Integrand size = 23, antiderivative size = 78 \[ \int \cosh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {1}{2} (a-5 b) (a+b)^2 x+\frac {(a+b)^3 \cosh (c+d x) \sinh (c+d x)}{2 d}+\frac {b^2 (3 a+2 b) \tanh (c+d x)}{d}+\frac {b^3 \tanh ^3(c+d x)}{3 d} \] Output:
1/2*(a-5*b)*(a+b)^2*x+1/2*(a+b)^3*cosh(d*x+c)*sinh(d*x+c)/d+b^2*(3*a+2*b)* tanh(d*x+c)/d+1/3*b^3*tanh(d*x+c)^3/d
Time = 1.94 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.88 \[ \int \cosh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {6 (a-5 b) (a+b)^2 (c+d x)+3 (a+b)^3 \sinh (2 (c+d x))+4 b^2 \left (9 a+7 b-b \text {sech}^2(c+d x)\right ) \tanh (c+d x)}{12 d} \] Input:
Integrate[Cosh[c + d*x]^2*(a + b*Tanh[c + d*x]^2)^3,x]
Output:
(6*(a - 5*b)*(a + b)^2*(c + d*x) + 3*(a + b)^3*Sinh[2*(c + d*x)] + 4*b^2*( 9*a + 7*b - b*Sech[c + d*x]^2)*Tanh[c + d*x])/(12*d)
Time = 0.32 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.12, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3042, 4158, 300, 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.
| \(\displaystyle \int \cosh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a-b \tan (i c+i d x)^2\right )^3}{\sec (i c+i d x)^2}dx\) |
| \(\Big \downarrow \) 4158 |
| \(\displaystyle \frac {\int \frac {\left (b \tanh ^2(c+d x)+a\right )^3}{\left (1-\tanh ^2(c+d x)\right )^2}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 300 |
| \(\displaystyle \frac {\int \left (\tanh ^2(c+d x) b^3+(3 a+2 b) b^2+\frac {3 b \tanh ^2(c+d x) (a+b)^2+(a-2 b) (a+b)^2}{\left (1-\tanh ^2(c+d x)\right )^2}\right )d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {1}{2} (a-5 b) (a+b)^2 \text {arctanh}(\tanh (c+d x))+b^2 (3 a+2 b) \tanh (c+d x)+\frac {(a+b)^3 \tanh (c+d x)}{2 \left (1-\tanh ^2(c+d x)\right )}+\frac {1}{3} b^3 \tanh ^3(c+d x)}{d}\) |
Input:
Int[Cosh[c + d*x]^2*(a + b*Tanh[c + d*x]^2)^3,x]
Output:
(((a - 5*b)*(a + b)^2*ArcTanh[Tanh[c + d*x]])/2 + b^2*(3*a + 2*b)*Tanh[c + d*x] + (b^3*Tanh[c + d*x]^3)/3 + ((a + b)^3*Tanh[c + d*x])/(2*(1 - Tanh[c + d*x]^2)))/d
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Int [PolynomialDivide[(a + b*x^2)^p, (c + d*x^2)^(-q), x], x] /; FreeQ[{a, b, c , d}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && ILtQ[q, 0] && GeQ[p, -q]
Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_ )])^(n_))^(p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Sim p[ff/(c^(m - 1)*f) Subst[Int[(c^2 + ff^2*x^2)^(m/2 - 1)*(a + b*(ff*x)^n)^ p, x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && I ntegerQ[m/2] && (IntegersQ[n, p] || IGtQ[m, 0] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])
Leaf count of result is larger than twice the leaf count of optimal. \(147\) vs. \(2(72)=144\).
