3.209 \(\int \sinh (c+d x) (a+b \sinh ^4(c+d x))^3 \, dx\)

Optimal. Leaf size=143 \[ \frac{b^2 (a+5 b) \cosh ^9(c+d x)}{3 d}-\frac{4 b^2 (3 a+5 b) \cosh ^7(c+d x)}{7 d}+\frac{3 b (a+b) (a+5 b) \cosh ^5(c+d x)}{5 d}-\frac{2 b (a+b)^2 \cosh ^3(c+d x)}{d}+\frac{(a+b)^3 \cosh (c+d x)}{d}+\frac{b^3 \cosh ^{13}(c+d x)}{13 d}-\frac{6 b^3 \cosh ^{11}(c+d x)}{11 d} \]

[Out]

((a + b)^3*Cosh[c + d*x])/d - (2*b*(a + b)^2*Cosh[c + d*x]^3)/d + (3*b*(a + b)*(a + 5*b)*Cosh[c + d*x]^5)/(5*d
) - (4*b^2*(3*a + 5*b)*Cosh[c + d*x]^7)/(7*d) + (b^2*(a + 5*b)*Cosh[c + d*x]^9)/(3*d) - (6*b^3*Cosh[c + d*x]^1
1)/(11*d) + (b^3*Cosh[c + d*x]^13)/(13*d)

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Rubi [A]  time = 0.15744, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {3215, 1090} \[ \frac{b^2 (a+5 b) \cosh ^9(c+d x)}{3 d}-\frac{4 b^2 (3 a+5 b) \cosh ^7(c+d x)}{7 d}+\frac{3 b (a+b) (a+5 b) \cosh ^5(c+d x)}{5 d}-\frac{2 b (a+b)^2 \cosh ^3(c+d x)}{d}+\frac{(a+b)^3 \cosh (c+d x)}{d}+\frac{b^3 \cosh ^{13}(c+d x)}{13 d}-\frac{6 b^3 \cosh ^{11}(c+d x)}{11 d} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[c + d*x]*(a + b*Sinh[c + d*x]^4)^3,x]

[Out]

((a + b)^3*Cosh[c + d*x])/d - (2*b*(a + b)^2*Cosh[c + d*x]^3)/d + (3*b*(a + b)*(a + 5*b)*Cosh[c + d*x]^5)/(5*d
) - (4*b^2*(3*a + 5*b)*Cosh[c + d*x]^7)/(7*d) + (b^2*(a + 5*b)*Cosh[c + d*x]^9)/(3*d) - (6*b^3*Cosh[c + d*x]^1
1)/(11*d) + (b^3*Cosh[c + d*x]^13)/(13*d)

Rule 3215

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Cos[e + f*x], x]}, -Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - 2*b*ff^2*x^2 + b*ff^4*x^4
)^p, x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rule 1090

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^2 + c*x^4)^p, x], x]
/; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \sinh (c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx &=\frac{\operatorname{Subst}\left (\int \left (a+b-2 b x^2+b x^4\right )^3 \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (a^3 \left (1+\frac{b \left (3 a^2+3 a b+b^2\right )}{a^3}\right )-6 b (a+b)^2 x^2+12 b^2 (a+b) \left (1+\frac{a+b}{4 b}\right ) x^4-8 b^3 \left (1+\frac{3 (a+b)}{2 b}\right ) x^6+12 b^3 \left (1+\frac{a+b}{4 b}\right ) x^8-6 b^3 x^{10}+b^3 x^{12}\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac{(a+b)^3 \cosh (c+d x)}{d}-\frac{2 b (a+b)^2 \cosh ^3(c+d x)}{d}+\frac{3 b (a+b) (a+5 b) \cosh ^5(c+d x)}{5 d}-\frac{4 b^2 (3 a+5 b) \cosh ^7(c+d x)}{7 d}+\frac{b^2 (a+5 b) \cosh ^9(c+d x)}{3 d}-\frac{6 b^3 \cosh ^{11}(c+d x)}{11 d}+\frac{b^3 \cosh ^{13}(c+d x)}{13 d}\\ \end{align*}

