\(\int (c+d x)^3 (a+b \tanh (e+f x))^3 \, dx\) [63]

Optimal result

Integrand size = 20, antiderivative size = 566 \[ \int (c+d x)^3 (a+b \tanh (e+f x))^3 \, dx=-\frac {3 b^3 d (c+d x)^2}{2 f^2}-\frac {3 a b^2 (c+d x)^3}{f}+\frac {b^3 (c+d x)^3}{2 f}+\frac {a^3 (c+d x)^4}{4 d}-\frac {3 a^2 b (c+d x)^4}{4 d}+\frac {3 a b^2 (c+d x)^4}{4 d}-\frac {b^3 (c+d x)^4}{4 d}+\frac {3 b^3 d^2 (c+d x) \log \left (1+e^{2 (e+f x)}\right )}{f^3}+\frac {9 a b^2 d (c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )}{f^2}+\frac {3 a^2 b (c+d x)^3 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {b^3 (c+d x)^3 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {3 b^3 d^3 \operatorname {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{2 f^4}+\frac {9 a b^2 d^2 (c+d x) \operatorname {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{f^3}+\frac {9 a^2 b d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{2 f^2}+\frac {3 b^3 d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{2 f^2}-\frac {9 a b^2 d^3 \operatorname {PolyLog}\left (3,-e^{2 (e+f x)}\right )}{2 f^4}-\frac {9 a^2 b d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{2 (e+f x)}\right )}{2 f^3}-\frac {3 b^3 d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{2 (e+f x)}\right )}{2 f^3}+\frac {9 a^2 b d^3 \operatorname {PolyLog}\left (4,-e^{2 (e+f x)}\right )}{4 f^4}+\frac {3 b^3 d^3 \operatorname {PolyLog}\left (4,-e^{2 (e+f x)}\right )}{4 f^4}-\frac {3 b^3 d (c+d x)^2 \tanh (e+f x)}{2 f^2}-\frac {3 a b^2 (c+d x)^3 \tanh (e+f x)}{f}-\frac {b^3 (c+d x)^3 \tanh ^2(e+f x)}{2 f} \] Output:

-3/2*b^3*d*(d*x+c)^2/f^2-3*a*b^2*(d*x+c)^3/f+1/2*b^3*(d*x+c)^3/f+1/4*a^3*( 
d*x+c)^4/d-3/4*a^2*b*(d*x+c)^4/d+3/4*a*b^2*(d*x+c)^4/d-1/4*b^3*(d*x+c)^4/d 
+3*b^3*d^2*(d*x+c)*ln(1+exp(2*f*x+2*e))/f^3+9*a*b^2*d*(d*x+c)^2*ln(1+exp(2 
*f*x+2*e))/f^2+3*a^2*b*(d*x+c)^3*ln(1+exp(2*f*x+2*e))/f+b^3*(d*x+c)^3*ln(1 
+exp(2*f*x+2*e))/f+3/2*b^3*d^3*polylog(2,-exp(2*f*x+2*e))/f^4+9*a*b^2*d^2* 
(d*x+c)*polylog(2,-exp(2*f*x+2*e))/f^3+9/2*a^2*b*d*(d*x+c)^2*polylog(2,-ex 
p(2*f*x+2*e))/f^2+3/2*b^3*d*(d*x+c)^2*polylog(2,-exp(2*f*x+2*e))/f^2-9/2*a 
*b^2*d^3*polylog(3,-exp(2*f*x+2*e))/f^4-9/2*a^2*b*d^2*(d*x+c)*polylog(3,-e 
xp(2*f*x+2*e))/f^3-3/2*b^3*d^2*(d*x+c)*polylog(3,-exp(2*f*x+2*e))/f^3+9/4* 
a^2*b*d^3*polylog(4,-exp(2*f*x+2*e))/f^4+3/4*b^3*d^3*polylog(4,-exp(2*f*x+ 
2*e))/f^4-3/2*b^3*d*(d*x+c)^2*tanh(f*x+e)/f^2-3*a*b^2*(d*x+c)^3*tanh(f*x+e 
)/f-1/2*b^3*(d*x+c)^3*tanh(f*x+e)^2/f
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(2010\) vs. \(2(566)=1132\).

