Integrand size = 12, antiderivative size = 43 \[ \int \frac {W\left (\frac {a}{x}\right )^2}{x^2} \, dx=-\frac {4}{x}+\frac {4}{x W\left (\frac {a}{x}\right )}+\frac {2 W\left (\frac {a}{x}\right )}{x}-\frac {W\left (\frac {a}{x}\right )^2}{x} \] Output:
-4/x+4/x/LambertW(a/x)+2*LambertW(a/x)/x-LambertW(a/x)^2/x
Time = 0.01 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.95 \[ \int \frac {W\left (\frac {a}{x}\right )^2}{x^2} \, dx=-\frac {-4+4 W\left (\frac {a}{x}\right )-2 W\left (\frac {a}{x}\right )^2+W\left (\frac {a}{x}\right )^3}{x W\left (\frac {a}{x}\right )} \] Input:
Integrate[ProductLog[a/x]^2/x^2,x]
Output:
-((-4 + 4*ProductLog[a/x] - 2*ProductLog[a/x]^2 + ProductLog[a/x]^3)/(x*Pr oductLog[a/x]))
Time = 0.53 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.12, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {7172, 7205, 7205, 7199, 7176}
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.
\(\displaystyle \int \frac {W\left (\frac {a}{x}\right )^2}{x^2} \, dx\) |
\(\Big \downarrow \) 7172 |
\(\displaystyle -2 \int \frac {W\left (\frac {a}{x}\right )^2}{x^2 \left (W\left (\frac {a}{x}\right )+1\right )}dx-\frac {W\left (\frac {a}{x}\right )^2}{x}\) |
\(\Big \downarrow \) 7205 |
\(\displaystyle -2 \left (-2 \int \frac {W\left (\frac {a}{x}\right )}{x^2 \left (W\left (\frac {a}{x}\right )+1\right )}dx-\frac {W\left (\frac {a}{x}\right )}{x}\right )-\frac {W\left (\frac {a}{x}\right )^2}{x}\) |
\(\Big \downarrow \) 7205 |
\(\displaystyle -2 \left (-2 \left (-\int \frac {1}{x^2 \left (W\left (\frac {a}{x}\right )+1\right )}dx-\frac {1}{x}\right )-\frac {W\left (\frac {a}{x}\right )}{x}\right )-\frac {W\left (\frac {a}{x}\right )^2}{x}\) |
\(\Big \downarrow \) 7199 |
\(\displaystyle -2 \left (-2 \left (\int \frac {1}{W\left (\frac {a}{x}\right )+1}d\frac {1}{x}-\frac {1}{x}\right )-\frac {W\left (\frac {a}{x}\right )}{x}\right )-\frac {W\left (\frac {a}{x}\right )^2}{x}\) |
\(\Big \downarrow \) 7176 |
\(\displaystyle -\frac {W\left (\frac {a}{x}\right )^2}{x}-2 \left (-2 \left (\frac {1}{x W\left (\frac {a}{x}\right )}-\frac {1}{x}\right )-\frac {W\left (\frac {a}{x}\right )}{x}\right )\) |
Input:
Int[ProductLog[a/x]^2/x^2,x]
Output:
-(ProductLog[a/x]^2/x) - 2*(-2*(-x^(-1) + 1/(x*ProductLog[a/x])) - Product Log[a/x]/x)
Int[(x_)^(m_.)*((c_.)*ProductLog[(a_.)*(x_)^(n_.)])^(p_.), x_Symbol] :> Sim p[x^(m + 1)*((c*ProductLog[a*x^n])^p/(m + 1)), x] - Simp[n*(p/(m + 1)) In t[x^m*((c*ProductLog[a*x^n])^p/(1 + ProductLog[a*x^n])), x], x] /; FreeQ[{a , c, m, n, p}, x] && NeQ[m, -1] && ((IntegerQ[p - 1/2] && IGtQ[2*Simplify[p + (m + 1)/n], 0]) || ( !IntegerQ[p - 1/2] && IGtQ[Simplify[p + (m + 1)/n] + 1, 0]))
Int[((d_) + (d_.)*ProductLog[(a_.) + (b_.)*(x_)])^(-1), x_Symbol] :> Simp[( a + b*x)/(b*d*ProductLog[a + b*x]), x] /; FreeQ[{a, b, d}, x]
Int[(x_)^(m_.)/((d_) + (d_.)*ProductLog[(a_.)*(x_)^(n_)]), x_Symbol] :> -Su bst[Int[1/(x^(m + 2)*(d + d*ProductLog[a/x^n])), x], x, 1/x] /; FreeQ[{a, d }, x] && IntegerQ[m] && ILtQ[n, 0] && NeQ[m, -1]
Int[((x_)^(m_.)*((c_.)*ProductLog[(a_.)*(x_)^(n_.)])^(p_.))/((d_) + (d_.)*P roductLog[(a_.)*(x_)^(n_.)]), x_Symbol] :> Simp[c*x^(m + 1)*((c*ProductLog[ a*x^n])^(p - 1)/(d*(m + 1))), x] - Simp[c*((m + n*(p - 1) + 1)/(m + 1)) I nt[x^m*((c*ProductLog[a*x^n])^(p - 1)/(d + d*ProductLog[a*x^n])), x], x] /; FreeQ[{a, c, d, m, n, p}, x] && NeQ[m, -1] && GtQ[Simplify[p + (m + 1)/n], 1]
