Integrand size = 14, antiderivative size = 70 \[ \int \frac {1}{x^2 \sqrt {c W(a x)}} \, dx=-\frac {2 a \sqrt {\pi } \text {erf}\left (\frac {\sqrt {c W(a x)}}{\sqrt {c}}\right )}{3 \sqrt {c}}-\frac {2}{3 x \sqrt {c W(a x)}}-\frac {2 \sqrt {c W(a x)}}{3 c x} \] Output:
-2/3*a*Pi^(1/2)*erf((c*LambertW(a*x))^(1/2)/c^(1/2))/c^(1/2)-2/3/x/(c*Lamb ertW(a*x))^(1/2)-2/3*(c*LambertW(a*x))^(1/2)/c/x
Time = 0.01 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.69 \[ \int \frac {1}{x^2 \sqrt {c W(a x)}} \, dx=-\frac {2 \left (1+a \sqrt {\pi } x \text {erf}\left (\sqrt {W(a x)}\right ) \sqrt {W(a x)}+W(a x)\right )}{3 x \sqrt {c W(a x)}} \] Input:
Integrate[1/(x^2*Sqrt[c*ProductLog[a*x]]),x]
Output:
(-2*(1 + a*Sqrt[Pi]*x*Erf[Sqrt[ProductLog[a*x]]]*Sqrt[ProductLog[a*x]] + P roductLog[a*x]))/(3*x*Sqrt[c*ProductLog[a*x]])
Time = 0.54 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.01, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {7173, 7206, 7203}
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.
\(\displaystyle \int \frac {1}{x^2 \sqrt {c W(a x)}} \, dx\) |
\(\Big \downarrow \) 7173 |
\(\displaystyle \frac {\int \frac {\sqrt {c W(a x)}}{x^2 (W(a x)+1)}dx}{3 c}-\frac {2}{3 x \sqrt {c W(a x)}}\) |
\(\Big \downarrow \) 7206 |
\(\displaystyle \frac {-\frac {2 \int \frac {(c W(a x))^{3/2}}{x^2 (W(a x)+1)}dx}{c}-\frac {2 \sqrt {c W(a x)}}{x}}{3 c}-\frac {2}{3 x \sqrt {c W(a x)}}\) |
\(\Big \downarrow \) 7203 |
\(\displaystyle \frac {-2 \sqrt {\pi } a \sqrt {c} \text {erf}\left (\frac {\sqrt {c W(a x)}}{\sqrt {c}}\right )-\frac {2 \sqrt {c W(a x)}}{x}}{3 c}-\frac {2}{3 x \sqrt {c W(a x)}}\) |
Input:
Int[1/(x^2*Sqrt[c*ProductLog[a*x]]),x]
Output:
-2/(3*x*Sqrt[c*ProductLog[a*x]]) + (-2*a*Sqrt[c]*Sqrt[Pi]*Erf[Sqrt[c*Produ ctLog[a*x]]/Sqrt[c]] - (2*Sqrt[c*ProductLog[a*x]])/x)/(3*c)
Int[(x_)^(m_.)*((c_.)*ProductLog[(a_.)*(x_)^(n_.)])^(p_.), x_Symbol] :> Sim p[x^(m + 1)*((c*ProductLog[a*x^n])^p/(m + n*p + 1)), x] + Simp[n*(p/(c*(m + n*p + 1))) Int[x^m*((c*ProductLog[a*x^n])^(p + 1)/(1 + ProductLog[a*x^n] )), x], x] /; FreeQ[{a, c, m, n, p}, x] && (EqQ[m, -1] || (IntegerQ[p - 1/2 ] && ILtQ[Simplify[p + (m + 1)/n] - 1/2, 0]) || ( !IntegerQ[p - 1/2] && ILt Q[Simplify[p + (m + 1)/n], 0]))
Int[((x_)^(m_.)*((c_.)*ProductLog[(a_.)*(x_)^(n_.)])^(p_))/((d_) + (d_.)*Pr oductLog[(a_.)*(x_)^(n_.)]), x_Symbol] :> Simp[a^(p - 1/2)*c^(p - 1/2)*Rt[P i*(c/(p - 1/2)), 2]*(Erf[Sqrt[c*ProductLog[a*x^n]]/Rt[c/(p - 1/2), 2]]/(d*n )), x] /; FreeQ[{a, c, d, m, n}, x] && NeQ[m, -1] && IntegerQ[p - 1/2] && E qQ[m + n*(p - 1/2), -1] && PosQ[c/(p - 1/2)]
Int[((x_)^(m_.)*((c_.)*ProductLog[(a_.)*(x_)^(n_.)])^(p_.))/((d_) + (d_.)*P roductLog[(a_.)*(x_)^(n_.)]), x_Symbol] :> Simp[x^(m + 1)*((c*ProductLog[a* x^n])^p/(d*(m + n*p + 1))), x] - Simp[(m + 1)/(c*(m + n*p + 1)) Int[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] && LtQ[Simplify[p + (m + 1)/n], 0]
Leaf count of result is larger than twice the leaf count of optimal. \(106\) vs. \(2(52)=104\).
