\(\int \text {csch}^3(c+d x) (a+b \text {sech}^2(c+d x)) \, dx\) [7]

Optimal result

Integrand size = 21, antiderivative size = 54 \[ \int \text {csch}^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right ) \, dx=\frac {(a+3 b) \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {(a+b) \coth (c+d x) \text {csch}(c+d x)}{2 d}-\frac {b \text {sech}(c+d x)}{d} \] Output:

1/2*(a+3*b)*arctanh(cosh(d*x+c))/d-1/2*(a+b)*coth(d*x+c)*csch(d*x+c)/d-b*s 
ech(d*x+c)/d
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(169\) vs. \(2(54)=108\).

Time = 0.26 (sec) , antiderivative size = 169, normalized size of antiderivative = 3.13 \[ \int \text {csch}^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right ) \, dx=-\frac {a \text {csch}^2\left (\frac {1}{2} (c+d x)\right )}{8 d}-\frac {b \text {csch}^2\left (\frac {1}{2} (c+d x)\right )}{8 d}+\frac {a \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}+\frac {3 b \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}-\frac {a \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}-\frac {3 b \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}-\frac {a \text {sech}^2\left (\frac {1}{2} (c+d x)\right )}{8 d}-\frac {b \text {sech}^2\left (\frac {1}{2} (c+d x)\right )}{8 d}-\frac {b \text {sech}(c+d x)}{d} \] Input:

Integrate[Csch[c + d*x]^3*(a + b*Sech[c + d*x]^2),x]
 

Output:

-1/8*(a*Csch[(c + d*x)/2]^2)/d - (b*Csch[(c + d*x)/2]^2)/(8*d) + (a*Log[Co 
sh[(c + d*x)/2]])/(2*d) + (3*b*Log[Cosh[(c + d*x)/2]])/(2*d) - (a*Log[Sinh 
[(c + d*x)/2]])/(2*d) - (3*b*Log[Sinh[(c + d*x)/2]])/(2*d) - (a*Sech[(c + 
d*x)/2]^2)/(8*d) - (b*Sech[(c + d*x)/2]^2)/(8*d) - (b*Sech[c + d*x])/d
 

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.09, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 26, 4621, 361, 25, 359, 219}

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.

\(\displaystyle \int \text {csch}^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right ) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int -\frac {i \left (a+b \sec (i c+i d x)^2\right )}{\sin (i c+i d x)^3}dx\)

\(\Big \downarrow \) 26

\(\displaystyle -i \int \frac {b \sec (i c+i d x)^2+a}{\sin (i c+i d x)^3}dx\)

\(\Big \downarrow \) 4621

\(\displaystyle \frac {\int \frac {\left (a \cosh ^2(c+d x)+b\right ) \text {sech}^2(c+d x)}{\left (1-\cosh ^2(c+d x)\right )^2}d\cosh (c+d x)}{d}\)

\(\Big \downarrow \) 361

\(\displaystyle \frac {\frac {(a+b) \cosh (c+d x)}{2 \left (1-\cosh ^2(c+d x)\right )}-\frac {1}{2} \int -\frac {\left ((a+b) \cosh ^2(c+d x)+2 b\right ) \text {sech}^2(c+d x)}{1-\cosh ^2(c+d x)}d\cosh (c+d x)}{d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {1}{2} \int \frac {\left ((a+b) \cosh ^2(c+d x)+2 b\right ) \text {sech}^2(c+d x)}{1-\cosh ^2(c+d x)}d\cosh (c+d x)+\frac {(a+b) \cosh (c+d x)}{2 \left (1-\cosh ^2(c+d x)\right )}}{d}\)

\(\Big \downarrow \) 359

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

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {1}{2} ((a+3 b) \text {arctanh}(\cosh (c+d x))-2 b \text {sech}(c+d x))+\frac {(a+b) \cosh (c+d x)}{2 \left (1-\cosh ^2(c+d x)\right )}}{d}\)

Input:

Int[Csch[c + d*x]^3*(a + b*Sech[c + d*x]^2),x]
 

Output:

(((a + b)*Cosh[c + d*x])/(2*(1 - Cosh[c + d*x]^2)) + ((a + 3*b)*ArcTanh[Co 
sh[c + d*x]] - 2*b*Sech[c + d*x])/2)/d
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 219

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

rule 359

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

rule 361

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

rule 3042

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

rule 4621

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

Maple [A] (verified)

