\(\int \frac {\log (c (d+e x^n)^p)}{x (f+g x^{-n})^2} \, dx\) [376]

Optimal result

Integrand size = 27, antiderivative size = 156 \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^{-n}\right )^2} \, dx=\frac {e g p \log \left (d+e x^n\right )}{f^2 (d f-e g) n}+\frac {g \log \left (c \left (d+e x^n\right )^p\right )}{f^2 n \left (g+f x^n\right )}-\frac {e g p \log \left (g+f x^n\right )}{f^2 (d f-e g) n}+\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (-\frac {e \left (g+f x^n\right )}{d f-e g}\right )}{f^2 n}+\frac {p \operatorname {PolyLog}\left (2,\frac {f \left (d+e x^n\right )}{d f-e g}\right )}{f^2 n} \] Output:

e*g*p*ln(d+e*x^n)/f^2/(d*f-e*g)/n+g*ln(c*(d+e*x^n)^p)/f^2/n/(g+f*x^n)-e*g* 
p*ln(g+f*x^n)/f^2/(d*f-e*g)/n+ln(c*(d+e*x^n)^p)*ln(-e*(g+f*x^n)/(d*f-e*g)) 
/f^2/n+p*polylog(2,f*(d+e*x^n)/(d*f-e*g))/f^2/n
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(433\) vs. \(2(156)=312\).

Time = 1.46 (sec) , antiderivative size = 433, normalized size of antiderivative = 2.78 \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^{-n}\right )^2} \, dx=\frac {g p \log \left (f-f x^{-n}\right )+f p x^n \log \left (f-f x^{-n}\right )-g n p \log (x) \log \left (f-f x^{-n}\right )-f n p x^n \log (x) \log \left (f-f x^{-n}\right )-p \log \left (e+d x^{-n}\right ) \left (-f x^n+\left (g+f x^n\right ) \log \left (f-f x^{-n}\right )\right )-f x^n \log \left (c \left (d+e x^n\right )^p\right )+g \log \left (f-f x^{-n}\right ) \log \left (c \left (d+e x^n\right )^p\right )+f x^n \log \left (f-f x^{-n}\right ) \log \left (c \left (d+e x^n\right )^p\right )+g n p \log (x) \log \left (1+\frac {f x^n}{g}\right )+f n p x^n \log (x) \log \left (1+\frac {f x^n}{g}\right )+p \left (g+f x^n\right ) \operatorname {PolyLog}\left (2,-\frac {f x^n}{g}\right )}{f^2 n \left (g+f x^n\right )}-\frac {p \left (-\frac {d f \log \left (e+d x^{-n}\right )}{d f-e g}+\frac {f x^n \log \left (e+d x^{-n}\right )}{g+f x^n}+\log \left (-\frac {d x^{-n}}{e}\right ) \log \left (e+d x^{-n}\right )+\frac {d f \log \left (f+g x^{-n}\right )}{d f-e g}-\log \left (e+d x^{-n}\right ) \log \left (\frac {d \left (f+g x^{-n}\right )}{d f-e g}\right )-\operatorname {PolyLog}\left (2,-\frac {g \left (e+d x^{-n}\right )}{d f-e g}\right )+\operatorname {PolyLog}\left (2,1+\frac {d x^{-n}}{e}\right )\right )}{f^2 n} \] Input:

Integrate[Log[c*(d + e*x^n)^p]/(x*(f + g/x^n)^2),x]
 

Output:

(g*p*Log[f - f/x^n] + f*p*x^n*Log[f - f/x^n] - g*n*p*Log[x]*Log[f - f/x^n] 
 - f*n*p*x^n*Log[x]*Log[f - f/x^n] - p*Log[e + d/x^n]*(-(f*x^n) + (g + f*x 
^n)*Log[f - f/x^n]) - f*x^n*Log[c*(d + e*x^n)^p] + g*Log[f - f/x^n]*Log[c* 
(d + e*x^n)^p] + f*x^n*Log[f - f/x^n]*Log[c*(d + e*x^n)^p] + g*n*p*Log[x]* 
Log[1 + (f*x^n)/g] + f*n*p*x^n*Log[x]*Log[1 + (f*x^n)/g] + p*(g + f*x^n)*P 
olyLog[2, -((f*x^n)/g)])/(f^2*n*(g + f*x^n)) - (p*(-((d*f*Log[e + d/x^n])/ 
(d*f - e*g)) + (f*x^n*Log[e + d/x^n])/(g + f*x^n) + Log[-(d/(e*x^n))]*Log[ 
e + d/x^n] + (d*f*Log[f + g/x^n])/(d*f - e*g) - Log[e + d/x^n]*Log[(d*(f + 
 g/x^n))/(d*f - e*g)] - PolyLog[2, -((g*(e + d/x^n))/(d*f - e*g))] + PolyL 
og[2, 1 + d/(e*x^n)]))/(f^2*n)
 

