\(\int \frac {(c+d x)^2}{(a+b \coth (e+f x))^2} \, dx\) [58]

Optimal result

Integrand size = 20, antiderivative size = 475 \[ \int \frac {(c+d x)^2}{(a+b \coth (e+f x))^2} \, dx=-\frac {2 b^2 (c+d x)^2}{\left (a^2-b^2\right )^2 f}+\frac {2 b^2 (c+d x)^2}{(a-b) (a+b)^2 \left (a-b-(a+b) e^{2 e+2 f x}\right ) f}+\frac {(c+d x)^3}{3 (a-b)^2 d}+\frac {2 b^2 d (c+d x) \log \left (1-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^2}-\frac {2 b (c+d x)^2 \log \left (1-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f}+\frac {2 b^2 (c+d x)^2 \log \left (1-\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f}+\frac {b^2 d^2 \operatorname {PolyLog}\left (2,\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^3}-\frac {2 b d (c+d x) \operatorname {PolyLog}\left (2,\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f^2}+\frac {2 b^2 d (c+d x) \operatorname {PolyLog}\left (2,\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^2}+\frac {b d^2 \operatorname {PolyLog}\left (3,\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{(a-b)^2 (a+b) f^3}-\frac {b^2 d^2 \operatorname {PolyLog}\left (3,\frac {(a+b) e^{2 e+2 f x}}{a-b}\right )}{\left (a^2-b^2\right )^2 f^3} \] Output:

-2*b^2*(d*x+c)^2/(a^2-b^2)^2/f+2*b^2*(d*x+c)^2/(a-b)/(a+b)^2/(a-b-(a+b)*ex 
p(2*f*x+2*e))/f+1/3*(d*x+c)^3/(a-b)^2/d+2*b^2*d*(d*x+c)*ln(1-(a+b)*exp(2*f 
*x+2*e)/(a-b))/(a^2-b^2)^2/f^2-2*b*(d*x+c)^2*ln(1-(a+b)*exp(2*f*x+2*e)/(a- 
b))/(a-b)^2/(a+b)/f+2*b^2*(d*x+c)^2*ln(1-(a+b)*exp(2*f*x+2*e)/(a-b))/(a^2- 
b^2)^2/f+b^2*d^2*polylog(2,(a+b)*exp(2*f*x+2*e)/(a-b))/(a^2-b^2)^2/f^3-2*b 
*d*(d*x+c)*polylog(2,(a+b)*exp(2*f*x+2*e)/(a-b))/(a-b)^2/(a+b)/f^2+2*b^2*d 
*(d*x+c)*polylog(2,(a+b)*exp(2*f*x+2*e)/(a-b))/(a^2-b^2)^2/f^2+b*d^2*polyl 
og(3,(a+b)*exp(2*f*x+2*e)/(a-b))/(a-b)^2/(a+b)/f^3-b^2*d^2*polylog(3,(a+b) 
*exp(2*f*x+2*e)/(a-b))/(a^2-b^2)^2/f^3
 

Mathematica [A] (warning: unable to verify)

Time = 3.21 (sec) , antiderivative size = 519, normalized size of antiderivative = 1.09 \[ \int \frac {(c+d x)^2}{(a+b \coth (e+f x))^2} \, dx=\frac {24 b c f^2 (-b d+a c f) x+\frac {24 (a-b) b c f^2 (-b d+a c f) x}{a \left (-1+e^{2 e}\right )+b \left (1+e^{2 e}\right )}+\frac {12 (a-b) b d f^2 (-b d+2 a c f) x^2}{a \left (-1+e^{2 e}\right )+b \left (1+e^{2 e}\right )}+\frac {8 a (a-b) b d^2 f^3 x^3}{a \left (-1+e^{2 e}\right )+b \left (1+e^{2 e}\right )}+12 b d f (b d-2 a c f) x \log \left (1+\frac {(-a+b) e^{-2 (e+f x)}}{a+b}\right )-12 a b d^2 f^2 x^2 \log \left (1+\frac {(-a+b) e^{-2 (e+f x)}}{a+b}\right )+12 b c f (b d-a c f) \log \left (a-b-(a+b) e^{2 (e+f x)}\right )-6 b d (b d-2 a c f) \operatorname {PolyLog}\left (2,\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )+6 a b d^2 \left (2 f x \operatorname {PolyLog}\left (2,\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )+\operatorname {PolyLog}\left (3,\frac {(a-b) e^{-2 (e+f x)}}{a+b}\right )\right )-\frac {(a-b) (a+b) f^2 \left (\left (a^2+b^2\right ) f x \left (3 c^2+3 c d x+d^2 x^2\right ) \cosh (f x)-\left (a^2-b^2\right ) f x \left (3 c^2+3 c d x+d^2 x^2\right ) \cosh (2 e+f x)+2 b \left (-3 b (c+d x)^2+a f x \left (3 c^2+3 c d x+d^2 x^2\right )\right ) \sinh (f x)\right )}{(b \cosh (e)+a \sinh (e)) (b \cosh (e+f x)+a \sinh (e+f x))}}{6 (a-b)^2 (a+b)^2 f^3} \] Input:

