\(\int \frac {W(a x^{\frac {1}{1-p}})^p}{1+W(a x^{\frac {1}{1-p}})} \, dx\) [359]

Optimal result

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)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

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)
 

Rubi [A] (verified)

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)
 

Defintions of rubi rules used

rule 7187

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]
 

Maple [A] (verified)

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)))
 

Fricas [F]

\[ \int \frac {W\left (a x^{\frac {1}{1-p}}\right )^p}{1+W\left (a x^{\frac {1}{1-p}}\right )} \, dx=\int { \frac {\operatorname {W}({\frac {a}{x^{\left (\frac {1}{p - 1}\right )}}})^{p}}{\operatorname {W}({\frac {a}{x^{\left (\frac {1}{p - 1}\right )}}}) + 1} \,d x } \] 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)
 

Sympy [F(-1)]

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
 

Maxima [F]

\[ \int \frac {W\left (a x^{\frac {1}{1-p}}\right )^p}{1+W\left (a x^{\frac {1}{1-p}}\right )} \, dx=\int { \frac {\operatorname {W}({\frac {a}{x^{\left (\frac {1}{p - 1}\right )}}})^{p}}{\operatorname {W}({\frac {a}{x^{\left (\frac {1}{p - 1}\right )}}}) + 1} \,d x } \] 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 
)
 

Giac [F]

\[ \int \frac {W\left (a x^{\frac {1}{1-p}}\right )^p}{1+W\left (a x^{\frac {1}{1-p}}\right )} \, dx=\int { \frac {\operatorname {W}({\frac {a}{x^{\left (\frac {1}{p - 1}\right )}}})^{p}}{\operatorname {W}({\frac {a}{x^{\left (\frac {1}{p - 1}\right )}}}) + 1} \,d x } \] 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 
)
 

Mupad [F(-1)]

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)
 

Reduce [F]

\[ \int \frac {W\left (a x^{\frac {1}{1-p}}\right )^p}{1+W\left (a x^{\frac {1}{1-p}}\right )} \, dx =\text {Too large to display} \] 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/...