\(\int (a+b \log (c x^n))^3 \log (d (e+f x^2)^m) \, dx\) [118]

Optimal result

Integrand size = 25, antiderivative size = 873 \[ \int \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right ) \, dx =\text {Too large to display} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.86 (sec) , antiderivative size = 2302, normalized size of antiderivative = 2.64 \[ \int \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right ) \, dx=\text {Result too large to show} \] Input:

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

Output:

(-2*a^3*Sqrt[f]*m*x + 12*a^2*b*Sqrt[f]*m*n*x - 36*a*b^2*Sqrt[f]*m*n^2*x + 
48*b^3*Sqrt[f]*m*n^3*x + 2*a^3*Sqrt[e]*m*ArcTan[(Sqrt[f]*x)/Sqrt[e]] - 6*a 
^2*b*Sqrt[e]*m*n*ArcTan[(Sqrt[f]*x)/Sqrt[e]] + 12*a*b^2*Sqrt[e]*m*n^2*ArcT 
an[(Sqrt[f]*x)/Sqrt[e]] - 12*b^3*Sqrt[e]*m*n^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]] 
 - 6*a^2*b*Sqrt[e]*m*n*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x] + 12*a*b^2*Sqrt[ 
e]*m*n^2*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x] - 12*b^3*Sqrt[e]*m*n^3*ArcTan[ 
(Sqrt[f]*x)/Sqrt[e]]*Log[x] + 6*a*b^2*Sqrt[e]*m*n^2*ArcTan[(Sqrt[f]*x)/Sqr 
t[e]]*Log[x]^2 - 6*b^3*Sqrt[e]*m*n^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]^2 
- 2*b^3*Sqrt[e]*m*n^3*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]^3 - 6*a^2*b*Sqrt[ 
f]*m*x*Log[c*x^n] + 24*a*b^2*Sqrt[f]*m*n*x*Log[c*x^n] - 36*b^3*Sqrt[f]*m*n 
^2*x*Log[c*x^n] + 6*a^2*b*Sqrt[e]*m*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^n] 
 - 12*a*b^2*Sqrt[e]*m*n*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^n] + 12*b^3*Sq 
rt[e]*m*n^2*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^n] - 12*a*b^2*Sqrt[e]*m*n* 
ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]*Log[c*x^n] + 12*b^3*Sqrt[e]*m*n^2*ArcTa 
n[(Sqrt[f]*x)/Sqrt[e]]*Log[x]*Log[c*x^n] + 6*b^3*Sqrt[e]*m*n^2*ArcTan[(Sqr 
t[f]*x)/Sqrt[e]]*Log[x]^2*Log[c*x^n] - 6*a*b^2*Sqrt[f]*m*x*Log[c*x^n]^2 + 
12*b^3*Sqrt[f]*m*n*x*Log[c*x^n]^2 + 6*a*b^2*Sqrt[e]*m*ArcTan[(Sqrt[f]*x)/S 
qrt[e]]*Log[c*x^n]^2 - 6*b^3*Sqrt[e]*m*n*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c 
*x^n]^2 - 6*b^3*Sqrt[e]*m*n*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[x]*Log[c*x^n]^ 
2 - 2*b^3*Sqrt[f]*m*x*Log[c*x^n]^3 + 2*b^3*Sqrt[e]*m*ArcTan[(Sqrt[f]*x)...
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.78 (sec) , antiderivative size = 988, normalized size of antiderivative = 1.13, 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.

\(\displaystyle \int \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right ) \, dx\)

\(\Big \downarrow \) 2818

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

\(\Big \downarrow \) 6

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

6*a*b^2*n^2*x*Log[d*(e + f*x^2)^m] - 6*b^3*n^3*x*Log[d*(e + f*x^2)^m] + 6* 
b^3*n^2*x*Log[c*x^n]*Log[d*(e + f*x^2)^m] - 3*b*n*x*(a + b*Log[c*x^n])^2*L 
og[d*(e + f*x^2)^m] + x*(a + b*Log[c*x^n])^3*Log[d*(e + f*x^2)^m] - 2*f*m* 
((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*Sqrt[e]*n^2*(a - b*n)*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/f^(3/2) + (18*b^3*n^2 
*x*Log[c*x^n])/f - (6*b^3*Sqrt[e]*n^2*ArcTan[(Sqrt[f]*x)/Sqrt[e]]*Log[c*x^ 
n])/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*Sqrt[-e]*n*(a + b*Log[c*x^n])^2*Log[1 - (Sqrt[f]*x)/Sqrt[-e]])/(2 
*f^(3/2)) + (Sqrt[-e]*(a + b*Log[c*x^n])^3*Log[1 - (Sqrt[f]*x)/Sqrt[-e]])/ 
(2*f^(3/2)) + (3*b*Sqrt[-e]*n*(a + b*Log[c*x^n])^2*Log[1 + (Sqrt[f]*x)/Sqr 
t[-e]])/(2*f^(3/2)) - (Sqrt[-e]*(a + b*Log[c*x^n])^3*Log[1 + (Sqrt[f]*x)/S 
qrt[-e]])/(2*f^(3/2)) + (3*b^2*Sqrt[-e]*n^2*(a + b*Log[c*x^n])*PolyLog[2, 
-((Sqrt[f]*x)/Sqrt[-e])])/f^(3/2) - (3*b*Sqrt[-e]*n*(a + b*Log[c*x^n])^2*P 
olyLog[2, -((Sqrt[f]*x)/Sqrt[-e])])/(2*f^(3/2)) - (3*b^2*Sqrt[-e]*n^2*(a + 
 b*Log[c*x^n])*PolyLog[2, (Sqrt[f]*x)/Sqrt[-e]])/f^(3/2) + (3*b*Sqrt[-e]*n 
*(a + b*Log[c*x^n])^2*PolyLog[2, (Sqrt[f]*x)/Sqrt[-e]])/(2*f^(3/2)) + ((3* 
I)*b^3*Sqrt[e]*n^3*PolyLog[2, ((-I)*Sqrt[f]*x)/Sqrt[e]])/f^(3/2) - ((3*I)* 
b^3*Sqrt[e]*n^3*PolyLog[2, (I*Sqrt[f]*x)/Sqrt[e]])/f^(3/2) - (3*b^3*Sqrt[- 
e]*n^3*PolyLog[3, -((Sqrt[f]*x)/Sqrt[-e])])/f^(3/2) + (3*b^2*Sqrt[-e]*n^2* 
(a + b*Log[c*x^n])*PolyLog[3, -((Sqrt[f]*x)/Sqrt[-e])])/f^(3/2) + (3*b^...
 

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (e+f x^2\right )^m\right ) \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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