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

Output:

(24*b*c*f^2*(-(b*d) + a*c*f)*x + (24*(a - b)*b*c*f^2*(-(b*d) + a*c*f)*x)/( 
a*(-1 + E^(2*e)) + b*(1 + E^(2*e))) + (12*(a - b)*b*d*f^2*(-(b*d) + 2*a*c* 
f)*x^2)/(a*(-1 + E^(2*e)) + b*(1 + E^(2*e))) + (8*a*(a - b)*b*d^2*f^3*x^3) 
/(a*(-1 + E^(2*e)) + b*(1 + E^(2*e))) + 12*b*d*f*(b*d - 2*a*c*f)*x*Log[1 + 
 (-a + b)/((a + b)*E^(2*(e + f*x)))] - 12*a*b*d^2*f^2*x^2*Log[1 + (-a + b) 
/((a + b)*E^(2*(e + f*x)))] + 12*b*c*f*(b*d - a*c*f)*Log[a - b - (a + b)*E 
^(2*(e + f*x))] - 6*b*d*(b*d - 2*a*c*f)*PolyLog[2, (a - b)/((a + b)*E^(2*( 
e + f*x)))] + 6*a*b*d^2*(2*f*x*PolyLog[2, (a - b)/((a + b)*E^(2*(e + f*x)) 
)] + PolyLog[3, (a - b)/((a + b)*E^(2*(e + f*x)))]) - ((a - b)*(a + b)*f^2 
*((a^2 + b^2)*f*x*(3*c^2 + 3*c*d*x + d^2*x^2)*Cosh[f*x] - (a^2 - b^2)*f*x* 
(3*c^2 + 3*c*d*x + d^2*x^2)*Cosh[2*e + f*x] + 2*b*(-3*b*(c + d*x)^2 + a*f* 
x*(3*c^2 + 3*c*d*x + d^2*x^2))*Sinh[f*x]))/((b*Cosh[e] + a*Sinh[e])*(b*Cos 
h[e + f*x] + a*Sinh[e + f*x])))/(6*(a - b)^2*(a + b)^2*f^3)
 

Rubi [A] (verified)

Time = 2.91 (sec) , antiderivative size = 475, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {3042, 4217, 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.

\(\displaystyle \int \frac {(c+d x)^2}{(a+b \coth (e+f x))^2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(c+d x)^2}{\left (a-i b \tan \left (i e+i f x+\frac {\pi }{2}\right )\right )^2}dx\)

