\(\int \frac {x (a+b \log (c (d+e x)^n))^2}{(f+g x^2)^2} \, dx\) [322]

Optimal result

Integrand size = 27, antiderivative size = 430 \[ \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx=\frac {e^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g \left (e^2 f+d^2 g\right )}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g \left (f+g x^2\right )}-\frac {b e \left (e f+d \sqrt {-f} \sqrt {g}\right ) n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 f g \left (e^2 f+d^2 g\right )}-\frac {b e \left (e f-d \sqrt {-f} \sqrt {g}\right ) n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 f g \left (e^2 f+d^2 g\right )}-\frac {b^2 e \left (e \sqrt {-f}+d \sqrt {g}\right ) n^2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 \sqrt {-f} g \left (e^2 f+d^2 g\right )}-\frac {b^2 e \left (e f+d \sqrt {-f} \sqrt {g}\right ) n^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 f g \left (e^2 f+d^2 g\right )} \] Output:

1/2*e^2*(a+b*ln(c*(e*x+d)^n))^2/g/(d^2*g+e^2*f)-1/2*(a+b*ln(c*(e*x+d)^n))^ 
2/g/(g*x^2+f)-1/2*b*e*(e*f+d*(-f)^(1/2)*g^(1/2))*n*(a+b*ln(c*(e*x+d)^n))*l 
n(e*((-f)^(1/2)-g^(1/2)*x)/(e*(-f)^(1/2)+d*g^(1/2)))/f/g/(d^2*g+e^2*f)-1/2 
*b*e*(e*f-d*(-f)^(1/2)*g^(1/2))*n*(a+b*ln(c*(e*x+d)^n))*ln(e*((-f)^(1/2)+g 
^(1/2)*x)/(e*(-f)^(1/2)-d*g^(1/2)))/f/g/(d^2*g+e^2*f)-1/2*b^2*e*(e*(-f)^(1 
/2)+d*g^(1/2))*n^2*polylog(2,-g^(1/2)*(e*x+d)/(e*(-f)^(1/2)-d*g^(1/2)))/(- 
f)^(1/2)/g/(d^2*g+e^2*f)-1/2*b^2*e*(e*f+d*(-f)^(1/2)*g^(1/2))*n^2*polylog( 
2,g^(1/2)*(e*x+d)/(e*(-f)^(1/2)+d*g^(1/2)))/f/g/(d^2*g+e^2*f)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.79 (sec) , antiderivative size = 590, normalized size of antiderivative = 1.37 \[ \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx=\frac {-\frac {2 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x^2}+\frac {2 b n \left (-a+b n \log (d+e x)-b \log \left (c (d+e x)^n\right )\right ) \left (2 \sqrt {f} g \left (d^2-e^2 x^2\right ) \log (d+e x)+e \left (f+g x^2\right ) \left (\left (e \sqrt {f}+i d \sqrt {g}\right ) \log \left (i \sqrt {f}-\sqrt {g} x\right )+\left (e \sqrt {f}-i d \sqrt {g}\right ) \log \left (i \sqrt {f}+\sqrt {g} x\right )\right )\right )}{\sqrt {f} \left (e^2 f+d^2 g\right ) \left (f+g x^2\right )}+\frac {i b^2 n^2 \left (\frac {-\sqrt {g} (d+e x) \log ^2(d+e x)+2 e \left (i \sqrt {f}+\sqrt {g} x\right ) \log (d+e x) \log \left (\frac {e \left (\sqrt {f}-i \sqrt {g} x\right )}{e \sqrt {f}+i d \sqrt {g}}\right )+2 e \left (i \sqrt {f}+\sqrt {g} x\right ) \operatorname {PolyLog}\left (2,\frac {i \sqrt {g} (d+e x)}{e \sqrt {f}+i d \sqrt {g}}\right )}{\left (e \sqrt {f}+i d \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}+\frac {\log (d+e x) \left (\sqrt {g} (d+e x) \log (d+e x)+2 i e \left (\sqrt {f}+i \sqrt {g} x\right ) \log \left (\frac {e \left (\sqrt {f}+i \sqrt {g} x\right )}{e \sqrt {f}-i d \sqrt {g}}\right )\right )+2 i e \left (\sqrt {f}+i \sqrt {g} x\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{i e \sqrt {f}+d \sqrt {g}}\right )}{\left (e \sqrt {f}-i d \sqrt {g}\right ) \left (\sqrt {f}+i \sqrt {g} x\right )}\right )}{\sqrt {f}}}{4 g} \] Input:

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

Output:

((-2*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2)/(f + g*x^2) + (2*b*n 
*(-a + b*n*Log[d + e*x] - b*Log[c*(d + e*x)^n])*(2*Sqrt[f]*g*(d^2 - e^2*x^ 
2)*Log[d + e*x] + e*(f + g*x^2)*((e*Sqrt[f] + I*d*Sqrt[g])*Log[I*Sqrt[f] - 
 Sqrt[g]*x] + (e*Sqrt[f] - I*d*Sqrt[g])*Log[I*Sqrt[f] + Sqrt[g]*x])))/(Sqr 
t[f]*(e^2*f + d^2*g)*(f + g*x^2)) + (I*b^2*n^2*((-(Sqrt[g]*(d + e*x)*Log[d 
 + e*x]^2) + 2*e*(I*Sqrt[f] + Sqrt[g]*x)*Log[d + e*x]*Log[(e*(Sqrt[f] - I* 
Sqrt[g]*x))/(e*Sqrt[f] + I*d*Sqrt[g])] + 2*e*(I*Sqrt[f] + Sqrt[g]*x)*PolyL 
og[2, (I*Sqrt[g]*(d + e*x))/(e*Sqrt[f] + I*d*Sqrt[g])])/((e*Sqrt[f] + I*d* 
Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x)) + (Log[d + e*x]*(Sqrt[g]*(d + e*x)*Log[d 
 + e*x] + (2*I)*e*(Sqrt[f] + I*Sqrt[g]*x)*Log[(e*(Sqrt[f] + I*Sqrt[g]*x))/ 
(e*Sqrt[f] - I*d*Sqrt[g])]) + (2*I)*e*(Sqrt[f] + I*Sqrt[g]*x)*PolyLog[2, ( 
Sqrt[g]*(d + e*x))/(I*e*Sqrt[f] + d*Sqrt[g])])/((e*Sqrt[f] - I*d*Sqrt[g])* 
(Sqrt[f] + I*Sqrt[g]*x))))/Sqrt[f])/(4*g)
 

Rubi [A] (verified)

Time = 1.48 (sec) , antiderivative size = 409, normalized size of antiderivative = 0.95, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2860, 2865, 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.

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

\(\Big \downarrow \) 2860

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

\(\Big \downarrow \) 2865

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {b e n \left (\frac {e \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 b n \left (d^2 g+e^2 f\right )}-\frac {\left (d \sqrt {-f} \sqrt {g}+e f\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{d \sqrt {g}+e \sqrt {-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 f \left (d^2 g+e^2 f\right )}-\frac {\left (e f-d \sqrt {-f} \sqrt {g}\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 f \left (d^2 g+e^2 f\right )}-\frac {b n \left (e f-d \sqrt {-f} \sqrt {g}\right ) \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 f \left (d^2 g+e^2 f\right )}-\frac {b n \left (d \sqrt {-f} \sqrt {g}+e f\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right )}{2 f \left (d^2 g+e^2 f\right )}\right )}{g}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g \left (f+g x^2\right )}\)

Input:

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

Output:

-1/2*(a + b*Log[c*(d + e*x)^n])^2/(g*(f + g*x^2)) + (b*e*n*((e*(a + b*Log[ 
c*(d + e*x)^n])^2)/(2*b*(e^2*f + d^2*g)*n) - ((e*f + d*Sqrt[-f]*Sqrt[g])*( 
a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*S 
qrt[g])])/(2*f*(e^2*f + d^2*g)) - ((e*f - d*Sqrt[-f]*Sqrt[g])*(a + b*Log[c 
*(d + e*x)^n])*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/( 
2*f*(e^2*f + d^2*g)) - (b*(e*f - d*Sqrt[-f]*Sqrt[g])*n*PolyLog[2, -((Sqrt[ 
g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(2*f*(e^2*f + d^2*g)) - (b*(e*f 
+ d*Sqrt[-f]*Sqrt[g])*n*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqr 
t[g])])/(2*f*(e^2*f + d^2*g))))/g
 

Defintions of rubi rules used

rule 2009

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

rule 2860

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

rule 2865

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Sy 
mbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, 
Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunctionQ[ 
RFx, x] && IntegerQ[p]
 

Maple [C] (warning: unable to verify)

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

Time = 3.07 (sec) , antiderivative size = 1231, normalized size of antiderivative = 2.86

method	result	size
risch	\(\text {Expression too large to display}\)	\(1231\)

Input:

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

Output:

