\(\int \frac {1}{x^2 \sqrt {c W(a x)}} \, dx\) [76]

Optimal result
Mathematica [A] (verified)
Rubi [A] (verified)
Maple [B] (verified)
Fricas [F]
Sympy [F]
Maxima [F]
Giac [F]
Mupad [F(-1)]
Reduce [F]

Optimal result

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
 

Mathematica [A] (verified)

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]])
 

Rubi [A] (verified)

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)
 

Defintions of rubi rules used

rule 7173
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]))
 

rule 7203
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)]
 

rule 7206
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]
 
Maple [B] (verified)

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)))))
 

Fricas [F]

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

Sympy [F]

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

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

Output:

Integral(1/(x**2*sqrt(c*LambertW(a*x))), x)
 

Maxima [F]

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

Giac [F]

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

Mupad [F(-1)]

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)
 

Reduce [F]

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