Time = 7.72 (sec) , antiderivative size = 2010, normalized size of antiderivative = 3.55 \[ \int (c+d x)^3 (a+b \tanh (e+f x))^3 \, dx=\text {Result too large to show} \] Input:

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

Output:

(b*E^(2*e)*(-24*b^2*c*d^2*x - 72*a*b*c^2*d*f*x - 24*a^2*c^3*f^2*x - 8*b^2* 
c^3*f^2*x - 12*b^2*d^3*x^2 - 72*a*b*c*d^2*f*x^2 - 36*a^2*c^2*d*f^2*x^2 - 1 
2*b^2*c^2*d*f^2*x^2 - 24*a*b*d^3*f*x^3 - 24*a^2*c*d^2*f^2*x^3 - 8*b^2*c*d^ 
2*f^2*x^3 - 6*a^2*d^3*f^2*x^4 - 2*b^2*d^3*f^2*x^4 + 36*a*b*c^2*d*Log[1 + E 
^(2*(e + f*x))] + (36*a*b*c^2*d*Log[1 + E^(2*(e + f*x))])/E^(2*e) + (12*b^ 
2*c*d^2*Log[1 + E^(2*(e + f*x))])/f + (12*b^2*c*d^2*Log[1 + E^(2*(e + f*x) 
)])/(E^(2*e)*f) + 12*a^2*c^3*f*Log[1 + E^(2*(e + f*x))] + 4*b^2*c^3*f*Log[ 
1 + E^(2*(e + f*x))] + (12*a^2*c^3*f*Log[1 + E^(2*(e + f*x))])/E^(2*e) + ( 
4*b^2*c^3*f*Log[1 + E^(2*(e + f*x))])/E^(2*e) + 72*a*b*c*d^2*x*Log[1 + E^( 
2*(e + f*x))] + (72*a*b*c*d^2*x*Log[1 + E^(2*(e + f*x))])/E^(2*e) + (12*b^ 
2*d^3*x*Log[1 + E^(2*(e + f*x))])/f + (12*b^2*d^3*x*Log[1 + E^(2*(e + f*x) 
)])/(E^(2*e)*f) + 36*a^2*c^2*d*f*x*Log[1 + E^(2*(e + f*x))] + 12*b^2*c^2*d 
*f*x*Log[1 + E^(2*(e + f*x))] + (36*a^2*c^2*d*f*x*Log[1 + E^(2*(e + f*x))] 
)/E^(2*e) + (12*b^2*c^2*d*f*x*Log[1 + E^(2*(e + f*x))])/E^(2*e) + 36*a*b*d 
^3*x^2*Log[1 + E^(2*(e + f*x))] + (36*a*b*d^3*x^2*Log[1 + E^(2*(e + f*x))] 
)/E^(2*e) + 36*a^2*c*d^2*f*x^2*Log[1 + E^(2*(e + f*x))] + 12*b^2*c*d^2*f*x 
^2*Log[1 + E^(2*(e + f*x))] + (36*a^2*c*d^2*f*x^2*Log[1 + E^(2*(e + f*x))] 
)/E^(2*e) + (12*b^2*c*d^2*f*x^2*Log[1 + E^(2*(e + f*x))])/E^(2*e) + 12*a^2 
*d^3*f*x^3*Log[1 + E^(2*(e + f*x))] + 4*b^2*d^3*f*x^3*Log[1 + E^(2*(e + f* 
x))] + (12*a^2*d^3*f*x^3*Log[1 + E^(2*(e + f*x))])/E^(2*e) + (4*b^2*d^3...
 

