\(\int (c+d x)^2 (a+b \coth (e+f x))^3 \, dx\) [48]

Optimal result

Integrand size = 20, antiderivative size = 392 \[ \int (c+d x)^2 (a+b \coth (e+f x))^3 \, dx=-\frac {3 a b^2 (c+d x)^2}{f}+\frac {b^3 (c+d x)^2}{2 f}+\frac {a^3 (c+d x)^3}{3 d}-\frac {a^2 b (c+d x)^3}{d}+\frac {a b^2 (c+d x)^3}{d}-\frac {b^3 (c+d x)^3}{3 d}-\frac {b^3 d (c+d x) \coth (e+f x)}{f^2}-\frac {3 a b^2 (c+d x)^2 \coth (e+f x)}{f}-\frac {b^3 (c+d x)^2 \coth ^2(e+f x)}{2 f}+\frac {6 a b^2 d (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f^2}+\frac {3 a^2 b (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {b^3 (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {b^3 d^2 \log (\sinh (e+f x))}{f^3}+\frac {3 a b^2 d^2 \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^3}+\frac {3 a^2 b d (c+d x) \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^2}+\frac {b^3 d (c+d x) \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^2}-\frac {3 a^2 b d^2 \operatorname {PolyLog}\left (3,e^{2 (e+f x)}\right )}{2 f^3}-\frac {b^3 d^2 \operatorname {PolyLog}\left (3,e^{2 (e+f x)}\right )}{2 f^3} \] Output:

-3*a*b^2*(d*x+c)^2/f+1/2*b^3*(d*x+c)^2/f+1/3*a^3*(d*x+c)^3/d-a^2*b*(d*x+c) 
^3/d+a*b^2*(d*x+c)^3/d-1/3*b^3*(d*x+c)^3/d-b^3*d*(d*x+c)*coth(f*x+e)/f^2-3 
*a*b^2*(d*x+c)^2*coth(f*x+e)/f-1/2*b^3*(d*x+c)^2*coth(f*x+e)^2/f+6*a*b^2*d 
*(d*x+c)*ln(1-exp(2*f*x+2*e))/f^2+3*a^2*b*(d*x+c)^2*ln(1-exp(2*f*x+2*e))/f 
+b^3*(d*x+c)^2*ln(1-exp(2*f*x+2*e))/f+b^3*d^2*ln(sinh(f*x+e))/f^3+3*a*b^2* 
d^2*polylog(2,exp(2*f*x+2*e))/f^3+3*a^2*b*d*(d*x+c)*polylog(2,exp(2*f*x+2* 
e))/f^2+b^3*d*(d*x+c)*polylog(2,exp(2*f*x+2*e))/f^2-3/2*a^2*b*d^2*polylog( 
3,exp(2*f*x+2*e))/f^3-1/2*b^3*d^2*polylog(3,exp(2*f*x+2*e))/f^3
 

Mathematica [A] (verified)

Time = 4.86 (sec) , antiderivative size = 585, normalized size of antiderivative = 1.49 \[ \int (c+d x)^2 (a+b \coth (e+f x))^3 \, dx=\frac {-\frac {8 b e^{2 e} f x \left (9 a b d f (2 c+d x)+3 a^2 f^2 \left (3 c^2+3 c d x+d^2 x^2\right )+b^2 \left (3 c^2 f^2+3 c d f^2 x+d^2 \left (3+f^2 x^2\right )\right )\right )}{-1+e^{2 e}}+12 b \left (6 a b d f (c+d x)+3 a^2 f^2 (c+d x)^2+b^2 \left (c^2 f^2+2 c d f^2 x+d^2 \left (1+f^2 x^2\right )\right )\right ) \log \left (1-e^{2 (e+f x)}\right )+12 b d \left (3 a b d+3 a^2 f (c+d x)+b^2 f (c+d x)\right ) \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )-6 b \left (3 a^2+b^2\right ) d^2 \operatorname {PolyLog}\left (3,e^{2 (e+f x)}\right )+f \text {csch}(e) \text {csch}^2(e+f x) \left (-2 b \left (9 a b f (c+d x)^2+3 a^2 f^2 x \left (3 c^2+3 c d x+d^2 x^2\right )+b^2 \left (3 c^2 f^2 x+d^2 x \left (3+f^2 x^2\right )+3 c \left (d+d f^2 x^2\right )\right )\right ) \cosh (e)+b \left (18 a b f (c+d x)^2+3 a^2 f^2 x \left (3 c^2+3 c d x+d^2 x^2\right )+b^2 \left (3 c^2 f^2 x+3 c d \left (2+f^2 x^2\right )+d^2 x \left (6+f^2 x^2\right )\right )\right ) \cosh (e+2 f x)+f \left (b \left (3 a^2+b^2\right ) f x \left (3 c^2+3 c d x+d^2 x^2\right ) \cosh (3 e+2 f x)-2 \left (3 b^3 (c+d x)^2+a^3 f x \left (3 c^2+3 c d x+d^2 x^2\right )+3 a b^2 f x \left (3 c^2+3 c d x+d^2 x^2\right )-a \left (a^2+3 b^2\right ) f x \left (3 c^2+3 c d x+d^2 x^2\right ) \cosh (2 (e+f x))\right ) \sinh (e)\right )\right )}{12 f^3} \] Input:

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

Output:

((-8*b*E^(2*e)*f*x*(9*a*b*d*f*(2*c + d*x) + 3*a^2*f^2*(3*c^2 + 3*c*d*x + d 
^2*x^2) + b^2*(3*c^2*f^2 + 3*c*d*f^2*x + d^2*(3 + f^2*x^2))))/(-1 + E^(2*e 
)) + 12*b*(6*a*b*d*f*(c + d*x) + 3*a^2*f^2*(c + d*x)^2 + b^2*(c^2*f^2 + 2* 
c*d*f^2*x + d^2*(1 + f^2*x^2)))*Log[1 - E^(2*(e + f*x))] + 12*b*d*(3*a*b*d 
 + 3*a^2*f*(c + d*x) + b^2*f*(c + d*x))*PolyLog[2, E^(2*(e + f*x))] - 6*b* 
(3*a^2 + b^2)*d^2*PolyLog[3, E^(2*(e + f*x))] + f*Csch[e]*Csch[e + f*x]^2* 
(-2*b*(9*a*b*f*(c + d*x)^2 + 3*a^2*f^2*x*(3*c^2 + 3*c*d*x + d^2*x^2) + b^2 
*(3*c^2*f^2*x + d^2*x*(3 + f^2*x^2) + 3*c*(d + d*f^2*x^2)))*Cosh[e] + b*(1 
8*a*b*f*(c + d*x)^2 + 3*a^2*f^2*x*(3*c^2 + 3*c*d*x + d^2*x^2) + b^2*(3*c^2 
*f^2*x + 3*c*d*(2 + f^2*x^2) + d^2*x*(6 + f^2*x^2)))*Cosh[e + 2*f*x] + f*( 
b*(3*a^2 + b^2)*f*x*(3*c^2 + 3*c*d*x + d^2*x^2)*Cosh[3*e + 2*f*x] - 2*(3*b 
^3*(c + d*x)^2 + a^3*f*x*(3*c^2 + 3*c*d*x + d^2*x^2) + 3*a*b^2*f*x*(3*c^2 
+ 3*c*d*x + d^2*x^2) - a*(a^2 + 3*b^2)*f*x*(3*c^2 + 3*c*d*x + d^2*x^2)*Cos 
h[2*(e + f*x)])*Sinh[e])))/(12*f^3)
 

Rubi [A] (verified)

Time = 1.04 (sec) , antiderivative size = 392, 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, 4205, 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.

\(\displaystyle \int (c+d x)^2 (a+b \coth (e+f x))^3 \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (c+d x)^2 \left (a-i b \tan \left (i e+i f x+\frac {\pi }{2}\right )\right )^3dx\)

\(\Big \downarrow \) 4205

\(\displaystyle \int \left (a^3 (c+d x)^2+3 a^2 b (c+d x)^2 \coth (e+f x)+3 a b^2 (c+d x)^2 \coth ^2(e+f x)+b^3 (c+d x)^2 \coth ^3(e+f x)\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {a^3 (c+d x)^3}{3 d}+\frac {3 a^2 b d (c+d x) \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^2}+\frac {3 a^2 b (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f}-\frac {a^2 b (c+d x)^3}{d}-\frac {3 a^2 b d^2 \operatorname {PolyLog}\left (3,e^{2 (e+f x)}\right )}{2 f^3}+\frac {6 a b^2 d (c+d x) \log \left (1-e^{2 (e+f x)}\right )}{f^2}-\frac {3 a b^2 (c+d x)^2 \coth (e+f x)}{f}-\frac {3 a b^2 (c+d x)^2}{f}+\frac {a b^2 (c+d x)^3}{d}+\frac {3 a b^2 d^2 \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^3}+\frac {b^3 d (c+d x) \operatorname {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^2}-\frac {b^3 d (c+d x) \coth (e+f x)}{f^2}+\frac {b^3 (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f}-\frac {b^3 (c+d x)^2 \coth ^2(e+f x)}{2 f}+\frac {b^3 (c+d x)^2}{2 f}-\frac {b^3 (c+d x)^3}{3 d}-\frac {b^3 d^2 \operatorname {PolyLog}\left (3,e^{2 (e+f x)}\right )}{2 f^3}+\frac {b^3 d^2 \log (\sinh (e+f x))}{f^3}\)

