\(\int \frac {(a+b \log (c x^n))^2 \log (d (\frac {1}{d}+f x^2))}{x^4} \, dx\) [45]

Optimal result

Integrand size = 28, antiderivative size = 490 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^4} \, dx=-\frac {52 b^2 d f n^2}{27 x}-\frac {4}{27} b^2 d^{3/2} f^{3/2} n^2 \arctan \left (\sqrt {d} \sqrt {f} x\right )-\frac {16 b d f n \left (a+b \log \left (c x^n\right )\right )}{9 x}-\frac {4}{9} b d^{3/2} f^{3/2} n \arctan \left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {2 d f \left (a+b \log \left (c x^n\right )\right )^2}{3 x}-\frac {2}{3} d^{3/2} f^{3/2} \arctan \left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {2 b^2 n^2 \log \left (1+d f x^2\right )}{27 x^3}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{9 x^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{3 x^3}-\frac {2}{9} b^2 (-d)^{3/2} f^{3/2} n^2 \operatorname {PolyLog}\left (2,-\sqrt {-d} \sqrt {f} x\right )-\frac {2}{3} b (-d)^{3/2} f^{3/2} n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\sqrt {-d} \sqrt {f} x\right )+\frac {2}{9} b^2 (-d)^{3/2} f^{3/2} n^2 \operatorname {PolyLog}\left (2,\sqrt {-d} \sqrt {f} x\right )+\frac {2}{3} b (-d)^{3/2} f^{3/2} n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,\sqrt {-d} \sqrt {f} x\right )+\frac {2}{3} b^2 (-d)^{3/2} f^{3/2} n^2 \operatorname {PolyLog}\left (3,-\sqrt {-d} \sqrt {f} x\right )-\frac {2}{3} b^2 (-d)^{3/2} f^{3/2} n^2 \operatorname {PolyLog}\left (3,\sqrt {-d} \sqrt {f} x\right ) \] Output:

-52/27*b^2*d*f*n^2/x-4/27*b^2*d^(3/2)*f^(3/2)*n^2*arctan(d^(1/2)*f^(1/2)*x 
)-16/9*b*d*f*n*(a+b*ln(c*x^n))/x-4/9*b*d^(3/2)*f^(3/2)*n*arctan(d^(1/2)*f^ 
(1/2)*x)*(a+b*ln(c*x^n))-2/3*d*f*(a+b*ln(c*x^n))^2/x-2/3*d^(3/2)*f^(3/2)*a 
rctan(d^(1/2)*f^(1/2)*x)*(a+b*ln(c*x^n))^2-2/27*b^2*n^2*ln(d*f*x^2+1)/x^3- 
2/9*b*n*(a+b*ln(c*x^n))*ln(d*f*x^2+1)/x^3-1/3*(a+b*ln(c*x^n))^2*ln(d*f*x^2 
+1)/x^3-2/9*b^2*(-d)^(3/2)*f^(3/2)*n^2*polylog(2,-(-d)^(1/2)*f^(1/2)*x)-2/ 
3*b*(-d)^(3/2)*f^(3/2)*n*(a+b*ln(c*x^n))*polylog(2,-(-d)^(1/2)*f^(1/2)*x)+ 
2/9*b^2*(-d)^(3/2)*f^(3/2)*n^2*polylog(2,(-d)^(1/2)*f^(1/2)*x)+2/3*b*(-d)^ 
(3/2)*f^(3/2)*n*(a+b*ln(c*x^n))*polylog(2,(-d)^(1/2)*f^(1/2)*x)+2/3*b^2*(- 
d)^(3/2)*f^(3/2)*n^2*polylog(3,-(-d)^(1/2)*f^(1/2)*x)-2/3*b^2*(-d)^(3/2)*f 
^(3/2)*n^2*polylog(3,(-d)^(1/2)*f^(1/2)*x)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.67 (sec) , antiderivative size = 585, normalized size of antiderivative = 1.19 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^4} \, dx=\frac {1}{27} \left (-2 d^{3/2} f^{3/2} \arctan \left (\sqrt {d} \sqrt {f} x\right ) \left (9 a^2+6 a b n+2 b^2 n^2+18 a b \left (-n \log (x)+\log \left (c x^n\right )\right )+6 b^2 n \left (-n \log (x)+\log \left (c x^n\right )\right )+9 b^2 \left (-n \log (x)+\log \left (c x^n\right )\right )^2\right )-\frac {2 d f \left (9 a^2+6 a b n+2 b^2 n^2+9 b^2 n^2 \log ^2(x)+6 b (3 a+b n) \log \left (c x^n\right )+9 b^2 \log ^2\left (c x^n\right )-6 b n \log (x) \left (3 a+b n+3 b \log \left (c x^n\right )\right )\right )}{x}-\frac {\left (9 a^2+6 a b n+2 b^2 n^2+6 b (3 a+b n) \log \left (c x^n\right )+9 b^2 \log ^2\left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{x^3}+\frac {6 i b d f n \left (3 a+b n-3 b n \log (x)+3 b \log \left (c x^n\right )\right ) \left (2 i+2 i \log (x)+\sqrt {d} \sqrt {f} x \left (\log (x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )+\operatorname {PolyLog}\left (2,-i \sqrt {d} \sqrt {f} x\right )\right )-\sqrt {d} \sqrt {f} x \left (\log (x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )+\operatorname {PolyLog}\left (2,i \sqrt {d} \sqrt {f} x\right )\right )\right )}{x}+\frac {9 i b^2 d f n^2 \left (4 i+4 i \log (x)+2 i \log ^2(x)+\sqrt {d} \sqrt {f} x \left (\log ^2(x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )+2 \log (x) \operatorname {PolyLog}\left (2,-i \sqrt {d} \sqrt {f} x\right )-2 \operatorname {PolyLog}\left (3,-i \sqrt {d} \sqrt {f} x\right )\right )-\sqrt {d} \sqrt {f} x \left (\log ^2(x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )+2 \log (x) \operatorname {PolyLog}\left (2,i \sqrt {d} \sqrt {f} x\right )-2 \operatorname {PolyLog}\left (3,i \sqrt {d} \sqrt {f} x\right )\right )\right )}{x}\right ) \] Input:

