\(\int \frac {\coth (c+d x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) [479]

Optimal result

Integrand size = 27, antiderivative size = 72 \[ \int \frac {\coth (c+d x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {b \text {csch}(c+d x)}{a^2 d}-\frac {\text {csch}^2(c+d x)}{2 a d}+\frac {b^2 \log (\sinh (c+d x))}{a^3 d}-\frac {b^2 \log (a+b \sinh (c+d x))}{a^3 d} \] Output:

b*csch(d*x+c)/a^2/d-1/2*csch(d*x+c)^2/a/d+b^2*ln(sinh(d*x+c))/a^3/d-b^2*ln 
(a+b*sinh(d*x+c))/a^3/d
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.83 \[ \int \frac {\coth (c+d x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {2 a b \text {csch}(c+d x)-a^2 \text {csch}^2(c+d x)+2 b^2 (\log (\sinh (c+d x))-\log (a+b \sinh (c+d x)))}{2 a^3 d} \] Input:

Integrate[(Coth[c + d*x]*Csch[c + d*x]^2)/(a + b*Sinh[c + d*x]),x]
 

Output:

(2*a*b*Csch[c + d*x] - a^2*Csch[c + d*x]^2 + 2*b^2*(Log[Sinh[c + d*x]] - L 
og[a + b*Sinh[c + d*x]]))/(2*a^3*d)
 

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.94, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {3042, 26, 3312, 26, 27, 54, 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.

\(\displaystyle \int \frac {\coth (c+d x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3312

\(\displaystyle -\frac {i \int \frac {i \text {csch}^3(c+d x)}{a+b \sinh (c+d x)}d(b \sinh (c+d x))}{b d}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {\int \frac {\text {csch}^3(c+d x)}{a+b \sinh (c+d x)}d(b \sinh (c+d x))}{b d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b^2 \int \frac {\text {csch}^3(c+d x)}{b^3 (a+b \sinh (c+d x))}d(b \sinh (c+d x))}{d}\)

\(\Big \downarrow \) 54

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {b^2 \left (\frac {\log (b \sinh (c+d x))}{a^3}-\frac {\log (a+b \sinh (c+d x))}{a^3}+\frac {\text {csch}(c+d x)}{a^2 b}-\frac {\text {csch}^2(c+d x)}{2 a b^2}\right )}{d}\)

Input:

Int[(Coth[c + d*x]*Csch[c + d*x]^2)/(a + b*Sinh[c + d*x]),x]
 

Output:

(b^2*(Csch[c + d*x]/(a^2*b) - Csch[c + d*x]^2/(2*a*b^2) + Log[b*Sinh[c + d 
*x]]/a^3 - Log[a + b*Sinh[c + d*x]]/a^3))/d
 

Defintions of rubi rules used

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 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 54

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E 
xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && 
 ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

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

rule 3312

Int[cos[(e_.) + (f_.)*(x_)]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*(( 
c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(b*f)   Su 
bst[Int[(a + x)^m*(c + (d/b)*x)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, 
b, c, d, e, f, m, n}, x]
 

Maple [A] (verified)

Time = 0.43 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.75

method	result	size
derivativedivides	\(-\frac {\operatorname {csch}\left (d x +c \right )^{2}}{2 a d}+\frac {b \,\operatorname {csch}\left (d x +c \right )}{a^{2} d}-\frac {b^{2} \ln \left (a \,\operatorname {csch}\left (d x +c \right )+b \right )}{d \,a^{3}}\)	\(54\)
default	\(-\frac {\operatorname {csch}\left (d x +c \right )^{2}}{2 a d}+\frac {b \,\operatorname {csch}\left (d x +c \right )}{a^{2} d}-\frac {b^{2} \ln \left (a \,\operatorname {csch}\left (d x +c \right )+b \right )}{d \,a^{3}}\)	\(54\)
risch	\(-\frac {2 \,{\mathrm e}^{d x +c} \left (-b \,{\mathrm e}^{2 d x +2 c}+a \,{\mathrm e}^{d x +c}+b \right )}{a^{2} d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}+\frac {b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{a^{3} d}-\frac {b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{b}-1\right )}{a^{3} d}\)	\(108\)

Input:

int(coth(d*x+c)*csch(d*x+c)^2/(a+b*sinh(d*x+c)),x,method=_RETURNVERBOSE)
 

Output:

-1/2*csch(d*x+c)^2/a/d+b*csch(d*x+c)/a^2/d-1/d*b^2/a^3*ln(a*csch(d*x+c)+b)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 545 vs. \(2 (70) = 140\).

