3.150 \(\int \sinh ^3(c+d x) (a+b \sinh ^3(c+d x))^2 \, dx\)

Optimal. Leaf size=192 \[ \frac{a^2 \cosh ^3(c+d x)}{3 d}-\frac{a^2 \cosh (c+d x)}{d}+\frac{a b \sinh ^5(c+d x) \cosh (c+d x)}{3 d}-\frac{5 a b \sinh ^3(c+d x) \cosh (c+d x)}{12 d}+\frac{5 a b \sinh (c+d x) \cosh (c+d x)}{8 d}-\frac{5 a b x}{8}+\frac{b^2 \cosh ^9(c+d x)}{9 d}-\frac{4 b^2 \cosh ^7(c+d x)}{7 d}+\frac{6 b^2 \cosh ^5(c+d x)}{5 d}-\frac{4 b^2 \cosh ^3(c+d x)}{3 d}+\frac{b^2 \cosh (c+d x)}{d} \]

[Out]

(-5*a*b*x)/8 - (a^2*Cosh[c + d*x])/d + (b^2*Cosh[c + d*x])/d + (a^2*Cosh[c + d*x]^3)/(3*d) - (4*b^2*Cosh[c + d
*x]^3)/(3*d) + (6*b^2*Cosh[c + d*x]^5)/(5*d) - (4*b^2*Cosh[c + d*x]^7)/(7*d) + (b^2*Cosh[c + d*x]^9)/(9*d) + (
5*a*b*Cosh[c + d*x]*Sinh[c + d*x])/(8*d) - (5*a*b*Cosh[c + d*x]*Sinh[c + d*x]^3)/(12*d) + (a*b*Cosh[c + d*x]*S
inh[c + d*x]^5)/(3*d)

________________________________________________________________________________________

Rubi [A]  time = 0.179161, antiderivative size = 192, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {3220, 2633, 2635, 8} \[ \frac{a^2 \cosh ^3(c+d x)}{3 d}-\frac{a^2 \cosh (c+d x)}{d}+\frac{a b \sinh ^5(c+d x) \cosh (c+d x)}{3 d}-\frac{5 a b \sinh ^3(c+d x) \cosh (c+d x)}{12 d}+\frac{5 a b \sinh (c+d x) \cosh (c+d x)}{8 d}-\frac{5 a b x}{8}+\frac{b^2 \cosh ^9(c+d x)}{9 d}-\frac{4 b^2 \cosh ^7(c+d x)}{7 d}+\frac{6 b^2 \cosh ^5(c+d x)}{5 d}-\frac{4 b^2 \cosh ^3(c+d x)}{3 d}+\frac{b^2 \cosh (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*a*b*x)/8 - (a^2*Cosh[c + d*x])/d + (b^2*Cosh[c + d*x])/d + (a^2*Cosh[c + d*x]^3)/(3*d) - (4*b^2*Cosh[c + d
*x]^3)/(3*d) + (6*b^2*Cosh[c + d*x]^5)/(5*d) - (4*b^2*Cosh[c + d*x]^7)/(7*d) + (b^2*Cosh[c + d*x]^9)/(9*d) + (
5*a*b*Cosh[c + d*x]*Sinh[c + d*x])/(8*d) - (5*a*b*Cosh[c + d*x]*Sinh[c + d*x]^3)/(12*d) + (a*b*Cosh[c + d*x]*S
inh[c + d*x]^5)/(3*d)

Rule 3220

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_.), x_Symbol] :> Int[ExpandTr
ig[sin[e + f*x]^m*(a + b*sin[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, e, f}, x] && IntegersQ[m, p] && (EqQ[n, 4]
|| GtQ[p, 0] || (EqQ[p, -1] && IntegerQ[n]))

