\(\int \text {csch}(c+d x) (a+b \text {sech}^2(c+d x))^2 \, dx\) [13]

Optimal result

Integrand size = 21, antiderivative size = 52 \[ \int \text {csch}(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx=-\frac {(a+b)^2 \text {arctanh}(\cosh (c+d x))}{d}+\frac {b (2 a+b) \text {sech}(c+d x)}{d}+\frac {b^2 \text {sech}^3(c+d x)}{3 d} \] Output:

-(a+b)^2*arctanh(cosh(d*x+c))/d+b*(2*a+b)*sech(d*x+c)/d+1/3*b^2*sech(d*x+c 
)^3/d
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(108\) vs. \(2(52)=104\).

Time = 1.87 (sec) , antiderivative size = 108, normalized size of antiderivative = 2.08 \[ \int \text {csch}(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx=-\frac {4 \left (b+a \cosh ^2(c+d x)\right )^2 \left (-b^2-3 b (2 a+b) \cosh ^2(c+d x)+3 (a+b)^2 \cosh ^3(c+d x) \left (\log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )\right )\right ) \text {sech}^3(c+d x)}{3 d (a+2 b+a \cosh (2 (c+d x)))^2} \] Input:

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

Output:

(-4*(b + a*Cosh[c + d*x]^2)^2*(-b^2 - 3*b*(2*a + b)*Cosh[c + d*x]^2 + 3*(a 
 + b)^2*Cosh[c + d*x]^3*(Log[Cosh[(c + d*x)/2]] - Log[Sinh[(c + d*x)/2]])) 
*Sech[c + d*x]^3)/(3*d*(a + 2*b + a*Cosh[2*(c + d*x)])^2)
 

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.92, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3042, 26, 4621, 364, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4621

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

\(\Big \downarrow \) 364

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-(((a + b)^2*ArcTanh[Cosh[c + d*x]] - b*(2*a + b)*Sech[c + d*x] - (b^2*Sec 
h[c + d*x]^3)/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 364

Int[(((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_))/((c_) + (d_.)*(x_)^2), 
x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*((a + b*x^2)^p/(c + d*x^2)), x], x 
] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && (In 
tegerQ[m] || IGtQ[2*(m + 1), 0] ||  !RationalQ[m])
 

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 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 = 22.76 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.38

method	result	size
derivativedivides	\(\frac {-2 a^{2} \operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )+2 a b \left (\frac {1}{\cosh \left (d x +c \right )}-2 \,\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )+b^{2} \left (\frac {1}{3 \cosh \left (d x +c \right )^{3}}+\frac {1}{\cosh \left (d x +c \right )}-2 \,\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )}{d}\)	\(72\)
default	\(\frac {-2 a^{2} \operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )+2 a b \left (\frac {1}{\cosh \left (d x +c \right )}-2 \,\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )+b^{2} \left (\frac {1}{3 \cosh \left (d x +c \right )^{3}}+\frac {1}{\cosh \left (d x +c \right )}-2 \,\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )}{d}\)	\(72\)
risch	\(\frac {2 b \,{\mathrm e}^{d x +c} \left (6 \,{\mathrm e}^{4 d x +4 c} a +3 \,{\mathrm e}^{4 d x +4 c} b +12 a \,{\mathrm e}^{2 d x +2 c}+10 b \,{\mathrm e}^{2 d x +2 c}+6 a +3 b \right )}{3 d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{3}}+\frac {\ln \left ({\mathrm e}^{d x +c}-1\right ) a^{2}}{d}+\frac {2 \ln \left ({\mathrm e}^{d x +c}-1\right ) a b}{d}+\frac {\ln \left ({\mathrm e}^{d x +c}-1\right ) b^{2}}{d}-\frac {\ln \left ({\mathrm e}^{d x +c}+1\right ) a^{2}}{d}-\frac {2 \ln \left ({\mathrm e}^{d x +c}+1\right ) a b}{d}-\frac {\ln \left ({\mathrm e}^{d x +c}+1\right ) b^{2}}{d}\)	\(180\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1148 vs. \(2 (50) = 100\).

Time = 0.22 (sec) , antiderivative size = 1148, normalized size of antiderivative = 22.08 \[ \int \text {csch}(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx=\text {Too large to display} \] Input:

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

Output:

1/3*(6*(2*a*b + b^2)*cosh(d*x + c)^5 + 30*(2*a*b + b^2)*cosh(d*x + c)*sinh 
(d*x + c)^4 + 6*(2*a*b + b^2)*sinh(d*x + c)^5 + 4*(6*a*b + 5*b^2)*cosh(d*x 
 + c)^3 + 4*(15*(2*a*b + b^2)*cosh(d*x + c)^2 + 6*a*b + 5*b^2)*sinh(d*x + 
c)^3 + 12*(5*(2*a*b + b^2)*cosh(d*x + c)^3 + (6*a*b + 5*b^2)*cosh(d*x + c) 
)*sinh(d*x + c)^2 + 6*(2*a*b + b^2)*cosh(d*x + c) - 3*((a^2 + 2*a*b + b^2) 
*cosh(d*x + c)^6 + 6*(a^2 + 2*a*b + b^2)*cosh(d*x + c)*sinh(d*x + c)^5 + ( 
a^2 + 2*a*b + b^2)*sinh(d*x + c)^6 + 3*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^4 
 + 3*(5*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^2 + a^2 + 2*a*b + b^2)*sinh(d*x 
+ c)^4 + 4*(5*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^3 + 3*(a^2 + 2*a*b + b^2)* 
cosh(d*x + c))*sinh(d*x + c)^3 + 3*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^2 + 3 
*(5*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^4 + 6*(a^2 + 2*a*b + b^2)*cosh(d*x + 
 c)^2 + a^2 + 2*a*b + b^2)*sinh(d*x + c)^2 + a^2 + 2*a*b + b^2 + 6*((a^2 + 
 2*a*b + b^2)*cosh(d*x + c)^5 + 2*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^3 + (a 
^2 + 2*a*b + b^2)*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d 
*x + c) + 1) + 3*((a^2 + 2*a*b + b^2)*cosh(d*x + c)^6 + 6*(a^2 + 2*a*b + b 
^2)*cosh(d*x + c)*sinh(d*x + c)^5 + (a^2 + 2*a*b + b^2)*sinh(d*x + c)^6 + 
3*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^4 + 3*(5*(a^2 + 2*a*b + b^2)*cosh(d*x 
+ c)^2 + a^2 + 2*a*b + b^2)*sinh(d*x + c)^4 + 4*(5*(a^2 + 2*a*b + b^2)*cos 
h(d*x + c)^3 + 3*(a^2 + 2*a*b + b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 3*(a 
^2 + 2*a*b + b^2)*cosh(d*x + c)^2 + 3*(5*(a^2 + 2*a*b + b^2)*cosh(d*x +...
 

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 197 vs. \(2 (50) = 100\).

Time = 0.04 (sec) , antiderivative size = 197, normalized size of antiderivative = 3.79 \[ \int \text {csch}(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx=-\frac {1}{3} \, b^{2} {\left (\frac {3 \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {3 \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{d} - \frac {2 \, {\left (3 \, e^{\left (-d x - c\right )} + 10 \, e^{\left (-3 \, d x - 3 \, c\right )} + 3 \, e^{\left (-5 \, d x - 5 \, c\right )}\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}}\right )} - 2 \, a b {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} - \frac {2 \, e^{\left (-d x - c\right )}}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}}\right )} + \frac {a^{2} \log \left (\tanh \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} \] Input:

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

Output:

-1/3*b^2*(3*log(e^(-d*x - c) + 1)/d - 3*log(e^(-d*x - c) - 1)/d - 2*(3*e^( 
-d*x - c) + 10*e^(-3*d*x - 3*c) + 3*e^(-5*d*x - 5*c))/(d*(3*e^(-2*d*x - 2* 
c) + 3*e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c) + 1))) - 2*a*b*(log(e^(-d*x - c 
) + 1)/d - log(e^(-d*x - c) - 1)/d - 2*e^(-d*x - c)/(d*(e^(-2*d*x - 2*c) + 
 1))) + a^2*log(tanh(1/2*d*x + 1/2*c))/d
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 139 vs. \(2 (50) = 100\).