Time = 0.11 (sec) , antiderivative size = 545, normalized size of antiderivative = 7.57 \[ \int \frac {\coth (c+d x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {2 \, a b \cosh \left (d x + c\right )^{3} + 2 \, a b \sinh \left (d x + c\right )^{3} - 2 \, a^{2} \cosh \left (d x + c\right )^{2} - 2 \, a b \cosh \left (d x + c\right ) + 2 \, {\left (3 \, a b \cosh \left (d x + c\right ) - a^{2}\right )} \sinh \left (d x + c\right )^{2} - {\left (b^{2} \cosh \left (d x + c\right )^{4} + 4 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + b^{2} \sinh \left (d x + c\right )^{4} - 2 \, b^{2} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, b^{2} \cosh \left (d x + c\right )^{2} - b^{2}\right )} \sinh \left (d x + c\right )^{2} + b^{2} + 4 \, {\left (b^{2} \cosh \left (d x + c\right )^{3} - b^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )} \log \left (\frac {2 \, {\left (b \sinh \left (d x + c\right ) + a\right )}}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + {\left (b^{2} \cosh \left (d x + c\right )^{4} + 4 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + b^{2} \sinh \left (d x + c\right )^{4} - 2 \, b^{2} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, b^{2} \cosh \left (d x + c\right )^{2} - b^{2}\right )} \sinh \left (d x + c\right )^{2} + b^{2} + 4 \, {\left (b^{2} \cosh \left (d x + c\right )^{3} - b^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )} \log \left (\frac {2 \, \sinh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + 2 \, {\left (3 \, a b \cosh \left (d x + c\right )^{2} - 2 \, a^{2} \cosh \left (d x + c\right ) - a b\right )} \sinh \left (d x + c\right )}{a^{3} d \cosh \left (d x + c\right )^{4} + 4 \, a^{3} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a^{3} d \sinh \left (d x + c\right )^{4} - 2 \, a^{3} d \cosh \left (d x + c\right )^{2} + a^{3} d + 2 \, {\left (3 \, a^{3} d \cosh \left (d x + c\right )^{2} - a^{3} d\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (a^{3} d \cosh \left (d x + c\right )^{3} - a^{3} d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )} \] Input:

integrate(coth(d*x+c)*csch(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm="fricas 
")
 

Output:

(2*a*b*cosh(d*x + c)^3 + 2*a*b*sinh(d*x + c)^3 - 2*a^2*cosh(d*x + c)^2 - 2 
*a*b*cosh(d*x + c) + 2*(3*a*b*cosh(d*x + c) - a^2)*sinh(d*x + c)^2 - (b^2* 
cosh(d*x + c)^4 + 4*b^2*cosh(d*x + c)*sinh(d*x + c)^3 + b^2*sinh(d*x + c)^ 
4 - 2*b^2*cosh(d*x + c)^2 + 2*(3*b^2*cosh(d*x + c)^2 - b^2)*sinh(d*x + c)^ 
2 + b^2 + 4*(b^2*cosh(d*x + c)^3 - b^2*cosh(d*x + c))*sinh(d*x + c))*log(2 
*(b*sinh(d*x + c) + a)/(cosh(d*x + c) - sinh(d*x + c))) + (b^2*cosh(d*x + 
c)^4 + 4*b^2*cosh(d*x + c)*sinh(d*x + c)^3 + b^2*sinh(d*x + c)^4 - 2*b^2*c 
osh(d*x + c)^2 + 2*(3*b^2*cosh(d*x + c)^2 - b^2)*sinh(d*x + c)^2 + b^2 + 4 
*(b^2*cosh(d*x + c)^3 - b^2*cosh(d*x + c))*sinh(d*x + c))*log(2*sinh(d*x + 
 c)/(cosh(d*x + c) - sinh(d*x + c))) + 2*(3*a*b*cosh(d*x + c)^2 - 2*a^2*co 
sh(d*x + c) - a*b)*sinh(d*x + c))/(a^3*d*cosh(d*x + c)^4 + 4*a^3*d*cosh(d* 
x + c)*sinh(d*x + c)^3 + a^3*d*sinh(d*x + c)^4 - 2*a^3*d*cosh(d*x + c)^2 + 
 a^3*d + 2*(3*a^3*d*cosh(d*x + c)^2 - a^3*d)*sinh(d*x + c)^2 + 4*(a^3*d*co 
sh(d*x + c)^3 - a^3*d*cosh(d*x + c))*sinh(d*x + c))
 

