\(\int (a+b \log (c x^n))^3 \log (d (\frac {1}{d}+f x^2)) \, dx\) [50]

Optimal result

Integrand size = 25, antiderivative size = 836 \[ \int \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx =\text {Too large to display} \] Output:

-6*b^2*n^2*(a+b*ln(c*x^n))*polylog(3,-(-d)^(1/2)*f^(1/2)*x)/(-d)^(1/2)/f^( 
1/2)+3*b*n*(a+b*ln(c*x^n))^2*polylog(2,-(-d)^(1/2)*f^(1/2)*x)/(-d)^(1/2)/f 
^(1/2)-6*b^2*n^2*(a+b*ln(c*x^n))*polylog(2,-(-d)^(1/2)*f^(1/2)*x)/(-d)^(1/ 
2)/f^(1/2)+6*b^2*n^2*(a+b*ln(c*x^n))*polylog(3,(-d)^(1/2)*f^(1/2)*x)/(-d)^ 
(1/2)/f^(1/2)+6*b^2*n^2*(a+b*ln(c*x^n))*polylog(2,(-d)^(1/2)*f^(1/2)*x)/(- 
d)^(1/2)/f^(1/2)-3*b*n*(a+b*ln(c*x^n))^2*polylog(2,(-d)^(1/2)*f^(1/2)*x)/( 
-d)^(1/2)/f^(1/2)+12*b^2*n^2*(-b*n+a)*arctan(d^(1/2)*f^(1/2)*x)/d^(1/2)/f^ 
(1/2)+12*b^3*n^2*arctan(d^(1/2)*f^(1/2)*x)*ln(c*x^n)/d^(1/2)/f^(1/2)-6*b*n 
*arctan(d^(1/2)*f^(1/2)*x)*(a+b*ln(c*x^n))^2/d^(1/2)/f^(1/2)+x*(a+b*ln(c*x 
^n))^3*ln(d*f*x^2+1)-12*b^2*n^2*(-b*n+a)*x-6*b^3*n^3*x*ln(d*f*x^2+1)-24*a* 
b^2*n^2*x-36*b^3*n^2*x*ln(c*x^n)+12*b*n*x*(a+b*ln(c*x^n))^2-2*x*(a+b*ln(c* 
x^n))^3+6*b^3*n^3*polylog(4,-(-d)^(1/2)*f^(1/2)*x)/(-d)^(1/2)/f^(1/2)+6*b^ 
3*n^3*polylog(3,-(-d)^(1/2)*f^(1/2)*x)/(-d)^(1/2)/f^(1/2)+6*b^3*n^3*polylo 
g(2,-(-d)^(1/2)*f^(1/2)*x)/(-d)^(1/2)/f^(1/2)-6*b^3*n^3*polylog(4,(-d)^(1/ 
2)*f^(1/2)*x)/(-d)^(1/2)/f^(1/2)-6*b^3*n^3*polylog(3,(-d)^(1/2)*f^(1/2)*x) 
/(-d)^(1/2)/f^(1/2)-6*b^3*n^3*polylog(2,(-d)^(1/2)*f^(1/2)*x)/(-d)^(1/2)/f 
^(1/2)+6*a*b^2*n^2*x*ln(d*f*x^2+1)+6*b^3*n^2*x*ln(c*x^n)*ln(d*f*x^2+1)-3*b 
*n*x*(a+b*ln(c*x^n))^2*ln(d*f*x^2+1)+2*arctan(d^(1/2)*f^(1/2)*x)*(a+b*ln(c 
*x^n))^3/d^(1/2)/f^(1/2)+36*b^3*n^3*x
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.96 (sec) , antiderivative size = 1027, normalized size of antiderivative = 1.23 \[ \int \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

