\(\int \frac {(a+b \log (c x^n))^3 \log (d (\frac {1}{d}+f x^2))}{x^3} \, dx\) [49]

Optimal result

Integrand size = 28, antiderivative size = 425 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^3} \, dx=\frac {3}{4} b^3 d f n^3 \log (x)-\frac {3}{4} b^2 d f n^2 \log \left (1+\frac {1}{d f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {3}{4} b d f n \log \left (1+\frac {1}{d f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{2} d f \log \left (1+\frac {1}{d f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {3}{8} b^3 d f n^3 \log \left (1+d f x^2\right )-\frac {3 b^3 n^3 \log \left (1+d f x^2\right )}{8 x^2}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{4 x^2}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+d f x^2\right )}{4 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+d f x^2\right )}{2 x^2}+\frac {3}{8} b^3 d f n^3 \operatorname {PolyLog}\left (2,-\frac {1}{d f x^2}\right )+\frac {3}{4} b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {1}{d f x^2}\right )+\frac {3}{4} b d f n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {1}{d f x^2}\right )+\frac {3}{8} b^3 d f n^3 \operatorname {PolyLog}\left (3,-\frac {1}{d f x^2}\right )+\frac {3}{4} b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-\frac {1}{d f x^2}\right )+\frac {3}{8} b^3 d f n^3 \operatorname {PolyLog}\left (4,-\frac {1}{d f x^2}\right ) \] Output:

3/4*b^3*d*f*n^3*ln(x)-3/4*b^2*d*f*n^2*ln(1+1/d/f/x^2)*(a+b*ln(c*x^n))-3/4* 
b*d*f*n*ln(1+1/d/f/x^2)*(a+b*ln(c*x^n))^2-1/2*d*f*ln(1+1/d/f/x^2)*(a+b*ln( 
c*x^n))^3-3/8*b^3*d*f*n^3*ln(d*f*x^2+1)-3/8*b^3*n^3*ln(d*f*x^2+1)/x^2-3/4* 
b^2*n^2*(a+b*ln(c*x^n))*ln(d*f*x^2+1)/x^2-3/4*b*n*(a+b*ln(c*x^n))^2*ln(d*f 
*x^2+1)/x^2-1/2*(a+b*ln(c*x^n))^3*ln(d*f*x^2+1)/x^2+3/8*b^3*d*f*n^3*polylo 
g(2,-1/d/f/x^2)+3/4*b^2*d*f*n^2*(a+b*ln(c*x^n))*polylog(2,-1/d/f/x^2)+3/4* 
b*d*f*n*(a+b*ln(c*x^n))^2*polylog(2,-1/d/f/x^2)+3/8*b^3*d*f*n^3*polylog(3, 
-1/d/f/x^2)+3/4*b^2*d*f*n^2*(a+b*ln(c*x^n))*polylog(3,-1/d/f/x^2)+3/8*b^3* 
d*f*n^3*polylog(4,-1/d/f/x^2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.45 (sec) , antiderivative size = 940, normalized size of antiderivative = 2.21 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^3} \, dx =\text {Too large to display} \] Input:

Integrate[((a + b*Log[c*x^n])^3*Log[d*(d^(-1) + f*x^2)])/x^3,x]
 

Output:

