\(\int \frac {1}{x^2 (1+W(\frac {a}{x^2}))} \, dx\) [334]

Optimal result

Integrand size = 14, antiderivative size = 49 \[ \int \frac {1}{x^2 \left (1+W\left (\frac {a}{x^2}\right )\right )} \, dx=-\frac {e^{\frac {1}{2} W\left (\frac {a}{x^2}\right )} x \Gamma \left (\frac {1}{2},-\frac {1}{2} W\left (\frac {a}{x^2}\right )\right ) \sqrt {-W\left (\frac {a}{x^2}\right )}}{\sqrt {2} a} \] Output:

-1/2*exp(1/2*LambertW(a/x^2))*x*Pi^(1/2)*erfc(1/2*(-2*LambertW(a/x^2))^(1/ 
2))*(-LambertW(a/x^2))^(1/2)*2^(1/2)/a
                                                                                    
                                                                                    
 

Mathematica [F]

\[ \int \frac {1}{x^2 \left (1+W\left (\frac {a}{x^2}\right )\right )} \, dx=\int \frac {1}{x^2 \left (1+W\left (\frac {a}{x^2}\right )\right )} \, dx \] Input:

Integrate[1/(x^2*(1 + ProductLog[a/x^2])),x]
 

Output:

Integrate[1/(x^2*(1 + ProductLog[a/x^2])), x]
 

Rubi [F]

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.

\(\displaystyle \int \frac {1}{x^2 \left (W\left (\frac {a}{x^2}\right )+1\right )} \, dx\)

\(\Big \downarrow \) 7199

\(\displaystyle -\int \frac {1}{W\left (\frac {a}{x^2}\right )+1}d\frac {1}{x}\)

\(\Big \downarrow \) 7299

\(\displaystyle -\int \frac {1}{W\left (\frac {a}{x^2}\right )+1}d\frac {1}{x}\)

Input:

Int[1/(x^2*(1 + ProductLog[a/x^2])),x]
 

Output:

$Aborted
 

Maple [F]

\[\int \frac {1}{x^{2} \left (1+\operatorname {LambertW}\left (\frac {a}{x^{2}}\right )\right )}d x\]

Input:

int(1/x^2/(1+LambertW(a/x^2)),x)
 

Output:

int(1/x^2/(1+LambertW(a/x^2)),x)
 

Fricas [F]

\[ \int \frac {1}{x^2 \left (1+W\left (\frac {a}{x^2}\right )\right )} \, dx=\int { \frac {1}{x^{2} {\left (\operatorname {W}({\frac {a}{x^{2}}}) + 1\right )}} \,d x } \] Input:

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

Output:

integral(1/(x^2*lambert_w(a/x^2) + x^2), x)
 

Sympy [F]

\[ \int \frac {1}{x^2 \left (1+W\left (\frac {a}{x^2}\right )\right )} \, dx=\int \frac {1}{x^{2} \left (W\left (\frac {a}{x^{2}}\right ) + 1\right )}\, dx \] Input:

integrate(1/x**2/(1+LambertW(a/x**2)),x)
 

Output:

Integral(1/(x**2*(LambertW(a/x**2) + 1)), x)
 

Maxima [F]

\[ \int \frac {1}{x^2 \left (1+W\left (\frac {a}{x^2}\right )\right )} \, dx=\int { \frac {1}{x^{2} {\left (\operatorname {W}({\frac {a}{x^{2}}}) + 1\right )}} \,d x } \] Input:

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

Output:

integrate(1/(x^2*(lambert_w(a/x^2) + 1)), x)
 

Giac [F]

\[ \int \frac {1}{x^2 \left (1+W\left (\frac {a}{x^2}\right )\right )} \, dx=\int { \frac {1}{x^{2} {\left (\operatorname {W}({\frac {a}{x^{2}}}) + 1\right )}} \,d x } \] Input:

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

Output:

integrate(1/(x^2*(lambert_w(a/x^2) + 1)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^2 \left (1+W\left (\frac {a}{x^2}\right )\right )} \, dx=\int \frac {1}{x^2\,\left (\mathrm {LambertW}\left (\frac {a}{x^2}\right )+1\right )} \,d x \] Input:

int(1/(x^2*(LambertW(a/x^2) + 1)),x)
 

Output:

int(1/(x^2*(LambertW(a/x^2) + 1)), x)
 

Reduce [F]

\[ \int \frac {1}{x^2 \left (1+W\left (\frac {a}{x^2}\right )\right )} \, dx=\int \frac {1}{\textit {lambert\_w} \left (\frac {a}{x^{2}}\right ) x^{2}+x^{2}}d x \] Input:

int(1/x^2/(1+Lambert_W(a/x^2)),x)
 

Output:

int(1/(lambert_w(a/x**2)*x**2 + x**2),x)