Integrand size = 21, antiderivative size = 110 \[ \int \sinh ^4(c+d x) \left (a+b \tanh ^3(c+d x)\right ) \, dx=\frac {3 a x}{8}-\frac {3 b \cosh ^2(c+d x)}{2 d}+\frac {b \cosh ^4(c+d x)}{4 d}+\frac {3 b \log (\cosh (c+d x))}{d}-\frac {3 a \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac {a \cosh (c+d x) \sinh ^3(c+d x)}{4 d}-\frac {b \tanh ^2(c+d x)}{2 d} \] Output:
3/8*a*x-3/2*b*cosh(d*x+c)^2/d+1/4*b*cosh(d*x+c)^4/d+3*b*ln(cosh(d*x+c))/d- 3/8*a*cosh(d*x+c)*sinh(d*x+c)/d+1/4*a*cosh(d*x+c)*sinh(d*x+c)^3/d-1/2*b*ta nh(d*x+c)^2/d
Time = 0.13 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.84 \[ \int \sinh ^4(c+d x) \left (a+b \tanh ^3(c+d x)\right ) \, dx=\frac {3 a (c+d x)}{8 d}+\frac {b \left (12 \log (\cosh (c+d x))+2 \text {sech}^2(c+d x)-4 \sinh ^2(c+d x)+\sinh ^4(c+d x)\right )}{4 d}-\frac {a \sinh (2 (c+d x))}{4 d}+\frac {a \sinh (4 (c+d x))}{32 d} \] Input:
Integrate[Sinh[c + d*x]^4*(a + b*Tanh[c + d*x]^3),x]
Output:
(3*a*(c + d*x))/(8*d) + (b*(12*Log[Cosh[c + d*x]] + 2*Sech[c + d*x]^2 - 4* Sinh[c + d*x]^2 + Sinh[c + d*x]^4))/(4*d) - (a*Sinh[2*(c + d*x)])/(4*d) + (a*Sinh[4*(c + d*x)])/(32*d)
Time = 0.75 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.21, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3042, 4146, 2335, 25, 2335, 27, 523, 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.
| \(\displaystyle \int \sinh ^4(c+d x) \left (a+b \tanh ^3(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin (i c+i d x)^4 \left (a+i b \tan (i c+i d x)^3\right )dx\) |
| \(\Big \downarrow \) 4146 |
| \(\displaystyle \frac {\int \frac {\tanh ^4(c+d x) \left (b \tanh ^3(c+d x)+a\right )}{\left (1-\tanh ^2(c+d x)\right )^3}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 2335 |
| \(\displaystyle \frac {\frac {1}{4} \int -\frac {\tanh ^3(c+d x) \left (4 b \tanh ^2(c+d x)+a \tanh (c+d x)+4 b\right )}{\left (1-\tanh ^2(c+d x)\right )^2}d\tanh (c+d x)+\frac {\tanh ^4(c+d x) (a \tanh (c+d x)+b)}{4 \left (1-\tanh ^2(c+d x)\right )^2}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\tanh ^4(c+d x) (a \tanh (c+d x)+b)}{4 \left (1-\tanh ^2(c+d x)\right )^2}-\frac {1}{4} \int \frac {\tanh ^3(c+d x) \left (4 b \tanh ^2(c+d x)+a \tanh (c+d x)+4 b\right )}{\left (1-\tanh ^2(c+d x)\right )^2}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 2335 |
| \(\displaystyle \frac {\frac {1}{4} \left (-\frac {1}{2} \int -\frac {3 \tanh ^2(c+d x) (a+8 b \tanh (c+d x))}{1-\tanh ^2(c+d x)}d\tanh (c+d x)-\frac {\tanh ^3(c+d x) (a+8 b \tanh (c+d x))}{2 \left (1-\tanh ^2(c+d x)\right )}\right )+\frac {\tanh ^4(c+d x) (a \tanh (c+d x)+b)}{4 \left (1-\tanh ^2(c+d x)\right )^2}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {3}{2} \int \frac {\tanh ^2(c+d x) (a+8 b \tanh (c+d x))}{1-\tanh ^2(c+d x)}d\tanh (c+d x)-\frac {\tanh ^3(c+d x) (a+8 b \tanh (c+d x))}{2 \left (1-\tanh ^2(c+d x)\right )}\right )+\frac {\tanh ^4(c+d x) (a \tanh (c+d x)+b)}{4 \left (1-\tanh ^2(c+d x)\right )^2}}{d}\) |