(2*d*f*Log[x]*(4*a^3 + 6*a^2*b*n + 6*a*b^2*n^2 + 3*b^3*n^3 + 12*a^2*b*(-(n 
*Log[x]) + Log[c*x^n]) + 12*a*b^2*n*(-(n*Log[x]) + Log[c*x^n]) + 6*b^3*n^2 
*(-(n*Log[x]) + Log[c*x^n]) + 12*a*b^2*(-(n*Log[x]) + Log[c*x^n])^2 + 6*b^ 
3*n*(-(n*Log[x]) + Log[c*x^n])^2 + 4*b^3*(-(n*Log[x]) + Log[c*x^n])^3) - ( 
(4*a^3 + 6*a^2*b*n + 6*a*b^2*n^2 + 3*b^3*n^3 + 6*b*(2*a^2 + 2*a*b*n + b^2* 
n^2)*Log[c*x^n] + 6*b^2*(2*a + b*n)*Log[c*x^n]^2 + 4*b^3*Log[c*x^n]^3)*Log 
[1 + d*f*x^2])/x^2 - d*f*(4*a^3 + 6*a^2*b*n + 6*a*b^2*n^2 + 3*b^3*n^3 + 12 
*a^2*b*(-(n*Log[x]) + Log[c*x^n]) + 12*a*b^2*n*(-(n*Log[x]) + Log[c*x^n]) 
+ 6*b^3*n^2*(-(n*Log[x]) + Log[c*x^n]) + 12*a*b^2*(-(n*Log[x]) + Log[c*x^n 
])^2 + 6*b^3*n*(-(n*Log[x]) + Log[c*x^n])^2 + 4*b^3*(-(n*Log[x]) + Log[c*x 
^n])^3)*Log[1 + d*f*x^2] + 6*b*d*f*n*(2*a^2 + 2*a*b*n + b^2*n^2 + 4*a*b*(- 
(n*Log[x]) + Log[c*x^n]) + 2*b^2*n*(-(n*Log[x]) + Log[c*x^n]) + 2*b^2*(-(n 
*Log[x]) + Log[c*x^n])^2)*(Log[x]*(Log[x] - Log[1 - I*Sqrt[d]*Sqrt[f]*x] - 
 Log[1 + I*Sqrt[d]*Sqrt[f]*x]) - PolyLog[2, (-I)*Sqrt[d]*Sqrt[f]*x] - Poly 
Log[2, I*Sqrt[d]*Sqrt[f]*x]) + 12*b^2*d*f*n^2*(2*a + b*n - 2*b*n*Log[x] + 
2*b*Log[c*x^n])*(Log[x]^3/3 - (Log[x]^2*Log[1 - I*Sqrt[d]*Sqrt[f]*x])/2 - 
(Log[x]^2*Log[1 + I*Sqrt[d]*Sqrt[f]*x])/2 - Log[x]*PolyLog[2, (-I)*Sqrt[d] 
*Sqrt[f]*x] - Log[x]*PolyLog[2, I*Sqrt[d]*Sqrt[f]*x] + PolyLog[3, (-I)*Sqr 
t[d]*Sqrt[f]*x] + PolyLog[3, I*Sqrt[d]*Sqrt[f]*x]) + 2*b^3*d*f*n^3*(Log[x] 
^4 - 2*Log[x]^3*Log[1 - I*Sqrt[d]*Sqrt[f]*x] - 2*Log[x]^3*Log[1 + I*Sqr...
 

Rubi [A] (verified)

Time = 0.91 (sec) , antiderivative size = 418, normalized size of antiderivative = 0.98, 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.

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

\(\Big \downarrow \) 2825

\(\displaystyle -2 f \int \left (-\frac {3 b^3 d n^3}{8 x \left (d f x^2+1\right )}-\frac {3 b^2 d \left (a+b \log \left (c x^n\right )\right ) n^2}{4 x \left (d f x^2+1\right )}-\frac {3 b d \left (a+b \log \left (c x^n\right )\right )^2 n}{4 x \left (d f x^2+1\right )}-\frac {d \left (a+b \log \left (c x^n\right )\right )^3}{2 x \left (d f x^2+1\right )}\right )dx-\frac {3 b^2 n^2 \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )}{4 x^2}-\frac {3 b n \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}-\frac {\log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}-\frac {3 b^3 n^3 \log \left (d f x^2+1\right )}{8 x^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3 b^2 n^2 \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )}{4 x^2}-2 f \left (-\frac {3}{8} b^2 d n^2 \operatorname {PolyLog}\left (2,-\frac {1}{d f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {3}{8} b^2 d n^2 \operatorname {PolyLog}\left (3,-\frac {1}{d f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {3}{8} b^2 d n^2 \log \left (\frac {1}{d f x^2}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {3}{8} b d n \operatorname {PolyLog}\left (2,-\frac {1}{d f x^2}\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {3}{8} b d n \log \left (\frac {1}{d f x^2}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {1}{4} d \log \left (\frac {1}{d f x^2}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {3}{16} b^3 d n^3 \operatorname {PolyLog}\left (2,-\frac {1}{d f x^2}\right )-\frac {3}{16} b^3 d n^3 \operatorname {PolyLog}\left (3,-\frac {1}{d f x^2}\right )-\frac {3}{16} b^3 d n^3 \operatorname {PolyLog}\left (4,-\frac {1}{d f x^2}\right )+\frac {3}{16} b^3 d n^3 \log \left (d f x^2+1\right )-\frac {3}{8} b^3 d n^3 \log (x)\right )-\frac {3 b n \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}-\frac {\log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}-\frac {3 b^3 n^3 \log \left (d f x^2+1\right )}{8 x^2}\)

