3.43 \(\int \frac{(c+d x)^3}{(a+a \tanh (e+f x))^3} \, dx\)

Optimal. Leaf size=336 \[ -\frac{d^2 (c+d x) e^{-6 e-6 f x}}{288 a^3 f^3}-\frac{9 d^2 (c+d x) e^{-4 e-4 f x}}{256 a^3 f^3}-\frac{9 d^2 (c+d x) e^{-2 e-2 f x}}{32 a^3 f^3}-\frac{d (c+d x)^2 e^{-6 e-6 f x}}{96 a^3 f^2}-\frac{9 d (c+d x)^2 e^{-4 e-4 f x}}{128 a^3 f^2}-\frac{9 d (c+d x)^2 e^{-2 e-2 f x}}{32 a^3 f^2}-\frac{(c+d x)^3 e^{-6 e-6 f x}}{48 a^3 f}-\frac{3 (c+d x)^3 e^{-4 e-4 f x}}{32 a^3 f}-\frac{3 (c+d x)^3 e^{-2 e-2 f x}}{16 a^3 f}+\frac{(c+d x)^4}{32 a^3 d}-\frac{d^3 e^{-6 e-6 f x}}{1728 a^3 f^4}-\frac{9 d^3 e^{-4 e-4 f x}}{1024 a^3 f^4}-\frac{9 d^3 e^{-2 e-2 f x}}{64 a^3 f^4} \]

[Out]

-(d^3*E^(-6*e - 6*f*x))/(1728*a^3*f^4) - (9*d^3*E^(-4*e - 4*f*x))/(1024*a^3*f^4) - (9*d^3*E^(-2*e - 2*f*x))/(6
4*a^3*f^4) - (d^2*E^(-6*e - 6*f*x)*(c + d*x))/(288*a^3*f^3) - (9*d^2*E^(-4*e - 4*f*x)*(c + d*x))/(256*a^3*f^3)
- (9*d^2*E^(-2*e - 2*f*x)*(c + d*x))/(32*a^3*f^3) - (d*E^(-6*e - 6*f*x)*(c + d*x)^2)/(96*a^3*f^2) - (9*d*E^(-
4*e - 4*f*x)*(c + d*x)^2)/(128*a^3*f^2) - (9*d*E^(-2*e - 2*f*x)*(c + d*x)^2)/(32*a^3*f^2) - (E^(-6*e - 6*f*x)*
(c + d*x)^3)/(48*a^3*f) - (3*E^(-4*e - 4*f*x)*(c + d*x)^3)/(32*a^3*f) - (3*E^(-2*e - 2*f*x)*(c + d*x)^3)/(16*a
^3*f) + (c + d*x)^4/(32*a^3*d)

________________________________________________________________________________________

Rubi [A]  time = 0.378999, antiderivative size = 336, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {3729, 2176, 2194} \[ -\frac{d^2 (c+d x) e^{-6 e-6 f x}}{288 a^3 f^3}-\frac{9 d^2 (c+d x) e^{-4 e-4 f x}}{256 a^3 f^3}-\frac{9 d^2 (c+d x) e^{-2 e-2 f x}}{32 a^3 f^3}-\frac{d (c+d x)^2 e^{-6 e-6 f x}}{96 a^3 f^2}-\frac{9 d (c+d x)^2 e^{-4 e-4 f x}}{128 a^3 f^2}-\frac{9 d (c+d x)^2 e^{-2 e-2 f x}}{32 a^3 f^2}-\frac{(c+d x)^3 e^{-6 e-6 f x}}{48 a^3 f}-\frac{3 (c+d x)^3 e^{-4 e-4 f x}}{32 a^3 f}-\frac{3 (c+d x)^3 e^{-2 e-2 f x}}{16 a^3 f}+\frac{(c+d x)^4}{32 a^3 d}-\frac{d^3 e^{-6 e-6 f x}}{1728 a^3 f^4}-\frac{9 d^3 e^{-4 e-4 f x}}{1024 a^3 f^4}-\frac{9 d^3 e^{-2 e-2 f x}}{64 a^3 f^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(d^3*E^(-6*e - 6*f*x))/(1728*a^3*f^4) - (9*d^3*E^(-4*e - 4*f*x))/(1024*a^3*f^4) - (9*d^3*E^(-2*e - 2*f*x))/(6
4*a^3*f^4) - (d^2*E^(-6*e - 6*f*x)*(c + d*x))/(288*a^3*f^3) - (9*d^2*E^(-4*e - 4*f*x)*(c + d*x))/(256*a^3*f^3)
- (9*d^2*E^(-2*e - 2*f*x)*(c + d*x))/(32*a^3*f^3) - (d*E^(-6*e - 6*f*x)*(c + d*x)^2)/(96*a^3*f^2) - (9*d*E^(-
4*e - 4*f*x)*(c + d*x)^2)/(128*a^3*f^2) - (9*d*E^(-2*e - 2*f*x)*(c + d*x)^2)/(32*a^3*f^2) - (E^(-6*e - 6*f*x)*
(c + d*x)^3)/(48*a^3*f) - (3*E^(-4*e - 4*f*x)*(c + d*x)^3)/(32*a^3*f) - (3*E^(-2*e - 2*f*x)*(c + d*x)^3)/(16*a
^3*f) + (c + d*x)^4/(32*a^3*d)