Integrate[((a + b*Log[c*x^n])^2*Log[d*(d^(-1) + f*x^2)])/x^4,x]
 

Output:

(-2*d^(3/2)*f^(3/2)*ArcTan[Sqrt[d]*Sqrt[f]*x]*(9*a^2 + 6*a*b*n + 2*b^2*n^2 
 + 18*a*b*(-(n*Log[x]) + Log[c*x^n]) + 6*b^2*n*(-(n*Log[x]) + Log[c*x^n]) 
+ 9*b^2*(-(n*Log[x]) + Log[c*x^n])^2) - (2*d*f*(9*a^2 + 6*a*b*n + 2*b^2*n^ 
2 + 9*b^2*n^2*Log[x]^2 + 6*b*(3*a + b*n)*Log[c*x^n] + 9*b^2*Log[c*x^n]^2 - 
 6*b*n*Log[x]*(3*a + b*n + 3*b*Log[c*x^n])))/x - ((9*a^2 + 6*a*b*n + 2*b^2 
*n^2 + 6*b*(3*a + b*n)*Log[c*x^n] + 9*b^2*Log[c*x^n]^2)*Log[1 + d*f*x^2])/ 
x^3 + ((6*I)*b*d*f*n*(3*a + b*n - 3*b*n*Log[x] + 3*b*Log[c*x^n])*(2*I + (2 
*I)*Log[x] + Sqrt[d]*Sqrt[f]*x*(Log[x]*Log[1 + I*Sqrt[d]*Sqrt[f]*x] + Poly 
Log[2, (-I)*Sqrt[d]*Sqrt[f]*x]) - Sqrt[d]*Sqrt[f]*x*(Log[x]*Log[1 - I*Sqrt 
[d]*Sqrt[f]*x] + PolyLog[2, I*Sqrt[d]*Sqrt[f]*x])))/x + ((9*I)*b^2*d*f*n^2 
*(4*I + (4*I)*Log[x] + (2*I)*Log[x]^2 + Sqrt[d]*Sqrt[f]*x*(Log[x]^2*Log[1 
+ I*Sqrt[d]*Sqrt[f]*x] + 2*Log[x]*PolyLog[2, (-I)*Sqrt[d]*Sqrt[f]*x] - 2*P 
olyLog[3, (-I)*Sqrt[d]*Sqrt[f]*x]) - Sqrt[d]*Sqrt[f]*x*(Log[x]^2*Log[1 - I 
*Sqrt[d]*Sqrt[f]*x] + 2*Log[x]*PolyLog[2, I*Sqrt[d]*Sqrt[f]*x] - 2*PolyLog 
[3, I*Sqrt[d]*Sqrt[f]*x])))/x)/27
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.08 (sec) , antiderivative size = 544, normalized size of antiderivative = 1.11, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2825, 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.

