\(\int \frac {\log (c (d+e x^n)^p)}{x (f+g x^{-2 n})^2} \, dx\) [377]

Optimal result

Integrand size = 27, antiderivative size = 377 \[ \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^{-2 n}\right )^2} \, dx=-\frac {d e \sqrt {g} p \arctan \left (\frac {\sqrt {f} x^n}{\sqrt {g}}\right )}{2 f^{3/2} \left (d^2 f+e^2 g\right ) n}-\frac {e^2 g p \log \left (d+e x^n\right )}{2 f^2 \left (d^2 f+e^2 g\right ) n}+\frac {g \log \left (c \left (d+e x^n\right )^p\right )}{2 f^2 n \left (g+f x^{2 n}\right )}+\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac {e \left (\sqrt {g}-\sqrt {-f} x^n\right )}{d \sqrt {-f}+e \sqrt {g}}\right )}{2 f^2 n}+\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (-\frac {e \left (\sqrt {g}+\sqrt {-f} x^n\right )}{d \sqrt {-f}-e \sqrt {g}}\right )}{2 f^2 n}+\frac {e^2 g p \log \left (g+f x^{2 n}\right )}{4 f^2 \left (d^2 f+e^2 g\right ) n}+\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt {-f} \left (d+e x^n\right )}{d \sqrt {-f}-e \sqrt {g}}\right )}{2 f^2 n}+\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt {-f} \left (d+e x^n\right )}{d \sqrt {-f}+e \sqrt {g}}\right )}{2 f^2 n} \] Output:

-1/2*d*e*g^(1/2)*p*arctan(f^(1/2)*x^n/g^(1/2))/f^(3/2)/(d^2*f+e^2*g)/n-1/2 
*e^2*g*p*ln(d+e*x^n)/f^2/(d^2*f+e^2*g)/n+1/2*g*ln(c*(d+e*x^n)^p)/f^2/n/(g+ 
f*x^(2*n))+1/2*ln(c*(d+e*x^n)^p)*ln(e*(g^(1/2)-(-f)^(1/2)*x^n)/(d*(-f)^(1/ 
2)+e*g^(1/2)))/f^2/n+1/2*ln(c*(d+e*x^n)^p)*ln(-e*(g^(1/2)+(-f)^(1/2)*x^n)/ 
(d*(-f)^(1/2)-e*g^(1/2)))/f^2/n+1/4*e^2*g*p*ln(g+f*x^(2*n))/f^2/(d^2*f+e^2 
*g)/n+1/2*p*polylog(2,(-f)^(1/2)*(d+e*x^n)/(d*(-f)^(1/2)-e*g^(1/2)))/f^2/n 
+1/2*p*polylog(2,(-f)^(1/2)*(d+e*x^n)/(d*(-f)^(1/2)+e*g^(1/2)))/f^2/n
 

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 1.38 (sec) , antiderivative size = 357, normalized size of antiderivative = 0.95, 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.

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

\(\Big \downarrow \) 2005

\(\displaystyle \int \frac {x^{4 n-1} \log \left (c \left (d+e x^n\right )^p\right )}{\left (f x^{2 n}+g\right )^2}dx\)

\(\Big \downarrow \) 2925

\(\displaystyle \frac {\int \frac {x^{3 n} \log \left (c \left (e x^n+d\right )^p\right )}{\left (f x^{2 n}+g\right )^2}dx^n}{n}\)

\(\Big \downarrow \) 2863

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {d e \sqrt {g} p \arctan \left (\frac {\sqrt {f} x^n}{\sqrt {g}}\right )}{2 f^{3/2} \left (d^2 f+e^2 g\right )}+\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac {e \left (\sqrt {g}-\sqrt {-f} x^n\right )}{d \sqrt {-f}+e \sqrt {g}}\right )}{2 f^2}+\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (-\frac {e \left (\sqrt {-f} x^n+\sqrt {g}\right )}{d \sqrt {-f}-e \sqrt {g}}\right )}{2 f^2}+\frac {g \log \left (c \left (d+e x^n\right )^p\right )}{2 f^2 \left (f x^{2 n}+g\right )}+\frac {e^2 g p \log \left (f x^{2 n}+g\right )}{4 f^2 \left (d^2 f+e^2 g\right )}-\frac {e^2 g p \log \left (d+e x^n\right )}{2 f^2 \left (d^2 f+e^2 g\right )}+\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt {-f} \left (e x^n+d\right )}{d \sqrt {-f}-e \sqrt {g}}\right )}{2 f^2}+\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt {-f} \left (e x^n+d\right )}{\sqrt {-f} d+e \sqrt {g}}\right )}{2 f^2}}{n}\)

