\(\int \text {sech}^5(c+d x) (a+b \sinh ^2(c+d x)) \, dx\) [255]

Optimal result

Integrand size = 21, antiderivative size = 70 \[ \int \text {sech}^5(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \, dx=\frac {(3 a+b) \arctan (\sinh (c+d x))}{8 d}+\frac {(3 a+b) \text {sech}(c+d x) \tanh (c+d x)}{8 d}+\frac {(a-b) \text {sech}^3(c+d x) \tanh (c+d x)}{4 d} \] Output:

1/8*(3*a+b)*arctan(sinh(d*x+c))/d+1/8*(3*a+b)*sech(d*x+c)*tanh(d*x+c)/d+1/ 
4*(a-b)*sech(d*x+c)^3*tanh(d*x+c)/d
 

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.64 \[ \int \text {sech}^5(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \, dx=\frac {3 a \arctan (\sinh (c+d x))}{8 d}+\frac {b \arctan (\sinh (c+d x))}{8 d}+\frac {3 a \text {sech}(c+d x) \tanh (c+d x)}{8 d}+\frac {b \text {sech}(c+d x) \tanh (c+d x)}{8 d}+\frac {a \text {sech}^3(c+d x) \tanh (c+d x)}{4 d}-\frac {b \text {sech}^3(c+d x) \tanh (c+d x)}{4 d} \] Input:

Integrate[Sech[c + d*x]^5*(a + b*Sinh[c + d*x]^2),x]
 

Output:

(3*a*ArcTan[Sinh[c + d*x]])/(8*d) + (b*ArcTan[Sinh[c + d*x]])/(8*d) + (3*a 
*Sech[c + d*x]*Tanh[c + d*x])/(8*d) + (b*Sech[c + d*x]*Tanh[c + d*x])/(8*d 
) + (a*Sech[c + d*x]^3*Tanh[c + d*x])/(4*d) - (b*Sech[c + d*x]^3*Tanh[c + 
d*x])/(4*d)
 

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.07, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3042, 3669, 298, 215, 216}

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.

\(\displaystyle \int \text {sech}^5(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {a-b \sin (i c+i d x)^2}{\cos (i c+i d x)^5}dx\)

\(\Big \downarrow \) 3669

\(\displaystyle \frac {\int \frac {b \sinh ^2(c+d x)+a}{\left (\sinh ^2(c+d x)+1\right )^3}d\sinh (c+d x)}{d}\)

\(\Big \downarrow \) 298

\(\displaystyle \frac {\frac {1}{4} (3 a+b) \int \frac {1}{\left (\sinh ^2(c+d x)+1\right )^2}d\sinh (c+d x)+\frac {(a-b) \sinh (c+d x)}{4 \left (\sinh ^2(c+d x)+1\right )^2}}{d}\)

\(\Big \downarrow \) 215

\(\displaystyle \frac {\frac {1}{4} (3 a+b) \left (\frac {1}{2} \int \frac {1}{\sinh ^2(c+d x)+1}d\sinh (c+d x)+\frac {\sinh (c+d x)}{2 \left (\sinh ^2(c+d x)+1\right )}\right )+\frac {(a-b) \sinh (c+d x)}{4 \left (\sinh ^2(c+d x)+1\right )^2}}{d}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {\frac {1}{4} (3 a+b) \left (\frac {1}{2} \arctan (\sinh (c+d x))+\frac {\sinh (c+d x)}{2 \left (\sinh ^2(c+d x)+1\right )}\right )+\frac {(a-b) \sinh (c+d x)}{4 \left (\sinh ^2(c+d x)+1\right )^2}}{d}\)

Input:

Int[Sech[c + d*x]^5*(a + b*Sinh[c + d*x]^2),x]
 

Output:

(((a - b)*Sinh[c + d*x])/(4*(1 + Sinh[c + d*x]^2)^2) + ((3*a + b)*(ArcTan[ 
Sinh[c + d*x]]/2 + Sinh[c + d*x]/(2*(1 + Sinh[c + d*x]^2))))/4)/d
 