Rule 2633

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \sinh ^3(c+d x) \left (a+b \sinh ^3(c+d x)\right )^2 \, dx &=i \int \left (-i a^2 \sinh ^3(c+d x)-2 i a b \sinh ^6(c+d x)-i b^2 \sinh ^9(c+d x)\right ) \, dx\\ &=a^2 \int \sinh ^3(c+d x) \, dx+(2 a b) \int \sinh ^6(c+d x) \, dx+b^2 \int \sinh ^9(c+d x) \, dx\\ &=\frac{a b \cosh (c+d x) \sinh ^5(c+d x)}{3 d}-\frac{1}{3} (5 a b) \int \sinh ^4(c+d x) \, dx-\frac{a^2 \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cosh (c+d x)\right )}{d}+\frac{b^2 \operatorname{Subst}\left (\int \left (1-4 x^2+6 x^4-4 x^6+x^8\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac{a^2 \cosh (c+d x)}{d}+\frac{b^2 \cosh (c+d x)}{d}+\frac{a^2 \cosh ^3(c+d x)}{3 d}-\frac{4 b^2 \cosh ^3(c+d x)}{3 d}+\frac{6 b^2 \cosh ^5(c+d x)}{5 d}-\frac{4 b^2 \cosh ^7(c+d x)}{7 d}+\frac{b^2 \cosh ^9(c+d x)}{9 d}-\frac{5 a b \cosh (c+d x) \sinh ^3(c+d x)}{12 d}+\frac{a b \cosh (c+d x) \sinh ^5(c+d x)}{3 d}+\frac{1}{4} (5 a b) \int \sinh ^2(c+d x) \, dx\\ &=-\frac{a^2 \cosh (c+d x)}{d}+\frac{b^2 \cosh (c+d x)}{d}+\frac{a^2 \cosh ^3(c+d x)}{3 d}-\frac{4 b^2 \cosh ^3(c+d x)}{3 d}+\frac{6 b^2 \cosh ^5(c+d x)}{5 d}-\frac{4 b^2 \cosh ^7(c+d x)}{7 d}+\frac{b^2 \cosh ^9(c+d x)}{9 d}+\frac{5 a b \cosh (c+d x) \sinh (c+d x)}{8 d}-\frac{5 a b \cosh (c+d x) \sinh ^3(c+d x)}{12 d}+\frac{a b \cosh (c+d x) \sinh ^5(c+d x)}{3 d}-\frac{1}{8} (5 a b) \int 1 \, dx\\ &=-\frac{5}{8} a b x-\frac{a^2 \cosh (c+d x)}{d}+\frac{b^2 \cosh (c+d x)}{d}+\frac{a^2 \cosh ^3(c+d x)}{3 d}-\frac{4 b^2 \cosh ^3(c+d x)}{3 d}+\frac{6 b^2 \cosh ^5(c+d x)}{5 d}-\frac{4 b^2 \cosh ^7(c+d x)}{7 d}+\frac{b^2 \cosh ^9(c+d x)}{9 d}+\frac{5 a b \cosh (c+d x) \sinh (c+d x)}{8 d}-\frac{5 a b \cosh (c+d x) \sinh ^3(c+d x)}{12 d}+\frac{a b \cosh (c+d x) \sinh ^5(c+d x)}{3 d}\\ \end{align*}

Mathematica [A]  time = 0.683032, size = 125, normalized size = 0.65 \[ \frac{-1890 \left (32 a^2-21 b^2\right ) \cosh (c+d x)+420 \left (16 a^2-21 b^2\right ) \cosh (3 (c+d x))+b (-840 a (-45 \sinh (2 (c+d x))+9 \sinh (4 (c+d x))-\sinh (6 (c+d x))+60 c+60 d x)+2268 b \cosh (5 (c+d x))-405 b \cosh (7 (c+d x))+35 b \cosh (9 (c+d x)))}{80640 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-1890*(32*a^2 - 21*b^2)*Cosh[c + d*x] + 420*(16*a^2 - 21*b^2)*Cosh[3*(c + d*x)] + b*(2268*b*Cosh[5*(c + d*x)]
- 405*b*Cosh[7*(c + d*x)] + 35*b*Cosh[9*(c + d*x)] - 840*a*(60*c + 60*d*x - 45*Sinh[2*(c + d*x)] + 9*Sinh[4*(
c + d*x)] - Sinh[6*(c + d*x)])))/(80640*d)