Rule 3729

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

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rubi steps

\begin{align*} \int \frac{(c+d x)^3}{(a+a \tanh (e+f x))^3} \, dx &=\int \left (\frac{(c+d x)^3}{8 a^3}+\frac{e^{-6 e-6 f x} (c+d x)^3}{8 a^3}+\frac{3 e^{-4 e-4 f x} (c+d x)^3}{8 a^3}+\frac{3 e^{-2 e-2 f x} (c+d x)^3}{8 a^3}\right ) \, dx\\ &=\frac{(c+d x)^4}{32 a^3 d}+\frac{\int e^{-6 e-6 f x} (c+d x)^3 \, dx}{8 a^3}+\frac{3 \int e^{-4 e-4 f x} (c+d x)^3 \, dx}{8 a^3}+\frac{3 \int e^{-2 e-2 f x} (c+d x)^3 \, dx}{8 a^3}\\ &=-\frac{e^{-6 e-6 f x} (c+d x)^3}{48 a^3 f}-\frac{3 e^{-4 e-4 f x} (c+d x)^3}{32 a^3 f}-\frac{3 e^{-2 e-2 f x} (c+d x)^3}{16 a^3 f}+\frac{(c+d x)^4}{32 a^3 d}+\frac{d \int e^{-6 e-6 f x} (c+d x)^2 \, dx}{16 a^3 f}+\frac{(9 d) \int e^{-4 e-4 f x} (c+d x)^2 \, dx}{32 a^3 f}+\frac{(9 d) \int e^{-2 e-2 f x} (c+d x)^2 \, dx}{16 a^3 f}\\ &=-\frac{d e^{-6 e-6 f x} (c+d x)^2}{96 a^3 f^2}-\frac{9 d e^{-4 e-4 f x} (c+d x)^2}{128 a^3 f^2}-\frac{9 d e^{-2 e-2 f x} (c+d x)^2}{32 a^3 f^2}-\frac{e^{-6 e-6 f x} (c+d x)^3}{48 a^3 f}-\frac{3 e^{-4 e-4 f x} (c+d x)^3}{32 a^3 f}-\frac{3 e^{-2 e-2 f x} (c+d x)^3}{16 a^3 f}+\frac{(c+d x)^4}{32 a^3 d}+\frac{d^2 \int e^{-6 e-6 f x} (c+d x) \, dx}{48 a^3 f^2}+\frac{\left (9 d^2\right ) \int e^{-4 e-4 f x} (c+d x) \, dx}{64 a^3 f^2}+\frac{\left (9 d^2\right ) \int e^{-2 e-2 f x} (c+d x) \, dx}{16 a^3 f^2}\\ &=-\frac{d^2 e^{-6 e-6 f x} (c+d x)}{288 a^3 f^3}-\frac{9 d^2 e^{-4 e-4 f x} (c+d x)}{256 a^3 f^3}-\frac{9 d^2 e^{-2 e-2 f x} (c+d x)}{32 a^3 f^3}-\frac{d e^{-6 e-6 f x} (c+d x)^2}{96 a^3 f^2}-\frac{9 d e^{-4 e-4 f x} (c+d x)^2}{128 a^3 f^2}-\frac{9 d e^{-2 e-2 f x} (c+d x)^2}{32 a^3 f^2}-\frac{e^{-6 e-6 f x} (c+d x)^3}{48 a^3 f}-\frac{3 e^{-4 e-4 f x} (c+d x)^3}{32 a^3 f}-\frac{3 e^{-2 e-2 f x} (c+d x)^3}{16 a^3 f}+\frac{(c+d x)^4}{32 a^3 d}+\frac{d^3 \int e^{-6 e-6 f x} \, dx}{288 a^3 f^3}+\frac{\left (9 d^3\right ) \int e^{-4 e-4 f x} \, dx}{256 a^3 f^3}+\frac{\left (9 d^3\right ) \int e^{-2 e-2 f x} \, dx}{32 a^3 f^3}\\ &=-\frac{d^3 e^{-6 e-6 f x}}{1728 a^3 f^4}-\frac{9 d^3 e^{-4 e-4 f x}}{1024 a^3 f^4}-\frac{9 d^3 e^{-2 e-2 f x}}{64 a^3 f^4}-\frac{d^2 e^{-6 e-6 f x} (c+d x)}{288 a^3 f^3}-\frac{9 d^2 e^{-4 e-4 f x} (c+d x)}{256 a^3 f^3}-\frac{9 d^2 e^{-2 e-2 f x} (c+d x)}{32 a^3 f^3}-\frac{d e^{-6 e-6 f x} (c+d x)^2}{96 a^3 f^2}-\frac{9 d e^{-4 e-4 f x} (c+d x)^2}{128 a^3 f^2}-\frac{9 d e^{-2 e-2 f x} (c+d x)^2}{32 a^3 f^2}-\frac{e^{-6 e-6 f x} (c+d x)^3}{48 a^3 f}-\frac{3 e^{-4 e-4 f x} (c+d x)^3}{32 a^3 f}-\frac{3 e^{-2 e-2 f x} (c+d x)^3}{16 a^3 f}+\frac{(c+d x)^4}{32 a^3 d}\\ \end{align*}