Defintions of rubi rules used

rule 215

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) 
/(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1))   Int[(a + b*x^2)^(p + 1 
), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 
*p])
 

rule 216

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

rule 298

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[(-( 
b*c - a*d))*x*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] - Simp[(a*d - b*c*( 
2*p + 3))/(2*a*b*(p + 1))   Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, 
 c, d, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/2 + p, 0])
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3669

Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^ 
(p_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff/f   S 
ubst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e + f*x] 
/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
 

Maple [A] (verified)

Time = 33.96 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.41

method	result	size
derivativedivides	\(\frac {a \left (\left (\frac {\operatorname {sech}\left (d x +c \right )^{3}}{4}+\frac {3 \,\operatorname {sech}\left (d x +c \right )}{8}\right ) \tanh \left (d x +c \right )+\frac {3 \arctan \left ({\mathrm e}^{d x +c}\right )}{4}\right )+b \left (-\frac {\sinh \left (d x +c \right )}{3 \cosh \left (d x +c \right )^{4}}+\frac {\left (\frac {\operatorname {sech}\left (d x +c \right )^{3}}{4}+\frac {3 \,\operatorname {sech}\left (d x +c \right )}{8}\right ) \tanh \left (d x +c \right )}{3}+\frac {\arctan \left ({\mathrm e}^{d x +c}\right )}{4}\right )}{d}\)	\(99\)
default	\(\frac {a \left (\left (\frac {\operatorname {sech}\left (d x +c \right )^{3}}{4}+\frac {3 \,\operatorname {sech}\left (d x +c \right )}{8}\right ) \tanh \left (d x +c \right )+\frac {3 \arctan \left ({\mathrm e}^{d x +c}\right )}{4}\right )+b \left (-\frac {\sinh \left (d x +c \right )}{3 \cosh \left (d x +c \right )^{4}}+\frac {\left (\frac {\operatorname {sech}\left (d x +c \right )^{3}}{4}+\frac {3 \,\operatorname {sech}\left (d x +c \right )}{8}\right ) \tanh \left (d x +c \right )}{3}+\frac {\arctan \left ({\mathrm e}^{d x +c}\right )}{4}\right )}{d}\)	\(99\)
risch	\(\frac {{\mathrm e}^{d x +c} \left (3 \,{\mathrm e}^{6 d x +6 c} a +{\mathrm e}^{6 d x +6 c} b +11 \,{\mathrm e}^{4 d x +4 c} a -7 b \,{\mathrm e}^{4 d x +4 c}-11 \,{\mathrm e}^{2 d x +2 c} a +7 \,{\mathrm e}^{2 d x +2 c} b -3 a -b \right )}{4 d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{4}}+\frac {3 i \ln \left ({\mathrm e}^{d x +c}+i\right ) a}{8 d}+\frac {i \ln \left ({\mathrm e}^{d x +c}+i\right ) b}{8 d}-\frac {3 i \ln \left ({\mathrm e}^{d x +c}-i\right ) a}{8 d}-\frac {i \ln \left ({\mathrm e}^{d x +c}-i\right ) b}{8 d}\)	\(172\)

Input:

int(sech(d*x+c)^5*(a+b*sinh(d*x+c)^2),x,method=_RETURNVERBOSE)
 

Output:

1/d*(a*((1/4*sech(d*x+c)^3+3/8*sech(d*x+c))*tanh(d*x+c)+3/4*arctan(exp(d*x 
+c)))+b*(-1/3*sinh(d*x+c)/cosh(d*x+c)^4+1/3*(1/4*sech(d*x+c)^3+3/8*sech(d* 
x+c))*tanh(d*x+c)+1/4*arctan(exp(d*x+c))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1046 vs. \(2 (64) = 128\).