Time = 13.17 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.90
| method | result | size |
| derivativedivides | \(\frac {a^{3} \left (\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 a^{2} b \left (\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}\right )+3 b^{2} a \left (\frac {\sinh \left (d x +c \right )^{3}}{2 \cosh \left (d x +c \right )}-\frac {3 d x}{2}-\frac {3 c}{2}+\frac {3 \tanh \left (d x +c \right )}{2}\right )+b^{3} \left (\frac {\sinh \left (d x +c \right )^{5}}{2 \cosh \left (d x +c \right )^{3}}-\frac {5 d x}{2}-\frac {5 c}{2}+\frac {5 \tanh \left (d x +c \right )}{2}+\frac {5 \tanh \left (d x +c \right )^{3}}{6}\right )}{d}\) | \(148\) |
| default | \(\frac {a^{3} \left (\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 a^{2} b \left (\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}\right )+3 b^{2} a \left (\frac {\sinh \left (d x +c \right )^{3}}{2 \cosh \left (d x +c \right )}-\frac {3 d x}{2}-\frac {3 c}{2}+\frac {3 \tanh \left (d x +c \right )}{2}\right )+b^{3} \left (\frac {\sinh \left (d x +c \right )^{5}}{2 \cosh \left (d x +c \right )^{3}}-\frac {5 d x}{2}-\frac {5 c}{2}+\frac {5 \tanh \left (d x +c \right )}{2}+\frac {5 \tanh \left (d x +c \right )^{3}}{6}\right )}{d}\) | \(148\) |
| risch | \(\frac {a^{3} x}{2}-\frac {3 a^{2} b x}{2}-\frac {9 a \,b^{2} x}{2}-\frac {5 b^{3} x}{2}+\frac {{\mathrm e}^{2 d x +2 c} a^{3}}{8 d}+\frac {3 \,{\mathrm e}^{2 d x +2 c} a^{2} b}{8 d}+\frac {3 \,{\mathrm e}^{2 d x +2 c} b^{2} a}{8 d}+\frac {{\mathrm e}^{2 d x +2 c} b^{3}}{8 d}-\frac {{\mathrm e}^{-2 d x -2 c} a^{3}}{8 d}-\frac {3 \,{\mathrm e}^{-2 d x -2 c} a^{2} b}{8 d}-\frac {3 \,{\mathrm e}^{-2 d x -2 c} b^{2} a}{8 d}-\frac {{\mathrm e}^{-2 d x -2 c} b^{3}}{8 d}-\frac {2 b^{2} \left (9 \,{\mathrm e}^{4 d x +4 c} a +9 b \,{\mathrm e}^{4 d x +4 c}+18 \,{\mathrm e}^{2 d x +2 c} a +12 \,{\mathrm e}^{2 d x +2 c} b +9 a +7 b \right )}{3 d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{3}}\) | \(244\) |
Input:
int(cosh(d*x+c)^2*(a+tanh(d*x+c)^2*b)^3,x,method=_RETURNVERBOSE)
Output:
1/d*(a^3*(1/2*cosh(d*x+c)*sinh(d*x+c)+1/2*d*x+1/2*c)+3*a^2*b*(1/2*cosh(d*x +c)*sinh(d*x+c)-1/2*d*x-1/2*c)+3*b^2*a*(1/2*sinh(d*x+c)^3/cosh(d*x+c)-3/2* d*x-3/2*c+3/2*tanh(d*x+c))+b^3*(1/2*sinh(d*x+c)^5/cosh(d*x+c)^3-5/2*d*x-5/ 2*c+5/2*tanh(d*x+c)+5/6*tanh(d*x+c)^3))
Leaf count of result is larger than twice the leaf count of optimal. 369 vs. \(2 (72) = 144\).