Mathematica [A]  time = 2.60011, size = 615, normalized size = 1.83 \[ \frac{\text{sech}^3(e+f x) \left (-243 \left (8 c^2 d f^2 (12 f x+5)+32 c^3 f^3+4 c d^2 f \left (24 f^2 x^2+20 f x+9\right )+d^3 \left (32 f^3 x^3+40 f^2 x^2+36 f x+17\right )\right ) \cosh (e+f x)+16 \left (18 c^2 d f^2 \left (18 f^2 x^2-6 f x-1\right )+36 c^3 f^3 (6 f x-1)+6 c d^2 f \left (36 f^3 x^3-18 f^2 x^2-6 f x-1\right )+d^3 \left (54 f^4 x^4-36 f^3 x^3-18 f^2 x^2-6 f x-1\right )\right ) \cosh (3 (e+f x))+5184 c^2 d f^4 x^2 \sinh (3 (e+f x))-7776 c^2 d f^3 x \sinh (e+f x)+1728 c^2 d f^3 x \sinh (3 (e+f x))-5832 c^2 d f^2 \sinh (e+f x)+288 c^2 d f^2 \sinh (3 (e+f x))+3456 c^3 f^4 x \sinh (3 (e+f x))-2592 c^3 f^3 \sinh (e+f x)+576 c^3 f^3 \sinh (3 (e+f x))+3456 c d^2 f^4 x^3 \sinh (3 (e+f x))-7776 c d^2 f^3 x^2 \sinh (e+f x)+1728 c d^2 f^3 x^2 \sinh (3 (e+f x))-11664 c d^2 f^2 x \sinh (e+f x)+576 c d^2 f^2 x \sinh (3 (e+f x))-6804 c d^2 f \sinh (e+f x)+96 c d^2 f \sinh (3 (e+f x))+864 d^3 f^4 x^4 \sinh (3 (e+f x))-2592 d^3 f^3 x^3 \sinh (e+f x)+576 d^3 f^3 x^3 \sinh (3 (e+f x))-5832 d^3 f^2 x^2 \sinh (e+f x)+288 d^3 f^2 x^2 \sinh (3 (e+f x))-6804 d^3 f x \sinh (e+f x)+96 d^3 f x \sinh (3 (e+f x))-3645 d^3 \sinh (e+f x)+16 d^3 \sinh (3 (e+f x))\right )}{27648 a^3 f^4 (\tanh (e+f x)+1)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sech[e + f*x]^3*(-243*(32*c^3*f^3 + 8*c^2*d*f^2*(5 + 12*f*x) + 4*c*d^2*f*(9 + 20*f*x + 24*f^2*x^2) + d^3*(17
+ 36*f*x + 40*f^2*x^2 + 32*f^3*x^3))*Cosh[e + f*x] + 16*(36*c^3*f^3*(-1 + 6*f*x) + 18*c^2*d*f^2*(-1 - 6*f*x +
18*f^2*x^2) + 6*c*d^2*f*(-1 - 6*f*x - 18*f^2*x^2 + 36*f^3*x^3) + d^3*(-1 - 6*f*x - 18*f^2*x^2 - 36*f^3*x^3 + 5
4*f^4*x^4))*Cosh[3*(e + f*x)] - 3645*d^3*Sinh[e + f*x] - 6804*c*d^2*f*Sinh[e + f*x] - 5832*c^2*d*f^2*Sinh[e +
f*x] - 2592*c^3*f^3*Sinh[e + f*x] - 6804*d^3*f*x*Sinh[e + f*x] - 11664*c*d^2*f^2*x*Sinh[e + f*x] - 7776*c^2*d*
f^3*x*Sinh[e + f*x] - 5832*d^3*f^2*x^2*Sinh[e + f*x] - 7776*c*d^2*f^3*x^2*Sinh[e + f*x] - 2592*d^3*f^3*x^3*Sin
h[e + f*x] + 16*d^3*Sinh[3*(e + f*x)] + 96*c*d^2*f*Sinh[3*(e + f*x)] + 288*c^2*d*f^2*Sinh[3*(e + f*x)] + 576*c
^3*f^3*Sinh[3*(e + f*x)] + 96*d^3*f*x*Sinh[3*(e + f*x)] + 576*c*d^2*f^2*x*Sinh[3*(e + f*x)] + 1728*c^2*d*f^3*x
*Sinh[3*(e + f*x)] + 3456*c^3*f^4*x*Sinh[3*(e + f*x)] + 288*d^3*f^2*x^2*Sinh[3*(e + f*x)] + 1728*c*d^2*f^3*x^2
*Sinh[3*(e + f*x)] + 5184*c^2*d*f^4*x^2*Sinh[3*(e + f*x)] + 576*d^3*f^3*x^3*Sinh[3*(e + f*x)] + 3456*c*d^2*f^4
*x^3*Sinh[3*(e + f*x)] + 864*d^3*f^4*x^4*Sinh[3*(e + f*x)]))/(27648*a^3*f^4*(1 + Tanh[e + f*x])^3)