Rubi [A] (verified)

Time = 1.48 (sec) , antiderivative size = 566, 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.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int (c+d x)^3 (a-i b \tan (i e+i f x))^3dx\)

\(\Big \downarrow \) 4205

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {a^3 (c+d x)^4}{4 d}-\frac {9 a^2 b d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{2 (e+f x)}\right )}{2 f^3}+\frac {9 a^2 b d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{2 f^2}+\frac {3 a^2 b (c+d x)^3 \log \left (e^{2 (e+f x)}+1\right )}{f}-\frac {3 a^2 b (c+d x)^4}{4 d}+\frac {9 a^2 b d^3 \operatorname {PolyLog}\left (4,-e^{2 (e+f x)}\right )}{4 f^4}+\frac {9 a b^2 d^2 (c+d x) \operatorname {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{f^3}+\frac {9 a b^2 d (c+d x)^2 \log \left (e^{2 (e+f x)}+1\right )}{f^2}-\frac {3 a b^2 (c+d x)^3 \tanh (e+f x)}{f}-\frac {3 a b^2 (c+d x)^3}{f}+\frac {3 a b^2 (c+d x)^4}{4 d}-\frac {9 a b^2 d^3 \operatorname {PolyLog}\left (3,-e^{2 (e+f x)}\right )}{2 f^4}-\frac {3 b^3 d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{2 (e+f x)}\right )}{2 f^3}+\frac {3 b^3 d^2 (c+d x) \log \left (e^{2 (e+f x)}+1\right )}{f^3}+\frac {3 b^3 d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{2 f^2}-\frac {3 b^3 d (c+d x)^2 \tanh (e+f x)}{2 f^2}+\frac {b^3 (c+d x)^3 \log \left (e^{2 (e+f x)}+1\right )}{f}-\frac {b^3 (c+d x)^3 \tanh ^2(e+f x)}{2 f}-\frac {3 b^3 d (c+d x)^2}{2 f^2}+\frac {b^3 (c+d x)^3}{2 f}-\frac {b^3 (c+d x)^4}{4 d}+\frac {3 b^3 d^3 \operatorname {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{2 f^4}+\frac {3 b^3 d^3 \operatorname {PolyLog}\left (4,-e^{2 (e+f x)}\right )}{4 f^4}\)

Input:

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

Output:

(-3*b^3*d*(c + d*x)^2)/(2*f^2) - (3*a*b^2*(c + d*x)^3)/f + (b^3*(c + d*x)^ 
3)/(2*f) + (a^3*(c + d*x)^4)/(4*d) - (3*a^2*b*(c + d*x)^4)/(4*d) + (3*a*b^ 
2*(c + d*x)^4)/(4*d) - (b^3*(c + d*x)^4)/(4*d) + (3*b^3*d^2*(c + d*x)*Log[ 
1 + E^(2*(e + f*x))])/f^3 + (9*a*b^2*d*(c + d*x)^2*Log[1 + E^(2*(e + f*x)) 
])/f^2 + (3*a^2*b*(c + d*x)^3*Log[1 + E^(2*(e + f*x))])/f + (b^3*(c + d*x) 
^3*Log[1 + E^(2*(e + f*x))])/f + (3*b^3*d^3*PolyLog[2, -E^(2*(e + f*x))])/ 
(2*f^4) + (9*a*b^2*d^2*(c + d*x)*PolyLog[2, -E^(2*(e + f*x))])/f^3 + (9*a^ 
2*b*d*(c + d*x)^2*PolyLog[2, -E^(2*(e + f*x))])/(2*f^2) + (3*b^3*d*(c + d* 
x)^2*PolyLog[2, -E^(2*(e + f*x))])/(2*f^2) - (9*a*b^2*d^3*PolyLog[3, -E^(2 
*(e + f*x))])/(2*f^4) - (9*a^2*b*d^2*(c + d*x)*PolyLog[3, -E^(2*(e + f*x)) 
])/(2*f^3) - (3*b^3*d^2*(c + d*x)*PolyLog[3, -E^(2*(e + f*x))])/(2*f^3) + 
(9*a^2*b*d^3*PolyLog[4, -E^(2*(e + f*x))])/(4*f^4) + (3*b^3*d^3*PolyLog[4, 
 -E^(2*(e + f*x))])/(4*f^4) - (3*b^3*d*(c + d*x)^2*Tanh[e + f*x])/(2*f^2) 
- (3*a*b^2*(c + d*x)^3*Tanh[e + f*x])/f - (b^3*(c + d*x)^3*Tanh[e + f*x]^2 
)/(2*f)
 

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. \(1833\) vs. \(2(534)=1068\).