(-2*Sqrt[d]*Sqrt[f]*x*(a^3 - 3*a^2*b*n + 6*a*b^2*n^2 - 6*b^3*n^3 + 6*a*b^2 
*n*(n*Log[x] - Log[c*x^n]) + 3*a^2*b*(-(n*Log[x]) + Log[c*x^n]) + 6*b^3*n^ 
2*(-(n*Log[x]) + Log[c*x^n]) + 3*a*b^2*(-(n*Log[x]) + Log[c*x^n])^2 - 3*b^ 
3*n*(-(n*Log[x]) + Log[c*x^n])^2 + b^3*(-(n*Log[x]) + Log[c*x^n])^3) + 2*A 
rcTan[Sqrt[d]*Sqrt[f]*x]*(a^3 - 3*a^2*b*n + 6*a*b^2*n^2 - 6*b^3*n^3 + 6*a* 
b^2*n*(n*Log[x] - Log[c*x^n]) + 3*a^2*b*(-(n*Log[x]) + Log[c*x^n]) + 6*b^3 
*n^2*(-(n*Log[x]) + Log[c*x^n]) + 3*a*b^2*(-(n*Log[x]) + Log[c*x^n])^2 - 3 
*b^3*n*(-(n*Log[x]) + Log[c*x^n])^2 + b^3*(-(n*Log[x]) + Log[c*x^n])^3) + 
Sqrt[d]*Sqrt[f]*x*(a^3 - 3*a^2*b*n + 6*a*b^2*n^2 - 6*b^3*n^3 + 3*b*(a^2 - 
2*a*b*n + 2*b^2*n^2)*Log[c*x^n] + 3*b^2*(a - b*n)*Log[c*x^n]^2 + b^3*Log[c 
*x^n]^3)*Log[1 + d*f*x^2] + 3*b*n*(a^2 - 2*a*b*n + 2*b^2*n^2 + 2*b^2*n*(n* 
Log[x] - Log[c*x^n]) + 2*a*b*(-(n*Log[x]) + Log[c*x^n]) + b^2*(-(n*Log[x]) 
 + Log[c*x^n])^2)*(-2*Sqrt[d]*Sqrt[f]*x*(-1 + Log[x]) - I*(Log[x]*Log[1 + 
I*Sqrt[d]*Sqrt[f]*x] + PolyLog[2, (-I)*Sqrt[d]*Sqrt[f]*x]) + I*(Log[x]*Log 
[1 - I*Sqrt[d]*Sqrt[f]*x] + PolyLog[2, I*Sqrt[d]*Sqrt[f]*x])) - 6*b^2*n^2* 
(a - b*n - b*n*Log[x] + b*Log[c*x^n])*(Sqrt[d]*Sqrt[f]*x*(2 - 2*Log[x] + L 
og[x]^2) + (I/2)*(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]) - (I/2 
)*(Log[x]^2*Log[1 - I*Sqrt[d]*Sqrt[f]*x] + 2*Log[x]*PolyLog[2, I*Sqrt[d]*S 
qrt[f]*x] - 2*PolyLog[3, I*Sqrt[d]*Sqrt[f]*x])) + 2*b^3*n^3*(-(Sqrt[d]*...
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.83 (sec) , antiderivative size = 972, normalized size of antiderivative = 1.16, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2818, 6, 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.

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

\(\Big \downarrow \) 2818

\(\displaystyle -2 f \int \left (\frac {6 d n^2 x^2 \log \left (c x^n\right ) b^3}{d f x^2+1}-\frac {6 d n^3 x^2 b^3}{d f x^2+1}+\frac {6 a d n^2 x^2 b^2}{d f x^2+1}-\frac {3 d n x^2 \left (a+b \log \left (c x^n\right )\right )^2 b}{d f x^2+1}+\frac {d x^2 \left (a+b \log \left (c x^n\right )\right )^3}{d f x^2+1}\right )dx+6 a b^2 n^2 x \log \left (d f x^2+1\right )-3 b n x \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^2+x \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^3+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (d f x^2+1\right )-6 b^3 n^3 x \log \left (d f x^2+1\right )\)