________________________________________________________________________________________

Maple [B]  time = 0.063, size = 4207, normalized size = 12.5 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/f^4/a^3*(d^3*(1/4*(f*x+e)^3*sinh(f*x+e)^2*cosh(f*x+e)^2+1/4*(f*x+e)^3*cosh(f*x+e)^2-3/16*(f*x+e)^2*sinh(f*x+
e)*cosh(f*x+e)^3-9/32*(f*x+e)^2*cosh(f*x+e)*sinh(f*x+e)-3/32*(f*x+e)^3+3/32*(f*x+e)*sinh(f*x+e)^2*cosh(f*x+e)^
2-3/128*cosh(f*x+e)^3*sinh(f*x+e)-45/256*cosh(f*x+e)*sinh(f*x+e)-45/256*f*x-45/256*e+3/8*(f*x+e)*cosh(f*x+e)^2
)-3*d^3*(1/4*(f*x+e)^3*sinh(f*x+e)*cosh(f*x+e)^3+3/8*(f*x+e)^3*cosh(f*x+e)*sinh(f*x+e)+3/32*(f*x+e)^4-3/16*(f*
x+e)^2*sinh(f*x+e)^2*cosh(f*x+e)^2-3/4*(f*x+e)^2*cosh(f*x+e)^2+3/32*(f*x+e)*sinh(f*x+e)*cosh(f*x+e)^3+45/64*(f
*x+e)*cosh(f*x+e)*sinh(f*x+e)+45/128*(f*x+e)^2-3/128*sinh(f*x+e)^2*cosh(f*x+e)^2-3/8*cosh(f*x+e)^2)+9*c^2*d*e*
f^2*((1/4*cosh(f*x+e)^3+3/8*cosh(f*x+e))*sinh(f*x+e)+3/8*f*x+3/8*e)-6*c*d^2*e*f*(1/4*(f*x+e)*sinh(f*x+e)^2*cos
h(f*x+e)^2+1/4*(f*x+e)*cosh(f*x+e)^2-1/16*cosh(f*x+e)^3*sinh(f*x+e)-3/32*cosh(f*x+e)*sinh(f*x+e)-3/32*f*x-3/32
*e)-9*c*d^2*e^2*f*((1/4*cosh(f*x+e)^3+3/8*cosh(f*x+e))*sinh(f*x+e)+3/8*f*x+3/8*e)+18*c*d^2*e*f*(1/4*(f*x+e)*si
nh(f*x+e)*cosh(f*x+e)^3+3/8*(f*x+e)*cosh(f*x+e)*sinh(f*x+e)+3/16*(f*x+e)^2-1/16*sinh(f*x+e)^2*cosh(f*x+e)^2-1/
4*cosh(f*x+e)^2)-3*c^2*d*e*f^2*(1/4*sinh(f*x+e)^2*cosh(f*x+e)^2+1/4*cosh(f*x+e)^2)+3*c*d^2*e^2*f*(1/4*sinh(f*x
+e)^2*cosh(f*x+e)^2+1/4*cosh(f*x+e)^2)+24*c*d^2*e*f*(1/6*(f*x+e)*sinh(f*x+e)^2*cosh(f*x+e)^4+1/6*(f*x+e)*sinh(
f*x+e)^2*cosh(f*x+e)^2+1/6*(f*x+e)*cosh(f*x+e)^2-1/36*sinh(f*x+e)*cosh(f*x+e)^5-5/144*cosh(f*x+e)^3*sinh(f*x+e
)-5/96*cosh(f*x+e)*sinh(f*x+e)-5/96*f*x-5/96*e)-12*c*d^2*e^2*f*(1/6*sinh(f*x+e)^2*cosh(f*x+e)^4+1/6*sinh(f*x+e
)^2*cosh(f*x+e)^2+1/6*cosh(f*x+e)^2)-24*c*d^2*e*f*(1/6*(f*x+e)*sinh(f*x+e)*cosh(f*x+e)^5+5/24*(f*x+e)*sinh(f*x
+e)*cosh(f*x+e)^3+5/16*(f*x+e)*cosh(f*x+e)*sinh(f*x+e)+5/32*(f*x+e)^2-1/36*sinh(f*x+e)^2*cosh(f*x+e)^4-23/288*
sinh(f*x+e)^2*cosh(f*x+e)^2-17/72*cosh(f*x+e)^2)+12*c^2*d*e*f^2*(1/6*sinh(f*x+e)^2*cosh(f*x+e)^4+1/6*sinh(f*x+
e)^2*cosh(f*x+e)^2+1/6*cosh(f*x+e)^2)-12*c^2*d*e*f^2*((1/6*cosh(f*x+e)^5+5/24*cosh(f*x+e)^3+5/16*cosh(f*x+e))*
sinh(f*x+e)+5/16*f*x+5/16*e)+12*c*d^2*e^2*f*((1/6*cosh(f*x+e)^5+5/24*cosh(f*x+e)^3+5/16*cosh(f*x+e))*sinh(f*x+
e)+5/16*f*x+5/16*e)+4*d^3*(1/6*(f*x+e)^3*sinh(f*x+e)*cosh(f*x+e)^5+5/24*(f*x+e)^3*sinh(f*x+e)*cosh(f*x+e)^3+5/
16*(f*x+e)^3*cosh(f*x+e)*sinh(f*x+e)+5/64*(f*x+e)^4-1/12*(f*x+e)^2*sinh(f*x+e)^2*cosh(f*x+e)^4-23/96*(f*x+e)^2
*sinh(f*x+e)^2*cosh(f*x+e)^2-17/24*(f*x+e)^2*cosh(f*x+e)^2+1/36*(f*x+e)*sinh(f*x+e)*cosh(f*x+e)^5+65/576*(f*x+
e)*sinh(f*x+e)*cosh(f*x+e)^3+245/384*(f*x+e)*cosh(f*x+e)*sinh(f*x+e)+245/768*(f*x+e)^2-1/216*sinh(f*x+e)^2*cos
h(f*x+e)^4-227/6912*sinh(f*x+e)^2*cosh(f*x+e)^2-19/54*cosh(f*x+e)^2)-4*d^3*(1/6*(f*x+e)^3*sinh(f*x+e)^2*cosh(f
*x+e)^4+1/6*(f*x+e)^3*sinh(f*x+e)^2*cosh(f*x+e)^2+1/6*(f*x+e)^3*cosh(f*x+e)^2-1/12*(f*x+e)^2*sinh(f*x+e)*cosh(
f*x+e)^5-5/48*(f*x+e)^2*sinh(f*x+e)*cosh(f*x+e)^3-5/32*(f*x+e)^2*cosh(f*x+e)*sinh(f*x+e)-5/96*(f*x+e)^3+1/36*(
f*x+e)*sinh(f*x+e)^2*cosh(f*x+e)^4+23/288*(f*x+e)*sinh(f*x+e)^2*cosh(f*x+e)^2+17/72*(f*x+e)*cosh(f*x+e)^2-1/21
6*sinh(f*x+e)*cosh(f*x+e)^5-65/3456*cosh(f*x+e)^3*sinh(f*x+e)-245/2304*cosh(f*x+e)*sinh(f*x+e)-245/2304*f*x-24
5/2304*e)-4*c^3*f^3*(1/6*sinh(f*x+e)^2*cosh(f*x+e)^4+1/6*sinh(f*x+e)^2*cosh(f*x+e)^2+1/6*cosh(f*x+e)^2)+4*d^3*
e^3*(1/6*sinh(f*x+e)^2*cosh(f*x+e)^4+1/6*sinh(f*x+e)^2*cosh(f*x+e)^2+1/6*cosh(f*x+e)^2)-12*e*d^3*(1/6*(f*x+e)^
2*sinh(f*x+e)*cosh(f*x+e)^5+5/24*(f*x+e)^2*sinh(f*x+e)*cosh(f*x+e)^3+5/16*(f*x+e)^2*cosh(f*x+e)*sinh(f*x+e)+5/
48*(f*x+e)^3-1/18*(f*x+e)*sinh(f*x+e)^2*cosh(f*x+e)^4-23/144*(f*x+e)*sinh(f*x+e)^2*cosh(f*x+e)^2-17/36*(f*x+e)
*cosh(f*x+e)^2+1/108*sinh(f*x+e)*cosh(f*x+e)^5+65/1728*cosh(f*x+e)^3*sinh(f*x+e)+245/1152*cosh(f*x+e)*sinh(f*x
+e)+245/1152*f*x+245/1152*e)+12*d^3*e^2*(1/6*(f*x+e)*sinh(f*x+e)*cosh(f*x+e)^5+5/24*(f*x+e)*sinh(f*x+e)*cosh(f