Time = 1.83 (sec) , antiderivative size = 1834, normalized size of antiderivative = 3.24

method	result	size
risch	\(\text {Expression too large to display}\)	\(1834\)

Input:

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

Output:

-9/2*a*b^2*d^3*polylog(3,-exp(2*f*x+2*e))/f^4+9/4*a^2*b*d^3*polylog(4,-exp 
(2*f*x+2*e))/f^4+3/2*b^3*d^3*polylog(2,-exp(2*f*x+2*e))/f^4+3/4*b^3*d^3*po 
lylog(4,-exp(2*f*x+2*e))/f^4+9/f^2*b^2*a*d^3*ln(1+exp(2*f*x+2*e))*x^2+9/f^ 
3*b^2*a*d^3*polylog(2,-exp(2*f*x+2*e))*x+6/f^4*b*e^3*a^2*d^3*ln(exp(f*x+e) 
)-6/f^3*b^3*e^2*c*d^2*ln(exp(f*x+e))-9/2/f^3*b*d^2*c*a^2*polylog(3,-exp(2* 
f*x+2*e))+3/f*b*a^2*d^3*ln(1+exp(2*f*x+2*e))*x^3+9/2/f^2*b*a^2*d^3*polylog 
(2,-exp(2*f*x+2*e))*x^2-9/2/f^3*b*a^2*d^3*polylog(3,-exp(2*f*x+2*e))*x+9/f 
^3*b^2*a*c*d^2*polylog(2,-exp(2*f*x+2*e))+9/2/f^2*b*d*c^2*a^2*polylog(2,-e 
xp(2*f*x+2*e))-3/f^4*b^3*e^2*d^3-3/f^2*b^3*d^3*x^2-3/2/f^4*b^3*d^3*e^4-2/f 
*b^3*c^3*ln(exp(f*x+e))+1/f*b^3*c^3*ln(1+exp(2*f*x+2*e))+1/4*d^3*a^3*x^4-1 
/4*d^3*b^3*x^4+1/4/d*c^4*a^3+1/4/d*c^4*b^3+b^2*(6*a*d^3*f*x^3*exp(2*f*x+2* 
e)+2*b*d^3*f*x^3*exp(2*f*x+2*e)+18*a*c*d^2*f*x^2*exp(2*f*x+2*e)+6*b*c*d^2* 
f*x^2*exp(2*f*x+2*e)+18*a*c^2*d*f*x*exp(2*f*x+2*e)+6*a*d^3*f*x^3+6*b*c^2*d 
*f*x*exp(2*f*x+2*e)+3*b*d^3*x^2*exp(2*f*x+2*e)+6*a*c^3*f*exp(2*f*x+2*e)+18 
*a*c*d^2*f*x^2+2*b*c^3*f*exp(2*f*x+2*e)+6*b*c*d^2*x*exp(2*f*x+2*e)+18*a*c^ 
2*d*f*x+3*b*c^2*d*exp(2*f*x+2*e)+3*b*d^3*x^2+6*a*c^3*f+6*b*c*d^2*x+3*b*c^2 
*d)/f^2/(1+exp(2*f*x+2*e))^2+6/f^2*b^3*c*d^2*e^2*x-6/f*b^3*d*c^2*e*x+18/f^ 
3*b^2*e^2*a*d^3*x-18/f*b^2*a*c*d^2*x^2+12/f^3*b*d^2*c*a^2*e^3-6/f^3*b*a^2* 
d^3*e^3*x-18/f^3*b^2*a*c*d^2*e^2-9/f^2*b*d*c^2*a^2*e^2+3/f*b^3*c*d^2*ln(1+ 
exp(2*f*x+2*e))*x^2+3/f^2*b^3*c*d^2*polylog(2,-exp(2*f*x+2*e))*x-18/f^2...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.27 (sec) , antiderivative size = 12909, normalized size of antiderivative = 22.81 \[ \int (c+d x)^3 (a+b \tanh (e+f x))^3 \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1297 vs. \(2 (530) = 1060\).