| \(\Big \downarrow \) 523 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {3}{2} \int \left (-a-8 b \tanh (c+d x)+\frac {a+8 b \tanh (c+d x)}{1-\tanh ^2(c+d x)}\right )d\tanh (c+d x)-\frac {\tanh ^3(c+d x) (a+8 b \tanh (c+d x))}{2 \left (1-\tanh ^2(c+d x)\right )}\right )+\frac {\tanh ^4(c+d x) (a \tanh (c+d x)+b)}{4 \left (1-\tanh ^2(c+d x)\right )^2}}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {3}{2} \left (a \text {arctanh}(\tanh (c+d x))-a \tanh (c+d x)-4 b \tanh ^2(c+d x)-4 b \log \left (1-\tanh ^2(c+d x)\right )\right )-\frac {\tanh ^3(c+d x) (a+8 b \tanh (c+d x))}{2 \left (1-\tanh ^2(c+d x)\right )}\right )+\frac {\tanh ^4(c+d x) (a \tanh (c+d x)+b)}{4 \left (1-\tanh ^2(c+d x)\right )^2}}{d}\) |
Input:
Int[Sinh[c + d*x]^4*(a + b*Tanh[c + d*x]^3),x]
Output:
((Tanh[c + d*x]^4*(b + a*Tanh[c + d*x]))/(4*(1 - Tanh[c + d*x]^2)^2) + (-1 /2*(Tanh[c + d*x]^3*(a + 8*b*Tanh[c + d*x]))/(1 - Tanh[c + d*x]^2) + (3*(a *ArcTanh[Tanh[c + d*x]] - 4*b*Log[1 - Tanh[c + d*x]^2] - a*Tanh[c + d*x] - 4*b*Tanh[c + d*x]^2))/2)/4)/d
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((x_)^(m_.)*((c_) + (d_.)*(x_)))/((a_) + (b_.)*(x_)^2), x_Symbol] :> In t[ExpandIntegrand[x^m*((c + d*x)/(a + b*x^2)), x], x] /; FreeQ[{a, b, c, d} , x] && IntegerQ[m]
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ {Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq , a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(c*x)^m*(a + b*x^2)^(p + 1)*((a*g - b*f*x)/(2*a*b*(p + 1))), x] + Simp[c/(2*a*b*(p + 1)) Int[(c*x)^(m - 1)*(a + b*x^2)^(p + 1)*ExpandToSu m[2*a*b*(p + 1)*x*Q - a*g*m + b*f*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && GtQ[m, 0]
Int[sin[(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[c*(ff^(m + 1)/f) Subst[Int[x^m*((a + b*(ff*x)^n)^p/(c^2 + ff^2*x^2)^(m/ 2 + 1)), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, e, f, n, p}, x ] && IntegerQ[m/2]
Time = 4.11 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.91
| method | result | size |
| derivativedivides | \(\frac {a \left (\left (\frac {\sinh \left (d x +c \right )^{3}}{4}-\frac {3 \sinh \left (d x +c \right )}{8}\right ) \cosh \left (d x +c \right )+\frac {3 d x}{8}+\frac {3 c}{8}\right )+b \left (\frac {\sinh \left (d x +c \right )^{6}}{4 \cosh \left (d x +c \right )^{2}}-\frac {3 \sinh \left (d x +c \right )^{4}}{4 \cosh \left (d x +c \right )^{2}}+3 \ln \left (\cosh \left (d x +c \right )\right )-\frac {3 \tanh \left (d x +c \right )^{2}}{2}\right )}{d}\) | \(100\) |
| default | \(\frac {a \left (\left (\frac {\sinh \left (d x +c \right )^{3}}{4}-\frac {3 \sinh \left (d x +c \right )}{8}\right ) \cosh \left (d x +c \right )+\frac {3 d x}{8}+\frac {3 c}{8}\right )+b \left (\frac {\sinh \left (d x +c \right )^{6}}{4 \cosh \left (d x +c \right )^{2}}-\frac {3 \sinh \left (d x +c \right )^{4}}{4 \cosh \left (d x +c \right )^{2}}+3 \ln \left (\cosh \left (d x +c \right )\right )-\frac {3 \tanh \left (d x +c \right )^{2}}{2}\right )}{d}\) | \(100\) |