Time = 0.12 (sec) , antiderivative size = 1046, normalized size of antiderivative = 14.94 \[ \int \text {sech}^5(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \, dx=\text {Too large to display} \] Input:

integrate(sech(d*x+c)^5*(a+b*sinh(d*x+c)^2),x, algorithm="fricas")
 

Output:

1/4*((3*a + b)*cosh(d*x + c)^7 + 7*(3*a + b)*cosh(d*x + c)*sinh(d*x + c)^6 
 + (3*a + b)*sinh(d*x + c)^7 + (11*a - 7*b)*cosh(d*x + c)^5 + (21*(3*a + b 
)*cosh(d*x + c)^2 + 11*a - 7*b)*sinh(d*x + c)^5 + 5*(7*(3*a + b)*cosh(d*x 
+ c)^3 + (11*a - 7*b)*cosh(d*x + c))*sinh(d*x + c)^4 - (11*a - 7*b)*cosh(d 
*x + c)^3 + (35*(3*a + b)*cosh(d*x + c)^4 + 10*(11*a - 7*b)*cosh(d*x + c)^ 
2 - 11*a + 7*b)*sinh(d*x + c)^3 + (21*(3*a + b)*cosh(d*x + c)^5 + 10*(11*a 
 - 7*b)*cosh(d*x + c)^3 - 3*(11*a - 7*b)*cosh(d*x + c))*sinh(d*x + c)^2 + 
((3*a + b)*cosh(d*x + c)^8 + 8*(3*a + b)*cosh(d*x + c)*sinh(d*x + c)^7 + ( 
3*a + b)*sinh(d*x + c)^8 + 4*(3*a + b)*cosh(d*x + c)^6 + 4*(7*(3*a + b)*co 
sh(d*x + c)^2 + 3*a + b)*sinh(d*x + c)^6 + 8*(7*(3*a + b)*cosh(d*x + c)^3 
+ 3*(3*a + b)*cosh(d*x + c))*sinh(d*x + c)^5 + 6*(3*a + b)*cosh(d*x + c)^4 
 + 2*(35*(3*a + b)*cosh(d*x + c)^4 + 30*(3*a + b)*cosh(d*x + c)^2 + 9*a + 
3*b)*sinh(d*x + c)^4 + 8*(7*(3*a + b)*cosh(d*x + c)^5 + 10*(3*a + b)*cosh( 
d*x + c)^3 + 3*(3*a + b)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(3*a + b)*cosh 
(d*x + c)^2 + 4*(7*(3*a + b)*cosh(d*x + c)^6 + 15*(3*a + b)*cosh(d*x + c)^ 
4 + 9*(3*a + b)*cosh(d*x + c)^2 + 3*a + b)*sinh(d*x + c)^2 + 8*((3*a + b)* 
cosh(d*x + c)^7 + 3*(3*a + b)*cosh(d*x + c)^5 + 3*(3*a + b)*cosh(d*x + c)^ 
3 + (3*a + b)*cosh(d*x + c))*sinh(d*x + c) + 3*a + b)*arctan(cosh(d*x + c) 
 + sinh(d*x + c)) - (3*a + b)*cosh(d*x + c) + (7*(3*a + b)*cosh(d*x + c)^6 
 + 5*(11*a - 7*b)*cosh(d*x + c)^4 - 3*(11*a - 7*b)*cosh(d*x + c)^2 - 3*...
 

Sympy [F]

\[ \int \text {sech}^5(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \, dx=\int \left (a + b \sinh ^{2}{\left (c + d x \right )}\right ) \operatorname {sech}^{5}{\left (c + d x \right )}\, dx \] Input:

integrate(sech(d*x+c)**5*(a+b*sinh(d*x+c)**2),x)
 

Output:

Integral((a + b*sinh(c + d*x)**2)*sech(c + d*x)**5, x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 228 vs. \(2 (64) = 128\).