Rubi [A] (verified)

Time = 0.78 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.93, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2005, 2925, 2863, 2009}

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.

\(\displaystyle \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^{-n}\right )^2} \, dx\)

\(\Big \downarrow \) 2005

\(\displaystyle \int \frac {x^{2 n-1} \log \left (c \left (d+e x^n\right )^p\right )}{\left (f x^n+g\right )^2}dx\)

\(\Big \downarrow \) 2925

\(\displaystyle \frac {\int \frac {x^n \log \left (c \left (e x^n+d\right )^p\right )}{\left (f x^n+g\right )^2}dx^n}{n}\)

\(\Big \downarrow \) 2863

\(\displaystyle \frac {\int \left (\frac {\log \left (c \left (e x^n+d\right )^p\right )}{f \left (f x^n+g\right )}-\frac {g \log \left (c \left (e x^n+d\right )^p\right )}{f \left (f x^n+g\right )^2}\right )dx^n}{n}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {g \log \left (c \left (d+e x^n\right )^p\right )}{f^2 \left (f x^n+g\right )}+\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (-\frac {e \left (f x^n+g\right )}{d f-e g}\right )}{f^2}+\frac {p \operatorname {PolyLog}\left (2,\frac {f \left (e x^n+d\right )}{d f-e g}\right )}{f^2}+\frac {e g p \log \left (d+e x^n\right )}{f^2 (d f-e g)}-\frac {e g p \log \left (f x^n+g\right )}{f^2 (d f-e g)}}{n}\)

Input:

Int[Log[c*(d + e*x^n)^p]/(x*(f + g/x^n)^2),x]
 

Output:

((e*g*p*Log[d + e*x^n])/(f^2*(d*f - e*g)) + (g*Log[c*(d + e*x^n)^p])/(f^2* 
(g + f*x^n)) - (e*g*p*Log[g + f*x^n])/(f^2*(d*f - e*g)) + (Log[c*(d + e*x^ 
n)^p]*Log[-((e*(g + f*x^n))/(d*f - e*g))])/f^2 + (p*PolyLog[2, (f*(d + e*x 
^n))/(d*f - e*g)])/f^2)/n
 

Defintions of rubi rules used

rule 2005

Int[(Fx_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[x^(m 
+ n*p)*(b + a/x^n)^p*Fx, x] /; FreeQ[{a, b, m, n}, x] && IntegerQ[p] && Neg 
Q[n]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2863

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_)) 
^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a 
 + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a, b, c 
, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]
 

rule 2925

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m 
_.)*((f_) + (g_.)*(x_)^(s_))^(r_.), x_Symbol] :> Simp[1/n   Subst[Int[x^(Si 
mplify[(m + 1)/n] - 1)*(f + g*x^(s/n))^r*(a + b*Log[c*(d + e*x)^p])^q, x], 
x, x^n], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q, r, s}, x] && Integer 
Q[r] && IntegerQ[s/n] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0 
] || IGtQ[q, 0])
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 16.21 (sec) , antiderivative size = 356, normalized size of antiderivative = 2.28

method	result	size
risch	\(\frac {\ln \left (\left (d +e \,x^{n}\right )^{p}\right ) \ln \left (g +f \,x^{n}\right )}{n \,f^{2}}+\frac {\ln \left (\left (d +e \,x^{n}\right )^{p}\right ) g}{n \,f^{2} \left (g +f \,x^{n}\right )}-\frac {p \operatorname {dilog}\left (\frac {\left (g +f \,x^{n}\right ) e +d f -e g}{d f -e g}\right )}{n \,f^{2}}-\frac {p \ln \left (g +f \,x^{n}\right ) \ln \left (\frac {\left (g +f \,x^{n}\right ) e +d f -e g}{d f -e g}\right )}{n \,f^{2}}+\frac {e p g \ln \left (\left (g +f \,x^{n}\right ) e +d f -e g \right )}{n \,f^{2} \left (d f -e g \right )}-\frac {e g p \ln \left (g +f \,x^{n}\right )}{f^{2} \left (d f -e g \right ) n}+\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) \operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \left (\frac {\ln \left (g +f \,x^{n}\right )}{n \,f^{2}}+\frac {g}{n \,f^{2} \left (g +f \,x^{n}\right )}\right )\)	\(356\)