Time = 0.10 (sec) , antiderivative size = 369, normalized size of antiderivative = 4.73 \[ \int \cosh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {3 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \sinh \left (d x + c\right )^{5} - 4 \, {\left (18 \, a b^{2} + 14 \, b^{3} - 3 \, {\left (a^{3} - 3 \, a^{2} b - 9 \, a b^{2} - 5 \, b^{3}\right )} d x\right )} \cosh \left (d x + c\right )^{3} - 12 \, {\left (18 \, a b^{2} + 14 \, b^{3} - 3 \, {\left (a^{3} - 3 \, a^{2} b - 9 \, a b^{2} - 5 \, b^{3}\right )} d x\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + {\left (9 \, a^{3} + 27 \, a^{2} b + 99 \, a b^{2} + 65 \, b^{3} + 30 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{3} - 12 \, {\left (18 \, a b^{2} + 14 \, b^{3} - 3 \, {\left (a^{3} - 3 \, a^{2} b - 9 \, a b^{2} - 5 \, b^{3}\right )} d x\right )} \cosh \left (d x + c\right ) + 3 \, {\left (5 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cosh \left (d x + c\right )^{4} + 2 \, a^{3} + 6 \, a^{2} b + 30 \, a b^{2} + 10 \, b^{3} + {\left (9 \, a^{3} + 27 \, a^{2} b + 99 \, a b^{2} + 65 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )}{24 \, {\left (d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + 3 \, d \cosh \left (d x + c\right )\right )}} \] Input:
integrate(cosh(d*x+c)^2*(a+b*tanh(d*x+c)^2)^3,x, algorithm="fricas")
Output:
1/24*(3*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*sinh(d*x + c)^5 - 4*(18*a*b^2 + 14 *b^3 - 3*(a^3 - 3*a^2*b - 9*a*b^2 - 5*b^3)*d*x)*cosh(d*x + c)^3 - 12*(18*a *b^2 + 14*b^3 - 3*(a^3 - 3*a^2*b - 9*a*b^2 - 5*b^3)*d*x)*cosh(d*x + c)*sin h(d*x + c)^2 + (9*a^3 + 27*a^2*b + 99*a*b^2 + 65*b^3 + 30*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^3 - 12*(18*a*b^2 + 14*b^3 - 3*(a^3 - 3*a^2*b - 9*a*b^2 - 5*b^3)*d*x)*cosh(d*x + c) + 3*(5*(a^3 + 3*a^ 2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^4 + 2*a^3 + 6*a^2*b + 30*a*b^2 + 10*b^3 + (9*a^3 + 27*a^2*b + 99*a*b^2 + 65*b^3)*cosh(d*x + c)^2)*sinh(d*x + c))/ (d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c)*sinh(d*x + c)^2 + 3*d*cosh(d*x + c) )
\[ \int \cosh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\int \left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{3} \cosh ^{2}{\left (c + d x \right )}\, dx \] Input:
integrate(cosh(d*x+c)**2*(a+b*tanh(d*x+c)**2)**3,x)
Output:
Integral((a + b*tanh(c + d*x)**2)**3*cosh(c + d*x)**2, x)
Leaf count of result is larger than twice the leaf count of optimal. 256 vs. \(2 (72) = 144\).
Time = 0.04 (sec) , antiderivative size = 256, normalized size of antiderivative = 3.28 \[ \int \cosh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {1}{8} \, a^{3} {\left (4 \, x + \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} - \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} - \frac {3}{8} \, a^{2} b {\left (4 \, x - \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} - \frac {1}{24} \, b^{3} {\left (\frac {60 \, {\left (d x + c\right )}}{d} + \frac {3 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac {121 \, e^{\left (-2 \, d x - 2 \, c\right )} + 201 \, e^{\left (-4 \, d x - 4 \, c\right )} + 147 \, e^{\left (-6 \, d x - 6 \, c\right )} + 3}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + 3 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-8 \, d x - 8 \, c\right )}\right )}}\right )} - \frac {3}{8} \, a b^{2} {\left (\frac {12 \, {\left (d x + c\right )}}{d} + \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac {17 \, e^{\left (-2 \, d x - 2 \, c\right )} + 1}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )}\right )}}\right )} \] Input:
integrate(cosh(d*x+c)^2*(a+b*tanh(d*x+c)^2)^3,x, algorithm="maxima")
Output:
1/8*a^3*(4*x + e^(2*d*x + 2*c)/d - e^(-2*d*x - 2*c)/d) - 3/8*a^2*b*(4*x - e^(2*d*x + 2*c)/d + e^(-2*d*x - 2*c)/d) - 1/24*b^3*(60*(d*x + c)/d + 3*e^( -2*d*x - 2*c)/d - (121*e^(-2*d*x - 2*c) + 201*e^(-4*d*x - 4*c) + 147*e^(-6 *d*x - 6*c) + 3)/(d*(e^(-2*d*x - 2*c) + 3*e^(-4*d*x - 4*c) + 3*e^(-6*d*x - 6*c) + e^(-8*d*x - 8*c)))) - 3/8*a*b^2*(12*(d*x + c)/d + e^(-2*d*x - 2*c) /d - (17*e^(-2*d*x - 2*c) + 1)/(d*(e^(-2*d*x - 2*c) + e^(-4*d*x - 4*c))))
Leaf count of result is larger than twice the leaf count of optimal. 265 vs. \(2 (72) = 144\).
