3.293 \(\int \cosh ^4(c+d x) (a+b \sinh ^2(c+d x))^2 \, dx\)

Optimal. Leaf size=159 \[ \frac{\left (48 a^2-16 a b+3 b^2\right ) \sinh (c+d x) \cosh ^3(c+d x)}{192 d}+\frac{\left (48 a^2-16 a b+3 b^2\right ) \sinh (c+d x) \cosh (c+d x)}{128 d}+\frac{1}{128} x \left (48 a^2-16 a b+3 b^2\right )+\frac{b (10 a-3 b) \sinh (c+d x) \cosh ^5(c+d x)}{48 d}+\frac{b \sinh (c+d x) \cosh ^7(c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )}{8 d} \]

[Out]

((48*a^2 - 16*a*b + 3*b^2)*x)/128 + ((48*a^2 - 16*a*b + 3*b^2)*Cosh[c + d*x]*Sinh[c + d*x])/(128*d) + ((48*a^2
- 16*a*b + 3*b^2)*Cosh[c + d*x]^3*Sinh[c + d*x])/(192*d) + ((10*a - 3*b)*b*Cosh[c + d*x]^5*Sinh[c + d*x])/(48
*d) + (b*Cosh[c + d*x]^7*Sinh[c + d*x]*(a - (a - b)*Tanh[c + d*x]^2))/(8*d)

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Rubi [A]  time = 0.174047, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {3191, 413, 385, 199, 206} \[ \frac{\left (48 a^2-16 a b+3 b^2\right ) \sinh (c+d x) \cosh ^3(c+d x)}{192 d}+\frac{\left (48 a^2-16 a b+3 b^2\right ) \sinh (c+d x) \cosh (c+d x)}{128 d}+\frac{1}{128} x \left (48 a^2-16 a b+3 b^2\right )+\frac{b (10 a-3 b) \sinh (c+d x) \cosh ^5(c+d x)}{48 d}+\frac{b \sinh (c+d x) \cosh ^7(c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )}{8 d} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[c + d*x]^4*(a + b*Sinh[c + d*x]^2)^2,x]

[Out]

((48*a^2 - 16*a*b + 3*b^2)*x)/128 + ((48*a^2 - 16*a*b + 3*b^2)*Cosh[c + d*x]*Sinh[c + d*x])/(128*d) + ((48*a^2
- 16*a*b + 3*b^2)*Cosh[c + d*x]^3*Sinh[c + d*x])/(192*d) + ((10*a - 3*b)*b*Cosh[c + d*x]^5*Sinh[c + d*x])/(48
*d) + (b*Cosh[c + d*x]^7*Sinh[c + d*x]*(a - (a - b)*Tanh[c + d*x]^2))/(8*d)

Rule 3191

Int[cos[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = FreeF
actors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[(a + (a + b)*ff^2*x^2)^p/(1 + ff^2*x^2)^(m/2 + p + 1), x], x, T
an[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]

Rule 413

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[((a*d - c*b)*x*(a + b*x^n)^
(p + 1)*(c + d*x^n)^(q - 1))/(a*b*n*(p + 1)), x] - Dist[1/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)
^(q - 2)*Simp[c*(a*d - c*b*(n*(p + 1) + 1)) + d*(a*d*(n*(q - 1) + 1) - b*c*(n*(p + q) + 1))*x^n, x], x], x] /;
FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, n, p, q
, x]

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> -Simp[((b*c - a*d)*x*(a + b*x^n)^(p +
1))/(a*b*n*(p + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /
; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \cosh ^4(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a-(a-b) x^2\right )^2}{\left (1-x^2\right )^5} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{b \cosh ^7(c+d x) \sinh (c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )}{8 d}-\frac{\operatorname{Subst}\left (\int \frac{-a (8 a-b)+(8 a-3 b) (a-b) x^2}{\left (1-x^2\right )^4} \, dx,x,\tanh (c+d x)\right )}{8 d}\\ &=\frac{(10 a-3 b) b \cosh ^5(c+d x) \sinh (c+d x)}{48 d}+\frac{b \cosh ^7(c+d x) \sinh (c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )}{8 d}+\frac{\left (48 a^2-16 a b+3 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{48 d}\\ &=\frac{\left (48 a^2-16 a b+3 b^2\right ) \cosh ^3(c+d x) \sinh (c+d x)}{192 d}+\frac{(10 a-3 b) b \cosh ^5(c+d x) \sinh (c+d x)}{48 d}+\frac{b \cosh ^7(c+d x) \sinh (c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )}{8 d}+\frac{\left (48 a^2-16 a b+3 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{64 d}\\ &=\frac{\left (48 a^2-16 a b+3 b^2\right ) \cosh (c+d x) \sinh (c+d x)}{128 d}+\frac{\left (48 a^2-16 a b+3 b^2\right ) \cosh ^3(c+d x) \sinh (c+d x)}{192 d}+\frac{(10 a-3 b) b \cosh ^5(c+d x) \sinh (c+d x)}{48 d}+\frac{b \cosh ^7(c+d x) \sinh (c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )}{8 d}+\frac{\left (48 a^2-16 a b+3 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{128 d}\\ &=\frac{1}{128} \left (48 a^2-16 a b+3 b^2\right ) x+\frac{\left (48 a^2-16 a b+3 b^2\right ) \cosh (c+d x) \sinh (c+d x)}{128 d}+\frac{\left (48 a^2-16 a b+3 b^2\right ) \cosh ^3(c+d x) \sinh (c+d x)}{192 d}+\frac{(10 a-3 b) b \cosh ^5(c+d x) \sinh (c+d x)}{48 d}+\frac{b \cosh ^7(c+d x) \sinh (c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )}{8 d}\\ \end{align*}