Input:

int(ln(c*(d+e*x^n)^p)/x/(f+g/(x^n))^2,x,method=_RETURNVERBOSE)
 

Output:

1/n*ln((d+e*x^n)^p)/f^2*ln(g+f*x^n)+1/n*ln((d+e*x^n)^p)*g/f^2/(g+f*x^n)-1/ 
n*p/f^2*dilog(((g+f*x^n)*e+d*f-e*g)/(d*f-e*g))-1/n*p/f^2*ln(g+f*x^n)*ln((( 
g+f*x^n)*e+d*f-e*g)/(d*f-e*g))+1/n*e*p/f^2*g/(d*f-e*g)*ln((g+f*x^n)*e+d*f- 
e*g)-e*g*p*ln(g+f*x^n)/f^2/(d*f-e*g)/n+(1/2*I*Pi*csgn(I*(d+e*x^n)^p)*csgn( 
I*c*(d+e*x^n)^p)^2-1/2*I*Pi*csgn(I*(d+e*x^n)^p)*csgn(I*c*(d+e*x^n)^p)*csgn 
(I*c)-1/2*I*Pi*csgn(I*c*(d+e*x^n)^p)^3+1/2*I*Pi*csgn(I*c*(d+e*x^n)^p)^2*cs 
gn(I*c)+ln(c))*(1/n/f^2*ln(g+f*x^n)+1/n*g/f^2/(g+f*x^n))
 

Fricas [F]

\[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^{-n}\right )^2} \, dx=\int { \frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{{\left (f + \frac {g}{x^{n}}\right )}^{2} x} \,d x } \] Input:

integrate(log(c*(d+e*x^n)^p)/x/(f+g/(x^n))^2,x, algorithm="fricas")
 

Output:

integral(log((e*x^n + d)^p*c)/(f^2*x + 2*f*g*x*x^n/x^(2*n) + g^2*x/x^(2*n) 
), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^{-n}\right )^2} \, dx=\text {Timed out} \] Input:

integrate(ln(c*(d+e*x**n)**p)/x/(f+g/(x**n))**2,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.34 \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^{-n}\right )^2} \, dx=e n p {\left (\frac {d \log \left (\frac {e x^{n} + d}{e}\right )}{d e f^{2} n^{2} - e^{2} f g n^{2}} - \frac {g \log \left (\frac {f x^{n} + g}{f}\right )}{d f^{3} n^{2} - e f^{2} g n^{2}} - \frac {\log \left (f x^{n} + g\right ) \log \left (\frac {e f x^{n} + e g}{d f - e g} + 1\right ) + {\rm Li}_2\left (-\frac {e f x^{n} + e g}{d f - e g}\right )}{e f^{2} n^{2}}\right )} - {\left (\frac {1}{f^{2} n + \frac {f g n}{x^{n}}} - \frac {\log \left (f + \frac {g}{x^{n}}\right )}{f^{2} n} + \frac {\log \left (\frac {1}{x^{n}}\right )}{f^{2} n}\right )} \log \left ({\left (e x^{n} + d\right )}^{p} c\right ) \] Input:

integrate(log(c*(d+e*x^n)^p)/x/(f+g/(x^n))^2,x, algorithm="maxima")
 

Output:

e*n*p*(d*log((e*x^n + d)/e)/(d*e*f^2*n^2 - e^2*f*g*n^2) - g*log((f*x^n + g 
)/f)/(d*f^3*n^2 - e*f^2*g*n^2) - (log(f*x^n + g)*log((e*f*x^n + e*g)/(d*f 
- e*g) + 1) + dilog(-(e*f*x^n + e*g)/(d*f - e*g)))/(e*f^2*n^2)) - (1/(f^2* 
n + f*g*n/x^n) - log(f + g/x^n)/(f^2*n) + log(1/(x^n))/(f^2*n))*log((e*x^n 
 + d)^p*c)
 

