]

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

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sympy [A]  time = 5.07, size = 20, normalized size = 0.87 \[ \frac {2 \left (\log {\relax (x )} + 1\right )^{\frac {3}{2}}}{3} - 2 \sqrt {\log {\relax (x )} + 1} \]

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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