Input:

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

Output:

(-3*a*b^2*(c + d*x)^2)/f + (b^3*(c + d*x)^2)/(2*f) + (a^3*(c + d*x)^3)/(3* 
d) - (a^2*b*(c + d*x)^3)/d + (a*b^2*(c + d*x)^3)/d - (b^3*(c + d*x)^3)/(3* 
d) - (b^3*d*(c + d*x)*Coth[e + f*x])/f^2 - (3*a*b^2*(c + d*x)^2*Coth[e + f 
*x])/f - (b^3*(c + d*x)^2*Coth[e + f*x]^2)/(2*f) + (6*a*b^2*d*(c + d*x)*Lo 
g[1 - E^(2*(e + f*x))])/f^2 + (3*a^2*b*(c + d*x)^2*Log[1 - E^(2*(e + f*x)) 
])/f + (b^3*(c + d*x)^2*Log[1 - E^(2*(e + f*x))])/f + (b^3*d^2*Log[Sinh[e 
+ f*x]])/f^3 + (3*a*b^2*d^2*PolyLog[2, E^(2*(e + f*x))])/f^3 + (3*a^2*b*d* 
(c + d*x)*PolyLog[2, E^(2*(e + f*x))])/f^2 + (b^3*d*(c + d*x)*PolyLog[2, E 
^(2*(e + f*x))])/f^2 - (3*a^2*b*d^2*PolyLog[3, E^(2*(e + f*x))])/(2*f^3) - 
 (b^3*d^2*PolyLog[3, E^(2*(e + f*x))])/(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 4205

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1585\) vs. \(2(380)=760\).

Time = 0.40 (sec) , antiderivative size = 1586, normalized size of antiderivative = 4.05

method	result	size
risch	\(\text {Expression too large to display}\)	\(1586\)

Input:

int((d*x+c)^2*(a+b*coth(f*x+e))^3,x,method=_RETURNVERBOSE)
 

Output:

-d^2*a^2*b*x^3+d^2*a*b^2*x^3+d*a^3*c*x^2-d*b^3*c*x^2+a^3*c^2*x+b^3*c^2*x+1 
/d*a^2*b*c^3+1/d*a*b^2*c^3+4/3/f^3*b^3*d^2*e^3+1/f*b^3*c^2*ln(exp(f*x+e)-1 
)-2/f*b^3*c^2*ln(exp(f*x+e))+1/f*b^3*c^2*ln(exp(f*x+e)+1)+1/f^3*b^3*d^2*ln 
(exp(f*x+e)-1)-2/f^3*b^3*d^2*ln(exp(f*x+e))+1/f^3*b^3*d^2*ln(exp(f*x+e)+1) 
-2/f^3*b^3*d^2*polylog(3,exp(f*x+e))-2/f^3*b^3*d^2*polylog(3,-exp(f*x+e))+ 
6/f*b*c*a^2*d*ln(1-exp(f*x+e))*x+6/f^2*b*c*a^2*d*ln(1-exp(f*x+e))*e+6/f*b* 
c*a^2*d*ln(exp(f*x+e)+1)*x-6/f^2*b*e*c*a^2*d*ln(exp(f*x+e)-1)+12/f^2*b*e*c 
*a^2*d*ln(exp(f*x+e))-2*b^2*(3*a*d^2*f*x^2*exp(2*f*x+2*e)+b*d^2*f*x^2*exp( 
2*f*x+2*e)+6*a*c*d*f*x*exp(2*f*x+2*e)+2*b*c*d*f*x*exp(2*f*x+2*e)+3*a*c^2*f 
*exp(2*f*x+2*e)-3*a*d^2*f*x^2+b*c^2*f*exp(2*f*x+2*e)+b*d^2*x*exp(2*f*x+2*e 
)-6*a*c*d*f*x+b*c*d*exp(2*f*x+2*e)-3*a*c^2*f-b*d^2*x-b*c*d)/f^2/(exp(2*f*x 
+2*e)-1)^2+4/f^3*b*a^2*d^2*e^3-2/f^2*b^3*d*c*e^2+2/f^2*b^3*d^2*e^2*x-6/f*b 
^2*a*d^2*x^2+6/f^3*b^2*a*d^2*polylog(2,-exp(f*x+e))+1/f^3*b^3*e^2*d^2*ln(e 
xp(f*x+e)-1)-2/f^3*b^3*e^2*d^2*ln(exp(f*x+e))-1/f^3*b^3*d^2*ln(1-exp(f*x+e 
))*e^2+3/f*b*a^2*c^2*ln(exp(f*x+e)-1)-6/f*b*a^2*c^2*ln(exp(f*x+e))+3/f*b*a 
^2*c^2*ln(exp(f*x+e)+1)-6/f^3*b*a^2*d^2*polylog(3,exp(f*x+e))-6/f^3*b*a^2* 
d^2*polylog(3,-exp(f*x+e))+2/f^2*b^3*d*c*polylog(2,exp(f*x+e))+2/f^2*b^3*d 
*c*polylog(2,-exp(f*x+e))+1/f*b^3*d^2*ln(1-exp(f*x+e))*x^2+2/f^2*b^3*d^2*p 
olylog(2,exp(f*x+e))*x+1/f*b^3*d^2*ln(exp(f*x+e)+1)*x^2+2/f^2*b^3*d^2*poly 
log(2,-exp(f*x+e))*x+6/f^3*b^2*a*d^2*polylog(2,exp(f*x+e))-6/f^3*b^2*a*...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6356 vs. \(2 (377) = 754\).

