Integrand size = 13, antiderivative size = 46 \[ \int \frac {\text {csch}^3(x)}{a+a \text {sech}(x)} \, dx=\frac {\text {arctanh}(\cosh (x))}{8 a}-\frac {\coth (x) \text {csch}(x)}{8 a}-\frac {\coth (x) \text {csch}^3(x)}{4 a}+\frac {\text {csch}^4(x)}{4 a} \] Output:
1/8*arctanh(cosh(x))/a-1/8*coth(x)*csch(x)/a-1/4*coth(x)*csch(x)^3/a+1/4*c sch(x)^4/a
Time = 0.12 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.28 \[ \int \frac {\text {csch}^3(x)}{a+a \text {sech}(x)} \, dx=\frac {\cosh ^2\left (\frac {x}{2}\right ) \left (-2 \text {csch}^2\left (\frac {x}{2}\right )+4 \log \left (\cosh \left (\frac {x}{2}\right )\right )-4 \log \left (\sinh \left (\frac {x}{2}\right )\right )+\text {sech}^4\left (\frac {x}{2}\right )\right ) \text {sech}(x)}{16 (a+a \text {sech}(x))} \] Input:
Integrate[Csch[x]^3/(a + a*Sech[x]),x]
Output:
(Cosh[x/2]^2*(-2*Csch[x/2]^2 + 4*Log[Cosh[x/2]] - 4*Log[Sinh[x/2]] + Sech[ x/2]^4)*Sech[x])/(16*(a + a*Sech[x]))
Result contains complex when optimal does not.
Time = 0.61 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.28, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.692, Rules used = {3042, 26, 4360, 26, 25, 3042, 26, 3314, 26, 3042, 26, 3086, 15, 3091, 26, 3042, 26, 4255, 26, 3042, 26, 4257}
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.
| \(\displaystyle \int \frac {\text {csch}^3(x)}{a \text {sech}(x)+a} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i}{\cos \left (-\frac {\pi }{2}+i x\right )^3 \left (a-a \csc \left (-\frac {\pi }{2}+i x\right )\right )}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {1}{\cos \left (i x-\frac {\pi }{2}\right )^3 \left (a-a \csc \left (i x-\frac {\pi }{2}\right )\right )}dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle -i \int -\frac {i \coth (x) \text {csch}^2(x)}{-\cosh (x) a-a}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\int -\frac {\coth (x) \text {csch}^2(x)}{\cosh (x) a+a}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \frac {\coth (x) \text {csch}^2(x)}{a \cosh (x)+a}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {i \sin \left (-\frac {\pi }{2}+i x\right )}{\cos \left (-\frac {\pi }{2}+i x\right )^3 \left (a-a \sin \left (-\frac {\pi }{2}+i x\right )\right )}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {\sin \left (i x-\frac {\pi }{2}\right )}{\cos \left (i x-\frac {\pi }{2}\right )^3 \left (a-a \sin \left (i x-\frac {\pi }{2}\right )\right )}dx\) |
| \(\Big \downarrow \) 3314 |
| \(\displaystyle i \left (\frac {\int -i \coth ^2(x) \text {csch}^3(x)dx}{a}+\frac {\int i \coth (x) \text {csch}^4(x)dx}{a}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {i \int \coth (x) \text {csch}^4(x)dx}{a}-\frac {i \int \coth ^2(x) \text {csch}^3(x)dx}{a}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (\frac {i \int -i \sec \left (i x-\frac {\pi }{2}\right )^4 \tan \left (i x-\frac {\pi }{2}\right )dx}{a}-\frac {i \int i \sec \left (i x-\frac {\pi }{2}\right )^3 \tan \left (i x-\frac {\pi }{2}\right )^2dx}{a}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {\int \sec \left (i x-\frac {\pi }{2}\right )^4 \tan \left (i x-\frac {\pi }{2}\right )dx}{a}+\frac {\int \sec \left (i x-\frac {\pi }{2}\right )^3 \tan \left (i x-\frac {\pi }{2}\right )^2dx}{a}\right )\) |
| \(\Big \downarrow \) 3086 |
| \(\displaystyle i \left (\frac {\int \sec \left (i x-\frac {\pi }{2}\right )^3 \tan \left (i x-\frac {\pi }{2}\right )^2dx}{a}-\frac {i \int i \text {csch}^3(x)d(-i \text {csch}(x))}{a}\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle i \left (\frac {\int \sec \left (i x-\frac {\pi }{2}\right )^3 \tan \left (i x-\frac {\pi }{2}\right )^2dx}{a}-\frac {i \text {csch}^4(x)}{4 a}\right )\) |
| \(\Big \downarrow \) 3091 |