\(\Big \downarrow \) 6

\(\displaystyle -2 f \int \left (\frac {6 d n^2 x^2 \log \left (c x^n\right ) b^3}{d f x^2+1}-\frac {3 d n x^2 \left (a+b \log \left (c x^n\right )\right )^2 b}{d f x^2+1}+\frac {d x^2 \left (a+b \log \left (c x^n\right )\right )^3}{d f x^2+1}+\frac {d \left (6 a b^2 n^2-6 b^3 n^3\right ) x^2}{d f x^2+1}\right )dx+6 a b^2 n^2 x \log \left (d f x^2+1\right )-3 b n x \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^2+x \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )^3+6 b^3 n^2 x \log \left (c x^n\right ) \log \left (d f x^2+1\right )-6 b^3 n^3 x \log \left (d f x^2+1\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle -6 n^3 x \log \left (d f x^2+1\right ) b^3+6 n^2 x \log \left (c x^n\right ) \log \left (d f x^2+1\right ) b^3+6 a n^2 x \log \left (d f x^2+1\right ) b^2-3 n x \left (a+b \log \left (c x^n\right )\right )^2 \log \left (d f x^2+1\right ) b+x \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d f x^2+1\right )-2 f \left (-\frac {18 n^3 x b^3}{f}+\frac {18 n^2 x \log \left (c x^n\right ) b^3}{f}-\frac {6 n^2 \arctan \left (\sqrt {d} \sqrt {f} x\right ) \log \left (c x^n\right ) b^3}{\sqrt {d} f^{3/2}}+\frac {3 i n^3 \operatorname {PolyLog}\left (2,-i \sqrt {d} \sqrt {f} x\right ) b^3}{\sqrt {d} f^{3/2}}-\frac {3 i n^3 \operatorname {PolyLog}\left (2,i \sqrt {d} \sqrt {f} x\right ) b^3}{\sqrt {d} f^{3/2}}-\frac {3 n^3 \operatorname {PolyLog}\left (3,-\sqrt {-d} \sqrt {f} x\right ) b^3}{\sqrt {-d} f^{3/2}}+\frac {3 n^3 \operatorname {PolyLog}\left (3,\sqrt {-d} \sqrt {f} x\right ) b^3}{\sqrt {-d} f^{3/2}}-\frac {3 n^3 \operatorname {PolyLog}\left (4,-\sqrt {-d} \sqrt {f} x\right ) b^3}{\sqrt {-d} f^{3/2}}+\frac {3 n^3 \operatorname {PolyLog}\left (4,\sqrt {-d} \sqrt {f} x\right ) b^3}{\sqrt {-d} f^{3/2}}+\frac {12 a n^2 x b^2}{f}+\frac {6 n^2 (a-b n) x b^2}{f}-\frac {6 n^2 (a-b n) \arctan \left (\sqrt {d} \sqrt {f} x\right ) b^2}{\sqrt {d} f^{3/2}}+\frac {3 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\sqrt {-d} \sqrt {f} x\right ) b^2}{\sqrt {-d} f^{3/2}}-\frac {3 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,\sqrt {-d} \sqrt {f} x\right ) b^2}{\sqrt {-d} f^{3/2}}+\frac {3 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-\sqrt {-d} \sqrt {f} x\right ) b^2}{\sqrt {-d} f^{3/2}}-\frac {3 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,\sqrt {-d} \sqrt {f} x\right ) b^2}{\sqrt {-d} f^{3/2}}-\frac {6 n x \left (a+b \log \left (c x^n\right )\right )^2 b}{f}-\frac {3 n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\sqrt {-d} \sqrt {f} x\right ) b}{2 \sqrt {-d} f^{3/2}}+\frac {3 n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (\sqrt {-d} \sqrt {f} x+1\right ) b}{2 \sqrt {-d} f^{3/2}}-\frac {3 n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\sqrt {-d} \sqrt {f} x\right ) b}{2 \sqrt {-d} f^{3/2}}+\frac {3 n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,\sqrt {-d} \sqrt {f} x\right ) b}{2 \sqrt {-d} f^{3/2}}+\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{f}+\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1-\sqrt {-d} \sqrt {f} x\right )}{2 \sqrt {-d} f^{3/2}}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (\sqrt {-d} \sqrt {f} x+1\right )}{2 \sqrt {-d} f^{3/2}}\right )\)

Input:

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

Output:

6*a*b^2*n^2*x*Log[1 + d*f*x^2] - 6*b^3*n^3*x*Log[1 + d*f*x^2] + 6*b^3*n^2* 
x*Log[c*x^n]*Log[1 + d*f*x^2] - 3*b*n*x*(a + b*Log[c*x^n])^2*Log[1 + d*f*x 
^2] + x*(a + b*Log[c*x^n])^3*Log[1 + d*f*x^2] - 2*f*((12*a*b^2*n^2*x)/f - 
(18*b^3*n^3*x)/f + (6*b^2*n^2*(a - b*n)*x)/f - (6*b^2*n^2*(a - b*n)*ArcTan 
[Sqrt[d]*Sqrt[f]*x])/(Sqrt[d]*f^(3/2)) + (18*b^3*n^2*x*Log[c*x^n])/f - (6* 
b^3*n^2*ArcTan[Sqrt[d]*Sqrt[f]*x]*Log[c*x^n])/(Sqrt[d]*f^(3/2)) - (6*b*n*x 
*(a + b*Log[c*x^n])^2)/f + (x*(a + b*Log[c*x^n])^3)/f - (3*b*n*(a + b*Log[ 
c*x^n])^2*Log[1 - Sqrt[-d]*Sqrt[f]*x])/(2*Sqrt[-d]*f^(3/2)) + ((a + b*Log[ 
c*x^n])^3*Log[1 - Sqrt[-d]*Sqrt[f]*x])/(2*Sqrt[-d]*f^(3/2)) + (3*b*n*(a + 
b*Log[c*x^n])^2*Log[1 + Sqrt[-d]*Sqrt[f]*x])/(2*Sqrt[-d]*f^(3/2)) - ((a + 
b*Log[c*x^n])^3*Log[1 + Sqrt[-d]*Sqrt[f]*x])/(2*Sqrt[-d]*f^(3/2)) + (3*b^2 
*n^2*(a + b*Log[c*x^n])*PolyLog[2, -(Sqrt[-d]*Sqrt[f]*x)])/(Sqrt[-d]*f^(3/ 
2)) - (3*b*n*(a + b*Log[c*x^n])^2*PolyLog[2, -(Sqrt[-d]*Sqrt[f]*x)])/(2*Sq 
rt[-d]*f^(3/2)) - (3*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[2, Sqrt[-d]*Sqrt[f 
]*x])/(Sqrt[-d]*f^(3/2)) + (3*b*n*(a + b*Log[c*x^n])^2*PolyLog[2, Sqrt[-d] 
*Sqrt[f]*x])/(2*Sqrt[-d]*f^(3/2)) + ((3*I)*b^3*n^3*PolyLog[2, (-I)*Sqrt[d] 
*Sqrt[f]*x])/(Sqrt[d]*f^(3/2)) - ((3*I)*b^3*n^3*PolyLog[2, I*Sqrt[d]*Sqrt[ 
f]*x])/(Sqrt[d]*f^(3/2)) - (3*b^3*n^3*PolyLog[3, -(Sqrt[-d]*Sqrt[f]*x)])/( 
Sqrt[-d]*f^(3/2)) + (3*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, -(Sqrt[-d]*Sq 
rt[f]*x)])/(Sqrt[-d]*f^(3/2)) + (3*b^3*n^3*PolyLog[3, Sqrt[-d]*Sqrt[f]*...
 