Mathematica [A]  time = 0.309086, size = 98, normalized size = 0.62 \[ \frac{24 \left (48 a^2-16 a b+3 b^2\right ) (c+d x)+24 \left (4 a^2+4 a b-b^2\right ) \sinh (4 (c+d x))+32 a b \sinh (6 (c+d x))+96 a (8 a-b) \sinh (2 (c+d x))+3 b^2 \sinh (8 (c+d x))}{3072 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[c + d*x]^4*(a + b*Sinh[c + d*x]^2)^2,x]

[Out]

(24*(48*a^2 - 16*a*b + 3*b^2)*(c + d*x) + 96*a*(8*a - b)*Sinh[2*(c + d*x)] + 24*(4*a^2 + 4*a*b - b^2)*Sinh[4*(
c + d*x)] + 32*a*b*Sinh[6*(c + d*x)] + 3*b^2*Sinh[8*(c + d*x)])/(3072*d)

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Maple [A]  time = 0.031, size = 172, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ({b}^{2} \left ({\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{3} \left ( \cosh \left ( dx+c \right ) \right ) ^{5}}{8}}-{\frac{\sinh \left ( dx+c \right ) \left ( \cosh \left ( dx+c \right ) \right ) ^{5}}{16}}+{\frac{\sinh \left ( dx+c \right ) }{16} \left ({\frac{ \left ( \cosh \left ( dx+c \right ) \right ) ^{3}}{4}}+{\frac{3\,\cosh \left ( dx+c \right ) }{8}} \right ) }+{\frac{3\,dx}{128}}+{\frac{3\,c}{128}} \right ) +2\,ab \left ( 1/6\,\sinh \left ( dx+c \right ) \left ( \cosh \left ( dx+c \right ) \right ) ^{5}-1/6\, \left ( 1/4\, \left ( \cosh \left ( dx+c \right ) \right ) ^{3}+3/8\,\cosh \left ( dx+c \right ) \right ) \sinh \left ( dx+c \right ) -1/16\,dx-c/16 \right ) +{a}^{2} \left ( \left ({\frac{ \left ( \cosh \left ( dx+c \right ) \right ) ^{3}}{4}}+{\frac{3\,\cosh \left ( dx+c \right ) }{8}} \right ) \sinh \left ( dx+c \right ) +{\frac{3\,dx}{8}}+{\frac{3\,c}{8}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(d*x+c)^4*(a+b*sinh(d*x+c)^2)^2,x)

[Out]

1/d*(b^2*(1/8*sinh(d*x+c)^3*cosh(d*x+c)^5-1/16*sinh(d*x+c)*cosh(d*x+c)^5+1/16*(1/4*cosh(d*x+c)^3+3/8*cosh(d*x+
c))*sinh(d*x+c)+3/128*d*x+3/128*c)+2*a*b*(1/6*sinh(d*x+c)*cosh(d*x+c)^5-1/6*(1/4*cosh(d*x+c)^3+3/8*cosh(d*x+c)
)*sinh(d*x+c)-1/16*d*x-1/16*c)+a^2*((1/4*cosh(d*x+c)^3+3/8*cosh(d*x+c))*sinh(d*x+c)+3/8*d*x+3/8*c))

