Integrand size = 12, antiderivative size = 67 \[ \int \sqrt {\frac {\log \left (\frac {1}{x}\right )}{x}} \, dx=2 x \sqrt {\frac {\log \left (\frac {1}{x}\right )}{x}}-\frac {\sqrt {2 \pi } \text {erf}\left (\frac {\sqrt {\log \left (\frac {1}{x}\right )}}{\sqrt {2}}\right ) \sqrt {\frac {\log \left (\frac {1}{x}\right )}{x}}}{\sqrt {\frac {1}{x}} \sqrt {\log \left (\frac {1}{x}\right )}} \] Output:
2*x*(ln(1/x)/x)^(1/2)-2^(1/2)*Pi^(1/2)*erf(1/2*ln(1/x)^(1/2)*2^(1/2))*(ln( 1/x)/x)^(1/2)/(1/x)^(1/2)/ln(1/x)^(1/2)
Time = 0.04 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.84 \[ \int \sqrt {\frac {\log \left (\frac {1}{x}\right )}{x}} \, dx=\left (2 x-\frac {\sqrt {2 \pi } \text {erf}\left (\frac {\sqrt {\log \left (\frac {1}{x}\right )}}{\sqrt {2}}\right )}{\sqrt {\frac {1}{x}} \sqrt {\log \left (\frac {1}{x}\right )}}\right ) \sqrt {\frac {\log \left (\frac {1}{x}\right )}{x}} \] Input:
Integrate[Sqrt[Log[x^(-1)]/x],x]
Output:
(2*x - (Sqrt[2*Pi]*Erf[Sqrt[Log[x^(-1)]]/Sqrt[2]])/(Sqrt[x^(-1)]*Sqrt[Log[ x^(-1)]]))*Sqrt[Log[x^(-1)]/x]
Time = 0.34 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.16, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {7270, 2742, 2747, 2611, 2634}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \sqrt {\frac {\log \left (\frac {1}{x}\right )}{x}} \, dx\) |
\(\Big \downarrow \) 7270 |
\(\displaystyle \frac {\sqrt {x} \sqrt {\frac {\log \left (\frac {1}{x}\right )}{x}} \int \frac {\sqrt {\log \left (\frac {1}{x}\right )}}{\sqrt {x}}dx}{\sqrt {\log \left (\frac {1}{x}\right )}}\) |
\(\Big \downarrow \) 2742 |
\(\displaystyle \frac {\sqrt {x} \sqrt {\frac {\log \left (\frac {1}{x}\right )}{x}} \left (\int \frac {1}{\sqrt {x} \sqrt {\log \left (\frac {1}{x}\right )}}dx+2 \sqrt {x} \sqrt {\log \left (\frac {1}{x}\right )}\right )}{\sqrt {\log \left (\frac {1}{x}\right )}}\) |
\(\Big \downarrow \) 2747 |
\(\displaystyle \frac {\sqrt {x} \sqrt {\frac {\log \left (\frac {1}{x}\right )}{x}} \left (2 \sqrt {x} \sqrt {\log \left (\frac {1}{x}\right )}-\sqrt {\frac {1}{x}} \sqrt {x} \int \frac {1}{\sqrt {\frac {1}{x}} \sqrt {\log \left (\frac {1}{x}\right )}}d\log \left (\frac {1}{x}\right )\right )}{\sqrt {\log \left (\frac {1}{x}\right )}}\) |
\(\Big \downarrow \) 2611 |
\(\displaystyle \frac {\sqrt {x} \sqrt {\frac {\log \left (\frac {1}{x}\right )}{x}} \left (2 \sqrt {x} \sqrt {\log \left (\frac {1}{x}\right )}-2 \sqrt {\frac {1}{x}} \sqrt {x} \int \frac {1}{\sqrt {\frac {1}{x}}}d\sqrt {\log \left (\frac {1}{x}\right )}\right )}{\sqrt {\log \left (\frac {1}{x}\right )}}\) |
\(\Big \downarrow \) 2634 |
\(\displaystyle \frac {\sqrt {x} \sqrt {\frac {\log \left (\frac {1}{x}\right )}{x}} \left (2 \sqrt {x} \sqrt {\log \left (\frac {1}{x}\right )}-\sqrt {2 \pi } \sqrt {\frac {1}{x}} \sqrt {x} \text {erf}\left (\frac {\sqrt {\log \left (\frac {1}{x}\right )}}{\sqrt {2}}\right )\right )}{\sqrt {\log \left (\frac {1}{x}\right )}}\) |