Mathematica [A]  time = 0.90112, size = 157, normalized size = 1.1 \[ \frac{-15015 b \left (1280 a^2+1344 a b+429 b^2\right ) \cosh (3 (c+d x))+3003 b \left (768 a^2+1728 a b+715 b^2\right ) \cosh (5 (c+d x))+60060 \left (1920 a^2 b+1024 a^3+1512 a b^2+429 b^3\right ) \cosh (c+d x)-4290 b^2 (216 a+143 b) \cosh (7 (c+d x))+10010 b^2 (8 a+13 b) \cosh (9 (c+d x))-17745 b^3 \cosh (11 (c+d x))+1155 b^3 \cosh (13 (c+d x))}{61501440 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[c + d*x]*(a + b*Sinh[c + d*x]^4)^3,x]

[Out]

(60060*(1024*a^3 + 1920*a^2*b + 1512*a*b^2 + 429*b^3)*Cosh[c + d*x] - 15015*b*(1280*a^2 + 1344*a*b + 429*b^2)*
Cosh[3*(c + d*x)] + 3003*b*(768*a^2 + 1728*a*b + 715*b^2)*Cosh[5*(c + d*x)] - 4290*b^2*(216*a + 143*b)*Cosh[7*
(c + d*x)] + 10010*b^2*(8*a + 13*b)*Cosh[9*(c + d*x)] - 17745*b^3*Cosh[11*(c + d*x)] + 1155*b^3*Cosh[13*(c + d
*x)])/(61501440*d)

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Maple [A]  time = 0.02, size = 176, normalized size = 1.2 \begin{align*}{\frac{1}{d} \left ({b}^{3} \left ({\frac{1024}{3003}}+{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{12}}{13}}-{\frac{12\, \left ( \sinh \left ( dx+c \right ) \right ) ^{10}}{143}}+{\frac{40\, \left ( \sinh \left ( dx+c \right ) \right ) ^{8}}{429}}-{\frac{320\, \left ( \sinh \left ( dx+c \right ) \right ) ^{6}}{3003}}+{\frac{128\, \left ( \sinh \left ( dx+c \right ) \right ) ^{4}}{1001}}-{\frac{512\, \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{3003}} \right ) \cosh \left ( dx+c \right ) +3\,a{b}^{2} \left ({\frac{128}{315}}+1/9\, \left ( \sinh \left ( dx+c \right ) \right ) ^{8}-{\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 ) +3\,{a}^{2}b \left ({\frac{8}{15}}+1/5\, \left ( \sinh \left ( dx+c \right ) \right ) ^{4}-{\frac{4\, \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{15}} \right ) \cosh \left ( dx+c \right ) +{a}^{3}\cosh \left ( dx+c \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(d*x+c)*(a+b*sinh(d*x+c)^4)^3,x)

[Out]

1/d*(b^3*(1024/3003+1/13*sinh(d*x+c)^12-12/143*sinh(d*x+c)^10+40/429*sinh(d*x+c)^8-320/3003*sinh(d*x+c)^6+128/
1001*sinh(d*x+c)^4-512/3003*sinh(d*x+c)^2)*cosh(d*x+c)+3*a*b^2*(128/315+1/9*sinh(d*x+c)^8-8/63*sinh(d*x+c)^6+1
6/105*sinh(d*x+c)^4-64/315*sinh(d*x+c)^2)*cosh(d*x+c)+3*a^2*b*(8/15+1/5*sinh(d*x+c)^4-4/15*sinh(d*x+c)^2)*cosh
(d*x+c)+a^3*cosh(d*x+c))