Defintions of rubi rules used

rule 6

Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v 
+ (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] &&  !FreeQ[Fx, x]
 

rule 2009

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

rule 2818

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_. 
)]*(b_.))^(p_.), x_Symbol] :> With[{u = IntHide[(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, r, m, n}, x] && IGtQ[p, 0] && Inte 
gerQ[m]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(2*sqrt(f)*sqrt(d)*atan((d*f*x)/(sqrt(f)*sqrt(d)))*a**3 - 6*sqrt(f)*sqrt(d 
)*atan((d*f*x)/(sqrt(f)*sqrt(d)))*a**2*b*n + 12*sqrt(f)*sqrt(d)*atan((d*f* 
x)/(sqrt(f)*sqrt(d)))*a*b**2*n**2 - 12*sqrt(f)*sqrt(d)*atan((d*f*x)/(sqrt( 
f)*sqrt(d)))*b**3*n**3 + 2*int(log(x**n*c)**3/(d*f*x**2 + 1),x)*b**3*d*f + 
 6*int(log(x**n*c)**2/(d*f*x**2 + 1),x)*a*b**2*d*f - 6*int(log(x**n*c)**2/ 
(d*f*x**2 + 1),x)*b**3*d*f*n + 6*int(log(x**n*c)/(d*f*x**2 + 1),x)*a**2*b* 
d*f - 12*int(log(x**n*c)/(d*f*x**2 + 1),x)*a*b**2*d*f*n + 12*int(log(x**n* 
c)/(d*f*x**2 + 1),x)*b**3*d*f*n**2 + log(d*f*x**2 + 1)*log(x**n*c)**3*b**3 
*d*f*x + 3*log(d*f*x**2 + 1)*log(x**n*c)**2*a*b**2*d*f*x - 3*log(d*f*x**2 
+ 1)*log(x**n*c)**2*b**3*d*f*n*x + 3*log(d*f*x**2 + 1)*log(x**n*c)*a**2*b* 
d*f*x - 6*log(d*f*x**2 + 1)*log(x**n*c)*a*b**2*d*f*n*x + 6*log(d*f*x**2 + 
1)*log(x**n*c)*b**3*d*f*n**2*x + log(d*f*x**2 + 1)*a**3*d*f*x - 3*log(d*f* 
x**2 + 1)*a**2*b*d*f*n*x + 6*log(d*f*x**2 + 1)*a*b**2*d*f*n**2*x - 6*log(d 
*f*x**2 + 1)*b**3*d*f*n**3*x - 2*log(x**n*c)**3*b**3*d*f*x - 6*log(x**n*c) 
**2*a*b**2*d*f*x + 12*log(x**n*c)**2*b**3*d*f*n*x - 6*log(x**n*c)*a**2*b*d 
*f*x + 24*log(x**n*c)*a*b**2*d*f*n*x - 36*log(x**n*c)*b**3*d*f*n**2*x - 2* 
a**3*d*f*x + 12*a**2*b*d*f*n*x - 36*a*b**2*d*f*n**2*x + 48*b**3*d*f*n**3*x 
)/(d*f)