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Maxima [A]  time = 1.11009, size = 304, normalized size = 1.91 \begin{align*} \frac{1}{64} \, a^{2}{\left (24 \, x + \frac{e^{\left (4 \, d x + 4 \, c\right )}}{d} + \frac{8 \, e^{\left (2 \, d x + 2 \, c\right )}}{d} - \frac{8 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac{e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} - \frac{1}{2048} \, b^{2}{\left (\frac{{\left (8 \, e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )} e^{\left (8 \, d x + 8 \, c\right )}}{d} - \frac{48 \,{\left (d x + c\right )}}{d} - \frac{8 \, e^{\left (-4 \, d x - 4 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )}}{d}\right )} + \frac{1}{192} \, a b{\left (\frac{{\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )} e^{\left (6 \, d x + 6 \, c\right )}}{d} - \frac{24 \,{\left (d x + c\right )}}{d} + \frac{3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} - e^{\left (-6 \, d x - 6 \, c\right )}}{d}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^4*(a+b*sinh(d*x+c)^2)^2,x, algorithm="maxima")

[Out]

1/64*a^2*(24*x + e^(4*d*x + 4*c)/d + 8*e^(2*d*x + 2*c)/d - 8*e^(-2*d*x - 2*c)/d - e^(-4*d*x - 4*c)/d) - 1/2048
*b^2*((8*e^(-4*d*x - 4*c) - 1)*e^(8*d*x + 8*c)/d - 48*(d*x + c)/d - (8*e^(-4*d*x - 4*c) - e^(-8*d*x - 8*c))/d)
+ 1/192*a*b*((3*e^(-2*d*x - 2*c) - 3*e^(-4*d*x - 4*c) + 1)*e^(6*d*x + 6*c)/d - 24*(d*x + c)/d + (3*e^(-2*d*x
- 2*c) - 3*e^(-4*d*x - 4*c) - e^(-6*d*x - 6*c))/d)

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Fricas [A]  time = 1.48994, size = 532, normalized size = 3.35 \begin{align*} \frac{3 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{7} + 3 \,{\left (7 \, b^{2} \cosh \left (d x + c\right )^{3} + 8 \, a b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{5} +{\left (21 \, b^{2} \cosh \left (d x + c\right )^{5} + 80 \, a b \cosh \left (d x + c\right )^{3} + 12 \,{\left (4 \, a^{2} + 4 \, a b - b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 3 \,{\left (48 \, a^{2} - 16 \, a b + 3 \, b^{2}\right )} d x + 3 \,{\left (b^{2} \cosh \left (d x + c\right )^{7} + 8 \, a b \cosh \left (d x + c\right )^{5} + 4 \,{\left (4 \, a^{2} + 4 \, a b - b^{2}\right )} \cosh \left (d x + c\right )^{3} + 8 \,{\left (8 \, a^{2} - a b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{384 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^4*(a+b*sinh(d*x+c)^2)^2,x, algorithm="fricas")

[Out]

1/384*(3*b^2*cosh(d*x + c)*sinh(d*x + c)^7 + 3*(7*b^2*cosh(d*x + c)^3 + 8*a*b*cosh(d*x + c))*sinh(d*x + c)^5 +
(21*b^2*cosh(d*x + c)^5 + 80*a*b*cosh(d*x + c)^3 + 12*(4*a^2 + 4*a*b - b^2)*cosh(d*x + c))*sinh(d*x + c)^3 +
3*(48*a^2 - 16*a*b + 3*b^2)*d*x + 3*(b^2*cosh(d*x + c)^7 + 8*a*b*cosh(d*x + c)^5 + 4*(4*a^2 + 4*a*b - b^2)*cos
h(d*x + c)^3 + 8*(8*a^2 - a*b)*cosh(d*x + c))*sinh(d*x + c))/d

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Sympy [A]  time = 12.8862, size = 481, normalized size = 3.03 \begin{align*} \begin{cases} \frac{3 a^{2} x \sinh ^{4}{\left (c + d x \right )}}{8} - \frac{3 a^{2} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4} + \frac{3 a^{2} x \cosh ^{4}{\left (c + d x \right )}}{8} - \frac{3 a^{2} \sinh ^{3}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{8 d} + \frac{5 a^{2} \sinh{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{8 d} + \frac{a b x \sinh ^{6}{\left (c + d x \right )}}{8} - \frac{3 a b x \sinh ^{4}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{8} + \frac{3 a b x \sinh ^{2}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{8} - \frac{a b x \cosh ^{6}{\left (c + d x \right )}}{8} - \frac{a b \sinh ^{5}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{8 d} + \frac{a b \sinh ^{3}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac{a b \sinh{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{8 d} + \frac{3 b^{2} x \sinh ^{8}{\left (c + d x \right )}}{128} - \frac{3 b^{2} x \sinh ^{6}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{32} + \frac{9 b^{2} x \sinh ^{4}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{64} - \frac{3 b^{2} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{6}{\left (c + d x \right )}}{32} + \frac{3 b^{2} x \cosh ^{8}{\left (c + d x \right )}}{128} - \frac{3 b^{2} \sinh ^{7}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{128 d} + \frac{11 b^{2} \sinh ^{5}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{128 d} + \frac{11 b^{2} \sinh ^{3}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{128 d} - \frac{3 b^{2} \sinh{\left (c + d x \right )} \cosh ^{7}{\left (c + d x \right )}}{128 d} & \text{for}\: d \neq 0 \\x \left (a + b \sinh ^{2}{\left (c \right )}\right )^{2} \cosh ^{4}{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)**4*(a+b*sinh(d*x+c)**2)**2,x)