Time = 0.05 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.53
method | result | size |
default | \(a \left (-\frac {2 \,{\mathrm e}^{-\operatorname {LambertW}\left (x a \right )}}{\sqrt {c \operatorname {LambertW}\left (x a \right )}}-\frac {2 \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {c \operatorname {LambertW}\left (x a \right )}}{\sqrt {c}}\right )}{\sqrt {c}}+2 c \left (-\frac {{\mathrm e}^{-\operatorname {LambertW}\left (x a \right )}}{3 \left (c \operatorname {LambertW}\left (x a \right )\right )^{\frac {3}{2}}}-\frac {2 \left (-\frac {{\mathrm e}^{-\operatorname {LambertW}\left (x a \right )}}{\sqrt {c \operatorname {LambertW}\left (x a \right )}}-\frac {\sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {c \operatorname {LambertW}\left (x a \right )}}{\sqrt {c}}\right )}{\sqrt {c}}\right )}{3 c}\right )\right )\) | \(107\) |
Input:
int(1/x^2/(c*LambertW(x*a))^(1/2),x,method=_RETURNVERBOSE)
Output:
a*(-2/(c*LambertW(x*a))^(1/2)*exp(-LambertW(x*a))-2/c^(1/2)*Pi^(1/2)*erf(( c*LambertW(x*a))^(1/2)/c^(1/2))+2*c*(-1/3/(c*LambertW(x*a))^(3/2)*exp(-Lam bertW(x*a))-2/3/c*(-1/(c*LambertW(x*a))^(1/2)*exp(-LambertW(x*a))-1/c^(1/2 )*Pi^(1/2)*erf((c*LambertW(x*a))^(1/2)/c^(1/2)))))
\[ \int \frac {1}{x^2 \sqrt {c W(a x)}} \, dx=\int { \frac {1}{\sqrt {c \operatorname {W}({a x})} x^{2}} \,d x } \] Input:
integrate(1/x^2/(c*lambert_w(a*x))^(1/2),x, algorithm="fricas")
Output:
integral(sqrt(c*lambert_w(a*x))/(c*x^2*lambert_w(a*x)), x)
\[ \int \frac {1}{x^2 \sqrt {c W(a x)}} \, dx=\int \frac {1}{x^{2} \sqrt {c W\left (a x\right )}}\, dx \] Input:
integrate(1/x**2/(c*LambertW(a*x))**(1/2),x)
Output:
Integral(1/(x**2*sqrt(c*LambertW(a*x))), x)
\[ \int \frac {1}{x^2 \sqrt {c W(a x)}} \, dx=\int { \frac {1}{\sqrt {c \operatorname {W}({a x})} x^{2}} \,d x } \] Input:
integrate(1/x^2/(c*lambert_w(a*x))^(1/2),x, algorithm="maxima")
Output:
integrate(1/(sqrt(c*lambert_w(a*x))*x^2), x)
\[ \int \frac {1}{x^2 \sqrt {c W(a x)}} \, dx=\int { \frac {1}{\sqrt {c \operatorname {W}({a x})} x^{2}} \,d x } \] Input:
integrate(1/x^2/(c*lambert_w(a*x))^(1/2),x, algorithm="giac")
Output:
integrate(1/(sqrt(c*lambert_w(a*x))*x^2), x)
Timed out. \[ \int \frac {1}{x^2 \sqrt {c W(a x)}} \, dx=\int \frac {1}{x^2\,\sqrt {c\,\mathrm {LambertW}\left (a\,x\right )}} \,d x \] Input:
int(1/(x^2*(c*LambertW(a*x))^(1/2)),x)
Output:
int(1/(x^2*(c*LambertW(a*x))^(1/2)), x)
\[ \int \frac {1}{x^2 \sqrt {c W(a x)}} \, dx=\frac {\sqrt {c}\, \left (\int \frac {\sqrt {\textit {lambert\_w} \left (a x \right )}}{\textit {lambert\_w} \left (a x \right ) x^{2}}d x \right )}{c} \] Input:
int(1/x^2/(c*Lambert_W(a*x))^(1/2),x)
Output:
(sqrt(c)*int(sqrt(lambert_w(a*x))/(lambert_w(a*x)*x**2),x))/c