Time = 0.22 (sec) , antiderivative size = 1297, normalized size of antiderivative = 2.29 \[ \int (c+d x)^3 (a+b \tanh (e+f x))^3 \, dx=\text {Too large to display} \] Input:

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

Output:

1/4*a^3*d^3*x^4 + a^3*c*d^2*x^3 + 3/2*a^3*c^2*d*x^2 + b^3*c^3*(x + e/f + l 
og(e^(-2*f*x - 2*e) + 1)/f + 2*e^(-2*f*x - 2*e)/(f*(2*e^(-2*f*x - 2*e) + e 
^(-4*f*x - 4*e) + 1))) + a^3*c^3*x + 3*a^2*b*c^3*log(cosh(f*x + e))/f + 1/ 
4*(24*a*b^2*c^3*f + 12*b^3*c^2*d + (3*a^2*b*d^3*f^2 + 3*a*b^2*d^3*f^2 + b^ 
3*d^3*f^2)*x^4 + 4*(3*a^2*b*c*d^2*f^2 + b^3*c*d^2*f^2 + 3*(c*d^2*f^2 + 2*d 
^3*f)*a*b^2)*x^3 + 6*(3*a^2*b*c^2*d*f^2 + 3*(c^2*d*f^2 + 4*c*d^2*f)*a*b^2 
+ (c^2*d*f^2 + 2*d^3)*b^3)*x^2 + 12*(2*b^3*c*d^2 + (c^3*f^2 + 6*c^2*d*f)*a 
*b^2)*x + (12*a*b^2*c^3*f^2*x*e^(4*e) + (3*a^2*b*d^3*f^2*e^(4*e) + 3*a*b^2 
*d^3*f^2*e^(4*e) + b^3*d^3*f^2*e^(4*e))*x^4 + 4*(3*a^2*b*c*d^2*f^2*e^(4*e) 
 + 3*a*b^2*c*d^2*f^2*e^(4*e) + b^3*c*d^2*f^2*e^(4*e))*x^3 + 6*(3*a^2*b*c^2 
*d*f^2*e^(4*e) + 3*a*b^2*c^2*d*f^2*e^(4*e) + b^3*c^2*d*f^2*e^(4*e))*x^2)*e 
^(4*f*x) + 2*(12*a*b^2*c^3*f*e^(2*e) + 6*b^3*c^2*d*e^(2*e) + (3*a^2*b*d^3* 
f^2*e^(2*e) + 3*a*b^2*d^3*f^2*e^(2*e) + b^3*d^3*f^2*e^(2*e))*x^4 + 4*(3*a^ 
2*b*c*d^2*f^2*e^(2*e) + 3*(c*d^2*f^2*e^(2*e) + d^3*f*e^(2*e))*a*b^2 + (c*d 
^2*f^2*e^(2*e) + d^3*f*e^(2*e))*b^3)*x^3 + 6*(3*a^2*b*c^2*d*f^2*e^(2*e) + 
3*(c^2*d*f^2*e^(2*e) + 2*c*d^2*f*e^(2*e))*a*b^2 + (c^2*d*f^2*e^(2*e) + 2*c 
*d^2*f*e^(2*e) + d^3*e^(2*e))*b^3)*x^2 + 12*((c^3*f^2*e^(2*e) + 3*c^2*d*f* 
e^(2*e))*a*b^2 + (c^2*d*f*e^(2*e) + c*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) - 6*(3*a*b^2*c^2*d*f + b^3 
*c*d^2)*x/f^2 + 3*(3*a*b^2*c^2*d*f + b^3*c*d^2)*log(e^(2*f*x + 2*e) + 1...
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 384*e**(4*e + 4*f*x)*int(x**3/(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**3*f**4 - 128*e**(4*e + 4*f*x)*int(x* 
*3/(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**3*f**4 - 1152*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*c*d**2*f**4 - 864*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**3*f**3 - 1152*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*b**2*d**3*f**3 - 3 
84*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*c*d**2*f**4 - 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)*b**3*d** 
3*f**3 - 1152*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**2*d*f**4 - 1728*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**2*f**3 - 1008*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**3*f**2 - 2304*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*c*d**2*f**3 - 1728*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**3*f**2 - 38 
4*e**(4*e + 4*f*x)*int(x/(e**(6*e + 6*f*x) + 3*e**(4*e + 4*f*x) + 3*e**...