*x+e)^3+5/16*(f*x+e)*cosh(f*x+e)*sinh(f*x+e)+5/32*(f*x+e)^2-1/36*sinh(f*x+e)^2*cosh(f*x+e)^4-23/288*sinh(f*x+e
)^2*cosh(f*x+e)^2-17/72*cosh(f*x+e)^2)+12*e*d^3*(1/6*(f*x+e)^2*sinh(f*x+e)^2*cosh(f*x+e)^4+1/6*(f*x+e)^2*sinh(
f*x+e)^2*cosh(f*x+e)^2+1/6*(f*x+e)^2*cosh(f*x+e)^2-1/18*(f*x+e)*sinh(f*x+e)*cosh(f*x+e)^5-5/72*(f*x+e)*sinh(f*
x+e)*cosh(f*x+e)^3-5/48*(f*x+e)*cosh(f*x+e)*sinh(f*x+e)-5/96*(f*x+e)^2+1/108*sinh(f*x+e)^2*cosh(f*x+e)^4+23/86
4*sinh(f*x+e)^2*cosh(f*x+e)^2+17/216*cosh(f*x+e)^2)-12*d^3*e^2*(1/6*(f*x+e)*sinh(f*x+e)^2*cosh(f*x+e)^4+1/6*(f
*x+e)*sinh(f*x+e)^2*cosh(f*x+e)^2+1/6*(f*x+e)*cosh(f*x+e)^2-1/36*sinh(f*x+e)*cosh(f*x+e)^5-5/144*cosh(f*x+e)^3
*sinh(f*x+e)-5/96*cosh(f*x+e)*sinh(f*x+e)-5/96*f*x-5/96*e)-4*d^3*e^3*((1/6*cosh(f*x+e)^5+5/24*cosh(f*x+e)^3+5/
16*cosh(f*x+e))*sinh(f*x+e)+5/16*f*x+5/16*e)+4*c^3*f^3*((1/6*cosh(f*x+e)^5+5/24*cosh(f*x+e)^3+5/16*cosh(f*x+e)
)*sinh(f*x+e)+5/16*f*x+5/16*e)+9*e*d^3*(1/4*(f*x+e)^2*sinh(f*x+e)*cosh(f*x+e)^3+3/8*(f*x+e)^2*cosh(f*x+e)*sinh
(f*x+e)+1/8*(f*x+e)^3-1/8*(f*x+e)*sinh(f*x+e)^2*cosh(f*x+e)^2-1/2*(f*x+e)*cosh(f*x+e)^2+1/32*cosh(f*x+e)^3*sin
h(f*x+e)+15/64*cosh(f*x+e)*sinh(f*x+e)+15/64*f*x+15/64*e)-3*e*d^3*(1/4*(f*x+e)^2*sinh(f*x+e)^2*cosh(f*x+e)^2+1
/4*(f*x+e)^2*cosh(f*x+e)^2-1/8*(f*x+e)*sinh(f*x+e)*cosh(f*x+e)^3-3/16*(f*x+e)*cosh(f*x+e)*sinh(f*x+e)-3/32*(f*
x+e)^2+1/32*sinh(f*x+e)^2*cosh(f*x+e)^2+1/8*cosh(f*x+e)^2)+3*d^3*e^2*(1/4*(f*x+e)*sinh(f*x+e)^2*cosh(f*x+e)^2+
1/4*(f*x+e)*cosh(f*x+e)^2-1/16*cosh(f*x+e)^3*sinh(f*x+e)-3/32*cosh(f*x+e)*sinh(f*x+e)-3/32*f*x-3/32*e)-d^3*e^3
*(1/4*sinh(f*x+e)^2*cosh(f*x+e)^2+1/4*cosh(f*x+e)^2)-3*c^3*f^3*((1/4*cosh(f*x+e)^3+3/8*cosh(f*x+e))*sinh(f*x+e
)+3/8*f*x+3/8*e)+3*d^3*e^3*((1/4*cosh(f*x+e)^3+3/8*cosh(f*x+e))*sinh(f*x+e)+3/8*f*x+3/8*e)-9*d^3*e^2*(1/4*(f*x
+e)*sinh(f*x+e)*cosh(f*x+e)^3+3/8*(f*x+e)*cosh(f*x+e)*sinh(f*x+e)+3/16*(f*x+e)^2-1/16*sinh(f*x+e)^2*cosh(f*x+e
)^2-1/4*cosh(f*x+e)^2)+c^3*f^3*(1/4*sinh(f*x+e)^2*cosh(f*x+e)^2+1/4*cosh(f*x+e)^2)+3*c*d^2*f*(1/4*(f*x+e)^2*si
nh(f*x+e)^2*cosh(f*x+e)^2+1/4*(f*x+e)^2*cosh(f*x+e)^2-1/8*(f*x+e)*sinh(f*x+e)*cosh(f*x+e)^3-3/16*(f*x+e)*cosh(
f*x+e)*sinh(f*x+e)-3/32*(f*x+e)^2+1/32*sinh(f*x+e)^2*cosh(f*x+e)^2+1/8*cosh(f*x+e)^2)+3*c^2*d*f^2*(1/4*(f*x+e)
*sinh(f*x+e)^2*cosh(f*x+e)^2+1/4*(f*x+e)*cosh(f*x+e)^2-1/16*cosh(f*x+e)^3*sinh(f*x+e)-3/32*cosh(f*x+e)*sinh(f*
x+e)-3/32*f*x-3/32*e)-9*c^2*d*f^2*(1/4*(f*x+e)*sinh(f*x+e)*cosh(f*x+e)^3+3/8*(f*x+e)*cosh(f*x+e)*sinh(f*x+e)+3
/16*(f*x+e)^2-1/16*sinh(f*x+e)^2*cosh(f*x+e)^2-1/4*cosh(f*x+e)^2)-9*c*d^2*f*(1/4*(f*x+e)^2*sinh(f*x+e)*cosh(f*
x+e)^3+3/8*(f*x+e)^2*cosh(f*x+e)*sinh(f*x+e)+1/8*(f*x+e)^3-1/8*(f*x+e)*sinh(f*x+e)^2*cosh(f*x+e)^2-1/2*(f*x+e)
*cosh(f*x+e)^2+1/32*cosh(f*x+e)^3*sinh(f*x+e)+15/64*cosh(f*x+e)*sinh(f*x+e)+15/64*f*x+15/64*e)-12*c^2*d*f^2*(1
/6*(f*x+e)*sinh(f*x+e)^2*cosh(f*x+e)^4+1/6*(f*x+e)*sinh(f*x+e)^2*cosh(f*x+e)^2+1/6*(f*x+e)*cosh(f*x+e)^2-1/36*
sinh(f*x+e)*cosh(f*x+e)^5-5/144*cosh(f*x+e)^3*sinh(f*x+e)-5/96*cosh(f*x+e)*sinh(f*x+e)-5/96*f*x-5/96*e)+12*c*d
^2*f*(1/6*(f*x+e)^2*sinh(f*x+e)*cosh(f*x+e)^5+5/24*(f*x+e)^2*sinh(f*x+e)*cosh(f*x+e)^3+5/16*(f*x+e)^2*cosh(f*x
+e)*sinh(f*x+e)+5/48*(f*x+e)^3-1/18*(f*x+e)*sinh(f*x+e)^2*cosh(f*x+e)^4-23/144*(f*x+e)*sinh(f*x+e)^2*cosh(f*x+
e)^2-17/36*(f*x+e)*cosh(f*x+e)^2+1/108*sinh(f*x+e)*cosh(f*x+e)^5+65/1728*cosh(f*x+e)^3*sinh(f*x+e)+245/1152*co
sh(f*x+e)*sinh(f*x+e)+245/1152*f*x+245/1152*e)+12*c^2*d*f^2*(1/6*(f*x+e)*sinh(f*x+e)*cosh(f*x+e)^5+5/24*(f*x+e
)*sinh(f*x+e)*cosh(f*x+e)^3+5/16*(f*x+e)*cosh(f*x+e)*sinh(f*x+e)+5/32*(f*x+e)^2-1/36*sinh(f*x+e)^2*cosh(f*x+e)
^4-23/288*sinh(f*x+e)^2*cosh(f*x+e)^2-17/72*cosh(f*x+e)^2)-12*c*d^2*f*(1/6*(f*x+e)^2*sinh(f*x+e)^2*cosh(f*x+e)
^4+1/6*(f*x+e)^2*sinh(f*x+e)^2*cosh(f*x+e)^2+1/6*(f*x+e)^2*cosh(f*x+e)^2-1/18*(f*x+e)*sinh(f*x+e)*cosh(f*x+e)^
5-5/72*(f*x+e)*sinh(f*x+e)*cosh(f*x+e)^3-5/48*(f*x+e)*cosh(f*x+e)*sinh(f*x+e)-5/96*(f*x+e)^2+1/108*sinh(f*x+e)
^2*cosh(f*x+e)^4+23/864*sinh(f*x+e)^2*cosh(f*x+e)^2+17/216*cosh(f*x+e)^2))