\(\Big \downarrow \) 4217

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(-2*b^2*(c + d*x)^2)/((a^2 - b^2)^2*f) + (2*b^2*(c + d*x)^2)/((a - b)*(a + 
 b)^2*(a - b - (a + b)*E^(2*e + 2*f*x))*f) + (c + d*x)^3/(3*(a - b)^2*d) + 
 (2*b^2*d*(c + d*x)*Log[1 - ((a + b)*E^(2*e + 2*f*x))/(a - b)])/((a^2 - b^ 
2)^2*f^2) - (2*b*(c + d*x)^2*Log[1 - ((a + b)*E^(2*e + 2*f*x))/(a - b)])/( 
(a - b)^2*(a + b)*f) + (2*b^2*(c + d*x)^2*Log[1 - ((a + b)*E^(2*e + 2*f*x) 
)/(a - b)])/((a^2 - b^2)^2*f) + (b^2*d^2*PolyLog[2, ((a + b)*E^(2*e + 2*f* 
x))/(a - b)])/((a^2 - b^2)^2*f^3) - (2*b*d*(c + d*x)*PolyLog[2, ((a + b)*E 
^(2*e + 2*f*x))/(a - b)])/((a - b)^2*(a + b)*f^2) + (2*b^2*d*(c + d*x)*Pol 
yLog[2, ((a + b)*E^(2*e + 2*f*x))/(a - b)])/((a^2 - b^2)^2*f^2) + (b*d^2*P 
olyLog[3, ((a + b)*E^(2*e + 2*f*x))/(a - b)])/((a - b)^2*(a + b)*f^3) - (b 
^2*d^2*PolyLog[3, ((a + b)*E^(2*e + 2*f*x))/(a - b)])/((a^2 - b^2)^2*f^3)
 

Defintions of rubi rules used

rule 2009

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

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4217

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), 
 x_Symbol] :> Int[ExpandIntegrand[(c + d*x)^m, (1/(a - I*b) - 2*I*(b/(a^2 + 
 b^2 + (a - I*b)^2*E^(2*I*(e + f*x)))))^(-n), x], x] /; FreeQ[{a, b, c, d, 
e, f}, x] && NeQ[a^2 + b^2, 0] && ILtQ[n, 0] && IGtQ[m, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1469\) vs. \(2(464)=928\).

Time = 0.21 (sec) , antiderivative size = 1470, normalized size of antiderivative = 3.09

method	result	size
risch	\(\text {Expression too large to display}\)	\(1470\)

Input:

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

Output:

-4/(a^2+2*a*b+b^2)/f^2/(a-b)^2*b*a*c*d*ln(1-(a+b)*exp(2*f*x+2*e)/(a-b))*e- 
4/(a^2+2*a*b+b^2)/f/(a-b)^2*b*a*c*d*ln(1-(a+b)*exp(2*f*x+2*e)/(a-b))*x+4/( 
a^2+2*a*b+b^2)/f^2/(a-b)^2*b*e*a*c*d*ln(exp(2*f*x+2*e)*a+exp(2*f*x+2*e)*b- 
a+b)-8/(a^2+2*a*b+b^2)/f^2/(a-b)^2*b*e*a*c*d*ln(exp(f*x+e))+8/(a^2+2*a*b+b 
^2)/f/(a-b)^2*b*a*c*d*e*x-2/(a^2+2*a*b+b^2)/f/(a-b)^2*b^2*d^2*x^2-2/(a^2+2 
*a*b+b^2)/f^3/(a-b)^2*b^2*d^2*e^2+1/(a^2+2*a*b+b^2)/f^3/(a-b)^2*b^2*d^2*po 
lylog(2,(a+b)*exp(2*f*x+2*e)/(a-b))-2/(a-b)/f/(a^2+2*a*b+b^2)*(d^2*x^2+2*c 
*d*x+c^2)*b^2/(exp(2*f*x+2*e)*a+exp(2*f*x+2*e)*b-a+b)+d/(a^2+2*a*b+b^2)*c* 
x^2+1/(a^2+2*a*b+b^2)*c^2*x+1/3*d^2/(a^2+2*a*b+b^2)*x^3+1/3/d/(a^2+2*a*b+b 
^2)*c^3-4/(a^2+2*a*b+b^2)/f^2/(a-b)^2*b*a*d^2*e^2*x-2/(a^2+2*a*b+b^2)/f^2/ 
(a-b)^2*b*a*c*d*polylog(2,(a+b)*exp(2*f*x+2*e)/(a-b))-2/(a^2+2*a*b+b^2)/f/ 
(a-b)^2*b*a*d^2*ln(1-(a+b)*exp(2*f*x+2*e)/(a-b))*x^2+2/(a^2+2*a*b+b^2)/f^3 
/(a-b)^2*b*a*d^2*ln(1-(a+b)*exp(2*f*x+2*e)/(a-b))*e^2-2/(a^2+2*a*b+b^2)/f^ 
2/(a-b)^2*b*a*d^2*polylog(2,(a+b)*exp(2*f*x+2*e)/(a-b))*x-2/(a^2+2*a*b+b^2 
)/f^3/(a-b)^2*b*e^2*a*d^2*ln(exp(2*f*x+2*e)*a+exp(2*f*x+2*e)*b-a+b)+4/(a^2 
+2*a*b+b^2)/f^3/(a-b)^2*b*e^2*a*d^2*ln(exp(f*x+e))+4/(a^2+2*a*b+b^2)/(a-b) 
^2*b*a*c*d*x^2+4/(a^2+2*a*b+b^2)/f^2/(a-b)^2*b*a*c*d*e^2+4/(a^2+2*a*b+b^2) 
/f^3/(a-b)^2*b^2*e*d^2*ln(exp(f*x+e))+2/(a^2+2*a*b+b^2)/f^2/(a-b)^2*b^2*c* 
d*ln(exp(2*f*x+2*e)*a+exp(2*f*x+2*e)*b-a+b)-4/(a^2+2*a*b+b^2)/f^2/(a-b)^2* 
b^2*c*d*ln(exp(f*x+e))-4/(a^2+2*a*b+b^2)/f^2/(a-b)^2*b^2*d^2*e*x+4/3/(a...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3702 vs. \(2 (460) = 920\).