\(\displaystyle \int \frac {\log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x^4} \, dx\)

\(\Big \downarrow \) 2825

\(\displaystyle -2 f \int \left (-\frac {2 b^2 d n^2}{27 x^2 \left (d f x^2+1\right )}-\frac {2 b d \left (a+b \log \left (c x^n\right )\right ) n}{9 x^2 \left (d f x^2+1\right )}-\frac {d \left (a+b \log \left (c x^n\right )\right )^2}{3 x^2 \left (d f x^2+1\right )}\right )dx-\frac {2 b n \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )}{9 x^3}-\frac {\log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{3 x^3}-\frac {2 b^2 n^2 \log \left (d f x^2+1\right )}{27 x^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle -2 f \left (\frac {2}{9} b d^{3/2} \sqrt {f} n \arctan \left (\sqrt {d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {1}{3} b (-d)^{3/2} \sqrt {f} n \operatorname {PolyLog}\left (2,-\sqrt {-d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{3} b (-d)^{3/2} \sqrt {f} n \operatorname {PolyLog}\left (2,\sqrt {-d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{6} (-d)^{3/2} \sqrt {f} \log \left (1-\sqrt {-d} \sqrt {f} x\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {1}{6} (-d)^{3/2} \sqrt {f} \log \left (\sqrt {-d} \sqrt {f} x+1\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {8 b d n \left (a+b \log \left (c x^n\right )\right )}{9 x}+\frac {d \left (a+b \log \left (c x^n\right )\right )^2}{3 x}+\frac {2}{27} b^2 d^{3/2} \sqrt {f} n^2 \arctan \left (\sqrt {d} \sqrt {f} x\right )-\frac {1}{9} i b^2 d^{3/2} \sqrt {f} n^2 \operatorname {PolyLog}\left (2,-i \sqrt {d} \sqrt {f} x\right )+\frac {1}{9} i b^2 d^{3/2} \sqrt {f} n^2 \operatorname {PolyLog}\left (2,i \sqrt {d} \sqrt {f} x\right )-\frac {1}{3} b^2 (-d)^{3/2} \sqrt {f} n^2 \operatorname {PolyLog}\left (3,-\sqrt {-d} \sqrt {f} x\right )+\frac {1}{3} b^2 (-d)^{3/2} \sqrt {f} n^2 \operatorname {PolyLog}\left (3,\sqrt {-d} \sqrt {f} x\right )+\frac {26 b^2 d n^2}{27 x}\right )-\frac {2 b n \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )}{9 x^3}-\frac {\log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{3 x^3}-\frac {2 b^2 n^2 \log \left (d f x^2+1\right )}{27 x^3}\)

Input:

Int[((a + b*Log[c*x^n])^2*Log[d*(d^(-1) + f*x^2)])/x^4,x]
 

Output:

(-2*b^2*n^2*Log[1 + d*f*x^2])/(27*x^3) - (2*b*n*(a + b*Log[c*x^n])*Log[1 + 
 d*f*x^2])/(9*x^3) - ((a + b*Log[c*x^n])^2*Log[1 + d*f*x^2])/(3*x^3) - 2*f 
*((26*b^2*d*n^2)/(27*x) + (2*b^2*d^(3/2)*Sqrt[f]*n^2*ArcTan[Sqrt[d]*Sqrt[f 
]*x])/27 + (8*b*d*n*(a + b*Log[c*x^n]))/(9*x) + (2*b*d^(3/2)*Sqrt[f]*n*Arc 
Tan[Sqrt[d]*Sqrt[f]*x]*(a + b*Log[c*x^n]))/9 + (d*(a + b*Log[c*x^n])^2)/(3 
*x) - ((-d)^(3/2)*Sqrt[f]*(a + b*Log[c*x^n])^2*Log[1 - Sqrt[-d]*Sqrt[f]*x] 
)/6 + ((-d)^(3/2)*Sqrt[f]*(a + b*Log[c*x^n])^2*Log[1 + Sqrt[-d]*Sqrt[f]*x] 
)/6 + (b*(-d)^(3/2)*Sqrt[f]*n*(a + b*Log[c*x^n])*PolyLog[2, -(Sqrt[-d]*Sqr 
t[f]*x)])/3 - (b*(-d)^(3/2)*Sqrt[f]*n*(a + b*Log[c*x^n])*PolyLog[2, Sqrt[- 
d]*Sqrt[f]*x])/3 - (I/9)*b^2*d^(3/2)*Sqrt[f]*n^2*PolyLog[2, (-I)*Sqrt[d]*S 
qrt[f]*x] + (I/9)*b^2*d^(3/2)*Sqrt[f]*n^2*PolyLog[2, I*Sqrt[d]*Sqrt[f]*x] 
- (b^2*(-d)^(3/2)*Sqrt[f]*n^2*PolyLog[3, -(Sqrt[-d]*Sqrt[f]*x)])/3 + (b^2* 
(-d)^(3/2)*Sqrt[f]*n^2*PolyLog[3, Sqrt[-d]*Sqrt[f]*x])/3)
 

Defintions of rubi rules used

rule 2009

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

rule 2825

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_. 
)]*(b_.))^(p_.)*((g_.)*(x_))^(q_.), x_Symbol] :> With[{u = IntHide[(g*x)^q* 
(a + b*Log[c*x^n])^p, x]}, Simp[Log[d*(e + f*x^m)^r]   u, x] - Simp[f*m*r 
 Int[x^(m - 1)/(e + f*x^m)   u, x], x]] /; FreeQ[{a, b, c, d, e, f, g, r, m 
, n, q}, x] && IGtQ[p, 0] && RationalQ[m] && RationalQ[q]
 

Maple [F]

\[\int \frac {{\left (a +b \ln \left (c \,x^{n}\right )\right )}^{2} \ln \left (d \left (\frac {1}{d}+f \,x^{2}\right )\right )}{x^{4}}d x\]

Input:

int((a+b*ln(c*x^n))^2*ln(d*(1/d+f*x^2))/x^4,x)
 

Output:

int((a+b*ln(c*x^n))^2*ln(d*(1/d+f*x^2))/x^4,x)
 

Fricas [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^4} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right )}{x^{4}} \,d x } \] Input:

integrate((a+b*log(c*x^n))^2*log(d*(1/d+f*x^2))/x^4,x, algorithm="fricas")
 

Output:

integral((b^2*log(d*f*x^2 + 1)*log(c*x^n)^2 + 2*a*b*log(d*f*x^2 + 1)*log(c 
*x^n) + a^2*log(d*f*x^2 + 1))/x^4, x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^4} \, dx=\text {Timed out} \] Input:

integrate((a+b*ln(c*x**n))**2*ln(d*(1/d+f*x**2))/x**4,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^4} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right )}{x^{4}} \,d x } \] Input:

integrate((a+b*log(c*x^n))^2*log(d*(1/d+f*x^2))/x^4,x, algorithm="maxima")
 

Output:

-1/27*(9*b^2*log(x^n)^2 + (2*n^2 + 6*n*log(c) + 9*log(c)^2)*b^2 + 6*a*b*(n 
 + 3*log(c)) + 9*a^2 + 6*(b^2*(n + 3*log(c)) + 3*a*b)*log(x^n))*log(d*f*x^ 
2 + 1)/x^3 + integrate(2/27*(9*b^2*d*f*log(x^n)^2 + 9*a^2*d*f + 6*(d*f*n + 
 3*d*f*log(c))*a*b + (2*d*f*n^2 + 6*d*f*n*log(c) + 9*d*f*log(c)^2)*b^2 + 6 
*(3*a*b*d*f + (d*f*n + 3*d*f*log(c))*b^2)*log(x^n))/(d*f*x^4 + x^2), x)
 

Giac [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^4} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right )}{x^{4}} \,d x } \] Input:

integrate((a+b*log(c*x^n))^2*log(d*(1/d+f*x^2))/x^4,x, algorithm="giac")
 