Time = 0.23 (sec) , antiderivative size = 265, normalized size of antiderivative = 3.40 \[ \int \cosh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {3 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 9 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 9 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 3 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 12 \, {\left (a^{3} - 3 \, a^{2} b - 9 \, a b^{2} - 5 \, b^{3}\right )} {\left (d x + c\right )} - 3 \, {\left (2 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} - 6 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 18 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 10 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )} - \frac {16 \, {\left (9 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 9 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 18 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 12 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 9 \, a b^{2} + 7 \, b^{3}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{3}}}{24 \, d} \] Input:
integrate(cosh(d*x+c)^2*(a+b*tanh(d*x+c)^2)^3,x, algorithm="giac")
Output:
1/24*(3*a^3*e^(2*d*x + 2*c) + 9*a^2*b*e^(2*d*x + 2*c) + 9*a*b^2*e^(2*d*x + 2*c) + 3*b^3*e^(2*d*x + 2*c) + 12*(a^3 - 3*a^2*b - 9*a*b^2 - 5*b^3)*(d*x + c) - 3*(2*a^3*e^(2*d*x + 2*c) - 6*a^2*b*e^(2*d*x + 2*c) - 18*a*b^2*e^(2* d*x + 2*c) - 10*b^3*e^(2*d*x + 2*c) + a^3 + 3*a^2*b + 3*a*b^2 + b^3)*e^(-2 *d*x - 2*c) - 16*(9*a*b^2*e^(4*d*x + 4*c) + 9*b^3*e^(4*d*x + 4*c) + 18*a*b ^2*e^(2*d*x + 2*c) + 12*b^3*e^(2*d*x + 2*c) + 9*a*b^2 + 7*b^3)/(e^(2*d*x + 2*c) + 1)^3)/d
Time = 0.28 (sec) , antiderivative size = 243, normalized size of antiderivative = 3.12 \[ \int \cosh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {{\mathrm {e}}^{2\,c+2\,d\,x}\,{\left (a+b\right )}^3}{8\,d}-\frac {\frac {2\,\left (b^3+3\,a\,b^2\right )}{3\,d}+\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (b^3+a\,b^2\right )}{d}}{2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1}-\frac {2\,\left (b^3+a\,b^2\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {{\mathrm {e}}^{-2\,c-2\,d\,x}\,{\left (a+b\right )}^3}{8\,d}-\frac {\frac {2\,\left (b^3+a\,b^2\right )}{d}+\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (b^3+3\,a\,b^2\right )}{3\,d}+\frac {2\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (b^3+a\,b^2\right )}{d}}{3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1}+\frac {x\,{\left (a+b\right )}^2\,\left (a-5\,b\right )}{2} \] Input:
int(cosh(c + d*x)^2*(a + b*tanh(c + d*x)^2)^3,x)
Output:
(exp(2*c + 2*d*x)*(a + b)^3)/(8*d) - ((2*(3*a*b^2 + b^3))/(3*d) + (2*exp(2 *c + 2*d*x)*(a*b^2 + b^3))/d)/(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1) - (2*(a*b^2 + b^3))/(d*(exp(2*c + 2*d*x) + 1)) - (exp(- 2*c - 2*d*x)*(a + b)^3)/(8*d) - ((2*(a*b^2 + b^3))/d + (4*exp(2*c + 2*d*x)*(3*a*b^2 + b^3))/ (3*d) + (2*exp(4*c + 4*d*x)*(a*b^2 + b^3))/d)/(3*exp(2*c + 2*d*x) + 3*exp( 4*c + 4*d*x) + exp(6*c + 6*d*x) + 1) + (x*(a + b)^2*(a - 5*b))/2
Time = 0.25 (sec) , antiderivative size = 606, normalized size of antiderivative = 7.77 \[ \int \cosh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {-180 e^{4 d x +4 c} a \,b^{2}-3 b^{3}+9 e^{10 d x +10 c} a^{2} b +9 e^{10 d x +10 c} a \,b^{2}+21 e^{8 d x +8 c} a^{2} b +69 e^{8 d x +8 c} a \,b^{2}-36 e^{4 d x +4 c} a^{2} b -9 a \,b^{2}-60 e^{4 d x +4 c} b^{3}-129 e^{2 d x +2 c} a \,b^{2}-3 a^{3}-75 e^{2 d x +2 c} b^{3}-108 e^{4 d x +4 c} a^{2} b d x -324 e^{4 d x +4 c} a \,b^{2} d x -36 e^{2 d x +2 c} a^{2} b d x -108 e^{2 d x +2 c} a \,b^{2} d x -33 e^{2 d x +2 c} a^{2} b +12 e^{8 d x +8 c} a^{3} d x -60 e^{8 d x +8 c} b^{3} d x +36 e^{6 d x +6 c} a^{3} d x -180 e^{6 d x +6 c} b^{3} d x -9 a^{2} b +3 e^{10 d x +10 c} a^{3}+3 e^{10 d x +10 c} b^{3}+7 e^{8 d x +8 c} a^{3}+55 e^{8 d x +8 c} b^{3}-12 e^{4 d x +4 c} a^{3}-11 e^{2 d x +2 c} a^{3}+36 e^{4 d x +4 c} a^{3} d x -180 e^{4 d x +4 c} b^{3} d x +12 e^{2 d x +2 c} a^{3} d x -60 e^{2 d x +2 c} b^{3} d x -36 e^{8 d x +8 c} a^{2} b d x -108 e^{8 d x +8 c} a \,b^{2} d x -108 e^{6 d x +6 c} a^{2} b d x -324 e^{6 d x +6 c} a \,b^{2} d x}{24 e^{2 d x +2 c} d \left (e^{6 d x +6 c}+3 e^{4 d x +4 c}+3 e^{2 d x +2 c}+1\right )} \] Input:
int(cosh(d*x+c)^2*(a+b*tanh(d*x+c)^2)^3,x)
Output:
(3*e**(10*c + 10*d*x)*a**3 + 9*e**(10*c + 10*d*x)*a**2*b + 9*e**(10*c + 10 *d*x)*a*b**2 + 3*e**(10*c + 10*d*x)*b**3 + 12*e**(8*c + 8*d*x)*a**3*d*x + 7*e**(8*c + 8*d*x)*a**3 - 36*e**(8*c + 8*d*x)*a**2*b*d*x + 21*e**(8*c + 8* d*x)*a**2*b - 108*e**(8*c + 8*d*x)*a*b**2*d*x + 69*e**(8*c + 8*d*x)*a*b**2 - 60*e**(8*c + 8*d*x)*b**3*d*x + 55*e**(8*c + 8*d*x)*b**3 + 36*e**(6*c + 6*d*x)*a**3*d*x - 108*e**(6*c + 6*d*x)*a**2*b*d*x - 324*e**(6*c + 6*d*x)*a *b**2*d*x - 180*e**(6*c + 6*d*x)*b**3*d*x + 36*e**(4*c + 4*d*x)*a**3*d*x - 12*e**(4*c + 4*d*x)*a**3 - 108*e**(4*c + 4*d*x)*a**2*b*d*x - 36*e**(4*c + 4*d*x)*a**2*b - 324*e**(4*c + 4*d*x)*a*b**2*d*x - 180*e**(4*c + 4*d*x)*a* b**2 - 180*e**(4*c + 4*d*x)*b**3*d*x - 60*e**(4*c + 4*d*x)*b**3 + 12*e**(2 *c + 2*d*x)*a**3*d*x - 11*e**(2*c + 2*d*x)*a**3 - 36*e**(2*c + 2*d*x)*a**2 *b*d*x - 33*e**(2*c + 2*d*x)*a**2*b - 108*e**(2*c + 2*d*x)*a*b**2*d*x - 12 9*e**(2*c + 2*d*x)*a*b**2 - 60*e**(2*c + 2*d*x)*b**3*d*x - 75*e**(2*c + 2* d*x)*b**3 - 3*a**3 - 9*a**2*b - 9*a*b**2 - 3*b**3)/(24*e**(2*c + 2*d*x)*d* (e**(6*c + 6*d*x) + 3*e**(4*c + 4*d*x) + 3*e**(2*c + 2*d*x) + 1))