________________________________________________________________________________________

Maple [A]  time = 0.02, size = 128, normalized size = 0.7 \begin{align*}{\frac{1}{d} \left ({b}^{2} \left ({\frac{128}{315}}+{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{8}}{9}}-{\frac{8\, \left ( \sinh \left ( dx+c \right ) \right ) ^{6}}{63}}+{\frac{16\, \left ( \sinh \left ( dx+c \right ) \right ) ^{4}}{105}}-{\frac{64\, \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{315}} \right ) \cosh \left ( dx+c \right ) +2\,ab \left ( \left ( 1/6\, \left ( \sinh \left ( dx+c \right ) \right ) ^{5}-{\frac{5\, \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{24}}+{\frac{5\,\sinh \left ( dx+c \right ) }{16}} \right ) \cosh \left ( dx+c \right ) -{\frac{5\,dx}{16}}-{\frac{5\,c}{16}} \right ) +{a}^{2} \left ( -{\frac{2}{3}}+{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) \cosh \left ( dx+c \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(d*x+c)^3*(a+b*sinh(d*x+c)^3)^2,x)

[Out]

1/d*(b^2*(128/315+1/9*sinh(d*x+c)^8-8/63*sinh(d*x+c)^6+16/105*sinh(d*x+c)^4-64/315*sinh(d*x+c)^2)*cosh(d*x+c)+
2*a*b*((1/6*sinh(d*x+c)^5-5/24*sinh(d*x+c)^3+5/16*sinh(d*x+c))*cosh(d*x+c)-5/16*d*x-5/16*c)+a^2*(-2/3+1/3*sinh
(d*x+c)^2)*cosh(d*x+c))

________________________________________________________________________________________

Maxima [A]  time = 1.18855, size = 367, normalized size = 1.91 \begin{align*} -\frac{1}{161280} \, b^{2}{\left (\frac{{\left (405 \, e^{\left (-2 \, d x - 2 \, c\right )} - 2268 \, e^{\left (-4 \, d x - 4 \, c\right )} + 8820 \, e^{\left (-6 \, d x - 6 \, c\right )} - 39690 \, e^{\left (-8 \, d x - 8 \, c\right )} - 35\right )} e^{\left (9 \, d x + 9 \, c\right )}}{d} - \frac{39690 \, e^{\left (-d x - c\right )} - 8820 \, e^{\left (-3 \, d x - 3 \, c\right )} + 2268 \, e^{\left (-5 \, d x - 5 \, c\right )} - 405 \, e^{\left (-7 \, d x - 7 \, c\right )} + 35 \, e^{\left (-9 \, d x - 9 \, c\right )}}{d}\right )} - \frac{1}{192} \, a b{\left (\frac{{\left (9 \, e^{\left (-2 \, d x - 2 \, c\right )} - 45 \, e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )} e^{\left (6 \, d x + 6 \, c\right )}}{d} + \frac{120 \,{\left (d x + c\right )}}{d} + \frac{45 \, e^{\left (-2 \, d x - 2 \, c\right )} - 9 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )}}{d}\right )} + \frac{1}{24} \, a^{2}{\left (\frac{e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac{9 \, e^{\left (d x + c\right )}}{d} - \frac{9 \, e^{\left (-d x - c\right )}}{d} + \frac{e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/161280*b^2*((405*e^(-2*d*x - 2*c) - 2268*e^(-4*d*x - 4*c) + 8820*e^(-6*d*x - 6*c) - 39690*e^(-8*d*x - 8*c)
- 35)*e^(9*d*x + 9*c)/d - (39690*e^(-d*x - c) - 8820*e^(-3*d*x - 3*c) + 2268*e^(-5*d*x - 5*c) - 405*e^(-7*d*x
- 7*c) + 35*e^(-9*d*x - 9*c))/d) - 1/192*a*b*((9*e^(-2*d*x - 2*c) - 45*e^(-4*d*x - 4*c) - 1)*e^(6*d*x + 6*c)/d
+ 120*(d*x + c)/d + (45*e^(-2*d*x - 2*c) - 9*e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c))/d) + 1/24*a^2*(e^(3*d*x + 3
*c)/d - 9*e^(d*x + c)/d - 9*e^(-d*x - c)/d + e^(-3*d*x - 3*c)/d)