-1/2*b^2*ln((e*x+d)^n)^2/g/(g*x^2+f)-1/2*b^2/g*n^2*e^2*ln(e*x+d)^2/(d^2*g+ 
e^2*f)+b^2/g*n*e^2/(d^2*g+e^2*f)*ln(e*x+d)*ln((e*x+d)^n)+1/2*b^2/g*n^2*e^2 
/(d^2*g+e^2*f)*ln(g*(e*x+d)^2-2*(e*x+d)*d*g+d^2*g+f*e^2)*ln(e*x+d)-1/2*b^2 
/g*n*e^2/(d^2*g+e^2*f)*ln(g*(e*x+d)^2-2*(e*x+d)*d*g+d^2*g+f*e^2)*ln((e*x+d 
)^n)-b^2*n^2*e/(d^2*g+e^2*f)*d/(g*f)^(1/2)*arctan(1/2*(2*g*(e*x+d)-2*d*g)/ 
e/(g*f)^(1/2))*ln(e*x+d)+b^2*n*e/(d^2*g+e^2*f)*d/(g*f)^(1/2)*arctan(1/2*(2 
*g*(e*x+d)-2*d*g)/e/(g*f)^(1/2))*ln((e*x+d)^n)-1/2*b^2/g*n^2*e^2/(d^2*g+e^ 
2*f)*ln(e*x+d)*ln((e*(-g*f)^(1/2)-g*(e*x+d)+d*g)/(e*(-g*f)^(1/2)+d*g))-1/2 
*b^2/g*n^2*e^2/(d^2*g+e^2*f)*ln(e*x+d)*ln((e*(-g*f)^(1/2)+g*(e*x+d)-d*g)/( 
e*(-g*f)^(1/2)-d*g))+1/2*b^2*n^2*e/(d^2*g+e^2*f)*ln(e*x+d)/(-g*f)^(1/2)*ln 
((e*(-g*f)^(1/2)-g*(e*x+d)+d*g)/(e*(-g*f)^(1/2)+d*g))*d-1/2*b^2*n^2*e/(d^2 
*g+e^2*f)*ln(e*x+d)/(-g*f)^(1/2)*ln((e*(-g*f)^(1/2)+g*(e*x+d)-d*g)/(e*(-g* 
f)^(1/2)-d*g))*d-1/2*b^2/g*n^2*e^2/(d^2*g+e^2*f)*dilog((e*(-g*f)^(1/2)-g*( 
e*x+d)+d*g)/(e*(-g*f)^(1/2)+d*g))-1/2*b^2/g*n^2*e^2/(d^2*g+e^2*f)*dilog((e 
*(-g*f)^(1/2)+g*(e*x+d)-d*g)/(e*(-g*f)^(1/2)-d*g))+1/2*b^2*n^2*e/(d^2*g+e^ 
2*f)/(-g*f)^(1/2)*dilog((e*(-g*f)^(1/2)-g*(e*x+d)+d*g)/(e*(-g*f)^(1/2)+d*g 
))*d-1/2*b^2*n^2*e/(d^2*g+e^2*f)/(-g*f)^(1/2)*dilog((e*(-g*f)^(1/2)+g*(e*x 
+d)-d*g)/(e*(-g*f)^(1/2)-d*g))*d+(I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d 
)^n)^2-I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)*csgn(I*c)-I*b*Pi*csgn( 
I*c*(e*x+d)^n)^3+I*b*Pi*csgn(I*c*(e*x+d)^n)^2*csgn(I*c)+2*b*ln(c)+2*a)*...
 

