Integrand size = 31, antiderivative size = 18 \[ \int \frac {W\left (a x^{\frac {1}{1-p}}\right )^p}{1+W\left (a x^{\frac {1}{1-p}}\right )} \, dx=x W\left (a x^{\frac {1}{1-p}}\right )^{-1+p} \] Output:
x*LambertW(a*x^(1/(1-p)))^(-1+p)
Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {W\left (a x^{\frac {1}{1-p}}\right )^p}{1+W\left (a x^{\frac {1}{1-p}}\right )} \, dx=x W\left (a x^{\frac {1}{1-p}}\right )^{-1+p} \] Input:
Integrate[ProductLog[a*x^(1 - p)^(-1)]^p/(1 + ProductLog[a*x^(1 - p)^(-1)] ),x]
Output:
x*ProductLog[a*x^(1 - p)^(-1)]^(-1 + p)
Time = 0.21 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {7187}
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.
\(\displaystyle \int \frac {W\left (a x^{\frac {1}{1-p}}\right )^p}{W\left (a x^{\frac {1}{1-p}}\right )+1} \, dx\) |
\(\Big \downarrow \) 7187 |
\(\displaystyle x W\left (a x^{\frac {1}{1-p}}\right )^{p-1}\) |
Input:
Int[ProductLog[a*x^(1 - p)^(-1)]^p/(1 + ProductLog[a*x^(1 - p)^(-1)]),x]
Output:
x*ProductLog[a*x^(1 - p)^(-1)]^(-1 + p)
Int[((c_.)*ProductLog[(a_.)*(x_)^(n_.)])^(p_.)/((d_) + (d_.)*ProductLog[(a_ .)*(x_)^(n_.)]), x_Symbol] :> Simp[c*x*((c*ProductLog[a*x^n])^(p - 1)/d), x ] /; FreeQ[{a, c, d, n, p}, x] && EqQ[n*(p - 1), -1]
Time = 0.24 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.72
method | result | size |
parallelrisch | \(\frac {x \operatorname {LambertW}\left (a \,x^{\frac {1}{1-p}}\right )^{p}}{\operatorname {LambertW}\left (a \,x^{\frac {1}{1-p}}\right )}\) | \(31\) |
Input:
int(LambertW(a*x^(1/(1-p)))^p/(1+LambertW(a*x^(1/(1-p)))),x,method=_RETURN VERBOSE)
Output:
x*LambertW(a*x^(1/(1-p)))^p/LambertW(a*x^(1/(1-p)))
\[ \int \frac {W\left (a x^{\frac {1}{1-p}}\right )^p}{1+W\left (a x^{\frac {1}{1-p}}\right )} \, dx=\int { \frac {\operatorname {W}({\frac {a}{x^{\left (\frac {1}{p - 1}\right )}}})^{p}}{\operatorname {W}({\frac {a}{x^{\left (\frac {1}{p - 1}\right )}}}) + 1} \,d x } \] Input:
integrate(lambert_w(a*x^(1/(1-p)))^p/(1+lambert_w(a*x^(1/(1-p)))),x, algor ithm="fricas")
Output:
integral(lambert_w(a/x^(1/(p - 1)))^p/(lambert_w(a/x^(1/(p - 1))) + 1), x)
Timed out. \[ \int \frac {W\left (a x^{\frac {1}{1-p}}\right )^p}{1+W\left (a x^{\frac {1}{1-p}}\right )} \, dx=\text {Timed out} \] Input:
integrate(LambertW(a*x**(1/(1-p)))**p/(1+LambertW(a*x**(1/(1-p)))),x)
Output:
Timed out
\[ \int \frac {W\left (a x^{\frac {1}{1-p}}\right )^p}{1+W\left (a x^{\frac {1}{1-p}}\right )} \, dx=\int { \frac {\operatorname {W}({\frac {a}{x^{\left (\frac {1}{p - 1}\right )}}})^{p}}{\operatorname {W}({\frac {a}{x^{\left (\frac {1}{p - 1}\right )}}}) + 1} \,d x } \] Input:
integrate(lambert_w(a*x^(1/(1-p)))^p/(1+lambert_w(a*x^(1/(1-p)))),x, algor ithm="maxima")
