\(\int \frac {1}{x^5 (1+W(a x^2))} \, dx\) [311]

Optimal result

Integrand size = 14, antiderivative size = 34 \[ \int \frac {1}{x^5 \left (1+W\left (a x^2\right )\right )} \, dx=-\frac {1}{4 x^4}+a^2 \operatorname {ExpIntegralEi}\left (-2 W\left (a x^2\right )\right )+\frac {W\left (a x^2\right )}{2 x^4} \] Output:

-1/4/x^4+a^2*Ei(-2*LambertW(a*x^2))+1/2*LambertW(a*x^2)/x^4
 

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^5 \left (1+W\left (a x^2\right )\right )} \, dx=-\frac {1}{4 x^4}+a^2 \operatorname {ExpIntegralEi}\left (-2 W\left (a x^2\right )\right )+\frac {W\left (a x^2\right )}{2 x^4} \] Input:

Integrate[1/(x^5*(1 + ProductLog[a*x^2])),x]
 

Output:

-1/4*1/x^4 + a^2*ExpIntegralEi[-2*ProductLog[a*x^2]] + ProductLog[a*x^2]/( 
2*x^4)
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {7283, 7196, 7206, 7202}

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.

\(\displaystyle \int \frac {1}{x^5 \left (W\left (a x^2\right )+1\right )} \, dx\)

\(\Big \downarrow \) 7283

\(\displaystyle \frac {1}{2} \int \frac {1}{x^6 \left (W\left (a x^2\right )+1\right )}dx^2\)

\(\Big \downarrow \) 7196

\(\displaystyle \frac {1}{2} \left (-\int \frac {W\left (a x^2\right )}{x^6 \left (W\left (a x^2\right )+1\right )}dx^2-\frac {1}{2 x^4}\right )\)

\(\Big \downarrow \) 7206

\(\displaystyle \frac {1}{2} \left (2 \int \frac {W\left (a x^2\right )^2}{x^6 \left (W\left (a x^2\right )+1\right )}dx^2+\frac {W\left (a x^2\right )}{x^4}-\frac {1}{2 x^4}\right )\)

\(\Big \downarrow \) 7202

\(\displaystyle \frac {1}{2} \left (2 a^2 \operatorname {ExpIntegralEi}\left (-2 W\left (a x^2\right )\right )+\frac {W\left (a x^2\right )}{x^4}-\frac {1}{2 x^4}\right )\)

Input:

Int[1/(x^5*(1 + ProductLog[a*x^2])),x]
 

Output:

(-1/2*1/x^4 + 2*a^2*ExpIntegralEi[-2*ProductLog[a*x^2]] + ProductLog[a*x^2 
]/x^4)/2
 

Defintions of rubi rules used

rule 7196

Int[(x_)^(m_.)/((d_) + (d_.)*ProductLog[(a_.)*(x_)]), x_Symbol] :> Simp[x^( 
m + 1)/(d*(m + 1)), x] - Int[x^m*(ProductLog[a*x]/(d + d*ProductLog[a*x])), 
 x] /; FreeQ[{a, d}, x] && LtQ[m, -1]
 

rule 7202

Int[((x_)^(m_.)*ProductLog[(a_.)*(x_)^(n_.)]^(p_.))/((d_) + (d_.)*ProductLo 
g[(a_.)*(x_)^(n_.)]), x_Symbol] :> Simp[a^p*(ExpIntegralEi[(-p)*ProductLog[ 
a*x^n]]/(d*n)), x] /; FreeQ[{a, d, m, n}, x] && IntegerQ[p] && EqQ[m + n*p, 
 -1]
 

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]
 

rule 7283

Int[(u_)*(x_)^(m_.), x_Symbol] :> With[{lst = PowerVariableExpn[u, m + 1, x 
]}, Simp[1/lst[[2]]   Subst[Int[NormalizeIntegrand[Simplify[lst[[1]]/x], x] 
, x], x, (lst[[3]]*x)^lst[[2]]], x] /;  !FalseQ[lst] && NeQ[lst[[2]], m + 1 
]] /; IntegerQ[m] && NeQ[m, -1] && NonsumQ[u] && (GtQ[m, 0] ||  !AlgebraicF 
unctionQ[u, x])
 

Maple [F]

\[\int \frac {1}{x^{5} \left (1+\operatorname {LambertW}\left (a \,x^{2}\right )\right )}d x\]

Input:

int(1/x^5/(1+LambertW(a*x^2)),x)
 

Output:

int(1/x^5/(1+LambertW(a*x^2)),x)
 

Fricas [F]

\[ \int \frac {1}{x^5 \left (1+W\left (a x^2\right )\right )} \, dx=\int { \frac {1}{x^{5} {\left (\operatorname {W}({a x^{2}}) + 1\right )}} \,d x } \] Input:

integrate(1/x^5/(1+lambert_w(a*x^2)),x, algorithm="fricas")
 

Output:

integral(1/(x^5*lambert_w(a*x^2) + x^5), x)
 

Sympy [F]

\[ \int \frac {1}{x^5 \left (1+W\left (a x^2\right )\right )} \, dx=\int \frac {1}{x^{5} \left (W\left (a x^{2}\right ) + 1\right )}\, dx \] Input:

integrate(1/x**5/(1+LambertW(a*x**2)),x)
 

Output:

Integral(1/(x**5*(LambertW(a*x**2) + 1)), x)
 

Maxima [F]

\[ \int \frac {1}{x^5 \left (1+W\left (a x^2\right )\right )} \, dx=\int { \frac {1}{x^{5} {\left (\operatorname {W}({a x^{2}}) + 1\right )}} \,d x } \] Input:

integrate(1/x^5/(1+lambert_w(a*x^2)),x, algorithm="maxima")
 

Output:

integrate(1/(x^5*(lambert_w(a*x^2) + 1)), x)
 

Giac [F]

\[ \int \frac {1}{x^5 \left (1+W\left (a x^2\right )\right )} \, dx=\int { \frac {1}{x^{5} {\left (\operatorname {W}({a x^{2}}) + 1\right )}} \,d x } \] Input:

integrate(1/x^5/(1+lambert_w(a*x^2)),x, algorithm="giac")
 

Output:

integrate(1/(x^5*(lambert_w(a*x^2) + 1)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^5 \left (1+W\left (a x^2\right )\right )} \, dx=\int \frac {1}{x^5\,\left (\mathrm {LambertW}\left (a\,x^2\right )+1\right )} \,d x \] Input:

int(1/(x^5*(LambertW(a*x^2) + 1)),x)
 

Output:

int(1/(x^5*(LambertW(a*x^2) + 1)), x)
 

Reduce [F]

\[ \int \frac {1}{x^5 \left (1+W\left (a x^2\right )\right )} \, dx=\frac {-4 \left (\int \frac {\textit {lambert\_w} \left (a \,x^{2}\right )}{\textit {lambert\_w} \left (a \,x^{2}\right ) x^{5}+x^{5}}d x \right ) x^{4}+12 \left (\int \frac {1}{\textit {lambert\_w} \left (a \,x^{2}\right ) x^{5}+x^{5}}d x \right ) x^{4}-1}{16 x^{4}} \] Input:

int(1/x^5/(1+Lambert_W(a*x^2)),x)
 

Output:

( - 4*int(lambert_w(a*x**2)/(lambert_w(a*x**2)*x**5 + x**5),x)*x**4 + 12*i 
nt(1/(lambert_w(a*x**2)*x**5 + x**5),x)*x**4 - 1)/(16*x**4)