\(\int \frac {\log (x)}{x \sqrt {1+\log (x)}} \, dx\) [66]

Optimal result

Integrand size = 14, antiderivative size = 23 \[ \int \frac {\log (x)}{x \sqrt {1+\log (x)}} \, dx=-2 \sqrt {1+\log (x)}+\frac {2}{3} (1+\log (x))^{3/2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2412, 45} \[ \int \frac {\log (x)}{x \sqrt {1+\log (x)}} \, dx=\frac {2}{3} (\log (x)+1)^{3/2}-2 \sqrt {\log (x)+1} \]

[In]

Int[Log[x]/(x*Sqrt[1 + Log[x]]),x]

[Out]

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

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+x}} \, dx,x,\log (x)\right ) \\ & = \text {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] (verified)

Time = 0.02 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.70 \[ \int \frac {\log (x)}{x \sqrt {1+\log (x)}} \, dx=\frac {2}{3} (-2+\log (x)) \sqrt {1+\log (x)} \]

[In]

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

[Out]

(2*(-2 + Log[x])*Sqrt[1 + Log[x]])/3

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.78

method	result	size
derivativedivides	\(\frac {2 \left (1+\ln \left (x \right )\right )^{\frac {3}{2}}}{3}-2 \sqrt {1+\ln \left (x \right )}\)	\(18\)
default	\(\frac {2 \left (1+\ln \left (x \right )\right )^{\frac {3}{2}}}{3}-2 \sqrt {1+\ln \left (x \right )}\)	\(18\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.52 \[ \int \frac {\log (x)}{x \sqrt {1+\log (x)}} \, dx=\frac {2}{3} \, \sqrt {\log \left (x\right ) + 1} {\left (\log \left (x\right ) - 2\right )} \]

[In]

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

[Out]

2/3*sqrt(log(x) + 1)*(log(x) - 2)

Sympy [A] (verification not implemented)

Time = 1.15 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {\log (x)}{x \sqrt {1+\log (x)}} \, dx=\frac {2 \left (\log {\left (x \right )} + 1\right )^{\frac {3}{2}}}{3} - 2 \sqrt {\log {\left (x \right )} + 1} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int \frac {\log (x)}{x \sqrt {1+\log (x)}} \, dx=\frac {2}{3} \, {\left (\log \left (x\right ) + 1\right )}^{\frac {3}{2}} - 2 \, \sqrt {\log \left (x\right ) + 1} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int \frac {\log (x)}{x \sqrt {1+\log (x)}} \, dx=\frac {2}{3} \, {\left (\log \left (x\right ) + 1\right )}^{\frac {3}{2}} - 2 \, \sqrt {\log \left (x\right ) + 1} \]

[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)

Mupad [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.57 \[ \int \frac {\log (x)}{x \sqrt {1+\log (x)}} \, dx=\sqrt {\ln \left (x\right )+1}\,\left (\frac {2\,\ln \left (x\right )}{3}-\frac {4}{3}\right ) \]

[In]

int(log(x)/(x*(log(x) + 1)^(1/2)),x)

[Out]

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