Integrand size = 14, antiderivative size = 22 \[ \int \frac {1}{x^3 \left (1+W\left (a x^2\right )\right )} \, dx=-\frac {1}{2 x^2}-\frac {1}{2} a \operatorname {ExpIntegralEi}\left (-W\left (a x^2\right )\right ) \] Output:
-1/2/x^2-1/2*a*Ei(-LambertW(a*x^2))
Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^3 \left (1+W\left (a x^2\right )\right )} \, dx=-\frac {1}{2 x^2}-\frac {1}{2} a \operatorname {ExpIntegralEi}\left (-W\left (a x^2\right )\right ) \] Input:
Integrate[1/(x^3*(1 + ProductLog[a*x^2])),x]
Output:
-1/2*1/x^2 - (a*ExpIntegralEi[-ProductLog[a*x^2]])/2
Time = 0.35 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {7283, 7196, 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.
\(\displaystyle \int \frac {1}{x^3 \left (W\left (a x^2\right )+1\right )} \, dx\) |
\(\Big \downarrow \) 7283 |
\(\displaystyle \frac {1}{2} \int \frac {1}{x^4 \left (W\left (a x^2\right )+1\right )}dx^2\) |
\(\Big \downarrow \) 7196 |
\(\displaystyle \frac {1}{2} \left (-\int \frac {W\left (a x^2\right )}{x^4 \left (W\left (a x^2\right )+1\right )}dx^2-\frac {1}{x^2}\right )\) |
\(\Big \downarrow \) 7202 |
\(\displaystyle \frac {1}{2} \left (-a \operatorname {ExpIntegralEi}\left (-W\left (a x^2\right )\right )-\frac {1}{x^2}\right )\) |
Input:
Int[1/(x^3*(1 + ProductLog[a*x^2])),x]
Output:
(-x^(-2) - a*ExpIntegralEi[-ProductLog[a*x^2]])/2
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]
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]
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])
\[\int \frac {1}{x^{3} \left (1+\operatorname {LambertW}\left (a \,x^{2}\right )\right )}d x\]
Input:
int(1/x^3/(1+LambertW(a*x^2)),x)
Output:
int(1/x^3/(1+LambertW(a*x^2)),x)
\[ \int \frac {1}{x^3 \left (1+W\left (a x^2\right )\right )} \, dx=\int { \frac {1}{x^{3} {\left (\operatorname {W}({a x^{2}}) + 1\right )}} \,d x } \] Input:
integrate(1/x^3/(1+lambert_w(a*x^2)),x, algorithm="fricas")
Output:
integral(1/(x^3*lambert_w(a*x^2) + x^3), x)
\[ \int \frac {1}{x^3 \left (1+W\left (a x^2\right )\right )} \, dx=\int \frac {1}{x^{3} \left (W\left (a x^{2}\right ) + 1\right )}\, dx \] Input:
integrate(1/x**3/(1+LambertW(a*x**2)),x)
Output:
Integral(1/(x**3*(LambertW(a*x**2) + 1)), x)
\[ \int \frac {1}{x^3 \left (1+W\left (a x^2\right )\right )} \, dx=\int { \frac {1}{x^{3} {\left (\operatorname {W}({a x^{2}}) + 1\right )}} \,d x } \] Input:
integrate(1/x^3/(1+lambert_w(a*x^2)),x, algorithm="maxima")
Output:
integrate(1/(x^3*(lambert_w(a*x^2) + 1)), x)
\[ \int \frac {1}{x^3 \left (1+W\left (a x^2\right )\right )} \, dx=\int { \frac {1}{x^{3} {\left (\operatorname {W}({a x^{2}}) + 1\right )}} \,d x } \] Input:
integrate(1/x^3/(1+lambert_w(a*x^2)),x, algorithm="giac")
Output:
integrate(1/(x^3*(lambert_w(a*x^2) + 1)), x)
Timed out. \[ \int \frac {1}{x^3 \left (1+W\left (a x^2\right )\right )} \, dx=\int \frac {1}{x^3\,\left (\mathrm {LambertW}\left (a\,x^2\right )+1\right )} \,d x \] Input:
int(1/(x^3*(LambertW(a*x^2) + 1)),x)
Output:
int(1/(x^3*(LambertW(a*x^2) + 1)), x)
\[ \int \frac {1}{x^3 \left (1+W\left (a x^2\right )\right )} \, dx=\frac {-2 \left (\int \frac {\textit {lambert\_w} \left (a \,x^{2}\right )}{\textit {lambert\_w} \left (a \,x^{2}\right ) x^{3}+x^{3}}d x \right ) x^{2}+6 \left (\int \frac {1}{\textit {lambert\_w} \left (a \,x^{2}\right ) x^{3}+x^{3}}d x \right ) x^{2}-1}{8 x^{2}} \] Input:
int(1/x^3/(1+Lambert_W(a*x^2)),x)
Output:
( - 2*int(lambert_w(a*x**2)/(lambert_w(a*x**2)*x**3 + x**3),x)*x**2 + 6*in t(1/(lambert_w(a*x**2)*x**3 + x**3),x)*x**2 - 1)/(8*x**2)