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Maxima [B]  time = 1.04483, size = 539, normalized size = 3.77 \begin{align*} -\frac{1}{24600576} \, b^{3}{\left (\frac{{\left (3549 \, e^{\left (-2 \, d x - 2 \, c\right )} - 26026 \, e^{\left (-4 \, d x - 4 \, c\right )} + 122694 \, e^{\left (-6 \, d x - 6 \, c\right )} - 429429 \, e^{\left (-8 \, d x - 8 \, c\right )} + 1288287 \, e^{\left (-10 \, d x - 10 \, c\right )} - 5153148 \, e^{\left (-12 \, d x - 12 \, c\right )} - 231\right )} e^{\left (13 \, d x + 13 \, c\right )}}{d} - \frac{5153148 \, e^{\left (-d x - c\right )} - 1288287 \, e^{\left (-3 \, d x - 3 \, c\right )} + 429429 \, e^{\left (-5 \, d x - 5 \, c\right )} - 122694 \, e^{\left (-7 \, d x - 7 \, c\right )} + 26026 \, e^{\left (-9 \, d x - 9 \, c\right )} - 3549 \, e^{\left (-11 \, d x - 11 \, c\right )} + 231 \, e^{\left (-13 \, d x - 13 \, c\right )}}{d}\right )} - \frac{1}{53760} \, a 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}{160} \, a^{2} b{\left (\frac{3 \, e^{\left (5 \, d x + 5 \, c\right )}}{d} - \frac{25 \, e^{\left (3 \, d x + 3 \, c\right )}}{d} + \frac{150 \, e^{\left (d x + c\right )}}{d} + \frac{150 \, e^{\left (-d x - c\right )}}{d} - \frac{25 \, e^{\left (-3 \, d x - 3 \, c\right )}}{d} + \frac{3 \, e^{\left (-5 \, d x - 5 \, c\right )}}{d}\right )} + \frac{a^{3} \cosh \left (d x + c\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)*(a+b*sinh(d*x+c)^4)^3,x, algorithm="maxima")

[Out]

-1/24600576*b^3*((3549*e^(-2*d*x - 2*c) - 26026*e^(-4*d*x - 4*c) + 122694*e^(-6*d*x - 6*c) - 429429*e^(-8*d*x
- 8*c) + 1288287*e^(-10*d*x - 10*c) - 5153148*e^(-12*d*x - 12*c) - 231)*e^(13*d*x + 13*c)/d - (5153148*e^(-d*x
- c) - 1288287*e^(-3*d*x - 3*c) + 429429*e^(-5*d*x - 5*c) - 122694*e^(-7*d*x - 7*c) + 26026*e^(-9*d*x - 9*c)
- 3549*e^(-11*d*x - 11*c) + 231*e^(-13*d*x - 13*c))/d) - 1/53760*a*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) + 1/160*a^2*b*(3*e^(
5*d*x + 5*c)/d - 25*e^(3*d*x + 3*c)/d + 150*e^(d*x + c)/d + 150*e^(-d*x - c)/d - 25*e^(-3*d*x - 3*c)/d + 3*e^(
-5*d*x - 5*c)/d) + a^3*cosh(d*x + c)/d

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Fricas [B]  time = 1.73622, size = 1620, normalized size = 11.33 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)*(a+b*sinh(d*x+c)^4)^3,x, algorithm="fricas")

[Out]

1/61501440*(1155*b^3*cosh(d*x + c)^13 + 15015*b^3*cosh(d*x + c)*sinh(d*x + c)^12 - 17745*b^3*cosh(d*x + c)^11
+ 15015*(22*b^3*cosh(d*x + c)^3 - 13*b^3*cosh(d*x + c))*sinh(d*x + c)^10 + 10010*(8*a*b^2 + 13*b^3)*cosh(d*x +
c)^9 + 45045*(33*b^3*cosh(d*x + c)^5 - 65*b^3*cosh(d*x + c)^3 + 2*(8*a*b^2 + 13*b^3)*cosh(d*x + c))*sinh(d*x
+ c)^8 - 4290*(216*a*b^2 + 143*b^3)*cosh(d*x + c)^7 + 30030*(66*b^3*cosh(d*x + c)^7 - 273*b^3*cosh(d*x + c)^5
+ 28*(8*a*b^2 + 13*b^3)*cosh(d*x + c)^3 - (216*a*b^2 + 143*b^3)*cosh(d*x + c))*sinh(d*x + c)^6 + 3003*(768*a^2
*b + 1728*a*b^2 + 715*b^3)*cosh(d*x + c)^5 + 15015*(55*b^3*cosh(d*x + c)^9 - 390*b^3*cosh(d*x + c)^7 + 84*(8*a
*b^2 + 13*b^3)*cosh(d*x + c)^5 - 10*(216*a*b^2 + 143*b^3)*cosh(d*x + c)^3 + (768*a^2*b + 1728*a*b^2 + 715*b^3)
*cosh(d*x + c))*sinh(d*x + c)^4 - 15015*(1280*a^2*b + 1344*a*b^2 + 429*b^3)*cosh(d*x + c)^3 + 15015*(6*b^3*cos
h(d*x + c)^11 - 65*b^3*cosh(d*x + c)^9 + 24*(8*a*b^2 + 13*b^3)*cosh(d*x + c)^7 - 6*(216*a*b^2 + 143*b^3)*cosh(
d*x + c)^5 + 2*(768*a^2*b + 1728*a*b^2 + 715*b^3)*cosh(d*x + c)^3 - 3*(1280*a^2*b + 1344*a*b^2 + 429*b^3)*cosh
(d*x + c))*sinh(d*x + c)^2 + 60060*(1024*a^3 + 1920*a^2*b + 1512*a*b^2 + 429*b^3)*cosh(d*x + c))/d