________________________________________________________________________________________

Fricas [B]  time = 1.91839, size = 973, normalized size = 5.07 \begin{align*} \frac{35 \, b^{2} \cosh \left (d x + c\right )^{9} + 315 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{8} - 405 \, b^{2} \cosh \left (d x + c\right )^{7} + 5040 \, a b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + 2268 \, b^{2} \cosh \left (d x + c\right )^{5} + 105 \,{\left (28 \, b^{2} \cosh \left (d x + c\right )^{3} - 27 \, b^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{6} + 315 \,{\left (14 \, b^{2} \cosh \left (d x + c\right )^{5} - 45 \, b^{2} \cosh \left (d x + c\right )^{3} + 36 \, b^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} - 50400 \, a b d x + 420 \,{\left (16 \, a^{2} - 21 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} + 3360 \,{\left (5 \, a b \cosh \left (d x + c\right )^{3} - 9 \, a b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 315 \,{\left (4 \, b^{2} \cosh \left (d x + c\right )^{7} - 27 \, b^{2} \cosh \left (d x + c\right )^{5} + 72 \, b^{2} \cosh \left (d x + c\right )^{3} + 4 \,{\left (16 \, a^{2} - 21 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} - 1890 \,{\left (32 \, a^{2} - 21 \, b^{2}\right )} \cosh \left (d x + c\right ) + 5040 \,{\left (a b \cosh \left (d x + c\right )^{5} - 6 \, a b \cosh \left (d x + c\right )^{3} + 15 \, a b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{80640 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/80640*(35*b^2*cosh(d*x + c)^9 + 315*b^2*cosh(d*x + c)*sinh(d*x + c)^8 - 405*b^2*cosh(d*x + c)^7 + 5040*a*b*c
osh(d*x + c)*sinh(d*x + c)^5 + 2268*b^2*cosh(d*x + c)^5 + 105*(28*b^2*cosh(d*x + c)^3 - 27*b^2*cosh(d*x + c))*
sinh(d*x + c)^6 + 315*(14*b^2*cosh(d*x + c)^5 - 45*b^2*cosh(d*x + c)^3 + 36*b^2*cosh(d*x + c))*sinh(d*x + c)^4
- 50400*a*b*d*x + 420*(16*a^2 - 21*b^2)*cosh(d*x + c)^3 + 3360*(5*a*b*cosh(d*x + c)^3 - 9*a*b*cosh(d*x + c))*
sinh(d*x + c)^3 + 315*(4*b^2*cosh(d*x + c)^7 - 27*b^2*cosh(d*x + c)^5 + 72*b^2*cosh(d*x + c)^3 + 4*(16*a^2 - 2
1*b^2)*cosh(d*x + c))*sinh(d*x + c)^2 - 1890*(32*a^2 - 21*b^2)*cosh(d*x + c) + 5040*(a*b*cosh(d*x + c)^5 - 6*a
*b*cosh(d*x + c)^3 + 15*a*b*cosh(d*x + c))*sinh(d*x + c))/d

________________________________________________________________________________________

Sympy [A]  time = 31.0145, size = 325, normalized size = 1.69 \begin{align*} \begin{cases} \frac{a^{2} \sinh ^{2}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{d} - \frac{2 a^{2} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac{5 a b x \sinh ^{6}{\left (c + d x \right )}}{8} - \frac{15 a b x \sinh ^{4}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{8} + \frac{15 a b x \sinh ^{2}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{8} - \frac{5 a b x \cosh ^{6}{\left (c + d x \right )}}{8} + \frac{11 a b \sinh ^{5}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{8 d} - \frac{5 a b \sinh ^{3}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac{5 a b \sinh{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{8 d} + \frac{b^{2} \sinh ^{8}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{d} - \frac{8 b^{2} \sinh ^{6}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac{16 b^{2} \sinh ^{4}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{5 d} - \frac{64 b^{2} \sinh ^{2}{\left (c + d x \right )} \cosh ^{7}{\left (c + d x \right )}}{35 d} + \frac{128 b^{2} \cosh ^{9}{\left (c + d x \right )}}{315 d} & \text{for}\: d \neq 0 \\x \left (a + b \sinh ^{3}{\left (c \right )}\right )^{2} \sinh ^{3}{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)**3*(a+b*sinh(d*x+c)**3)**2,x)