| risch | \(\frac {3 a x}{8}-3 b x +\frac {{\mathrm e}^{4 d x +4 c} a}{64 d}+\frac {{\mathrm e}^{4 d x +4 c} b}{64 d}-\frac {{\mathrm e}^{2 d x +2 c} a}{8 d}-\frac {5 \,{\mathrm e}^{2 d x +2 c} b}{16 d}+\frac {{\mathrm e}^{-2 d x -2 c} a}{8 d}-\frac {5 \,{\mathrm e}^{-2 d x -2 c} b}{16 d}-\frac {{\mathrm e}^{-4 d x -4 c} a}{64 d}+\frac {{\mathrm e}^{-4 d x -4 c} b}{64 d}-\frac {6 b c}{d}+\frac {2 b \,{\mathrm e}^{2 d x +2 c}}{d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{2}}+\frac {3 b \ln \left ({\mathrm e}^{2 d x +2 c}+1\right )}{d}\) | \(183\) |
Input:
int(sinh(d*x+c)^4*(a+b*tanh(d*x+c)^3),x,method=_RETURNVERBOSE)
Output:
1/d*(a*((1/4*sinh(d*x+c)^3-3/8*sinh(d*x+c))*cosh(d*x+c)+3/8*d*x+3/8*c)+b*( 1/4*sinh(d*x+c)^6/cosh(d*x+c)^2-3/4*sinh(d*x+c)^4/cosh(d*x+c)^2+3*ln(cosh( d*x+c))-3/2*tanh(d*x+c)^2))
Leaf count of result is larger than twice the leaf count of optimal. 1530 vs. \(2 (98) = 196\).
Time = 0.10 (sec) , antiderivative size = 1530, normalized size of antiderivative = 13.91 \[ \int \sinh ^4(c+d x) \left (a+b \tanh ^3(c+d x)\right ) \, dx=\text {Too large to display} \] Input:
integrate(sinh(d*x+c)^4*(a+b*tanh(d*x+c)^3),x, algorithm="fricas")
Output:
1/64*((a + b)*cosh(d*x + c)^12 + 12*(a + b)*cosh(d*x + c)*sinh(d*x + c)^11 + (a + b)*sinh(d*x + c)^12 - 6*(a + 3*b)*cosh(d*x + c)^10 + 6*(11*(a + b) *cosh(d*x + c)^2 - a - 3*b)*sinh(d*x + c)^10 + 20*(11*(a + b)*cosh(d*x + c )^3 - 3*(a + 3*b)*cosh(d*x + c))*sinh(d*x + c)^9 + 3*(8*(a - 8*b)*d*x - 5* a - 13*b)*cosh(d*x + c)^8 + 3*(165*(a + b)*cosh(d*x + c)^4 + 8*(a - 8*b)*d *x - 90*(a + 3*b)*cosh(d*x + c)^2 - 5*a - 13*b)*sinh(d*x + c)^8 + 24*(33*( a + b)*cosh(d*x + c)^5 - 30*(a + 3*b)*cosh(d*x + c)^3 + (8*(a - 8*b)*d*x - 5*a - 13*b)*cosh(d*x + c))*sinh(d*x + c)^7 + 8*(6*(a - 8*b)*d*x + 11*b)*c osh(d*x + c)^6 + 4*(231*(a + b)*cosh(d*x + c)^6 - 315*(a + 3*b)*cosh(d*x + c)^4 + 12*(a - 8*b)*d*x + 21*(8*(a - 8*b)*d*x - 5*a - 13*b)*cosh(d*x + c) ^2 + 22*b)*sinh(d*x + c)^6 + 24*(33*(a + b)*cosh(d*x + c)^7 - 63*(a + 3*b) *cosh(d*x + c)^5 + 7*(8*(a - 8*b)*d*x - 5*a - 13*b)*cosh(d*x + c)^3 + 2*(6 *(a - 8*b)*d*x + 11*b)*cosh(d*x + c))*sinh(d*x + c)^5 + 3*(8*(a - 8*b)*d*x + 5*a - 13*b)*cosh(d*x + c)^4 + 3*(165*(a + b)*cosh(d*x + c)^8 - 420*(a + 3*b)*cosh(d*x + c)^6 + 70*(8*(a - 8*b)*d*x - 5*a - 13*b)*cosh(d*x + c)^4 + 8*(a - 8*b)*d*x + 40*(6*(a - 8*b)*d*x + 11*b)*cosh(d*x + c)^2 + 5*a - 13 *b)*sinh(d*x + c)^4 + 4*(55*(a + b)*cosh(d*x + c)^9 - 180*(a + 3*b)*cosh(d *x + c)^7 + 42*(8*(a - 8*b)*d*x - 5*a - 13*b)*cosh(d*x + c)^5 + 40*(6*(a - 8*b)*d*x + 11*b)*cosh(d*x + c)^3 + 3*(8*(a - 8*b)*d*x + 5*a - 13*b)*cosh( d*x + c))*sinh(d*x + c)^3 + 6*(a - 3*b)*cosh(d*x + c)^2 + 6*(11*(a + b)...