Input:

Int[((a + b*Log[c*x^n])^3*Log[d*(d^(-1) + f*x^2)])/x^3,x]
 

Output:

(-3*b^3*n^3*Log[1 + d*f*x^2])/(8*x^2) - (3*b^2*n^2*(a + b*Log[c*x^n])*Log[ 
1 + d*f*x^2])/(4*x^2) - (3*b*n*(a + b*Log[c*x^n])^2*Log[1 + d*f*x^2])/(4*x 
^2) - ((a + b*Log[c*x^n])^3*Log[1 + d*f*x^2])/(2*x^2) - 2*f*((-3*b^3*d*n^3 
*Log[x])/8 + (3*b^2*d*n^2*Log[1 + 1/(d*f*x^2)]*(a + b*Log[c*x^n]))/8 + (3* 
b*d*n*Log[1 + 1/(d*f*x^2)]*(a + b*Log[c*x^n])^2)/8 + (d*Log[1 + 1/(d*f*x^2 
)]*(a + b*Log[c*x^n])^3)/4 + (3*b^3*d*n^3*Log[1 + d*f*x^2])/16 - (3*b^3*d* 
n^3*PolyLog[2, -(1/(d*f*x^2))])/16 - (3*b^2*d*n^2*(a + b*Log[c*x^n])*PolyL 
og[2, -(1/(d*f*x^2))])/8 - (3*b*d*n*(a + b*Log[c*x^n])^2*PolyLog[2, -(1/(d 
*f*x^2))])/8 - (3*b^3*d*n^3*PolyLog[3, -(1/(d*f*x^2))])/16 - (3*b^2*d*n^2* 
(a + b*Log[c*x^n])*PolyLog[3, -(1/(d*f*x^2))])/8 - (3*b^3*d*n^3*PolyLog[4, 
 -(1/(d*f*x^2))])/16)
 

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 [C] (warning: unable to verify)

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

Time = 46.48 (sec) , antiderivative size = 1171, normalized size of antiderivative = 2.76

method	result	size
risch	\(\text {Expression too large to display}\)	\(1171\)

Input:

int((a+b*ln(c*x^n))^3*ln(d*(1/d+f*x^2))/x^3,x,method=_RETURNVERBOSE)
 

Output:

-3/8*b^3*n^3*d*f*polylog(2,-d*f*x^2)+3/8*b^3*n^3*d*f*polylog(3,-d*f*x^2)-3 
/8*b^3*n^3*d*f*polylog(4,-d*f*x^2)-3/4*b^3*n^3*d*f*ln(x)^2+1/2*b^3*n^3*d*f 
*ln(x)^3-1/4*b^3*n^3*d*f*ln(x)^4-1/2*b^3/x^2*ln(d*f*x^2+1)*ln(x^n)^3-3/4*b 
^3*n*d*f*polylog(2,-d*f*x^2)*ln(x^n)^2+3/2*b^3*n^2*d*f*ln(x)*ln(x^n)-3/4*b 
^3*n^2*d*f*ln(d*f*x^2+1)*ln(x^n)-3/4*b^3*n^2*d*f*polylog(2,-d*f*x^2)*ln(x^ 
n)+3/4*(I*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)^2-I*Pi*b*csgn(I*x^n)*csgn(I*c*x^n 
)*csgn(I*c)-I*Pi*b*csgn(I*c*x^n)^3+I*Pi*b*csgn(I*c*x^n)^2*csgn(I*c)+2*b*ln 
(c)+2*a)^2*b*((ln(x^n)-n*ln(x))*(-1/2/x^2*ln(d*f*x^2+1)+d*f*(ln(x)-1/2*ln( 
d*f*x^2+1)))+n*((-1/4-1/2*ln(x))/x^2*ln(d*f*x^2+1)+1/2*d*f*ln(x)-1/4*d*f*l 
n(d*f*x^2+1)+1/2*d*f*ln(x)^2-1/2*d*f*ln(x)*ln(d*f*x^2+1)-1/4*d*f*polylog(2 
,-d*f*x^2)))+1/8*(I*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)^2-I*Pi*b*csgn(I*x^n)*cs 
gn(I*c*x^n)*csgn(I*c)-I*Pi*b*csgn(I*c*x^n)^3+I*Pi*b*csgn(I*c*x^n)^2*csgn(I 
*c)+2*b*ln(c)+2*a)^3*(-1/2/x^2*ln(d*f*x^2+1)+d*f*(ln(x)-1/2*ln(d*f*x^2+1)) 
)-3/2*b^3*d*f*ln(x)^2*ln(x^n)^2*n-3/2*b^3*d*f*ln(x)^2*ln(x^n)*n^2+3/2*b^3* 
n*d*f*ln(x)*ln(x^n)^2-3/4*b^3*n*d*f*ln(d*f*x^2+1)*ln(x^n)^2+3/4*b^3*n^2*d* 
f*polylog(3,-d*f*x^2)*ln(x^n)+b^3*d*f*ln(x)^3*ln(x^n)*n^2-3/4*b^3*n^2/x^2* 
ln(d*f*x^2+1)*ln(x^n)+b^3*d*f*ln(x)*ln(x^n)^3-1/2*b^3*d*f*ln(d*f*x^2+1)*ln 
(x^n)^3-3/4*b^3*n/x^2*ln(d*f*x^2+1)*ln(x^n)^2+3/2*(I*Pi*b*csgn(I*x^n)*csgn 
(I*c*x^n)^2-I*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-I*Pi*b*csgn(I*c*x^n 
)^3+I*Pi*b*csgn(I*c*x^n)^2*csgn(I*c)+2*b*ln(c)+2*a)*b^2*((ln(x^n)-n*ln(...
 

Fricas [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right )}{x^{3}} \,d x } \] Input:

integrate((a+b*log(c*x^n))^3*log(d*(1/d+f*x^2))/x^3,x, algorithm="fricas")
 

Output:

integral((b^3*log(d*f*x^2 + 1)*log(c*x^n)^3 + 3*a*b^2*log(d*f*x^2 + 1)*log 
(c*x^n)^2 + 3*a^2*b*log(d*f*x^2 + 1)*log(c*x^n) + a^3*log(d*f*x^2 + 1))/x^ 
3, x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^3} \, dx=\text {Timed out} \] Input:

integrate((a+b*ln(c*x**n))**3*ln(d*(1/d+f*x**2))/x**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right )}{x^{3}} \,d x } \] Input:

integrate((a+b*log(c*x^n))^3*log(d*(1/d+f*x^2))/x^3,x, algorithm="maxima")
 

Output:

-1/8*(4*b^3*log(x^n)^3 + 6*(n^2 + 2*n*log(c) + 2*log(c)^2)*a*b^2 + (3*n^3 
+ 6*n^2*log(c) + 6*n*log(c)^2 + 4*log(c)^3)*b^3 + 6*a^2*b*(n + 2*log(c)) + 
 4*a^3 + 6*(b^3*(n + 2*log(c)) + 2*a*b^2)*log(x^n)^2 + 6*((n^2 + 2*n*log(c 
) + 2*log(c)^2)*b^3 + 2*a*b^2*(n + 2*log(c)) + 2*a^2*b)*log(x^n))*log(d*f* 
x^2 + 1)/x^2 + integrate(1/4*(4*b^3*d*f*log(x^n)^3 + 4*a^3*d*f + 6*(d*f*n 
+ 2*d*f*log(c))*a^2*b + 6*(d*f*n^2 + 2*d*f*n*log(c) + 2*d*f*log(c)^2)*a*b^ 
2 + (3*d*f*n^3 + 6*d*f*n^2*log(c) + 6*d*f*n*log(c)^2 + 4*d*f*log(c)^3)*b^3 
 + 6*(2*a*b^2*d*f + (d*f*n + 2*d*f*log(c))*b^3)*log(x^n)^2 + 6*(2*a^2*b*d* 
f + 2*(d*f*n + 2*d*f*log(c))*a*b^2 + (d*f*n^2 + 2*d*f*n*log(c) + 2*d*f*log 
(c)^2)*b^3)*log(x^n))/(d*f*x^3 + x), x)
 