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Sympy [A]  time = 169.693, size = 377, normalized size = 2.64 \begin{align*} \begin{cases} \frac{a^{3} \cosh{\left (c + d x \right )}}{d} + \frac{3 a^{2} b \sinh ^{4}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{d} - \frac{4 a^{2} b \sinh ^{2}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{d} + \frac{8 a^{2} b \cosh ^{5}{\left (c + d x \right )}}{5 d} + \frac{3 a b^{2} \sinh ^{8}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{d} - \frac{8 a b^{2} \sinh ^{6}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{d} + \frac{48 a b^{2} \sinh ^{4}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{5 d} - \frac{192 a b^{2} \sinh ^{2}{\left (c + d x \right )} \cosh ^{7}{\left (c + d x \right )}}{35 d} + \frac{128 a b^{2} \cosh ^{9}{\left (c + d x \right )}}{105 d} + \frac{b^{3} \sinh ^{12}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{d} - \frac{4 b^{3} \sinh ^{10}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{d} + \frac{8 b^{3} \sinh ^{8}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{d} - \frac{64 b^{3} \sinh ^{6}{\left (c + d x \right )} \cosh ^{7}{\left (c + d x \right )}}{7 d} + \frac{128 b^{3} \sinh ^{4}{\left (c + d x \right )} \cosh ^{9}{\left (c + d x \right )}}{21 d} - \frac{512 b^{3} \sinh ^{2}{\left (c + d x \right )} \cosh ^{11}{\left (c + d x \right )}}{231 d} + \frac{1024 b^{3} \cosh ^{13}{\left (c + d x \right )}}{3003 d} & \text{for}\: d \neq 0 \\x \left (a + b \sinh ^{4}{\left (c \right )}\right )^{3} \sinh{\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)*(a+b*sinh(d*x+c)**4)**3,x)

[Out]

Piecewise((a**3*cosh(c + d*x)/d + 3*a**2*b*sinh(c + d*x)**4*cosh(c + d*x)/d - 4*a**2*b*sinh(c + d*x)**2*cosh(c
+ d*x)**3/d + 8*a**2*b*cosh(c + d*x)**5/(5*d) + 3*a*b**2*sinh(c + d*x)**8*cosh(c + d*x)/d - 8*a*b**2*sinh(c +
d*x)**6*cosh(c + d*x)**3/d + 48*a*b**2*sinh(c + d*x)**4*cosh(c + d*x)**5/(5*d) - 192*a*b**2*sinh(c + d*x)**2*
cosh(c + d*x)**7/(35*d) + 128*a*b**2*cosh(c + d*x)**9/(105*d) + b**3*sinh(c + d*x)**12*cosh(c + d*x)/d - 4*b**
3*sinh(c + d*x)**10*cosh(c + d*x)**3/d + 8*b**3*sinh(c + d*x)**8*cosh(c + d*x)**5/d - 64*b**3*sinh(c + d*x)**6
*cosh(c + d*x)**7/(7*d) + 128*b**3*sinh(c + d*x)**4*cosh(c + d*x)**9/(21*d) - 512*b**3*sinh(c + d*x)**2*cosh(c
+ d*x)**11/(231*d) + 1024*b**3*cosh(c + d*x)**13/(3003*d), Ne(d, 0)), (x*(a + b*sinh(c)**4)**3*sinh(c), True)
)