| \(\displaystyle i \left (\frac {\frac {1}{4} i \coth (x) \text {csch}^3(x)-\frac {1}{4} \int i \text {csch}^3(x)dx}{a}-\frac {i \text {csch}^4(x)}{4 a}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {\frac {1}{4} i \coth (x) \text {csch}^3(x)-\frac {1}{4} i \int \text {csch}^3(x)dx}{a}-\frac {i \text {csch}^4(x)}{4 a}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (\frac {\frac {1}{4} i \coth (x) \text {csch}^3(x)-\frac {1}{4} i \int -i \csc (i x)^3dx}{a}-\frac {i \text {csch}^4(x)}{4 a}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {\frac {1}{4} i \coth (x) \text {csch}^3(x)-\frac {1}{4} \int \csc (i x)^3dx}{a}-\frac {i \text {csch}^4(x)}{4 a}\right )\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle i \left (\frac {\frac {1}{4} \left (\frac {1}{2} i \coth (x) \text {csch}(x)-\frac {1}{2} \int -i \text {csch}(x)dx\right )+\frac {1}{4} i \coth (x) \text {csch}^3(x)}{a}-\frac {i \text {csch}^4(x)}{4 a}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {\frac {1}{4} \left (\frac {1}{2} i \int \text {csch}(x)dx+\frac {1}{2} i \coth (x) \text {csch}(x)\right )+\frac {1}{4} i \coth (x) \text {csch}^3(x)}{a}-\frac {i \text {csch}^4(x)}{4 a}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (\frac {\frac {1}{4} \left (\frac {1}{2} i \int i \csc (i x)dx+\frac {1}{2} i \coth (x) \text {csch}(x)\right )+\frac {1}{4} i \coth (x) \text {csch}^3(x)}{a}-\frac {i \text {csch}^4(x)}{4 a}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {\frac {1}{4} \left (\frac {1}{2} i \coth (x) \text {csch}(x)-\frac {1}{2} \int \csc (i x)dx\right )+\frac {1}{4} i \coth (x) \text {csch}^3(x)}{a}-\frac {i \text {csch}^4(x)}{4 a}\right )\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle i \left (\frac {\frac {1}{4} \left (\frac {1}{2} i \coth (x) \text {csch}(x)-\frac {1}{2} i \text {arctanh}(\cosh (x))\right )+\frac {1}{4} i \coth (x) \text {csch}^3(x)}{a}-\frac {i \text {csch}^4(x)}{4 a}\right )\) |
Input:
Int[Csch[x]^3/(a + a*Sech[x]),x]
Output:
I*(((-1/4*I)*Csch[x]^4)/a + ((I/4)*Coth[x]*Csch[x]^3 + ((-1/2*I)*ArcTanh[C osh[x]] + (I/2)*Coth[x]*Csch[x])/4)/a)
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_.), x_Symbol] :> Simp[a/f Subst[Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2 ), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2 ] && !(IntegerQ[m/2] && LtQ[0, m, n + 1])
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_), x_Symbol] :> Simp[b*(a*Sec[e + f*x])^m*((b*Tan[e + f*x])^(n - 1)/(f*(m + n - 1))), x] - Simp[b^2*((n - 1)/(m + n - 1)) Int[(a*Sec[e + f*x])^m*( b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] & & NeQ[m + n - 1, 0] && IntegersQ[2*m, 2*n]
Int[(cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.))/(( a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[1/a Int[Cos[e + f *x]^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Simp[1/(b*d) Int[Cos[e + f*x]^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n, p}, x] & & IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && IntegerQ[n] && (LtQ[0, n, (p + 1)/2] || (LeQ[p, -n] && LtQ[-n, 2*p - 3]) || (GtQ[n, 0] && LeQ[n, -p]))
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Cos[e + f*x])^p*((b + a*Sin[e + f*x])^m/Si n[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]
Time = 0.36 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.83
| method | result | size |
| default | \(\frac {\frac {\tanh \left (\frac {x}{2}\right )^{4}}{4}-\frac {\tanh \left (\frac {x}{2}\right )^{2}}{2}-\frac {1}{2 \tanh \left (\frac {x}{2}\right )^{2}}-\ln \left (\tanh \left (\frac {x}{2}\right )\right )}{8 a}\) | \(38\) |
| risch | \(-\frac {{\mathrm e}^{x} \left ({\mathrm e}^{4 x}+2 \,{\mathrm e}^{3 x}+10 \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x}+1\right )}{4 \left (1+{\mathrm e}^{x}\right )^{4} a \left ({\mathrm e}^{x}-1\right )^{2}}-\frac {\ln \left ({\mathrm e}^{x}-1\right )}{8 a}+\frac {\ln \left (1+{\mathrm e}^{x}\right )}{8 a}\) | \(63\) |
Input:
int(csch(x)^3/(a+a*sech(x)),x,method=_RETURNVERBOSE)
Output:
1/8/a*(1/4*tanh(1/2*x)^4-1/2*tanh(1/2*x)^2-1/2/tanh(1/2*x)^2-ln(tanh(1/2*x )))
Leaf count of result is larger than twice the leaf count of optimal. 630 vs. \(2 (38) = 76\).