Output:
integrate(lambert_w(a/x^(1/(p - 1)))^p/(lambert_w(a/x^(1/(p - 1))) + 1), x )
\[ \int \frac {W\left (a x^{\frac {1}{1-p}}\right )^p}{1+W\left (a x^{\frac {1}{1-p}}\right )} \, dx=\int { \frac {\operatorname {W}({\frac {a}{x^{\left (\frac {1}{p - 1}\right )}}})^{p}}{\operatorname {W}({\frac {a}{x^{\left (\frac {1}{p - 1}\right )}}}) + 1} \,d x } \] Input:
integrate(lambert_w(a*x^(1/(1-p)))^p/(1+lambert_w(a*x^(1/(1-p)))),x, algor ithm="giac")
Output:
integrate(lambert_w(a/x^(1/(p - 1)))^p/(lambert_w(a/x^(1/(p - 1))) + 1), x )
Timed out. \[ \int \frac {W\left (a x^{\frac {1}{1-p}}\right )^p}{1+W\left (a x^{\frac {1}{1-p}}\right )} \, dx=\int \frac {{\mathrm {LambertW}\left (\frac {a}{x^{\frac {1}{p-1}}}\right )}^p}{\mathrm {LambertW}\left (\frac {a}{x^{\frac {1}{p-1}}}\right )+1} \,d x \] Input:
int(LambertW(a/x^(1/(p - 1)))^p/(LambertW(a/x^(1/(p - 1))) + 1),x)
Output:
int(LambertW(a/x^(1/(p - 1)))^p/(LambertW(a/x^(1/(p - 1))) + 1), x)
\[ \int \frac {W\left (a x^{\frac {1}{1-p}}\right )^p}{1+W\left (a x^{\frac {1}{1-p}}\right )} \, dx =\text {Too large to display} \] Input:
int(Lambert_W(a*x^(1/(1-p)))^p/(1+Lambert_W(a*x^(1/(1-p)))),x)
Output:
(3*lambert_w(a/x**(1/(p - 1)))**p*x + int(lambert_w(a/x**(1/(p - 1)))**p/( x**(1/(p - 1))*e**lambert_w(a/x**(1/(p - 1)))*lambert_w(a/x**(1/(p - 1)))* *2*p**2 - 2*x**(1/(p - 1))*e**lambert_w(a/x**(1/(p - 1)))*lambert_w(a/x**( 1/(p - 1)))**2*p + x**(1/(p - 1))*e**lambert_w(a/x**(1/(p - 1)))*lambert_w (a/x**(1/(p - 1)))**2 + x**(1/(p - 1))*e**lambert_w(a/x**(1/(p - 1)))*lamb ert_w(a/x**(1/(p - 1)))*p**2 - 2*x**(1/(p - 1))*e**lambert_w(a/x**(1/(p - 1)))*lambert_w(a/x**(1/(p - 1)))*p + x**(1/(p - 1))*e**lambert_w(a/x**(1/( p - 1)))*lambert_w(a/x**(1/(p - 1)))),x)*a*p**2 - int(lambert_w(a/x**(1/(p - 1)))**p/(x**(1/(p - 1))*e**lambert_w(a/x**(1/(p - 1)))*lambert_w(a/x**( 1/(p - 1)))**2*p**2 - 2*x**(1/(p - 1))*e**lambert_w(a/x**(1/(p - 1)))*lamb ert_w(a/x**(1/(p - 1)))**2*p + x**(1/(p - 1))*e**lambert_w(a/x**(1/(p - 1) ))*lambert_w(a/x**(1/(p - 1)))**2 + x**(1/(p - 1))*e**lambert_w(a/x**(1/(p - 1)))*lambert_w(a/x**(1/(p - 1)))*p**2 - 2*x**(1/(p - 1))*e**lambert_w(a /x**(1/(p - 1)))*lambert_w(a/x**(1/(p - 1)))*p + x**(1/(p - 1))*e**lambert _w(a/x**(1/(p - 1)))*lambert_w(a/x**(1/(p - 1)))),x)*a*p + 2*int(lambert_w (a/x**(1/(p - 1)))**p/(x**(1/(p - 1))*e**lambert_w(a/x**(1/(p - 1)))*lambe rt_w(a/x**(1/(p - 1)))**2*p - x**(1/(p - 1))*e**lambert_w(a/x**(1/(p - 1)) )*lambert_w(a/x**(1/(p - 1)))**2 + x**(1/(p - 1))*e**lambert_w(a/x**(1/(p - 1)))*lambert_w(a/x**(1/(p - 1)))*p - x**(1/(p - 1))*e**lambert_w(a/x**(1 /(p - 1)))*lambert_w(a/x**(1/(p - 1)))),x)*a*p + int(lambert_w(a/x**(1/...