Time = 8.34 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.30

method	result	size
derivativedivides	\(\frac {a \left (-\frac {\operatorname {csch}\left (d x +c \right ) \coth \left (d x +c \right )}{2}+\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )+b \left (-\frac {1}{2 \sinh \left (d x +c \right )^{2} \cosh \left (d x +c \right )}-\frac {3}{2 \cosh \left (d x +c \right )}+3 \,\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )}{d}\)	\(70\)
default	\(\frac {a \left (-\frac {\operatorname {csch}\left (d x +c \right ) \coth \left (d x +c \right )}{2}+\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )+b \left (-\frac {1}{2 \sinh \left (d x +c \right )^{2} \cosh \left (d x +c \right )}-\frac {3}{2 \cosh \left (d x +c \right )}+3 \,\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )}{d}\)	\(70\)
risch	\(-\frac {{\mathrm e}^{d x +c} \left ({\mathrm e}^{4 d x +4 c} a +3 \,{\mathrm e}^{4 d x +4 c} b +2 a \,{\mathrm e}^{2 d x +2 c}-2 b \,{\mathrm e}^{2 d x +2 c}+a +3 b \right )}{d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2} \left ({\mathrm e}^{2 d x +2 c}+1\right )}-\frac {\ln \left ({\mathrm e}^{d x +c}-1\right ) a}{2 d}-\frac {3 \ln \left ({\mathrm e}^{d x +c}-1\right ) b}{2 d}+\frac {\ln \left ({\mathrm e}^{d x +c}+1\right ) a}{2 d}+\frac {3 \ln \left ({\mathrm e}^{d x +c}+1\right ) b}{2 d}\)	\(151\)

Input:

int(csch(d*x+c)^3*(a+b*sech(d*x+c)^2),x,method=_RETURNVERBOSE)
 

Output:

1/d*(a*(-1/2*csch(d*x+c)*coth(d*x+c)+arctanh(exp(d*x+c)))+b*(-1/2/sinh(d*x 
+c)^2/cosh(d*x+c)-3/2/cosh(d*x+c)+3*arctanh(exp(d*x+c))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 924 vs. \(2 (50) = 100\).

Time = 0.21 (sec) , antiderivative size = 924, normalized size of antiderivative = 17.11 \[ \int \text {csch}^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right ) \, dx=\text {Too large to display} \] Input:

integrate(csch(d*x+c)^3*(a+b*sech(d*x+c)^2),x, algorithm="fricas")
 

Output:

-1/2*(2*(a + 3*b)*cosh(d*x + c)^5 + 10*(a + 3*b)*cosh(d*x + c)*sinh(d*x + 
c)^4 + 2*(a + 3*b)*sinh(d*x + c)^5 + 4*(a - b)*cosh(d*x + c)^3 + 4*(5*(a + 
 3*b)*cosh(d*x + c)^2 + a - b)*sinh(d*x + c)^3 + 4*(5*(a + 3*b)*cosh(d*x + 
 c)^3 + 3*(a - b)*cosh(d*x + c))*sinh(d*x + c)^2 + 2*(a + 3*b)*cosh(d*x + 
c) - ((a + 3*b)*cosh(d*x + c)^6 + 6*(a + 3*b)*cosh(d*x + c)*sinh(d*x + c)^ 
5 + (a + 3*b)*sinh(d*x + c)^6 - (a + 3*b)*cosh(d*x + c)^4 + (15*(a + 3*b)* 
cosh(d*x + c)^2 - a - 3*b)*sinh(d*x + c)^4 + 4*(5*(a + 3*b)*cosh(d*x + c)^ 
3 - (a + 3*b)*cosh(d*x + c))*sinh(d*x + c)^3 - (a + 3*b)*cosh(d*x + c)^2 + 
 (15*(a + 3*b)*cosh(d*x + c)^4 - 6*(a + 3*b)*cosh(d*x + c)^2 - a - 3*b)*si 
nh(d*x + c)^2 + 2*(3*(a + 3*b)*cosh(d*x + c)^5 - 2*(a + 3*b)*cosh(d*x + c) 
^3 - (a + 3*b)*cosh(d*x + c))*sinh(d*x + c) + a + 3*b)*log(cosh(d*x + c) + 
 sinh(d*x + c) + 1) + ((a + 3*b)*cosh(d*x + c)^6 + 6*(a + 3*b)*cosh(d*x + 
c)*sinh(d*x + c)^5 + (a + 3*b)*sinh(d*x + c)^6 - (a + 3*b)*cosh(d*x + c)^4 
 + (15*(a + 3*b)*cosh(d*x + c)^2 - a - 3*b)*sinh(d*x + c)^4 + 4*(5*(a + 3* 
b)*cosh(d*x + c)^3 - (a + 3*b)*cosh(d*x + c))*sinh(d*x + c)^3 - (a + 3*b)* 
cosh(d*x + c)^2 + (15*(a + 3*b)*cosh(d*x + c)^4 - 6*(a + 3*b)*cosh(d*x + c 
)^2 - a - 3*b)*sinh(d*x + c)^2 + 2*(3*(a + 3*b)*cosh(d*x + c)^5 - 2*(a + 3 
*b)*cosh(d*x + c)^3 - (a + 3*b)*cosh(d*x + c))*sinh(d*x + c) + a + 3*b)*lo 
g(cosh(d*x + c) + sinh(d*x + c) - 1) + 2*(5*(a + 3*b)*cosh(d*x + c)^4 + 6* 
(a - b)*cosh(d*x + c)^2 + a + 3*b)*sinh(d*x + c))/(d*cosh(d*x + c)^6 + ...
 