Input:

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

Output:

(-1/2*(d*e*Sqrt[g]*p*ArcTan[(Sqrt[f]*x^n)/Sqrt[g]])/(f^(3/2)*(d^2*f + e^2* 
g)) - (e^2*g*p*Log[d + e*x^n])/(2*f^2*(d^2*f + e^2*g)) + (g*Log[c*(d + e*x 
^n)^p])/(2*f^2*(g + f*x^(2*n))) + (Log[c*(d + e*x^n)^p]*Log[(e*(Sqrt[g] - 
Sqrt[-f]*x^n))/(d*Sqrt[-f] + e*Sqrt[g])])/(2*f^2) + (Log[c*(d + e*x^n)^p]* 
Log[-((e*(Sqrt[g] + Sqrt[-f]*x^n))/(d*Sqrt[-f] - e*Sqrt[g]))])/(2*f^2) + ( 
e^2*g*p*Log[g + f*x^(2*n)])/(4*f^2*(d^2*f + e^2*g)) + (p*PolyLog[2, (Sqrt[ 
-f]*(d + e*x^n))/(d*Sqrt[-f] - e*Sqrt[g])])/(2*f^2) + (p*PolyLog[2, (Sqrt[ 
-f]*(d + e*x^n))/(d*Sqrt[-f] + e*Sqrt[g])])/(2*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 = 118.10 (sec) , antiderivative size = 561, normalized size of antiderivative = 1.49

method	result	size
risch	\(\frac {\ln \left (\left (d +e \,x^{n}\right )^{p}\right ) \ln \left (g +f \,x^{2 n}\right )}{2 n \,f^{2}}+\frac {\ln \left (\left (d +e \,x^{n}\right )^{p}\right ) g}{2 n \,f^{2} \left (g +f \,x^{2 n}\right )}-\frac {p \ln \left (d +e \,x^{n}\right ) \ln \left (g +f \,x^{2 n}\right )}{2 n \,f^{2}}+\frac {p \ln \left (d +e \,x^{n}\right ) \ln \left (\frac {e \sqrt {-g f}-f \left (d +e \,x^{n}\right )+d f}{e \sqrt {-g f}+d f}\right )}{2 n \,f^{2}}+\frac {p \ln \left (d +e \,x^{n}\right ) \ln \left (\frac {e \sqrt {-g f}+f \left (d +e \,x^{n}\right )-d f}{e \sqrt {-g f}-d f}\right )}{2 n \,f^{2}}+\frac {p \operatorname {dilog}\left (\frac {e \sqrt {-g f}-f \left (d +e \,x^{n}\right )+d f}{e \sqrt {-g f}+d f}\right )}{2 n \,f^{2}}+\frac {p \operatorname {dilog}\left (\frac {e \sqrt {-g f}+f \left (d +e \,x^{n}\right )-d f}{e \sqrt {-g f}-d f}\right )}{2 n \,f^{2}}-\frac {e^{2} g p \ln \left (d +e \,x^{n}\right )}{2 f^{2} \left (d^{2} f +e^{2} g \right ) n}+\frac {e^{2} g p \ln \left (g +f \,x^{2 n}\right )}{4 f^{2} \left (d^{2} f +e^{2} g \right ) n}-\frac {e p g d \arctan \left (\frac {x^{n} f}{\sqrt {g f}}\right )}{2 n f \left (d^{2} f +e^{2} g \right ) \sqrt {g f}}+\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) \operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \left (\frac {\ln \left (g +f \,x^{2 n}\right )}{2 n \,f^{2}}+\frac {g}{2 n \,f^{2} \left (g +f \,x^{2 n}\right )}\right )\)	\(561\)

Input:

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

Output:

1/2/n*ln((d+e*x^n)^p)/f^2*ln(g+f*(x^n)^2)+1/2/n*ln((d+e*x^n)^p)*g/f^2/(g+f 
*(x^n)^2)-1/2/n*p/f^2*ln(d+e*x^n)*ln(g+f*(x^n)^2)+1/2/n*p/f^2*ln(d+e*x^n)* 
ln((e*(-g*f)^(1/2)-f*(d+e*x^n)+d*f)/(e*(-g*f)^(1/2)+d*f))+1/2/n*p/f^2*ln(d 
+e*x^n)*ln((e*(-g*f)^(1/2)+f*(d+e*x^n)-d*f)/(e*(-g*f)^(1/2)-d*f))+1/2/n*p/ 
f^2*dilog((e*(-g*f)^(1/2)-f*(d+e*x^n)+d*f)/(e*(-g*f)^(1/2)+d*f))+1/2/n*p/f 
^2*dilog((e*(-g*f)^(1/2)+f*(d+e*x^n)-d*f)/(e*(-g*f)^(1/2)-d*f))-1/2*e^2*g* 
p*ln(d+e*x^n)/f^2/(d^2*f+e^2*g)/n+1/4/n*e^2*p*g/f^2/(d^2*f+e^2*g)*ln(g+f*( 
x^n)^2)-1/2/n*e*p*g/f/(d^2*f+e^2*g)*d/(g*f)^(1/2)*arctan(x^n*f/(g*f)^(1/2) 
)+(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*csgn(I*c)+ln(c))*(1/2/n/f^2*ln(g+f*(x^n 
)^2)+1/2/n*g/f^2/(g+f*(x^n)^2))
 

Fricas [F]

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

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

Output:

integral(log((e*x^n + d)^p*c)/(f^2*x + 2*f*g*x*x^(2*n)/x^(4*n) + g^2*x/x^( 
4*n)), x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(6*x**(2*n)*sqrt(g)*sqrt(f)*atan((x**n*f)/(sqrt(g)*sqrt(f)))*d*e**3*f*g*p* 
*2 + 6*sqrt(g)*sqrt(f)*atan((x**n*f)/(sqrt(g)*sqrt(f)))*d*e**3*g**2*p**2 + 
 2*x**(2*n)*int((x**(5*n)*log((x**n*e + d)**p*c))/(x**(5*n)*e*f**2*x + x** 
(4*n)*d*f**2*x + 2*x**(3*n)*e*f*g*x + 2*x**(2*n)*d*f*g*x + x**n*e*g**2*x + 
 d*g**2*x),x)*d**4*e*f**5*n*p + 8*x**(2*n)*int((x**(5*n)*log((x**n*e + d)* 
*p*c))/(x**(5*n)*e*f**2*x + x**(4*n)*d*f**2*x + 2*x**(3*n)*e*f*g*x + 2*x** 
(2*n)*d*f*g*x + x**n*e*g**2*x + d*g**2*x),x)*d**2*e**3*f**4*g*n*p + 6*x**( 
2*n)*int((x**(5*n)*log((x**n*e + d)**p*c))/(x**(5*n)*e*f**2*x + x**(4*n)*d 
*f**2*x + 2*x**(3*n)*e*f*g*x + 2*x**(2*n)*d*f*g*x + x**n*e*g**2*x + d*g**2 
*x),x)*e**5*f**3*g**2*n*p + 6*x**(2*n)*int((x**n*log((x**n*e + d)**p*c))/( 
x**(5*n)*e*f**2*x + x**(4*n)*d*f**2*x + 2*x**(3*n)*e*f*g*x + 2*x**(2*n)*d* 
f*g*x + x**n*e*g**2*x + d*g**2*x),x)*d**4*e*f**3*g**2*n*p + 8*x**(2*n)*int 
((x**n*log((x**n*e + d)**p*c))/(x**(5*n)*e*f**2*x + x**(4*n)*d*f**2*x + 2* 
x**(3*n)*e*f*g*x + 2*x**(2*n)*d*f*g*x + x**n*e*g**2*x + d*g**2*x),x)*d**2* 
e**3*f**2*g**3*n*p + 2*x**(2*n)*int((x**n*log((x**n*e + d)**p*c))/(x**(5*n 
)*e*f**2*x + x**(4*n)*d*f**2*x + 2*x**(3*n)*e*f*g*x + 2*x**(2*n)*d*f*g*x + 
 x**n*e*g**2*x + d*g**2*x),x)*e**5*f*g**4*n*p + 2*x**(2*n)*log(x**(2*n)*f 
+ g)*d**2*e**2*f**2*g*p**2 - x**(2*n)*log(x**(2*n)*f + g)*e**4*f*g**2*p**2 
 - 4*x**(2*n)*log(x**n*e + d)*d**4*f**3*p**2 - 20*x**(2*n)*log(x**n*e + d) 
*d**2*e**2*f**2*g*p**2 - 10*x**(2*n)*log(x**n*e + d)*e**4*f*g**2*p**2 -...