Time = 0.13 (sec) , antiderivative size = 228, normalized size of antiderivative = 3.26 \[ \int \text {sech}^5(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \, dx=-\frac {1}{4} \, a {\left (\frac {3 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {3 \, e^{\left (-d x - c\right )} + 11 \, e^{\left (-3 \, d x - 3 \, c\right )} - 11 \, e^{\left (-5 \, d x - 5 \, c\right )} - 3 \, e^{\left (-7 \, d x - 7 \, c\right )}}{d {\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} + 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-8 \, d x - 8 \, c\right )} + 1\right )}}\right )} - \frac {1}{4} \, b {\left (\frac {\arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {e^{\left (-d x - c\right )} - 7 \, e^{\left (-3 \, d x - 3 \, c\right )} + 7 \, e^{\left (-5 \, d x - 5 \, c\right )} - e^{\left (-7 \, d x - 7 \, c\right )}}{d {\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} + 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-8 \, d x - 8 \, c\right )} + 1\right )}}\right )} \] Input:

integrate(sech(d*x+c)^5*(a+b*sinh(d*x+c)^2),x, algorithm="maxima")
 

Output:

-1/4*a*(3*arctan(e^(-d*x - c))/d - (3*e^(-d*x - c) + 11*e^(-3*d*x - 3*c) - 
 11*e^(-5*d*x - 5*c) - 3*e^(-7*d*x - 7*c))/(d*(4*e^(-2*d*x - 2*c) + 6*e^(- 
4*d*x - 4*c) + 4*e^(-6*d*x - 6*c) + e^(-8*d*x - 8*c) + 1))) - 1/4*b*(arcta 
n(e^(-d*x - c))/d - (e^(-d*x - c) - 7*e^(-3*d*x - 3*c) + 7*e^(-5*d*x - 5*c 
) - e^(-7*d*x - 7*c))/(d*(4*e^(-2*d*x - 2*c) + 6*e^(-4*d*x - 4*c) + 4*e^(- 
6*d*x - 6*c) + e^(-8*d*x - 8*c) + 1)))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 153 vs. \(2 (64) = 128\).

Time = 0.14 (sec) , antiderivative size = 153, normalized size of antiderivative = 2.19 \[ \int \text {sech}^5(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \, dx=\frac {{\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} e^{\left (-d x - c\right )}\right )\right )} {\left (3 \, a + b\right )} + \frac {4 \, {\left (3 \, a {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 20 \, a {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 4 \, b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}\right )}}{{\left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )}^{2}}}{16 \, d} \] Input:

integrate(sech(d*x+c)^5*(a+b*sinh(d*x+c)^2),x, algorithm="giac")
 

Output:

1/16*((pi + 2*arctan(1/2*(e^(2*d*x + 2*c) - 1)*e^(-d*x - c)))*(3*a + b) + 
4*(3*a*(e^(d*x + c) - e^(-d*x - c))^3 + b*(e^(d*x + c) - e^(-d*x - c))^3 + 
 20*a*(e^(d*x + c) - e^(-d*x - c)) - 4*b*(e^(d*x + c) - e^(-d*x - c)))/((e 
^(d*x + c) - e^(-d*x - c))^2 + 4)^2)/d
 

Mupad [B] (verification not implemented)

Time = 1.99 (sec) , antiderivative size = 280, normalized size of antiderivative = 4.00 \[ \int \text {sech}^5(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \, dx=\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (3\,a\,\sqrt {d^2}+b\,\sqrt {d^2}\right )}{d\,\sqrt {9\,a^2+6\,a\,b+b^2}}\right )\,\sqrt {9\,a^2+6\,a\,b+b^2}}{4\,\sqrt {d^2}}-\frac {\frac {b\,{\mathrm {e}}^{5\,c+5\,d\,x}}{d}+\frac {2\,{\mathrm {e}}^{3\,c+3\,d\,x}\,\left (2\,a-b\right )}{d}+\frac {b\,{\mathrm {e}}^{c+d\,x}}{d}}{4\,{\mathrm {e}}^{2\,c+2\,d\,x}+6\,{\mathrm {e}}^{4\,c+4\,d\,x}+4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (3\,a+b\right )}{4\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (a-3\,b\right )}{2\,d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}-\frac {2\,{\mathrm {e}}^{c+d\,x}\,\left (a-b\right )}{d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1\right )} \] Input:

int((a + b*sinh(c + d*x)^2)/cosh(c + d*x)^5,x)
 

Output:

(atan((exp(d*x)*exp(c)*(3*a*(d^2)^(1/2) + b*(d^2)^(1/2)))/(d*(6*a*b + 9*a^ 
2 + b^2)^(1/2)))*(6*a*b + 9*a^2 + b^2)^(1/2))/(4*(d^2)^(1/2)) - ((b*exp(5* 
c + 5*d*x))/d + (2*exp(3*c + 3*d*x)*(2*a - b))/d + (b*exp(c + d*x))/d)/(4* 
exp(2*c + 2*d*x) + 6*exp(4*c + 4*d*x) + 4*exp(6*c + 6*d*x) + exp(8*c + 8*d 
*x) + 1) + (exp(c + d*x)*(3*a + b))/(4*d*(exp(2*c + 2*d*x) + 1)) + (exp(c 
+ d*x)*(a - 3*b))/(2*d*(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1)) - (2*e 
xp(c + d*x)*(a - b))/(d*(3*exp(2*c + 2*d*x) + 3*exp(4*c + 4*d*x) + exp(6*c 
 + 6*d*x) + 1))
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 341, normalized size of antiderivative = 4.87 \[ \int \text {sech}^5(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \, dx=\frac {3 e^{8 d x +8 c} \mathit {atan} \left (e^{d x +c}\right ) a +e^{8 d x +8 c} \mathit {atan} \left (e^{d x +c}\right ) b +12 e^{6 d x +6 c} \mathit {atan} \left (e^{d x +c}\right ) a +4 e^{6 d x +6 c} \mathit {atan} \left (e^{d x +c}\right ) b +18 e^{4 d x +4 c} \mathit {atan} \left (e^{d x +c}\right ) a +6 e^{4 d x +4 c} \mathit {atan} \left (e^{d x +c}\right ) b +12 e^{2 d x +2 c} \mathit {atan} \left (e^{d x +c}\right ) a +4 e^{2 d x +2 c} \mathit {atan} \left (e^{d x +c}\right ) b +3 \mathit {atan} \left (e^{d x +c}\right ) a +\mathit {atan} \left (e^{d x +c}\right ) b +3 e^{7 d x +7 c} a +e^{7 d x +7 c} b +11 e^{5 d x +5 c} a -7 e^{5 d x +5 c} b -11 e^{3 d x +3 c} a +7 e^{3 d x +3 c} b -3 e^{d x +c} a -e^{d x +c} b}{4 d \left (e^{8 d x +8 c}+4 e^{6 d x +6 c}+6 e^{4 d x +4 c}+4 e^{2 d x +2 c}+1\right )} \] Input:

int(sech(d*x+c)^5*(a+b*sinh(d*x+c)^2),x)
 

Output:

(3*e**(8*c + 8*d*x)*atan(e**(c + d*x))*a + e**(8*c + 8*d*x)*atan(e**(c + d 
*x))*b + 12*e**(6*c + 6*d*x)*atan(e**(c + d*x))*a + 4*e**(6*c + 6*d*x)*ata 
n(e**(c + d*x))*b + 18*e**(4*c + 4*d*x)*atan(e**(c + d*x))*a + 6*e**(4*c + 
 4*d*x)*atan(e**(c + d*x))*b + 12*e**(2*c + 2*d*x)*atan(e**(c + d*x))*a + 
4*e**(2*c + 2*d*x)*atan(e**(c + d*x))*b + 3*atan(e**(c + d*x))*a + atan(e* 
*(c + d*x))*b + 3*e**(7*c + 7*d*x)*a + e**(7*c + 7*d*x)*b + 11*e**(5*c + 5 
*d*x)*a - 7*e**(5*c + 5*d*x)*b - 11*e**(3*c + 3*d*x)*a + 7*e**(3*c + 3*d*x 
)*b - 3*e**(c + d*x)*a - e**(c + d*x)*b)/(4*d*(e**(8*c + 8*d*x) + 4*e**(6* 
c + 6*d*x) + 6*e**(4*c + 4*d*x) + 4*e**(2*c + 2*d*x) + 1))