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

-1/2*a*b*e*n*(e*log(g*x^2 + f)/(e^2*f*g + d^2*g^2) - 2*e*log(e*x + d)/(e^2 
*f*g + d^2*g^2) - 2*d*arctan(g*x/sqrt(f*g))/((e^2*f + d^2*g)*sqrt(f*g))) - 
 1/2*b^2*(log((e*x + d)^n)^2/(g^2*x^2 + f*g) - 2*integrate((e*g*x^2*log(c) 
^2 + d*g*x*log(c)^2 + (2*d*g*x*log(c) + e*f*n + (e*g*n + 2*e*g*log(c))*x^2 
)*log((e*x + d)^n))/(e*g^3*x^5 + d*g^3*x^4 + 2*e*f*g^2*x^3 + 2*d*f*g^2*x^2 
 + e*f^2*g*x + d*f^2*g), x)) - a*b*log((e*x + d)^n*c)/(g^2*x^2 + f*g) - 1/ 
2*a^2/(g^2*x^2 + f*g)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx=\frac {2 \sqrt {g}\, \sqrt {f}\, \mathit {atan} \left (\frac {g x}{\sqrt {g}\, \sqrt {f}}\right ) a b d e f n +2 \sqrt {g}\, \sqrt {f}\, \mathit {atan} \left (\frac {g x}{\sqrt {g}\, \sqrt {f}}\right ) a b d e g n \,x^{2}+2 \left (\int \frac {\mathrm {log}\left (\left (e x +d \right )^{n} c \right )}{e g \,x^{3}+d g \,x^{2}+e f x +d f}d x \right ) b^{2} d^{2} e \,f^{2} g n +2 \left (\int \frac {\mathrm {log}\left (\left (e x +d \right )^{n} c \right )}{e g \,x^{3}+d g \,x^{2}+e f x +d f}d x \right ) b^{2} d^{2} e f \,g^{2} n \,x^{2}+2 \left (\int \frac {\mathrm {log}\left (\left (e x +d \right )^{n} c \right )}{e g \,x^{3}+d g \,x^{2}+e f x +d f}d x \right ) b^{2} e^{3} f^{3} n +2 \left (\int \frac {\mathrm {log}\left (\left (e x +d \right )^{n} c \right )}{e g \,x^{3}+d g \,x^{2}+e f x +d f}d x \right ) b^{2} e^{3} f^{2} g n \,x^{2}-2 \,\mathrm {log}\left (e x +d \right ) a b \,d^{2} f g n -2 \,\mathrm {log}\left (e x +d \right ) a b \,d^{2} g^{2} n \,x^{2}-\mathrm {log}\left (g \,x^{2}+f \right ) a b \,e^{2} f^{2} n -\mathrm {log}\left (g \,x^{2}+f \right ) a b \,e^{2} f g n \,x^{2}-\mathrm {log}\left (\left (e x +d \right )^{n} c \right )^{2} b^{2} d^{2} f g -\mathrm {log}\left (\left (e x +d \right )^{n} c \right )^{2} b^{2} e^{2} f^{2}+2 \,\mathrm {log}\left (\left (e x +d \right )^{n} c \right ) a b \,d^{2} g^{2} x^{2}+2 \,\mathrm {log}\left (\left (e x +d \right )^{n} c \right ) a b \,e^{2} f g \,x^{2}+a^{2} d^{2} g^{2} x^{2}+a^{2} e^{2} f g \,x^{2}}{2 f g \left (d^{2} g^{2} x^{2}+e^{2} f g \,x^{2}+d^{2} f g +e^{2} f^{2}\right )} \] Input:

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

Output:

(2*sqrt(g)*sqrt(f)*atan((g*x)/(sqrt(g)*sqrt(f)))*a*b*d*e*f*n + 2*sqrt(g)*s 
qrt(f)*atan((g*x)/(sqrt(g)*sqrt(f)))*a*b*d*e*g*n*x**2 + 2*int(log((d + e*x 
)**n*c)/(d*f + d*g*x**2 + e*f*x + e*g*x**3),x)*b**2*d**2*e*f**2*g*n + 2*in 
t(log((d + e*x)**n*c)/(d*f + d*g*x**2 + e*f*x + e*g*x**3),x)*b**2*d**2*e*f 
*g**2*n*x**2 + 2*int(log((d + e*x)**n*c)/(d*f + d*g*x**2 + e*f*x + e*g*x** 
3),x)*b**2*e**3*f**3*n + 2*int(log((d + e*x)**n*c)/(d*f + d*g*x**2 + e*f*x 
 + e*g*x**3),x)*b**2*e**3*f**2*g*n*x**2 - 2*log(d + e*x)*a*b*d**2*f*g*n - 
2*log(d + e*x)*a*b*d**2*g**2*n*x**2 - log(f + g*x**2)*a*b*e**2*f**2*n - lo 
g(f + g*x**2)*a*b*e**2*f*g*n*x**2 - log((d + e*x)**n*c)**2*b**2*d**2*f*g - 
 log((d + e*x)**n*c)**2*b**2*e**2*f**2 + 2*log((d + e*x)**n*c)*a*b*d**2*g* 
*2*x**2 + 2*log((d + e*x)**n*c)*a*b*e**2*f*g*x**2 + a**2*d**2*g**2*x**2 + 
a**2*e**2*f*g*x**2)/(2*f*g*(d**2*f*g + d**2*g**2*x**2 + e**2*f**2 + e**2*f 
*g*x**2))