Sympy [F]

\[ \int \text {csch}^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right ) \, dx=\int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right ) \operatorname {csch}^{3}{\left (c + d x \right )}\, dx \] Input:

integrate(csch(d*x+c)**3*(a+b*sech(d*x+c)**2),x)
 

Output:

Integral((a + b*sech(c + d*x)**2)*csch(c + d*x)**3, x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 198 vs. \(2 (50) = 100\).

Time = 0.04 (sec) , antiderivative size = 198, normalized size of antiderivative = 3.67 \[ \int \text {csch}^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right ) \, dx=\frac {1}{2} \, b {\left (\frac {3 \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {3 \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (3 \, e^{\left (-d x - c\right )} - 2 \, e^{\left (-3 \, d x - 3 \, c\right )} + 3 \, e^{\left (-5 \, d x - 5 \, c\right )}\right )}}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} - e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}}\right )} + \frac {1}{2} \, a {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (e^{\left (-d x - c\right )} + e^{\left (-3 \, d x - 3 \, c\right )}\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} \] Input:

integrate(csch(d*x+c)^3*(a+b*sech(d*x+c)^2),x, algorithm="maxima")
 

Output:

1/2*b*(3*log(e^(-d*x - c) + 1)/d - 3*log(e^(-d*x - c) - 1)/d + 2*(3*e^(-d* 
x - c) - 2*e^(-3*d*x - 3*c) + 3*e^(-5*d*x - 5*c))/(d*(e^(-2*d*x - 2*c) + e 
^(-4*d*x - 4*c) - e^(-6*d*x - 6*c) - 1))) + 1/2*a*(log(e^(-d*x - c) + 1)/d 
 - log(e^(-d*x - c) - 1)/d + 2*(e^(-d*x - c) + e^(-3*d*x - 3*c))/(d*(2*e^( 
-2*d*x - 2*c) - e^(-4*d*x - 4*c) - 1)))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 142 vs. \(2 (50) = 100\).

Time = 0.12 (sec) , antiderivative size = 142, normalized size of antiderivative = 2.63 \[ \int \text {csch}^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right ) \, dx=\frac {{\left (a + 3 \, b\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} + 2\right ) - {\left (a + 3 \, b\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} - 2\right ) - \frac {4 \, {\left (a {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} + 3 \, b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} - 8 \, b\right )}}{{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 4 \, e^{\left (d x + c\right )} - 4 \, e^{\left (-d x - c\right )}}}{4 \, d} \] Input:

integrate(csch(d*x+c)^3*(a+b*sech(d*x+c)^2),x, algorithm="giac")
 

Output:

1/4*((a + 3*b)*log(e^(d*x + c) + e^(-d*x - c) + 2) - (a + 3*b)*log(e^(d*x 
+ c) + e^(-d*x - c) - 2) - 4*(a*(e^(d*x + c) + e^(-d*x - c))^2 + 3*b*(e^(d 
*x + c) + e^(-d*x - c))^2 - 8*b)/((e^(d*x + c) + e^(-d*x - c))^3 - 4*e^(d* 
x + c) - 4*e^(-d*x - c)))/d
 

