\(\int \frac {1}{x^2 (1+W(a x^2))} \, dx\) [315]

Optimal result

Integrand size = 14, antiderivative size = 54 \[ \int \frac {1}{x^2 \left (1+W\left (a x^2\right )\right )} \, dx=-\frac {1}{x}+\frac {e^{\frac {3}{2} W\left (a x^2\right )} \Gamma \left (\frac {1}{2},\frac {1}{2} W\left (a x^2\right )\right ) W\left (a x^2\right )^{3/2}}{\sqrt {2} a x^3} \] Output:

-1/x+1/2*2^(1/2)*exp(3/2*LambertW(a*x^2))*Pi^(1/2)*erfc(1/2*2^(1/2)*Lamber 
tW(a*x^2)^(1/2))*LambertW(a*x^2)^(3/2)/a/x^3
 

Mathematica [F]

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

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

\(\Big \downarrow \) 7299

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

Input:

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

Output:

$Aborted
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

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

Output:

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

Maxima [F]

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

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

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

Output:

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

Reduce [F]

\[ \int \frac {1}{x^2 \left (1+W\left (a x^2\right )\right )} \, dx=\frac {-\left (\int \frac {\textit {lambert\_w} \left (a \,x^{2}\right )}{\textit {lambert\_w} \left (a \,x^{2}\right ) x^{2}+x^{2}}d x \right ) x +3 \left (\int \frac {1}{\textit {lambert\_w} \left (a \,x^{2}\right ) x^{2}+x^{2}}d x \right ) x -1}{4 x} \] Input:

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

Output:

( - int(lambert_w(a*x**2)/(lambert_w(a*x**2)*x**2 + x**2),x)*x + 3*int(1/( 
lambert_w(a*x**2)*x**2 + x**2),x)*x - 1)/(4*x)