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Maxima [A]  time = 8.00915, size = 549, normalized size = 1.63 \begin{align*} \frac{1}{96} \, c^{3}{\left (\frac{12 \,{\left (f x + e\right )}}{a^{3} f} - \frac{18 \, e^{\left (-2 \, f x - 2 \, e\right )} + 9 \, e^{\left (-4 \, f x - 4 \, e\right )} + 2 \, e^{\left (-6 \, f x - 6 \, e\right )}}{a^{3} f}\right )} + \frac{{\left (72 \, f^{2} x^{2} e^{\left (6 \, e\right )} - 108 \,{\left (2 \, f x e^{\left (4 \, e\right )} + e^{\left (4 \, e\right )}\right )} e^{\left (-2 \, f x\right )} - 27 \,{\left (4 \, f x e^{\left (2 \, e\right )} + e^{\left (2 \, e\right )}\right )} e^{\left (-4 \, f x\right )} - 4 \,{\left (6 \, f x + 1\right )} e^{\left (-6 \, f x\right )}\right )} c^{2} d e^{\left (-6 \, e\right )}}{384 \, a^{3} f^{2}} + \frac{{\left (288 \, f^{3} x^{3} e^{\left (6 \, e\right )} - 648 \,{\left (2 \, f^{2} x^{2} e^{\left (4 \, e\right )} + 2 \, f x e^{\left (4 \, e\right )} + e^{\left (4 \, e\right )}\right )} e^{\left (-2 \, f x\right )} - 81 \,{\left (8 \, f^{2} x^{2} e^{\left (2 \, e\right )} + 4 \, f x e^{\left (2 \, e\right )} + e^{\left (2 \, e\right )}\right )} e^{\left (-4 \, f x\right )} - 8 \,{\left (18 \, f^{2} x^{2} + 6 \, f x + 1\right )} e^{\left (-6 \, f x\right )}\right )} c d^{2} e^{\left (-6 \, e\right )}}{2304 \, a^{3} f^{3}} + \frac{{\left (864 \, f^{4} x^{4} e^{\left (6 \, e\right )} - 1296 \,{\left (4 \, f^{3} x^{3} e^{\left (4 \, e\right )} + 6 \, f^{2} x^{2} e^{\left (4 \, e\right )} + 6 \, f x e^{\left (4 \, e\right )} + 3 \, e^{\left (4 \, e\right )}\right )} e^{\left (-2 \, f x\right )} - 81 \,{\left (32 \, f^{3} x^{3} e^{\left (2 \, e\right )} + 24 \, f^{2} x^{2} e^{\left (2 \, e\right )} + 12 \, f x e^{\left (2 \, e\right )} + 3 \, e^{\left (2 \, e\right )}\right )} e^{\left (-4 \, f x\right )} - 16 \,{\left (36 \, f^{3} x^{3} + 18 \, f^{2} x^{2} + 6 \, f x + 1\right )} e^{\left (-6 \, f x\right )}\right )} d^{3} e^{\left (-6 \, e\right )}}{27648 \, a^{3} f^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/96*c^3*(12*(f*x + e)/(a^3*f) - (18*e^(-2*f*x - 2*e) + 9*e^(-4*f*x - 4*e) + 2*e^(-6*f*x - 6*e))/(a^3*f)) + 1/
384*(72*f^2*x^2*e^(6*e) - 108*(2*f*x*e^(4*e) + e^(4*e))*e^(-2*f*x) - 27*(4*f*x*e^(2*e) + e^(2*e))*e^(-4*f*x) -
4*(6*f*x + 1)*e^(-6*f*x))*c^2*d*e^(-6*e)/(a^3*f^2) + 1/2304*(288*f^3*x^3*e^(6*e) - 648*(2*f^2*x^2*e^(4*e) + 2
*f*x*e^(4*e) + e^(4*e))*e^(-2*f*x) - 81*(8*f^2*x^2*e^(2*e) + 4*f*x*e^(2*e) + e^(2*e))*e^(-4*f*x) - 8*(18*f^2*x
^2 + 6*f*x + 1)*e^(-6*f*x))*c*d^2*e^(-6*e)/(a^3*f^3) + 1/27648*(864*f^4*x^4*e^(6*e) - 1296*(4*f^3*x^3*e^(4*e)
+ 6*f^2*x^2*e^(4*e) + 6*f*x*e^(4*e) + 3*e^(4*e))*e^(-2*f*x) - 81*(32*f^3*x^3*e^(2*e) + 24*f^2*x^2*e^(2*e) + 12
*f*x*e^(2*e) + 3*e^(2*e))*e^(-4*f*x) - 16*(36*f^3*x^3 + 18*f^2*x^2 + 6*f*x + 1)*e^(-6*f*x))*d^3*e^(-6*e)/(a^3*
f^4)

