\(\int \frac {1}{x^4 (1+W(\frac {a}{x^2}))} \, dx\) [335]

Optimal result

Integrand size = 14, antiderivative size = 77 \[ \int \frac {1}{x^4 \left (1+W\left (\frac {a}{x^2}\right )\right )} \, dx=-\frac {1}{3 x^3 W\left (\frac {a}{x^2}\right )}+\frac {e^{-\frac {1}{2} W\left (\frac {a}{x^2}\right )} \Gamma \left (\frac {1}{2},-\frac {3}{2} W\left (\frac {a}{x^2}\right )\right ) \sqrt {-W\left (\frac {a}{x^2}\right )}}{3 \sqrt {6} a x W\left (\frac {a}{x^2}\right )} \] Output:

-1/3/x^3/LambertW(a/x^2)+1/18*Pi^(1/2)*erfc(1/2*(-6*LambertW(a/x^2))^(1/2) 
)*(-LambertW(a/x^2))^(1/2)*6^(1/2)/a/exp(1/2*LambertW(a/x^2))/x/LambertW(a 
/x^2)
                                                                                    
                                                                                    
 

Mathematica [F]

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

Integrate[1/(x^4*(1 + ProductLog[a/x^2])),x]
 

Output:

Integrate[1/(x^4*(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.

\(\displaystyle \int \frac {1}{x^4 \left (W\left (\frac {a}{x^2}\right )+1\right )} \, dx\)

\(\Big \downarrow \) 7199

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

\(\Big \downarrow \) 7299

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

Input:

Int[1/(x^4*(1 + ProductLog[a/x^2])),x]
 

Output:

$Aborted
 

Maple [F]

\[\int \frac {1}{x^{4} \left (1+\operatorname {LambertW}\left (\frac {a}{x^{2}}\right )\right )}d x\]

Input:

int(1/x^4/(1+LambertW(a/x^2)),x)
 

Output:

int(1/x^4/(1+LambertW(a/x^2)),x)
 

Fricas [F]

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

integrate(1/x^4/(1+lambert_w(a/x^2)),x, algorithm="fricas")
                                                                                    
                                                                                    
 

Output:

integral(1/(x^4*lambert_w(a/x^2) + x^4), x)
 

Sympy [F]

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

integrate(1/x**4/(1+LambertW(a/x**2)),x)
 

Output:

Integral(1/(x**4*(LambertW(a/x**2) + 1)), x)
 

Maxima [F]

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

integrate(1/x^4/(1+lambert_w(a/x^2)),x, algorithm="maxima")
 

Output:

integrate(1/(x^4*(lambert_w(a/x^2) + 1)), x)
 

Giac [F]

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

integrate(1/x^4/(1+lambert_w(a/x^2)),x, algorithm="giac")
 

Output:

integrate(1/(x^4*(lambert_w(a/x^2) + 1)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^4 \left (1+W\left (\frac {a}{x^2}\right )\right )} \, dx=\int \frac {1}{x^4\,\left (\mathrm {LambertW}\left (\frac {a}{x^2}\right )+1\right )} \,d x \] Input:

int(1/(x^4*(LambertW(a/x^2) + 1)),x)
 

Output:

int(1/(x^4*(LambertW(a/x^2) + 1)), x)
 

Reduce [F]

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

int(1/x^4/(1+Lambert_W(a/x^2)),x)
 

Output:

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