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
\[ \int \frac {1}{x^2 \left (1+W\left (a x^2\right )\right )} \, dx=\int \frac {1}{x^2 \left (1+W\left (a x^2\right )\right )} \, dx \] Input:
Integrate[1/(x^2*(1 + ProductLog[a*x^2])),x]
Output:
Integrate[1/(x^2*(1 + ProductLog[a*x^2])), x]
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
\[\int \frac {1}{x^{2} \left (1+\operatorname {LambertW}\left (a \,x^{2}\right )\right )}d x\]
Input:
int(1/x^2/(1+LambertW(a*x^2)),x)
Output:
int(1/x^2/(1+LambertW(a*x^2)),x)
\[ \int \frac {1}{x^2 \left (1+W\left (a x^2\right )\right )} \, dx=\int { \frac {1}{x^{2} {\left (\operatorname {W}({a x^{2}}) + 1\right )}} \,d x } \] 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)
\[ \int \frac {1}{x^2 \left (1+W\left (a x^2\right )\right )} \, dx=\int \frac {1}{x^{2} \left (W\left (a x^{2}\right ) + 1\right )}\, dx \] Input:
integrate(1/x**2/(1+LambertW(a*x**2)),x)
Output:
Integral(1/(x**2*(LambertW(a*x**2) + 1)), x)
\[ \int \frac {1}{x^2 \left (1+W\left (a x^2\right )\right )} \, dx=\int { \frac {1}{x^{2} {\left (\operatorname {W}({a x^{2}}) + 1\right )}} \,d x } \] 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)
\[ \int \frac {1}{x^2 \left (1+W\left (a x^2\right )\right )} \, dx=\int { \frac {1}{x^{2} {\left (\operatorname {W}({a x^{2}}) + 1\right )}} \,d x } \] 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)
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)
\[ \int \frac {1}{x^2 \left (1+W\left (a x^2\right )\right )} \, dx=\frac {-\left (\int \frac {\textit {lambert\_w} \left (a \,x^{2}\right )}{\textit {lambert\_w} \left (a \,x^{2}\right ) x^{2}+x^{2}}d x \right ) x +3 \left (\int \frac {1}{\textit {lambert\_w} \left (a \,x^{2}\right ) x^{2}+x^{2}}d x \right ) x -1}{4 x} \] 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)