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Fricas [B]  time = 2.24762, size = 1872, normalized size = 5.57 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/27648*(16*(54*d^3*f^4*x^4 - 36*c^3*f^3 - 18*c^2*d*f^2 - 6*c*d^2*f + 36*(6*c*d^2*f^4 - d^3*f^3)*x^3 - d^3 + 1
8*(18*c^2*d*f^4 - 6*c*d^2*f^3 - d^3*f^2)*x^2 + 6*(36*c^3*f^4 - 18*c^2*d*f^3 - 6*c*d^2*f^2 - d^3*f)*x)*cosh(f*x
+ e)^3 + 48*(54*d^3*f^4*x^4 - 36*c^3*f^3 - 18*c^2*d*f^2 - 6*c*d^2*f + 36*(6*c*d^2*f^4 - d^3*f^3)*x^3 - d^3 +
18*(18*c^2*d*f^4 - 6*c*d^2*f^3 - d^3*f^2)*x^2 + 6*(36*c^3*f^4 - 18*c^2*d*f^3 - 6*c*d^2*f^2 - d^3*f)*x)*cosh(f*
x + e)*sinh(f*x + e)^2 + 16*(54*d^3*f^4*x^4 + 36*c^3*f^3 + 18*c^2*d*f^2 + 6*c*d^2*f + 36*(6*c*d^2*f^4 + d^3*f^
3)*x^3 + d^3 + 18*(18*c^2*d*f^4 + 6*c*d^2*f^3 + d^3*f^2)*x^2 + 6*(36*c^3*f^4 + 18*c^2*d*f^3 + 6*c*d^2*f^2 + d^
3*f)*x)*sinh(f*x + e)^3 - 243*(32*d^3*f^3*x^3 + 32*c^3*f^3 + 40*c^2*d*f^2 + 36*c*d^2*f + 17*d^3 + 8*(12*c*d^2*
f^3 + 5*d^3*f^2)*x^2 + 4*(24*c^2*d*f^3 + 20*c*d^2*f^2 + 9*d^3*f)*x)*cosh(f*x + e) - 3*(864*d^3*f^3*x^3 + 864*c
^3*f^3 + 1944*c^2*d*f^2 + 2268*c*d^2*f + 1215*d^3 + 648*(4*c*d^2*f^3 + 3*d^3*f^2)*x^2 - 16*(54*d^3*f^4*x^4 + 3
6*c^3*f^3 + 18*c^2*d*f^2 + 6*c*d^2*f + 36*(6*c*d^2*f^4 + d^3*f^3)*x^3 + d^3 + 18*(18*c^2*d*f^4 + 6*c*d^2*f^3 +
d^3*f^2)*x^2 + 6*(36*c^3*f^4 + 18*c^2*d*f^3 + 6*c*d^2*f^2 + d^3*f)*x)*cosh(f*x + e)^2 + 324*(8*c^2*d*f^3 + 12
*c*d^2*f^2 + 7*d^3*f)*x)*sinh(f*x + e))/(a^3*f^4*cosh(f*x + e)^3 + 3*a^3*f^4*cosh(f*x + e)^2*sinh(f*x + e) + 3
*a^3*f^4*cosh(f*x + e)*sinh(f*x + e)^2 + a^3*f^4*sinh(f*x + e)^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{c^{3}}{\tanh ^{3}{\left (e + f x \right )} + 3 \tanh ^{2}{\left (e + f x \right )} + 3 \tanh{\left (e + f x \right )} + 1}\, dx + \int \frac{d^{3} x^{3}}{\tanh ^{3}{\left (e + f x \right )} + 3 \tanh ^{2}{\left (e + f x \right )} + 3 \tanh{\left (e + f x \right )} + 1}\, dx + \int \frac{3 c d^{2} x^{2}}{\tanh ^{3}{\left (e + f x \right )} + 3 \tanh ^{2}{\left (e + f x \right )} + 3 \tanh{\left (e + f x \right )} + 1}\, dx + \int \frac{3 c^{2} d x}{\tanh ^{3}{\left (e + f x \right )} + 3 \tanh ^{2}{\left (e + f x \right )} + 3 \tanh{\left (e + f x \right )} + 1}\, dx}{a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(Integral(c**3/(tanh(e + f*x)**3 + 3*tanh(e + f*x)**2 + 3*tanh(e + f*x) + 1), x) + Integral(d**3*x**3/(tanh(e
+ f*x)**3 + 3*tanh(e + f*x)**2 + 3*tanh(e + f*x) + 1), x) + Integral(3*c*d**2*x**2/(tanh(e + f*x)**3 + 3*tanh(
e + f*x)**2 + 3*tanh(e + f*x) + 1), x) + Integral(3*c**2*d*x/(tanh(e + f*x)**3 + 3*tanh(e + f*x)**2 + 3*tanh(e
+ f*x) + 1), x))/a**3