Output:

integrate((b*log(c*x^n) + a)^2*log((f*x^2 + 1/d)*d)/x^4, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^4} \, dx=\int \frac {\ln \left (d\,\left (f\,x^2+\frac {1}{d}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x^4} \,d x \] Input:

int((log(d*(f*x^2 + 1/d))*(a + b*log(c*x^n))^2)/x^4,x)
 

Output:

int((log(d*(f*x^2 + 1/d))*(a + b*log(c*x^n))^2)/x^4, x)
 

Reduce [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^4} \, dx=\frac {-54 \sqrt {f}\, \sqrt {d}\, \mathit {atan} \left (\frac {d f x}{\sqrt {f}\, \sqrt {d}}\right ) a^{2} d f \,x^{3}-36 \sqrt {f}\, \sqrt {d}\, \mathit {atan} \left (\frac {d f x}{\sqrt {f}\, \sqrt {d}}\right ) a b d f n \,x^{3}-12 \sqrt {f}\, \sqrt {d}\, \mathit {atan} \left (\frac {d f x}{\sqrt {f}\, \sqrt {d}}\right ) b^{2} d f \,n^{2} x^{3}-54 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )^{2}}{d f \,x^{6}+x^{4}}d x \right ) b^{2} x^{3}-108 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{d f \,x^{6}+x^{4}}d x \right ) a b \,x^{3}-36 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{d f \,x^{6}+x^{4}}d x \right ) b^{2} n \,x^{3}-27 \,\mathrm {log}\left (d f \,x^{2}+1\right ) \mathrm {log}\left (x^{n} c \right )^{2} b^{2}-54 \,\mathrm {log}\left (d f \,x^{2}+1\right ) \mathrm {log}\left (x^{n} c \right ) a b -18 \,\mathrm {log}\left (d f \,x^{2}+1\right ) \mathrm {log}\left (x^{n} c \right ) b^{2} n -27 \,\mathrm {log}\left (d f \,x^{2}+1\right ) a^{2}-18 \,\mathrm {log}\left (d f \,x^{2}+1\right ) a b n -6 \,\mathrm {log}\left (d f \,x^{2}+1\right ) b^{2} n^{2}-18 \mathrm {log}\left (x^{n} c \right )^{2} b^{2}-36 \,\mathrm {log}\left (x^{n} c \right ) a b -24 \,\mathrm {log}\left (x^{n} c \right ) b^{2} n -54 a^{2} d f \,x^{2}-36 a b d f n \,x^{2}-12 a b n -12 b^{2} d f \,n^{2} x^{2}-8 b^{2} n^{2}}{81 x^{3}} \] Input:

int((a+b*log(c*x^n))^2*log(d*(1/d+f*x^2))/x^4,x)
 

Output:

( - 54*sqrt(f)*sqrt(d)*atan((d*f*x)/(sqrt(f)*sqrt(d)))*a**2*d*f*x**3 - 36* 
sqrt(f)*sqrt(d)*atan((d*f*x)/(sqrt(f)*sqrt(d)))*a*b*d*f*n*x**3 - 12*sqrt(f 
)*sqrt(d)*atan((d*f*x)/(sqrt(f)*sqrt(d)))*b**2*d*f*n**2*x**3 - 54*int(log( 
x**n*c)**2/(d*f*x**6 + x**4),x)*b**2*x**3 - 108*int(log(x**n*c)/(d*f*x**6 
+ x**4),x)*a*b*x**3 - 36*int(log(x**n*c)/(d*f*x**6 + x**4),x)*b**2*n*x**3 
- 27*log(d*f*x**2 + 1)*log(x**n*c)**2*b**2 - 54*log(d*f*x**2 + 1)*log(x**n 
*c)*a*b - 18*log(d*f*x**2 + 1)*log(x**n*c)*b**2*n - 27*log(d*f*x**2 + 1)*a 
**2 - 18*log(d*f*x**2 + 1)*a*b*n - 6*log(d*f*x**2 + 1)*b**2*n**2 - 18*log( 
x**n*c)**2*b**2 - 36*log(x**n*c)*a*b - 24*log(x**n*c)*b**2*n - 54*a**2*d*f 
*x**2 - 36*a*b*d*f*n*x**2 - 12*a*b*n - 12*b**2*d*f*n**2*x**2 - 8*b**2*n**2 
)/(81*x**3)