Giac [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f x^{2} + \frac {1}{d}\right )} d\right )}{x^{3}} \,d x } \] Input:

integrate((a+b*log(c*x^n))^3*log(d*(1/d+f*x^2))/x^3,x, algorithm="giac")
 

Output:

integrate((b*log(c*x^n) + a)^3*log((f*x^2 + 1/d)*d)/x^3, x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^3} \, dx=\int \frac {\ln \left (d\,\left (f\,x^2+\frac {1}{d}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3}{x^3} \,d x \] Input:

int((log(d*(f*x^2 + 1/d))*(a + b*log(c*x^n))^3)/x^3,x)
 

Output:

int((log(d*(f*x^2 + 1/d))*(a + b*log(c*x^n))^3)/x^3, x)
 

Reduce [F]

\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^3} \, dx =\text {Too large to display} \] Input:

int((a+b*log(c*x^n))^3*log(d*(1/d+f*x^2))/x^3,x)
 

Output:

( - 8*int(log(x**n*c)**3/(d*f*x**5 + x**3),x)*b**3*x**2 - 24*int(log(x**n* 
c)**2/(d*f*x**5 + x**3),x)*a*b**2*x**2 - 12*int(log(x**n*c)**2/(d*f*x**5 + 
 x**3),x)*b**3*n*x**2 - 24*int(log(x**n*c)/(d*f*x**5 + x**3),x)*a**2*b*x** 
2 - 24*int(log(x**n*c)/(d*f*x**5 + x**3),x)*a*b**2*n*x**2 - 12*int(log(x** 
n*c)/(d*f*x**5 + x**3),x)*b**3*n**2*x**2 - 4*log(d*f*x**2 + 1)*log(x**n*c) 
**3*b**3 - 12*log(d*f*x**2 + 1)*log(x**n*c)**2*a*b**2 - 6*log(d*f*x**2 + 1 
)*log(x**n*c)**2*b**3*n - 12*log(d*f*x**2 + 1)*log(x**n*c)*a**2*b - 12*log 
(d*f*x**2 + 1)*log(x**n*c)*a*b**2*n - 6*log(d*f*x**2 + 1)*log(x**n*c)*b**3 
*n**2 - 4*log(d*f*x**2 + 1)*a**3*d*f*x**2 - 4*log(d*f*x**2 + 1)*a**3 - 6*l 
og(d*f*x**2 + 1)*a**2*b*d*f*n*x**2 - 6*log(d*f*x**2 + 1)*a**2*b*n - 6*log( 
d*f*x**2 + 1)*a*b**2*d*f*n**2*x**2 - 6*log(d*f*x**2 + 1)*a*b**2*n**2 - 3*l 
og(d*f*x**2 + 1)*b**3*d*f*n**3*x**2 - 3*log(d*f*x**2 + 1)*b**3*n**3 - 4*lo 
g(x**n*c)**3*b**3 - 12*log(x**n*c)**2*a*b**2 - 12*log(x**n*c)**2*b**3*n - 
12*log(x**n*c)*a**2*b - 24*log(x**n*c)*a*b**2*n - 18*log(x**n*c)*b**3*n**2 
 + 8*log(x)*a**3*d*f*x**2 + 12*log(x)*a**2*b*d*f*n*x**2 + 12*log(x)*a*b**2 
*d*f*n**2*x**2 + 6*log(x)*b**3*d*f*n**3*x**2 - 6*a**2*b*n - 12*a*b**2*n**2 
 - 9*b**3*n**3)/(8*x**2)