[Out]

Piecewise((3*a**2*x*sinh(c + d*x)**4/8 - 3*a**2*x*sinh(c + d*x)**2*cosh(c + d*x)**2/4 + 3*a**2*x*cosh(c + d*x)
**4/8 - 3*a**2*sinh(c + d*x)**3*cosh(c + d*x)/(8*d) + 5*a**2*sinh(c + d*x)*cosh(c + d*x)**3/(8*d) + a*b*x*sinh
(c + d*x)**6/8 - 3*a*b*x*sinh(c + d*x)**4*cosh(c + d*x)**2/8 + 3*a*b*x*sinh(c + d*x)**2*cosh(c + d*x)**4/8 - a
*b*x*cosh(c + d*x)**6/8 - a*b*sinh(c + d*x)**5*cosh(c + d*x)/(8*d) + a*b*sinh(c + d*x)**3*cosh(c + d*x)**3/(3*
d) + a*b*sinh(c + d*x)*cosh(c + d*x)**5/(8*d) + 3*b**2*x*sinh(c + d*x)**8/128 - 3*b**2*x*sinh(c + d*x)**6*cosh
(c + d*x)**2/32 + 9*b**2*x*sinh(c + d*x)**4*cosh(c + d*x)**4/64 - 3*b**2*x*sinh(c + d*x)**2*cosh(c + d*x)**6/3
2 + 3*b**2*x*cosh(c + d*x)**8/128 - 3*b**2*sinh(c + d*x)**7*cosh(c + d*x)/(128*d) + 11*b**2*sinh(c + d*x)**5*c
osh(c + d*x)**3/(128*d) + 11*b**2*sinh(c + d*x)**3*cosh(c + d*x)**5/(128*d) - 3*b**2*sinh(c + d*x)*cosh(c + d*
x)**7/(128*d), Ne(d, 0)), (x*(a + b*sinh(c)**2)**2*cosh(c)**4, True))

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Giac [A]  time = 1.23409, size = 354, normalized size = 2.23 \begin{align*} \frac{3 \, b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 32 \, a b e^{\left (6 \, d x + 6 \, c\right )} + 96 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} + 96 \, a b e^{\left (4 \, d x + 4 \, c\right )} - 24 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 768 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} - 96 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 48 \,{\left (48 \, a^{2} - 16 \, a b + 3 \, b^{2}\right )}{\left (d x + c\right )} -{\left (2400 \, a^{2} e^{\left (8 \, d x + 8 \, c\right )} - 800 \, a b e^{\left (8 \, d x + 8 \, c\right )} + 150 \, b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 768 \, a^{2} e^{\left (6 \, d x + 6 \, c\right )} - 96 \, a b e^{\left (6 \, d x + 6 \, c\right )} + 96 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} + 96 \, a b e^{\left (4 \, d x + 4 \, c\right )} - 24 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 32 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 3 \, b^{2}\right )} e^{\left (-8 \, d x - 8 \, c\right )}}{6144 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^4*(a+b*sinh(d*x+c)^2)^2,x, algorithm="giac")

[Out]

1/6144*(3*b^2*e^(8*d*x + 8*c) + 32*a*b*e^(6*d*x + 6*c) + 96*a^2*e^(4*d*x + 4*c) + 96*a*b*e^(4*d*x + 4*c) - 24*
b^2*e^(4*d*x + 4*c) + 768*a^2*e^(2*d*x + 2*c) - 96*a*b*e^(2*d*x + 2*c) + 48*(48*a^2 - 16*a*b + 3*b^2)*(d*x + c
) - (2400*a^2*e^(8*d*x + 8*c) - 800*a*b*e^(8*d*x + 8*c) + 150*b^2*e^(8*d*x + 8*c) + 768*a^2*e^(6*d*x + 6*c) -
96*a*b*e^(6*d*x + 6*c) + 96*a^2*e^(4*d*x + 4*c) + 96*a*b*e^(4*d*x + 4*c) - 24*b^2*e^(4*d*x + 4*c) + 32*a*b*e^(
2*d*x + 2*c) + 3*b^2)*e^(-8*d*x - 8*c))/d