Time = 0.17 (sec) , antiderivative size = 3702, normalized size of antiderivative = 7.79 \[ \int \frac {(c+d x)^2}{(a+b \coth (e+f x))^2} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-2)]

Exception generated. \[ \int \frac {(c+d x)^2}{(a+b \coth (e+f x))^2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((d*x+c)**2/(a+b*coth(f*x+e))**2,x)
 

Output:

Exception raised: TypeError >> Invalid NaN comparison
 

Maxima [A] (verification not implemented)

Time = 1.03 (sec) , antiderivative size = 751, normalized size of antiderivative = 1.58 \[ \int \frac {(c+d x)^2}{(a+b \coth (e+f x))^2} \, dx =\text {Too large to display} \] Input:

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

Output:

-4*b^2*c*d*f*x/(a^4*f^2 - 2*a^2*b^2*f^2 + b^4*f^2) - (2*f^2*x^2*log(-(a*e^ 
(2*e) + b*e^(2*e))*e^(2*f*x)/(a - b) + 1) + 2*f*x*dilog((a*e^(2*e) + b*e^( 
2*e))*e^(2*f*x)/(a - b)) - polylog(3, (a*e^(2*e) + b*e^(2*e))*e^(2*f*x)/(a 
 - b)))*a*b*d^2/(a^4*f^3 - 2*a^2*b^2*f^3 + b^4*f^3) + 2*b^2*c*d*log((a*e^( 
2*e) + b*e^(2*e))*e^(2*f*x) - a + b)/(a^4*f^2 - 2*a^2*b^2*f^2 + b^4*f^2) - 
 c^2*(2*a*b*log(-(a - b)*e^(-2*f*x - 2*e) + a + b)/((a^4 - 2*a^2*b^2 + b^4 
)*f) + 2*b^2/((a^4 - 2*a^2*b^2 + b^4 - (a^4 - 2*a^3*b + 2*a*b^3 - b^4)*e^( 
-2*f*x - 2*e))*f) - (f*x + e)/((a^2 + 2*a*b + b^2)*f)) - (2*a*b*c*d*f - b^ 
2*d^2)*(2*f*x*log(-(a*e^(2*e) + b*e^(2*e))*e^(2*f*x)/(a - b) + 1) + dilog( 
(a*e^(2*e) + b*e^(2*e))*e^(2*f*x)/(a - b)))/(a^4*f^3 - 2*a^2*b^2*f^3 + b^4 
*f^3) + 2/3*(2*a*b*d^2*f^3*x^3 + 3*(2*a*b*c*d*f - b^2*d^2)*f^2*x^2)/(a^4*f 
^3 - 2*a^2*b^2*f^3 + b^4*f^3) + 1/3*(12*b^2*c*d*x + (a^2*d^2*f - 2*a*b*d^2 
*f + b^2*d^2*f)*x^3 + 3*(a^2*c*d*f - 2*a*b*c*d*f + (c*d*f + 2*d^2)*b^2)*x^ 
2 - ((a^2*d^2*f*e^(2*e) - b^2*d^2*f*e^(2*e))*x^3 + 3*(a^2*c*d*f*e^(2*e) - 
b^2*c*d*f*e^(2*e))*x^2)*e^(2*f*x))/(a^4*f - 2*a^2*b^2*f + b^4*f - (a^4*f*e 
^(2*e) + 2*a^3*b*f*e^(2*e) - 2*a*b^3*f*e^(2*e) - b^4*f*e^(2*e))*e^(2*f*x))
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(c+d x)^2}{(a+b \coth (e+f x))^2} \, dx=\text {too large to display} \] Input:

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

Output:

(12*e**(2*e + 2*f*x)*int(x**2/(e**(4*e + 4*f*x)*a**4 + 4*e**(4*e + 4*f*x)* 
a**3*b + 6*e**(4*e + 4*f*x)*a**2*b**2 + 4*e**(4*e + 4*f*x)*a*b**3 + e**(4* 
e + 4*f*x)*b**4 - 2*e**(2*e + 2*f*x)*a**4 - 4*e**(2*e + 2*f*x)*a**3*b + 4* 
e**(2*e + 2*f*x)*a*b**3 + 2*e**(2*e + 2*f*x)*b**4 + a**4 - 2*a**2*b**2 + b 
**4),x)*a**6*b*d**2*f**3 + 12*e**(2*e + 2*f*x)*int(x**2/(e**(4*e + 4*f*x)* 
a**4 + 4*e**(4*e + 4*f*x)*a**3*b + 6*e**(4*e + 4*f*x)*a**2*b**2 + 4*e**(4* 
e + 4*f*x)*a*b**3 + e**(4*e + 4*f*x)*b**4 - 2*e**(2*e + 2*f*x)*a**4 - 4*e* 
*(2*e + 2*f*x)*a**3*b + 4*e**(2*e + 2*f*x)*a*b**3 + 2*e**(2*e + 2*f*x)*b** 
4 + a**4 - 2*a**2*b**2 + b**4),x)*a**5*b**2*d**2*f**3 - 24*e**(2*e + 2*f*x 
)*int(x**2/(e**(4*e + 4*f*x)*a**4 + 4*e**(4*e + 4*f*x)*a**3*b + 6*e**(4*e 
+ 4*f*x)*a**2*b**2 + 4*e**(4*e + 4*f*x)*a*b**3 + e**(4*e + 4*f*x)*b**4 - 2 
*e**(2*e + 2*f*x)*a**4 - 4*e**(2*e + 2*f*x)*a**3*b + 4*e**(2*e + 2*f*x)*a* 
b**3 + 2*e**(2*e + 2*f*x)*b**4 + a**4 - 2*a**2*b**2 + b**4),x)*a**4*b**3*d 
**2*f**3 - 24*e**(2*e + 2*f*x)*int(x**2/(e**(4*e + 4*f*x)*a**4 + 4*e**(4*e 
 + 4*f*x)*a**3*b + 6*e**(4*e + 4*f*x)*a**2*b**2 + 4*e**(4*e + 4*f*x)*a*b** 
3 + e**(4*e + 4*f*x)*b**4 - 2*e**(2*e + 2*f*x)*a**4 - 4*e**(2*e + 2*f*x)*a 
**3*b + 4*e**(2*e + 2*f*x)*a*b**3 + 2*e**(2*e + 2*f*x)*b**4 + a**4 - 2*a** 
2*b**2 + b**4),x)*a**3*b**4*d**2*f**3 + 12*e**(2*e + 2*f*x)*int(x**2/(e**( 
4*e + 4*f*x)*a**4 + 4*e**(4*e + 4*f*x)*a**3*b + 6*e**(4*e + 4*f*x)*a**2*b* 
*2 + 4*e**(4*e + 4*f*x)*a*b**3 + e**(4*e + 4*f*x)*b**4 - 2*e**(2*e + 2*...