\(\int \frac {\log (c (d+e x^n)^p)}{x (f+g x^n)^2} \, dx\) [375]

Optimal result

Integrand size = 25, antiderivative size = 204 \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^n\right )^2} \, dx=-\frac {e p \log \left (d+e x^n\right )}{f (e f-d g) n}+\frac {\log \left (c \left (d+e x^n\right )^p\right )}{f n \left (f+g x^n\right )}+\frac {\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{f^2 n}+\frac {e p \log \left (f+g x^n\right )}{f (e f-d g) n}-\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac {e \left (f+g x^n\right )}{e f-d g}\right )}{f^2 n}-\frac {p \operatorname {PolyLog}\left (2,-\frac {g \left (d+e x^n\right )}{e f-d g}\right )}{f^2 n}+\frac {p \operatorname {PolyLog}\left (2,1+\frac {e x^n}{d}\right )}{f^2 n} \] Output:

-e*p*ln(d+e*x^n)/f/(-d*g+e*f)/n+ln(c*(d+e*x^n)^p)/f/n/(f+g*x^n)+ln(-e*x^n/ 
d)*ln(c*(d+e*x^n)^p)/f^2/n+e*p*ln(f+g*x^n)/f/(-d*g+e*f)/n-ln(c*(d+e*x^n)^p 
)*ln(e*(f+g*x^n)/(-d*g+e*f))/f^2/n-p*polylog(2,-g*(d+e*x^n)/(-d*g+e*f))/f^ 
2/n+p*polylog(2,1+e*x^n/d)/f^2/n
 

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.84 \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^n\right )^2} \, dx=\frac {-\frac {e f p \log \left (d+e x^n\right )}{e f-d g}+\frac {f \log \left (c \left (d+e x^n\right )^p\right )}{f+g x^n}+\log \left (-\frac {e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )+\frac {e f p \log \left (f+g x^n\right )}{e f-d g}-\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac {e \left (f+g x^n\right )}{e f-d g}\right )-p \operatorname {PolyLog}\left (2,\frac {g \left (d+e x^n\right )}{-e f+d g}\right )+p \operatorname {PolyLog}\left (2,1+\frac {e x^n}{d}\right )}{f^2 n} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.84 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.92, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {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.

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

\(\Big \downarrow \) 2925

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

\(\Big \downarrow \) 2863

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

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 = 6.14 (sec) , antiderivative size = 414, normalized size of antiderivative = 2.03

method	result	size
risch	\(-\frac {\ln \left (\left (d +e \,x^{n}\right )^{p}\right ) \ln \left (f +g \,x^{n}\right )}{n \,f^{2}}+\frac {\ln \left (\left (d +e \,x^{n}\right )^{p}\right )}{n f \left (f +g \,x^{n}\right )}+\frac {\ln \left (\left (d +e \,x^{n}\right )^{p}\right ) \ln \left (x^{n}\right )}{n \,f^{2}}-\frac {e p \ln \left (f +g \,x^{n}\right )}{n f \left (d g -e f \right )}+\frac {e p \ln \left (d +e \,x^{n}\right )}{n f \left (d g -e f \right )}-\frac {p \operatorname {dilog}\left (\frac {d +e \,x^{n}}{d}\right )}{n \,f^{2}}-\frac {p \ln \left (x^{n}\right ) \ln \left (\frac {d +e \,x^{n}}{d}\right )}{n \,f^{2}}+\frac {p \operatorname {dilog}\left (\frac {\left (f +g \,x^{n}\right ) e +d g -e f}{d g -e f}\right )}{n \,f^{2}}+\frac {p \ln \left (f +g \,x^{n}\right ) \ln \left (\frac {\left (f +g \,x^{n}\right ) e +d g -e f}{d g -e f}\right )}{n \,f^{2}}+\frac {\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) \operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \left (-\frac {\ln \left (f +g \,x^{n}\right )}{f^{2}}+\frac {1}{f \left (f +g \,x^{n}\right )}+\frac {\ln \left (x^{n}\right )}{f^{2}}\right )}{n}\)	\(414\)

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(f+g*x^n)+1/n*ln((d+e*x^n)^p)/f/(f+g*x^n)+1/n*l 
n((d+e*x^n)^p)/f^2*ln(x^n)-1/n*e*p/f/(d*g-e*f)*ln(f+g*x^n)+1/n*e*p/f/(d*g- 
e*f)*ln(d+e*x^n)-1/n*p/f^2*dilog((d+e*x^n)/d)-1/n*p/f^2*ln(x^n)*ln((d+e*x^ 
n)/d)+1/n*p/f^2*dilog(((f+g*x^n)*e+d*g-e*f)/(d*g-e*f))+1/n*p/f^2*ln(f+g*x^ 
n)*ln(((f+g*x^n)*e+d*g-e*f)/(d*g-e*f))+(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))/n*(-ln(f+g*x^n)/f^2+1/f/(f+g*x^n)+ln(x^n)/f^2)
 

Fricas [F]

\[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^n\right )^2} \, dx=\int { \frac {\log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{{\left (g x^{n} + f\right )}^{2} x} \,d x } \] 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)/(g^2*x*x^(2*n) + 2*f*g*x*x^n + f^2*x), 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 = 233, normalized size of antiderivative = 1.14 \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^n\right )^2} \, dx=-e n p {\left (\frac {\log \left (\frac {e x^{n} + d}{e}\right )}{e f^{2} n^{2} - d f g n^{2}} - \frac {\log \left (\frac {g x^{n} + f}{g}\right )}{e f^{2} n^{2} - d f g n^{2}} + \frac {\log \left (x^{n}\right ) \log \left (\frac {e x^{n}}{d} + 1\right ) + {\rm Li}_2\left (-\frac {e x^{n}}{d}\right )}{e f^{2} n^{2}} - \frac {\log \left (g x^{n} + f\right ) \log \left (-\frac {e g x^{n} + e f}{e f - d g} + 1\right ) + {\rm Li}_2\left (\frac {e g x^{n} + e f}{e f - d g}\right )}{e f^{2} n^{2}}\right )} + {\left (\frac {1}{f g n x^{n} + f^{2} n} - \frac {\log \left (g x^{n} + f\right )}{f^{2} n} + \frac {\log \left (x^{n}\right )}{f^{2} n}\right )} \log \left ({\left (e x^{n} + d\right )}^{p} c\right ) \] Input:

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

Output:

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

Giac [F]

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

Mupad [F(-1)]

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

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

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

Output:

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