Sympy [F]

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

integrate(coth(d*x+c)*csch(d*x+c)**2/(a+b*sinh(d*x+c)),x)
 

Output:

Integral(coth(c + d*x)*csch(c + d*x)**2/(a + b*sinh(c + d*x)), x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 161 vs. \(2 (70) = 140\).

Time = 0.04 (sec) , antiderivative size = 161, normalized size of antiderivative = 2.24 \[ \int \frac {\coth (c+d x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {2 \, {\left (b e^{\left (-d x - c\right )} - a e^{\left (-2 \, d x - 2 \, c\right )} - b e^{\left (-3 \, d x - 3 \, c\right )}\right )}}{{\left (2 \, a^{2} e^{\left (-2 \, d x - 2 \, c\right )} - a^{2} e^{\left (-4 \, d x - 4 \, c\right )} - a^{2}\right )} d} - \frac {b^{2} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{a^{3} d} + \frac {b^{2} \log \left (e^{\left (-d x - c\right )} + 1\right )}{a^{3} d} + \frac {b^{2} \log \left (e^{\left (-d x - c\right )} - 1\right )}{a^{3} d} \] Input:

integrate(coth(d*x+c)*csch(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm="maxima 
")
                                                                                    
                                                                                    
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 145 vs. \(2 (70) = 140\).

Time = 0.15 (sec) , antiderivative size = 145, normalized size of antiderivative = 2.01 \[ \int \frac {\coth (c+d x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {\frac {2 \, b^{2} \log \left ({\left | b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 2 \, a \right |}\right )}{a^{3}} - \frac {2 \, b^{2} \log \left ({\left | e^{\left (d x + c\right )} - e^{\left (-d x - c\right )} \right |}\right )}{a^{3}} + \frac {3 \, b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} - 4 \, a b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 4 \, a^{2}}{a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2}}}{2 \, d} \] Input:

integrate(coth(d*x+c)*csch(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm="giac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 2.08 (sec) , antiderivative size = 470, normalized size of antiderivative = 6.53 \[ \int \frac {\coth (c+d x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {\frac {2}{a\,d}-\frac {2\,b\,{\mathrm {e}}^{c+d\,x}}{a^2\,d}}{{\mathrm {e}}^{2\,c+2\,d\,x}-1}-\frac {\left (2\,\mathrm {atan}\left (-\frac {4\,a^3\,b^5\,\sqrt {-a^6\,d^2}+4\,a\,b^7\,\sqrt {-a^6\,d^2}-4\,b^8\,{\mathrm {e}}^{3\,c}\,{\mathrm {e}}^{3\,d\,x}\,\sqrt {-a^6\,d^2}+4\,b^8\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-a^6\,d^2}-8\,a\,b^7\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\,\sqrt {-a^6\,d^2}+4\,a^2\,b^6\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-a^6\,d^2}-8\,a^3\,b^5\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\,\sqrt {-a^6\,d^2}-4\,a^2\,b^6\,{\mathrm {e}}^{3\,c}\,{\mathrm {e}}^{3\,d\,x}\,\sqrt {-a^6\,d^2}}{4\,a^4\,b\,d\,{\left (b^4\right )}^{3/2}+4\,a^6\,b^3\,d\,\sqrt {b^4}}\right )+2\,\mathrm {atan}\left (\left (4\,a^4\,b^5\,d\,\sqrt {b^4}\,\sqrt {-a^6\,d^2}+4\,a^6\,b^3\,d\,\sqrt {b^4}\,\sqrt {-a^6\,d^2}\right )\,\left (\frac {1}{8\,a^5\,b^5\,d^2\,{\left (a^2+b^2\right )}^2}-{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (\frac {1}{16\,a^4\,b^6\,d^2\,{\left (a^2+b^2\right )}^2}-\frac {{\left (a^2+2\,b^2\right )}^2}{16\,a^8\,b^6\,d^2\,{\left (a^2+b^2\right )}^2}\right )+\frac {a^2+2\,b^2}{8\,a^7\,b^5\,d^2\,{\left (a^2+b^2\right )}^2}\right )\right )\right )\,\sqrt {b^4}}{\sqrt {-a^6\,d^2}}-\frac {2}{a\,d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \] Input:

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

Output:

- (2/(a*d) - (2*b*exp(c + d*x))/(a^2*d))/(exp(2*c + 2*d*x) - 1) - ((2*atan 
(-(4*a^3*b^5*(-a^6*d^2)^(1/2) + 4*a*b^7*(-a^6*d^2)^(1/2) - 4*b^8*exp(3*c)* 
exp(3*d*x)*(-a^6*d^2)^(1/2) + 4*b^8*exp(d*x)*exp(c)*(-a^6*d^2)^(1/2) - 8*a 
*b^7*exp(2*c)*exp(2*d*x)*(-a^6*d^2)^(1/2) + 4*a^2*b^6*exp(d*x)*exp(c)*(-a^ 
6*d^2)^(1/2) - 8*a^3*b^5*exp(2*c)*exp(2*d*x)*(-a^6*d^2)^(1/2) - 4*a^2*b^6* 
exp(3*c)*exp(3*d*x)*(-a^6*d^2)^(1/2))/(4*a^4*b*d*(b^4)^(3/2) + 4*a^6*b^3*d 
*(b^4)^(1/2))) + 2*atan((4*a^4*b^5*d*(b^4)^(1/2)*(-a^6*d^2)^(1/2) + 4*a^6* 
b^3*d*(b^4)^(1/2)*(-a^6*d^2)^(1/2))*(1/(8*a^5*b^5*d^2*(a^2 + b^2)^2) - exp 
(d*x)*exp(c)*(1/(16*a^4*b^6*d^2*(a^2 + b^2)^2) - (a^2 + 2*b^2)^2/(16*a^8*b 
^6*d^2*(a^2 + b^2)^2)) + (a^2 + 2*b^2)/(8*a^7*b^5*d^2*(a^2 + b^2)^2))))*(b 
^4)^(1/2))/(-a^6*d^2)^(1/2) - 2/(a*d*(exp(4*c + 4*d*x) - 2*exp(2*c + 2*d*x 
) + 1))
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 321, normalized size of antiderivative = 4.46 \[ \int \frac {\coth (c+d x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {e^{4 d x +4 c} \mathrm {log}\left (e^{d x +c}-1\right ) b^{2}+e^{4 d x +4 c} \mathrm {log}\left (e^{d x +c}+1\right ) b^{2}-e^{4 d x +4 c} \mathrm {log}\left (e^{2 d x +2 c} b +2 e^{d x +c} a -b \right ) b^{2}-e^{4 d x +4 c} a^{2}+2 e^{3 d x +3 c} a b -2 e^{2 d x +2 c} \mathrm {log}\left (e^{d x +c}-1\right ) b^{2}-2 e^{2 d x +2 c} \mathrm {log}\left (e^{d x +c}+1\right ) b^{2}+2 e^{2 d x +2 c} \mathrm {log}\left (e^{2 d x +2 c} b +2 e^{d x +c} a -b \right ) b^{2}-2 e^{d x +c} a b +\mathrm {log}\left (e^{d x +c}-1\right ) b^{2}+\mathrm {log}\left (e^{d x +c}+1\right ) b^{2}-\mathrm {log}\left (e^{2 d x +2 c} b +2 e^{d x +c} a -b \right ) b^{2}-a^{2}}{a^{3} d \left (e^{4 d x +4 c}-2 e^{2 d x +2 c}+1\right )} \] Input:

int(coth(d*x+c)*csch(d*x+c)^2/(a+b*sinh(d*x+c)),x)
 

Output:

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