Time = 0.07 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00
method | result | size |
parallelrisch | \(\frac {4-\operatorname {LambertW}\left (\frac {a}{x}\right )^{3}+2 \operatorname {LambertW}\left (\frac {a}{x}\right )^{2}-4 \operatorname {LambertW}\left (\frac {a}{x}\right )}{x \operatorname {LambertW}\left (\frac {a}{x}\right )}\) | \(43\) |
derivativedivides | \(-\frac {\frac {\operatorname {LambertW}\left (\frac {a}{x}\right )^{2} a}{x}-\frac {2 a \operatorname {LambertW}\left (\frac {a}{x}\right )}{x}+\frac {4 a}{x}-\frac {4 a}{x \operatorname {LambertW}\left (\frac {a}{x}\right )}}{a}\) | \(52\) |
default | \(-\frac {\frac {\operatorname {LambertW}\left (\frac {a}{x}\right )^{2} a}{x}-\frac {2 a \operatorname {LambertW}\left (\frac {a}{x}\right )}{x}+\frac {4 a}{x}-\frac {4 a}{x \operatorname {LambertW}\left (\frac {a}{x}\right )}}{a}\) | \(52\) |
Input:
int(LambertW(a/x)^2/x^2,x,method=_RETURNVERBOSE)
Output:
1/x*(4-LambertW(a/x)^3+2*LambertW(a/x)^2-4*LambertW(a/x))/LambertW(a/x)
Time = 0.07 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.05 \[ \int \frac {W\left (\frac {a}{x}\right )^2}{x^2} \, dx=-\frac {\operatorname {W}({\frac {a}{x}})^{3} - 2 \, \operatorname {W}({\frac {a}{x}})^{2} + 4 \, \operatorname {W}({\frac {a}{x}}) - 4}{x \operatorname {W}({\frac {a}{x}})} \] Input:
integrate(lambert_w(a/x)^2/x^2,x, algorithm="fricas")
Output:
-(lambert_w(a/x)^3 - 2*lambert_w(a/x)^2 + 4*lambert_w(a/x) - 4)/(x*lambert _w(a/x))
Time = 0.23 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.67 \[ \int \frac {W\left (\frac {a}{x}\right )^2}{x^2} \, dx=\begin {cases} - \frac {W^{2}\left (\frac {a}{x}\right )}{x} + \frac {2 W\left (\frac {a}{x}\right )}{x} - \frac {4}{x} + \frac {4}{x W\left (\frac {a}{x}\right )} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:
integrate(LambertW(a/x)**2/x**2,x)
Output:
Piecewise((-LambertW(a/x)**2/x + 2*LambertW(a/x)/x - 4/x + 4/(x*LambertW(a /x)), Ne(a, 0)), (0, True))
\[ \int \frac {W\left (\frac {a}{x}\right )^2}{x^2} \, dx=\int { \frac {\operatorname {W}({\frac {a}{x}})^{2}}{x^{2}} \,d x } \] Input:
integrate(lambert_w(a/x)^2/x^2,x, algorithm="maxima")
Output:
integrate(lambert_w(a/x)^2/x^2, x)
\[ \int \frac {W\left (\frac {a}{x}\right )^2}{x^2} \, dx=\int { \frac {\operatorname {W}({\frac {a}{x}})^{2}}{x^{2}} \,d x } \] Input:
integrate(lambert_w(a/x)^2/x^2,x, algorithm="giac")
Output:
integrate(lambert_w(a/x)^2/x^2, x)
Timed out. \[ \int \frac {W\left (\frac {a}{x}\right )^2}{x^2} \, dx=\int \frac {{\mathrm {LambertW}\left (\frac {a}{x}\right )}^2}{x^2} \,d x \] Input:
int(LambertW(a/x)^2/x^2,x)
Output:
int(LambertW(a/x)^2/x^2, x)
Time = 0.44 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.98 \[ \int \frac {W\left (\frac {a}{x}\right )^2}{x^2} \, dx=\frac {e^{\textit {lambert\_w} \left (\frac {a}{x}\right )} \left (-\textit {lambert\_w} \left (\frac {a}{x}\right )^{3}+2 \textit {lambert\_w} \left (\frac {a}{x}\right )^{2}-4 \textit {lambert\_w} \left (\frac {a}{x}\right )+4\right )}{a} \] Input:
int(Lambert_W(a/x)^2/x^2,x)
Output:
(e**lambert_w(a/x)*( - lambert_w(a/x)**3 + 2*lambert_w(a/x)**2 - 4*lambert _w(a/x) + 4))/a