Input:
Int[Sqrt[Log[x^(-1)]/x],x]
Output:
(Sqrt[x]*(-(Sqrt[2*Pi]*Sqrt[x^(-1)]*Sqrt[x]*Erf[Sqrt[Log[x^(-1)]]/Sqrt[2]] ) + 2*Sqrt[x]*Sqrt[Log[x^(-1)]])*Sqrt[Log[x^(-1)]/x])/Sqrt[Log[x^(-1)]]
Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] : > Simp[2/d Subst[Int[F^(g*(e - c*(f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d *x]], x] /; FreeQ[{F, c, d, e, f, g}, x] && !TrueQ[$UseGamma]
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt [Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F], 2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; Fr eeQ[{F, a, b, c, d}, x] && NegQ[b]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbo l] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^n])^p/(d*(m + 1))), x] - Simp[b*n* (p/(m + 1)) Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b , c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol ] :> Simp[(d*x)^(m + 1)/(d*n*(c*x^n)^((m + 1)/n)) Subst[Int[E^(((m + 1)/n )*x)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}, x]
Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Simp[a^IntPart[p ]*((a*v^m*w^n)^FracPart[p]/(v^(m*FracPart[p])*w^(n*FracPart[p]))) Int[u*v ^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] && !IntegerQ[p] && !Free Q[v, x] && !FreeQ[w, x]
\[\int \sqrt {\frac {\ln \left (\frac {1}{x}\right )}{x}}d x\]
Input:
int((ln(1/x)/x)^(1/2),x)
Output:
int((ln(1/x)/x)^(1/2),x)
Exception generated. \[ \int \sqrt {\frac {\log \left (\frac {1}{x}\right )}{x}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((log(1/x)/x)^(1/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \sqrt {\frac {\log \left (\frac {1}{x}\right )}{x}} \, dx=\int \sqrt {\frac {\log {\left (\frac {1}{x} \right )}}{x}}\, dx \] Input:
integrate((ln(1/x)/x)**(1/2),x)
Output:
Integral(sqrt(log(1/x)/x), x)
\[ \int \sqrt {\frac {\log \left (\frac {1}{x}\right )}{x}} \, dx=\int { \sqrt {\frac {\log \left (\frac {1}{x}\right )}{x}} \,d x } \] Input:
integrate((log(1/x)/x)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(-log(x)/x), x)
\[ \int \sqrt {\frac {\log \left (\frac {1}{x}\right )}{x}} \, dx=\int { \sqrt {\frac {\log \left (\frac {1}{x}\right )}{x}} \,d x } \] Input:
integrate((log(1/x)/x)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(log(1/x)/x), x)
Timed out. \[ \int \sqrt {\frac {\log \left (\frac {1}{x}\right )}{x}} \, dx=\int \sqrt {\frac {\ln \left (\frac {1}{x}\right )}{x}} \,d x \] Input:
int((log(1/x)/x)^(1/2),x)
Output:
int((log(1/x)/x)^(1/2), x)
\[ \int \sqrt {\frac {\log \left (\frac {1}{x}\right )}{x}} \, dx=i \left (2 \sqrt {x}\, \sqrt {\mathrm {log}\left (x \right )}-\left (\int \frac {\sqrt {x}\, \sqrt {\mathrm {log}\left (x \right )}}{\mathrm {log}\left (x \right ) x}d x \right )\right ) \] Input:
int((log(1/x)/x)^(1/2),x)
Output:
i*(2*sqrt(x)*sqrt(log(x)) - int((sqrt(x)*sqrt(log(x)))/(log(x)*x),x))