\[ \int \sinh ^4(c+d x) \left (a+b \tanh ^3(c+d x)\right ) \, dx=\int \left (a + b \tanh ^{3}{\left (c + d x \right )}\right ) \sinh ^{4}{\left (c + d x \right )}\, dx \] Input:
integrate(sinh(d*x+c)**4*(a+b*tanh(d*x+c)**3),x)
Output:
Integral((a + b*tanh(c + d*x)**3)*sinh(c + d*x)**4, x)
Time = 0.12 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.76 \[ \int \sinh ^4(c+d x) \left (a+b \tanh ^3(c+d x)\right ) \, dx=\frac {1}{64} \, a {\left (24 \, x + \frac {e^{\left (4 \, d x + 4 \, c\right )}}{d} - \frac {8 \, e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {8 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac {e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} + \frac {1}{64} \, b {\left (\frac {192 \, {\left (d x + c\right )}}{d} - \frac {20 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )}}{d} + \frac {192 \, \log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{d} - \frac {18 \, e^{\left (-2 \, d x - 2 \, c\right )} + 39 \, e^{\left (-4 \, d x - 4 \, c\right )} - 108 \, e^{\left (-6 \, d x - 6 \, c\right )} - 1}{d {\left (e^{\left (-4 \, d x - 4 \, c\right )} + 2 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-8 \, d x - 8 \, c\right )}\right )}}\right )} \] Input:
integrate(sinh(d*x+c)^4*(a+b*tanh(d*x+c)^3),x, algorithm="maxima")
Output:
1/64*a*(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) + 1/64*b*(192*(d*x + c)/d - (20*e^(-2*d*x - 2*c) - e^(-4*d*x - 4*c))/d + 192*log(e^(-2*d*x - 2*c) + 1)/d - (18*e^(-2*d*x - 2*c) + 39*e^(-4*d*x - 4*c) - 108*e^(-6*d*x - 6*c) - 1)/(d*(e^(-4*d*x - 4* c) + 2*e^(-6*d*x - 6*c) + e^(-8*d*x - 8*c))))
Leaf count of result is larger than twice the leaf count of optimal. 203 vs. \(2 (98) = 196\).
Time = 0.16 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.85 \[ \int \sinh ^4(c+d x) \left (a+b \tanh ^3(c+d x)\right ) \, dx=\frac {24 \, {\left (d x + c\right )} {\left (a - 8 \, b\right )} + a e^{\left (4 \, d x + 4 \, c\right )} + b e^{\left (4 \, d x + 4 \, c\right )} - 8 \, a e^{\left (2 \, d x + 2 \, c\right )} - 20 \, b e^{\left (2 \, d x + 2 \, c\right )} + 192 \, b \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right ) - \frac {9 \, a e^{\left (8 \, d x + 8 \, c\right )} + 72 \, b e^{\left (8 \, d x + 8 \, c\right )} + 10 \, a e^{\left (6 \, d x + 6 \, c\right )} + 36 \, b e^{\left (6 \, d x + 6 \, c\right )} - 6 \, a e^{\left (4 \, d x + 4 \, c\right )} + 111 \, b e^{\left (4 \, d x + 4 \, c\right )} - 6 \, a e^{\left (2 \, d x + 2 \, c\right )} + 18 \, b e^{\left (2 \, d x + 2 \, c\right )} + a - b}{{\left (e^{\left (4 \, d x + 4 \, c\right )} + e^{\left (2 \, d x + 2 \, c\right )}\right )}^{2}}}{64 \, d} \] Input:
integrate(sinh(d*x+c)^4*(a+b*tanh(d*x+c)^3),x, algorithm="giac")
Output:
1/64*(24*(d*x + c)*(a - 8*b) + a*e^(4*d*x + 4*c) + b*e^(4*d*x + 4*c) - 8*a *e^(2*d*x + 2*c) - 20*b*e^(2*d*x + 2*c) + 192*b*log(e^(2*d*x + 2*c) + 1) - (9*a*e^(8*d*x + 8*c) + 72*b*e^(8*d*x + 8*c) + 10*a*e^(6*d*x + 6*c) + 36*b *e^(6*d*x + 6*c) - 6*a*e^(4*d*x + 4*c) + 111*b*e^(4*d*x + 4*c) - 6*a*e^(2* d*x + 2*c) + 18*b*e^(2*d*x + 2*c) + a - b)/(e^(4*d*x + 4*c) + e^(2*d*x + 2 *c))^2)/d
Time = 0.30 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.42 \[ \int \sinh ^4(c+d x) \left (a+b \tanh ^3(c+d x)\right ) \, dx=x\,\left (\frac {3\,a}{8}-3\,b\right )+\frac {2\,b}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}+\frac {{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (a+b\right )}{64\,d}-\frac {2\,b}{d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}-\frac {{\mathrm {e}}^{-4\,c-4\,d\,x}\,\left (a-b\right )}{64\,d}+\frac {3\,b\,\ln \left ({\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+1\right )}{d}+\frac {{\mathrm {e}}^{-2\,c-2\,d\,x}\,\left (2\,a-5\,b\right )}{16\,d}-\frac {{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (2\,a+5\,b\right )}{16\,d} \] Input:
int(sinh(c + d*x)^4*(a + b*tanh(c + d*x)^3),x)
Output:
x*((3*a)/8 - 3*b) + (2*b)/(d*(exp(2*c + 2*d*x) + 1)) + (exp(4*c + 4*d*x)*( a + b))/(64*d) - (2*b)/(d*(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1)) - ( exp(- 4*c - 4*d*x)*(a - b))/(64*d) + (3*b*log(exp(2*c)*exp(2*d*x) + 1))/d + (exp(- 2*c - 2*d*x)*(2*a - 5*b))/(16*d) - (exp(2*c + 2*d*x)*(2*a + 5*b)) /(16*d)
Time = 0.23 (sec) , antiderivative size = 344, normalized size of antiderivative = 3.13 \[ \int \sinh ^4(c+d x) \left (a+b \tanh ^3(c+d x)\right ) \, dx=\frac {e^{12 d x +12 c} a +e^{12 d x +12 c} b -6 e^{10 d x +10 c} a -18 e^{10 d x +10 c} b +192 e^{8 d x +8 c} \mathrm {log}\left (e^{2 d x +2 c}+1\right ) b +24 e^{8 d x +8 c} a d x -15 e^{8 d x +8 c} a -192 e^{8 d x +8 c} b d x -83 e^{8 d x +8 c} b +384 e^{6 d x +6 c} \mathrm {log}\left (e^{2 d x +2 c}+1\right ) b +48 e^{6 d x +6 c} a d x -384 e^{6 d x +6 c} b d x +192 e^{4 d x +4 c} \mathrm {log}\left (e^{2 d x +2 c}+1\right ) b +24 e^{4 d x +4 c} a d x +15 e^{4 d x +4 c} a -192 e^{4 d x +4 c} b d x -83 e^{4 d x +4 c} b +6 e^{2 d x +2 c} a -18 e^{2 d x +2 c} b -a +b}{64 e^{4 d x +4 c} d \left (e^{4 d x +4 c}+2 e^{2 d x +2 c}+1\right )} \] Input:
int(sinh(d*x+c)^4*(a+b*tanh(d*x+c)^3),x)
Output:
(e**(12*c + 12*d*x)*a + e**(12*c + 12*d*x)*b - 6*e**(10*c + 10*d*x)*a - 18 *e**(10*c + 10*d*x)*b + 192*e**(8*c + 8*d*x)*log(e**(2*c + 2*d*x) + 1)*b + 24*e**(8*c + 8*d*x)*a*d*x - 15*e**(8*c + 8*d*x)*a - 192*e**(8*c + 8*d*x)* b*d*x - 83*e**(8*c + 8*d*x)*b + 384*e**(6*c + 6*d*x)*log(e**(2*c + 2*d*x) + 1)*b + 48*e**(6*c + 6*d*x)*a*d*x - 384*e**(6*c + 6*d*x)*b*d*x + 192*e**( 4*c + 4*d*x)*log(e**(2*c + 2*d*x) + 1)*b + 24*e**(4*c + 4*d*x)*a*d*x + 15* e**(4*c + 4*d*x)*a - 192*e**(4*c + 4*d*x)*b*d*x - 83*e**(4*c + 4*d*x)*b + 6*e**(2*c + 2*d*x)*a - 18*e**(2*c + 2*d*x)*b - a + b)/(64*e**(4*c + 4*d*x) *d*(e**(4*c + 4*d*x) + 2*e**(2*c + 2*d*x) + 1))