Mupad [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 160, normalized size of antiderivative = 2.96 \[ \int \text {csch}^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right ) \, dx=\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (a\,\sqrt {-d^2}+3\,b\,\sqrt {-d^2}\right )}{d\,\sqrt {a^2+6\,a\,b+9\,b^2}}\right )\,\sqrt {a^2+6\,a\,b+9\,b^2}}{\sqrt {-d^2}}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (a+b\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {2\,{\mathrm {e}}^{c+d\,x}\,\left (a+b\right )}{d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {2\,b\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \] Input:

int((a + b/cosh(c + d*x)^2)/sinh(c + d*x)^3,x)
 

Output:

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

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 440, normalized size of antiderivative = 8.15 \[ \int \text {csch}^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right ) \, dx=\frac {-e^{6 d x +6 c} \mathrm {log}\left (e^{d x +c}-1\right ) a -3 e^{6 d x +6 c} \mathrm {log}\left (e^{d x +c}-1\right ) b +e^{6 d x +6 c} \mathrm {log}\left (e^{d x +c}+1\right ) a +3 e^{6 d x +6 c} \mathrm {log}\left (e^{d x +c}+1\right ) b -2 e^{5 d x +5 c} a -6 e^{5 d x +5 c} b +e^{4 d x +4 c} \mathrm {log}\left (e^{d x +c}-1\right ) a +3 e^{4 d x +4 c} \mathrm {log}\left (e^{d x +c}-1\right ) b -e^{4 d x +4 c} \mathrm {log}\left (e^{d x +c}+1\right ) a -3 e^{4 d x +4 c} \mathrm {log}\left (e^{d x +c}+1\right ) b -4 e^{3 d x +3 c} a +4 e^{3 d x +3 c} b +e^{2 d x +2 c} \mathrm {log}\left (e^{d x +c}-1\right ) a +3 e^{2 d x +2 c} \mathrm {log}\left (e^{d x +c}-1\right ) b -e^{2 d x +2 c} \mathrm {log}\left (e^{d x +c}+1\right ) a -3 e^{2 d x +2 c} \mathrm {log}\left (e^{d x +c}+1\right ) b -2 e^{d x +c} a -6 e^{d x +c} b -\mathrm {log}\left (e^{d x +c}-1\right ) a -3 \,\mathrm {log}\left (e^{d x +c}-1\right ) b +\mathrm {log}\left (e^{d x +c}+1\right ) a +3 \,\mathrm {log}\left (e^{d x +c}+1\right ) b}{2 d \left (e^{6 d x +6 c}-e^{4 d x +4 c}-e^{2 d x +2 c}+1\right )} \] Input:

int(csch(d*x+c)^3*(a+b*sech(d*x+c)^2),x)
 

Output:

( - e**(6*c + 6*d*x)*log(e**(c + d*x) - 1)*a - 3*e**(6*c + 6*d*x)*log(e**( 
c + d*x) - 1)*b + e**(6*c + 6*d*x)*log(e**(c + d*x) + 1)*a + 3*e**(6*c + 6 
*d*x)*log(e**(c + d*x) + 1)*b - 2*e**(5*c + 5*d*x)*a - 6*e**(5*c + 5*d*x)* 
b + e**(4*c + 4*d*x)*log(e**(c + d*x) - 1)*a + 3*e**(4*c + 4*d*x)*log(e**( 
c + d*x) - 1)*b - e**(4*c + 4*d*x)*log(e**(c + d*x) + 1)*a - 3*e**(4*c + 4 
*d*x)*log(e**(c + d*x) + 1)*b - 4*e**(3*c + 3*d*x)*a + 4*e**(3*c + 3*d*x)* 
b + e**(2*c + 2*d*x)*log(e**(c + d*x) - 1)*a + 3*e**(2*c + 2*d*x)*log(e**( 
c + d*x) - 1)*b - e**(2*c + 2*d*x)*log(e**(c + d*x) + 1)*a - 3*e**(2*c + 2 
*d*x)*log(e**(c + d*x) + 1)*b - 2*e**(c + d*x)*a - 6*e**(c + d*x)*b - log( 
e**(c + d*x) - 1)*a - 3*log(e**(c + d*x) - 1)*b + log(e**(c + d*x) + 1)*a 
+ 3*log(e**(c + d*x) + 1)*b)/(2*d*(e**(6*c + 6*d*x) - e**(4*c + 4*d*x) - e 
**(2*c + 2*d*x) + 1))