Time = 0.16 (sec) , antiderivative size = 6356, normalized size of antiderivative = 16.21 \[ \int (c+d x)^2 (a+b \coth (e+f x))^3 \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

integrate((d*x+c)**2*(a+b*coth(f*x+e))**3,x)
 

Output:

Integral((a + b*coth(e + f*x))**3*(c + d*x)**2, x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 997 vs. \(2 (377) = 754\).

Time = 0.37 (sec) , antiderivative size = 997, normalized size of antiderivative = 2.54 \[ \int (c+d x)^2 (a+b \coth (e+f x))^3 \, dx=\text {Too large to display} \] Input:

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

Output:

1/3*a^3*d^2*x^3 + a^3*c*d*x^2 + a^3*c^2*x + 3*a^2*b*c^2*log(sinh(f*x + e)) 
/f + 1/3*(18*a*b^2*c^2*f + 6*b^3*c*d + (3*a^2*b*d^2*f^2 + 3*a*b^2*d^2*f^2 
+ b^3*d^2*f^2)*x^3 + 3*(3*a^2*b*c*d*f^2 + b^3*c*d*f^2 + 3*(c*d*f^2 + 2*d^2 
*f)*a*b^2)*x^2 + 3*(3*(c^2*f^2 + 4*c*d*f)*a*b^2 + (c^2*f^2 + 2*d^2)*b^3)*x 
 + ((3*a^2*b*d^2*f^2*e^(4*e) + 3*a*b^2*d^2*f^2*e^(4*e) + b^3*d^2*f^2*e^(4* 
e))*x^3 + 3*(3*a^2*b*c*d*f^2*e^(4*e) + 3*a*b^2*c*d*f^2*e^(4*e) + b^3*c*d*f 
^2*e^(4*e))*x^2 + 3*(3*a*b^2*c^2*f^2*e^(4*e) + b^3*c^2*f^2*e^(4*e))*x)*e^( 
4*f*x) - 2*(9*a*b^2*c^2*f*e^(2*e) + 3*(c^2*f*e^(2*e) + c*d*e^(2*e))*b^3 + 
(3*a^2*b*d^2*f^2*e^(2*e) + 3*a*b^2*d^2*f^2*e^(2*e) + b^3*d^2*f^2*e^(2*e))* 
x^3 + 3*(3*a^2*b*c*d*f^2*e^(2*e) + 3*(c*d*f^2*e^(2*e) + d^2*f*e^(2*e))*a*b 
^2 + (c*d*f^2*e^(2*e) + d^2*f*e^(2*e))*b^3)*x^2 + 3*(3*(c^2*f^2*e^(2*e) + 
2*c*d*f*e^(2*e))*a*b^2 + (c^2*f^2*e^(2*e) + 2*c*d*f*e^(2*e) + d^2*e^(2*e)) 
*b^3)*x)*e^(2*f*x))/(f^2*e^(4*f*x + 4*e) - 2*f^2*e^(2*f*x + 2*e) + f^2) - 
2*(6*a*b^2*c*d*f + (c^2*f^2 + d^2)*b^3)*x/f^2 + (3*a^2*b*d^2 + b^3*d^2)*(f 
^2*x^2*log(e^(f*x + e) + 1) + 2*f*x*dilog(-e^(f*x + e)) - 2*polylog(3, -e^ 
(f*x + e)))/f^3 + (3*a^2*b*d^2 + b^3*d^2)*(f^2*x^2*log(-e^(f*x + e) + 1) + 
 2*f*x*dilog(e^(f*x + e)) - 2*polylog(3, e^(f*x + e)))/f^3 + 2*(3*a^2*b*c* 
d*f + b^3*c*d*f + 3*a*b^2*d^2)*(f*x*log(e^(f*x + e) + 1) + dilog(-e^(f*x + 
 e)))/f^3 + 2*(3*a^2*b*c*d*f + b^3*c*d*f + 3*a*b^2*d^2)*(f*x*log(-e^(f*x + 
 e) + 1) + dilog(e^(f*x + e)))/f^3 + (6*a*b^2*c*d*f + (c^2*f^2 + d^2)*b...
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (c+d x)^2 (a+b \coth (e+f x))^3 \, dx=\text {too large to display} \] Input:

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

Output:

(288*e**(4*e + 4*f*x)*int(x**2/(e**(6*e + 6*f*x) - 3*e**(4*e + 4*f*x) + 3* 
e**(2*e + 2*f*x) - 1),x)*a**2*b*d**2*f**3 + 96*e**(4*e + 4*f*x)*int(x**2/( 
e**(6*e + 6*f*x) - 3*e**(4*e + 4*f*x) + 3*e**(2*e + 2*f*x) - 1),x)*b**3*d* 
*2*f**3 + 576*e**(4*e + 4*f*x)*int(x/(e**(6*e + 6*f*x) - 3*e**(4*e + 4*f*x 
) + 3*e**(2*e + 2*f*x) - 1),x)*a**2*b*c*d*f**3 + 432*e**(4*e + 4*f*x)*int( 
x/(e**(6*e + 6*f*x) - 3*e**(4*e + 4*f*x) + 3*e**(2*e + 2*f*x) - 1),x)*a**2 
*b*d**2*f**2 + 576*e**(4*e + 4*f*x)*int(x/(e**(6*e + 6*f*x) - 3*e**(4*e + 
4*f*x) + 3*e**(2*e + 2*f*x) - 1),x)*a*b**2*d**2*f**2 + 192*e**(4*e + 4*f*x 
)*int(x/(e**(6*e + 6*f*x) - 3*e**(4*e + 4*f*x) + 3*e**(2*e + 2*f*x) - 1),x 
)*b**3*c*d*f**3 + 144*e**(4*e + 4*f*x)*int(x/(e**(6*e + 6*f*x) - 3*e**(4*e 
 + 4*f*x) + 3*e**(2*e + 2*f*x) - 1),x)*b**3*d**2*f**2 + 144*e**(4*e + 4*f* 
x)*log(e**(e + f*x) - 1)*a**2*b*c**2*f**2 + 216*e**(4*e + 4*f*x)*log(e**(e 
 + f*x) - 1)*a**2*b*c*d*f + 126*e**(4*e + 4*f*x)*log(e**(e + f*x) - 1)*a** 
2*b*d**2 + 288*e**(4*e + 4*f*x)*log(e**(e + f*x) - 1)*a*b**2*c*d*f + 216*e 
**(4*e + 4*f*x)*log(e**(e + f*x) - 1)*a*b**2*d**2 + 48*e**(4*e + 4*f*x)*lo 
g(e**(e + f*x) - 1)*b**3*c**2*f**2 + 72*e**(4*e + 4*f*x)*log(e**(e + f*x) 
- 1)*b**3*c*d*f + 90*e**(4*e + 4*f*x)*log(e**(e + f*x) - 1)*b**3*d**2 + 14 
4*e**(4*e + 4*f*x)*log(e**(e + f*x) + 1)*a**2*b*c**2*f**2 + 216*e**(4*e + 
4*f*x)*log(e**(e + f*x) + 1)*a**2*b*c*d*f + 126*e**(4*e + 4*f*x)*log(e**(e 
 + f*x) + 1)*a**2*b*d**2 + 288*e**(4*e + 4*f*x)*log(e**(e + f*x) + 1)*a...