[Out]

Piecewise((a**2*sinh(c + d*x)**2*cosh(c + d*x)/d - 2*a**2*cosh(c + d*x)**3/(3*d) + 5*a*b*x*sinh(c + d*x)**6/8
- 15*a*b*x*sinh(c + d*x)**4*cosh(c + d*x)**2/8 + 15*a*b*x*sinh(c + d*x)**2*cosh(c + d*x)**4/8 - 5*a*b*x*cosh(c
+ d*x)**6/8 + 11*a*b*sinh(c + d*x)**5*cosh(c + d*x)/(8*d) - 5*a*b*sinh(c + d*x)**3*cosh(c + d*x)**3/(3*d) + 5
*a*b*sinh(c + d*x)*cosh(c + d*x)**5/(8*d) + b**2*sinh(c + d*x)**8*cosh(c + d*x)/d - 8*b**2*sinh(c + d*x)**6*co
sh(c + d*x)**3/(3*d) + 16*b**2*sinh(c + d*x)**4*cosh(c + d*x)**5/(5*d) - 64*b**2*sinh(c + d*x)**2*cosh(c + d*x
)**7/(35*d) + 128*b**2*cosh(c + d*x)**9/(315*d), Ne(d, 0)), (x*(a + b*sinh(c)**3)**2*sinh(c)**3, True))

________________________________________________________________________________________

Giac [A]  time = 1.26917, size = 369, normalized size = 1.92 \begin{align*} -\frac{100800 \,{\left (d x + c\right )} a b - 35 \, b^{2} e^{\left (9 \, d x + 9 \, c\right )} + 405 \, b^{2} e^{\left (7 \, d x + 7 \, c\right )} - 840 \, a b e^{\left (6 \, d x + 6 \, c\right )} - 2268 \, b^{2} e^{\left (5 \, d x + 5 \, c\right )} + 7560 \, a b e^{\left (4 \, d x + 4 \, c\right )} - 6720 \, a^{2} e^{\left (3 \, d x + 3 \, c\right )} + 8820 \, b^{2} e^{\left (3 \, d x + 3 \, c\right )} - 37800 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 60480 \, a^{2} e^{\left (d x + c\right )} - 39690 \, b^{2} e^{\left (d x + c\right )} +{\left (37800 \, a b e^{\left (7 \, d x + 7 \, c\right )} - 7560 \, a b e^{\left (5 \, d x + 5 \, c\right )} - 2268 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 840 \, a b e^{\left (3 \, d x + 3 \, c\right )} + 405 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 35 \, b^{2} + 1890 \,{\left (32 \, a^{2} - 21 \, b^{2}\right )} e^{\left (8 \, d x + 8 \, c\right )} - 420 \,{\left (16 \, a^{2} - 21 \, b^{2}\right )} e^{\left (6 \, d x + 6 \, c\right )}\right )} e^{\left (-9 \, d x - 9 \, c\right )}}{161280 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/161280*(100800*(d*x + c)*a*b - 35*b^2*e^(9*d*x + 9*c) + 405*b^2*e^(7*d*x + 7*c) - 840*a*b*e^(6*d*x + 6*c) -
2268*b^2*e^(5*d*x + 5*c) + 7560*a*b*e^(4*d*x + 4*c) - 6720*a^2*e^(3*d*x + 3*c) + 8820*b^2*e^(3*d*x + 3*c) - 3
7800*a*b*e^(2*d*x + 2*c) + 60480*a^2*e^(d*x + c) - 39690*b^2*e^(d*x + c) + (37800*a*b*e^(7*d*x + 7*c) - 7560*a
*b*e^(5*d*x + 5*c) - 2268*b^2*e^(4*d*x + 4*c) + 840*a*b*e^(3*d*x + 3*c) + 405*b^2*e^(2*d*x + 2*c) - 35*b^2 + 1
890*(32*a^2 - 21*b^2)*e^(8*d*x + 8*c) - 420*(16*a^2 - 21*b^2)*e^(6*d*x + 6*c))*e^(-9*d*x - 9*c))/d