Time = 0.13 (sec) , antiderivative size = 139, normalized size of antiderivative = 2.67 \[ \int \text {csch}(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx=-\frac {3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} + 2\right ) - 3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} - 2\right ) - \frac {4 \, {\left (6 \, a b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} + 3 \, b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} + 4 \, b^{2}\right )}}{{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3}}}{6 \, d} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 2.64 (sec) , antiderivative size = 232, normalized size of antiderivative = 4.46 \[ \int \text {csch}(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx=\frac {2\,{\mathrm {e}}^{c+d\,x}\,\left (b^2+2\,a\,b\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {8\,b^2\,{\mathrm {e}}^{c+d\,x}}{3\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1\right )}-\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (a^2\,\sqrt {-d^2}+b^2\,\sqrt {-d^2}+2\,a\,b\,\sqrt {-d^2}\right )}{d\,\sqrt {a^4+4\,a^3\,b+6\,a^2\,b^2+4\,a\,b^3+b^4}}\right )\,\sqrt {a^4+4\,a^3\,b+6\,a^2\,b^2+4\,a\,b^3+b^4}}{\sqrt {-d^2}}+\frac {8\,b^2\,{\mathrm {e}}^{c+d\,x}}{3\,d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 657, normalized size of antiderivative = 12.63 \[ \int \text {csch}(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx=\frac {6 e^{6 d x +6 c} \mathrm {log}\left (e^{d x +c}-1\right ) a b -6 e^{6 d x +6 c} \mathrm {log}\left (e^{d x +c}+1\right ) a b +18 e^{4 d x +4 c} \mathrm {log}\left (e^{d x +c}-1\right ) a b -18 e^{4 d x +4 c} \mathrm {log}\left (e^{d x +c}+1\right ) a b +18 e^{2 d x +2 c} \mathrm {log}\left (e^{d x +c}-1\right ) a b -18 e^{2 d x +2 c} \mathrm {log}\left (e^{d x +c}+1\right ) a b +20 e^{3 d x +3 c} b^{2}+12 e^{5 d x +5 c} a b +24 e^{3 d x +3 c} a b +12 e^{d x +c} a b +3 e^{6 d x +6 c} \mathrm {log}\left (e^{d x +c}-1\right ) a^{2}+3 e^{6 d x +6 c} \mathrm {log}\left (e^{d x +c}-1\right ) b^{2}-3 e^{6 d x +6 c} \mathrm {log}\left (e^{d x +c}+1\right ) a^{2}-3 e^{6 d x +6 c} \mathrm {log}\left (e^{d x +c}+1\right ) b^{2}+9 e^{4 d x +4 c} \mathrm {log}\left (e^{d x +c}-1\right ) a^{2}+9 e^{4 d x +4 c} \mathrm {log}\left (e^{d x +c}-1\right ) b^{2}-9 e^{4 d x +4 c} \mathrm {log}\left (e^{d x +c}+1\right ) a^{2}-9 e^{4 d x +4 c} \mathrm {log}\left (e^{d x +c}+1\right ) b^{2}+9 e^{2 d x +2 c} \mathrm {log}\left (e^{d x +c}-1\right ) a^{2}+9 e^{2 d x +2 c} \mathrm {log}\left (e^{d x +c}-1\right ) b^{2}-9 e^{2 d x +2 c} \mathrm {log}\left (e^{d x +c}+1\right ) a^{2}-9 e^{2 d x +2 c} \mathrm {log}\left (e^{d x +c}+1\right ) b^{2}+6 \,\mathrm {log}\left (e^{d x +c}-1\right ) a b -6 \,\mathrm {log}\left (e^{d x +c}+1\right ) a b +6 e^{5 d x +5 c} b^{2}+6 e^{d x +c} b^{2}+3 \,\mathrm {log}\left (e^{d x +c}-1\right ) a^{2}+3 \,\mathrm {log}\left (e^{d x +c}-1\right ) b^{2}-3 \,\mathrm {log}\left (e^{d x +c}+1\right ) a^{2}-3 \,\mathrm {log}\left (e^{d x +c}+1\right ) b^{2}}{3 d \left (e^{6 d x +6 c}+3 e^{4 d x +4 c}+3 e^{2 d x +2 c}+1\right )} \] Input:

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

Output:

(3*e**(6*c + 6*d*x)*log(e**(c + d*x) - 1)*a**2 + 6*e**(6*c + 6*d*x)*log(e* 
*(c + d*x) - 1)*a*b + 3*e**(6*c + 6*d*x)*log(e**(c + d*x) - 1)*b**2 - 3*e* 
*(6*c + 6*d*x)*log(e**(c + d*x) + 1)*a**2 - 6*e**(6*c + 6*d*x)*log(e**(c + 
 d*x) + 1)*a*b - 3*e**(6*c + 6*d*x)*log(e**(c + d*x) + 1)*b**2 + 12*e**(5* 
c + 5*d*x)*a*b + 6*e**(5*c + 5*d*x)*b**2 + 9*e**(4*c + 4*d*x)*log(e**(c + 
d*x) - 1)*a**2 + 18*e**(4*c + 4*d*x)*log(e**(c + d*x) - 1)*a*b + 9*e**(4*c 
 + 4*d*x)*log(e**(c + d*x) - 1)*b**2 - 9*e**(4*c + 4*d*x)*log(e**(c + d*x) 
 + 1)*a**2 - 18*e**(4*c + 4*d*x)*log(e**(c + d*x) + 1)*a*b - 9*e**(4*c + 4 
*d*x)*log(e**(c + d*x) + 1)*b**2 + 24*e**(3*c + 3*d*x)*a*b + 20*e**(3*c + 
3*d*x)*b**2 + 9*e**(2*c + 2*d*x)*log(e**(c + d*x) - 1)*a**2 + 18*e**(2*c + 
 2*d*x)*log(e**(c + d*x) - 1)*a*b + 9*e**(2*c + 2*d*x)*log(e**(c + d*x) - 
1)*b**2 - 9*e**(2*c + 2*d*x)*log(e**(c + d*x) + 1)*a**2 - 18*e**(2*c + 2*d 
*x)*log(e**(c + d*x) + 1)*a*b - 9*e**(2*c + 2*d*x)*log(e**(c + d*x) + 1)*b 
**2 + 12*e**(c + d*x)*a*b + 6*e**(c + d*x)*b**2 + 3*log(e**(c + d*x) - 1)* 
a**2 + 6*log(e**(c + d*x) - 1)*a*b + 3*log(e**(c + d*x) - 1)*b**2 - 3*log( 
e**(c + d*x) + 1)*a**2 - 6*log(e**(c + d*x) + 1)*a*b - 3*log(e**(c + d*x) 
+ 1)*b**2)/(3*d*(e**(6*c + 6*d*x) + 3*e**(4*c + 4*d*x) + 3*e**(2*c + 2*d*x 
) + 1))