Giac [F]

\[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^{-n}\right )^2} \, dx=\int { \frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{{\left (f + \frac {g}{x^{n}}\right )}^{2} x} \,d x } \] Input:

integrate(log(c*(d+e*x^n)^p)/x/(f+g/(x^n))^2,x, algorithm="giac")
 

Output:

integrate(log((e*x^n + d)^p*c)/((f + g/x^n)^2*x), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^{-n}\right )^2} \, dx=\int \frac {\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )}{x\,{\left (f+\frac {g}{x^n}\right )}^2} \,d x \] Input:

int(log(c*(d + e*x^n)^p)/(x*(f + g/x^n)^2),x)
 

Output:

int(log(c*(d + e*x^n)^p)/(x*(f + g/x^n)^2), x)
 

Reduce [F]

\[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^{-n}\right )^2} \, dx =\text {Too large to display} \] Input:

int(log(c*(d+e*x^n)^p)/x/(f+g/(x^n))^2,x)
 

Output:

(2*x**n*int((x**(2*n)*log((x**n*e + d)**p*c))/(x**(3*n)*e*f**2*x + x**(2*n 
)*d*f**2*x + 2*x**(2*n)*e*f*g*x + 2*x**n*d*f*g*x + x**n*e*g**2*x + d*g**2* 
x),x)*d**3*e*f**5*g*n*p - 6*x**n*int((x**(2*n)*log((x**n*e + d)**p*c))/(x* 
*(3*n)*e*f**2*x + x**(2*n)*d*f**2*x + 2*x**(2*n)*e*f*g*x + 2*x**n*d*f*g*x 
+ x**n*e*g**2*x + d*g**2*x),x)*d**2*e**2*f**4*g**2*n*p + 6*x**n*int((x**(2 
*n)*log((x**n*e + d)**p*c))/(x**(3*n)*e*f**2*x + x**(2*n)*d*f**2*x + 2*x** 
(2*n)*e*f*g*x + 2*x**n*d*f*g*x + x**n*e*g**2*x + d*g**2*x),x)*d*e**3*f**3* 
g**3*n*p - 2*x**n*int((x**(2*n)*log((x**n*e + d)**p*c))/(x**(3*n)*e*f**2*x 
 + x**(2*n)*d*f**2*x + 2*x**(2*n)*e*f*g*x + 2*x**n*d*f*g*x + x**n*e*g**2*x 
 + d*g**2*x),x)*e**4*f**2*g**4*n*p + 2*x**n*log(x**n*e + d)*d**3*f**4*p**2 
 - 6*x**n*log(x**n*e + d)*d**2*e*f**3*g*p**2 + 6*x**n*log(x**n*e + d)*d*e* 
*2*f**2*g**2*p**2 - 2*x**n*log(x**n*f + g)*e**3*f*g**3*p**2 + x**n*log((x* 
*n*e + d)**p*c)**2*d**2*e*f**3*g - x**n*log((x**n*e + d)**p*c)**2*d*e**2*f 
**2*g**2 - 2*x**n*log((x**n*e + d)**p*c)*d**3*f**4*p + 6*x**n*log((x**n*e 
+ d)**p*c)*d**2*e*f**3*g*p - 6*x**n*log((x**n*e + d)**p*c)*d*e**2*f**2*g** 
2*p + 2*x**n*log((x**n*e + d)**p*c)*e**3*f*g**3*p + 2*int((x**(2*n)*log((x 
**n*e + d)**p*c))/(x**(3*n)*e*f**2*x + x**(2*n)*d*f**2*x + 2*x**(2*n)*e*f* 
g*x + 2*x**n*d*f*g*x + x**n*e*g**2*x + d*g**2*x),x)*d**3*e*f**4*g**2*n*p - 
 6*int((x**(2*n)*log((x**n*e + d)**p*c))/(x**(3*n)*e*f**2*x + x**(2*n)*d*f 
**2*x + 2*x**(2*n)*e*f*g*x + 2*x**n*d*f*g*x + x**n*e*g**2*x + d*g**2*x)...