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Giac [A]  time = 1.21566, size = 774, normalized size = 2.3 \begin{align*} \frac{{\left (864 \, d^{3} f^{4} x^{4} e^{\left (6 \, f x + 6 \, e\right )} + 3456 \, c d^{2} f^{4} x^{3} e^{\left (6 \, f x + 6 \, e\right )} + 5184 \, c^{2} d f^{4} x^{2} e^{\left (6 \, f x + 6 \, e\right )} - 5184 \, d^{3} f^{3} x^{3} e^{\left (4 \, f x + 4 \, e\right )} - 2592 \, d^{3} f^{3} x^{3} e^{\left (2 \, f x + 2 \, e\right )} - 576 \, d^{3} f^{3} x^{3} + 3456 \, c^{3} f^{4} x e^{\left (6 \, f x + 6 \, e\right )} - 15552 \, c d^{2} f^{3} x^{2} e^{\left (4 \, f x + 4 \, e\right )} - 7776 \, c d^{2} f^{3} x^{2} e^{\left (2 \, f x + 2 \, e\right )} - 1728 \, c d^{2} f^{3} x^{2} - 15552 \, c^{2} d f^{3} x e^{\left (4 \, f x + 4 \, e\right )} - 7776 \, d^{3} f^{2} x^{2} e^{\left (4 \, f x + 4 \, e\right )} - 7776 \, c^{2} d f^{3} x e^{\left (2 \, f x + 2 \, e\right )} - 1944 \, d^{3} f^{2} x^{2} e^{\left (2 \, f x + 2 \, e\right )} - 1728 \, c^{2} d f^{3} x - 288 \, d^{3} f^{2} x^{2} - 5184 \, c^{3} f^{3} e^{\left (4 \, f x + 4 \, e\right )} - 15552 \, c d^{2} f^{2} x e^{\left (4 \, f x + 4 \, e\right )} - 2592 \, c^{3} f^{3} e^{\left (2 \, f x + 2 \, e\right )} - 3888 \, c d^{2} f^{2} x e^{\left (2 \, f x + 2 \, e\right )} - 576 \, c^{3} f^{3} - 576 \, c d^{2} f^{2} x - 7776 \, c^{2} d f^{2} e^{\left (4 \, f x + 4 \, e\right )} - 7776 \, d^{3} f x e^{\left (4 \, f x + 4 \, e\right )} - 1944 \, c^{2} d f^{2} e^{\left (2 \, f x + 2 \, e\right )} - 972 \, d^{3} f x e^{\left (2 \, f x + 2 \, e\right )} - 288 \, c^{2} d f^{2} - 96 \, d^{3} f x - 7776 \, c d^{2} f e^{\left (4 \, f x + 4 \, e\right )} - 972 \, c d^{2} f e^{\left (2 \, f x + 2 \, e\right )} - 96 \, c d^{2} f - 3888 \, d^{3} e^{\left (4 \, f x + 4 \, e\right )} - 243 \, d^{3} e^{\left (2 \, f x + 2 \, e\right )} - 16 \, d^{3}\right )} e^{\left (-6 \, f x - 6 \, e\right )}}{27648 \, a^{3} f^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/27648*(864*d^3*f^4*x^4*e^(6*f*x + 6*e) + 3456*c*d^2*f^4*x^3*e^(6*f*x + 6*e) + 5184*c^2*d*f^4*x^2*e^(6*f*x +
6*e) - 5184*d^3*f^3*x^3*e^(4*f*x + 4*e) - 2592*d^3*f^3*x^3*e^(2*f*x + 2*e) - 576*d^3*f^3*x^3 + 3456*c^3*f^4*x*
e^(6*f*x + 6*e) - 15552*c*d^2*f^3*x^2*e^(4*f*x + 4*e) - 7776*c*d^2*f^3*x^2*e^(2*f*x + 2*e) - 1728*c*d^2*f^3*x^
2 - 15552*c^2*d*f^3*x*e^(4*f*x + 4*e) - 7776*d^3*f^2*x^2*e^(4*f*x + 4*e) - 7776*c^2*d*f^3*x*e^(2*f*x + 2*e) -
1944*d^3*f^2*x^2*e^(2*f*x + 2*e) - 1728*c^2*d*f^3*x - 288*d^3*f^2*x^2 - 5184*c^3*f^3*e^(4*f*x + 4*e) - 15552*c
*d^2*f^2*x*e^(4*f*x + 4*e) - 2592*c^3*f^3*e^(2*f*x + 2*e) - 3888*c*d^2*f^2*x*e^(2*f*x + 2*e) - 576*c^3*f^3 - 5
76*c*d^2*f^2*x - 7776*c^2*d*f^2*e^(4*f*x + 4*e) - 7776*d^3*f*x*e^(4*f*x + 4*e) - 1944*c^2*d*f^2*e^(2*f*x + 2*e
) - 972*d^3*f*x*e^(2*f*x + 2*e) - 288*c^2*d*f^2 - 96*d^3*f*x - 7776*c*d^2*f*e^(4*f*x + 4*e) - 972*c*d^2*f*e^(2
*f*x + 2*e) - 96*c*d^2*f - 3888*d^3*e^(4*f*x + 4*e) - 243*d^3*e^(2*f*x + 2*e) - 16*d^3)*e^(-6*f*x - 6*e)/(a^3*
f^4)