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Giac [B]  time = 1.67781, size = 621, normalized size = 4.34 \begin{align*} \frac{1155 \, b^{3} e^{\left (13 \, d x + 13 \, c\right )} - 17745 \, b^{3} e^{\left (11 \, d x + 11 \, c\right )} + 80080 \, a b^{2} e^{\left (9 \, d x + 9 \, c\right )} + 130130 \, b^{3} e^{\left (9 \, d x + 9 \, c\right )} - 926640 \, a b^{2} e^{\left (7 \, d x + 7 \, c\right )} - 613470 \, b^{3} e^{\left (7 \, d x + 7 \, c\right )} + 2306304 \, a^{2} b e^{\left (5 \, d x + 5 \, c\right )} + 5189184 \, a b^{2} e^{\left (5 \, d x + 5 \, c\right )} + 2147145 \, b^{3} e^{\left (5 \, d x + 5 \, c\right )} - 19219200 \, a^{2} b e^{\left (3 \, d x + 3 \, c\right )} - 20180160 \, a b^{2} e^{\left (3 \, d x + 3 \, c\right )} - 6441435 \, b^{3} e^{\left (3 \, d x + 3 \, c\right )} + 61501440 \, a^{3} e^{\left (d x + c\right )} + 115315200 \, a^{2} b e^{\left (d x + c\right )} + 90810720 \, a b^{2} e^{\left (d x + c\right )} + 25765740 \, b^{3} e^{\left (d x + c\right )} +{\left (61501440 \, a^{3} e^{\left (12 \, d x + 12 \, c\right )} + 115315200 \, a^{2} b e^{\left (12 \, d x + 12 \, c\right )} + 90810720 \, a b^{2} e^{\left (12 \, d x + 12 \, c\right )} + 25765740 \, b^{3} e^{\left (12 \, d x + 12 \, c\right )} - 19219200 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} - 20180160 \, a b^{2} e^{\left (10 \, d x + 10 \, c\right )} - 6441435 \, b^{3} e^{\left (10 \, d x + 10 \, c\right )} + 2306304 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} + 5189184 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 2147145 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} - 926640 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 613470 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 80080 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 130130 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 17745 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 1155 \, b^{3}\right )} e^{\left (-13 \, d x - 13 \, c\right )}}{123002880 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)*(a+b*sinh(d*x+c)^4)^3,x, algorithm="giac")

[Out]

1/123002880*(1155*b^3*e^(13*d*x + 13*c) - 17745*b^3*e^(11*d*x + 11*c) + 80080*a*b^2*e^(9*d*x + 9*c) + 130130*b
^3*e^(9*d*x + 9*c) - 926640*a*b^2*e^(7*d*x + 7*c) - 613470*b^3*e^(7*d*x + 7*c) + 2306304*a^2*b*e^(5*d*x + 5*c)
+ 5189184*a*b^2*e^(5*d*x + 5*c) + 2147145*b^3*e^(5*d*x + 5*c) - 19219200*a^2*b*e^(3*d*x + 3*c) - 20180160*a*b
^2*e^(3*d*x + 3*c) - 6441435*b^3*e^(3*d*x + 3*c) + 61501440*a^3*e^(d*x + c) + 115315200*a^2*b*e^(d*x + c) + 90
810720*a*b^2*e^(d*x + c) + 25765740*b^3*e^(d*x + c) + (61501440*a^3*e^(12*d*x + 12*c) + 115315200*a^2*b*e^(12*
d*x + 12*c) + 90810720*a*b^2*e^(12*d*x + 12*c) + 25765740*b^3*e^(12*d*x + 12*c) - 19219200*a^2*b*e^(10*d*x + 1
0*c) - 20180160*a*b^2*e^(10*d*x + 10*c) - 6441435*b^3*e^(10*d*x + 10*c) + 2306304*a^2*b*e^(8*d*x + 8*c) + 5189
184*a*b^2*e^(8*d*x + 8*c) + 2147145*b^3*e^(8*d*x + 8*c) - 926640*a*b^2*e^(6*d*x + 6*c) - 613470*b^3*e^(6*d*x +
6*c) + 80080*a*b^2*e^(4*d*x + 4*c) + 130130*b^3*e^(4*d*x + 4*c) - 17745*b^3*e^(2*d*x + 2*c) + 1155*b^3)*e^(-1
3*d*x - 13*c))/d