Time = 0.09 (sec) , antiderivative size = 630, normalized size of antiderivative = 13.70 \[ \int \frac {\text {csch}^3(x)}{a+a \text {sech}(x)} \, dx=\text {Too large to display} \] Input:
integrate(csch(x)^3/(a+a*sech(x)),x, algorithm="fricas")
Output:
-1/8*(2*cosh(x)^5 + 2*(5*cosh(x) + 2)*sinh(x)^4 + 2*sinh(x)^5 + 4*cosh(x)^ 4 + 4*(5*cosh(x)^2 + 4*cosh(x) + 5)*sinh(x)^3 + 20*cosh(x)^3 + 4*(5*cosh(x )^3 + 6*cosh(x)^2 + 15*cosh(x) + 1)*sinh(x)^2 + 4*cosh(x)^2 - (cosh(x)^6 + 2*(3*cosh(x) + 1)*sinh(x)^5 + sinh(x)^6 + 2*cosh(x)^5 + (15*cosh(x)^2 + 1 0*cosh(x) - 1)*sinh(x)^4 - cosh(x)^4 + 4*(5*cosh(x)^3 + 5*cosh(x)^2 - cosh (x) - 1)*sinh(x)^3 - 4*cosh(x)^3 + (15*cosh(x)^4 + 20*cosh(x)^3 - 6*cosh(x )^2 - 12*cosh(x) - 1)*sinh(x)^2 - cosh(x)^2 + 2*(3*cosh(x)^5 + 5*cosh(x)^4 - 2*cosh(x)^3 - 6*cosh(x)^2 - cosh(x) + 1)*sinh(x) + 2*cosh(x) + 1)*log(c osh(x) + sinh(x) + 1) + (cosh(x)^6 + 2*(3*cosh(x) + 1)*sinh(x)^5 + sinh(x) ^6 + 2*cosh(x)^5 + (15*cosh(x)^2 + 10*cosh(x) - 1)*sinh(x)^4 - cosh(x)^4 + 4*(5*cosh(x)^3 + 5*cosh(x)^2 - cosh(x) - 1)*sinh(x)^3 - 4*cosh(x)^3 + (15 *cosh(x)^4 + 20*cosh(x)^3 - 6*cosh(x)^2 - 12*cosh(x) - 1)*sinh(x)^2 - cosh (x)^2 + 2*(3*cosh(x)^5 + 5*cosh(x)^4 - 2*cosh(x)^3 - 6*cosh(x)^2 - cosh(x) + 1)*sinh(x) + 2*cosh(x) + 1)*log(cosh(x) + sinh(x) - 1) + 2*(5*cosh(x)^4 + 8*cosh(x)^3 + 30*cosh(x)^2 + 4*cosh(x) + 1)*sinh(x) + 2*cosh(x))/(a*cos h(x)^6 + a*sinh(x)^6 + 2*a*cosh(x)^5 + 2*(3*a*cosh(x) + a)*sinh(x)^5 - a*c osh(x)^4 + (15*a*cosh(x)^2 + 10*a*cosh(x) - a)*sinh(x)^4 - 4*a*cosh(x)^3 + 4*(5*a*cosh(x)^3 + 5*a*cosh(x)^2 - a*cosh(x) - a)*sinh(x)^3 - a*cosh(x)^2 + (15*a*cosh(x)^4 + 20*a*cosh(x)^3 - 6*a*cosh(x)^2 - 12*a*cosh(x) - a)*si nh(x)^2 + 2*a*cosh(x) + 2*(3*a*cosh(x)^5 + 5*a*cosh(x)^4 - 2*a*cosh(x)^...
\[ \int \frac {\text {csch}^3(x)}{a+a \text {sech}(x)} \, dx=\frac {\int \frac {\operatorname {csch}^{3}{\left (x \right )}}{\operatorname {sech}{\left (x \right )} + 1}\, dx}{a} \] Input:
integrate(csch(x)**3/(a+a*sech(x)),x)
Output:
Integral(csch(x)**3/(sech(x) + 1), x)/a
Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (38) = 76\).
Time = 0.04 (sec) , antiderivative size = 99, normalized size of antiderivative = 2.15 \[ \int \frac {\text {csch}^3(x)}{a+a \text {sech}(x)} \, dx=-\frac {e^{\left (-x\right )} + 2 \, e^{\left (-2 \, x\right )} + 10 \, e^{\left (-3 \, x\right )} + 2 \, e^{\left (-4 \, x\right )} + e^{\left (-5 \, x\right )}}{4 \, {\left (2 \, a e^{\left (-x\right )} - a e^{\left (-2 \, x\right )} - 4 \, a e^{\left (-3 \, x\right )} - a e^{\left (-4 \, x\right )} + 2 \, a e^{\left (-5 \, x\right )} + a e^{\left (-6 \, x\right )} + a\right )}} + \frac {\log \left (e^{\left (-x\right )} + 1\right )}{8 \, a} - \frac {\log \left (e^{\left (-x\right )} - 1\right )}{8 \, a} \] Input:
integrate(csch(x)^3/(a+a*sech(x)),x, algorithm="maxima")
Output:
-1/4*(e^(-x) + 2*e^(-2*x) + 10*e^(-3*x) + 2*e^(-4*x) + e^(-5*x))/(2*a*e^(- x) - a*e^(-2*x) - 4*a*e^(-3*x) - a*e^(-4*x) + 2*a*e^(-5*x) + a*e^(-6*x) + a) + 1/8*log(e^(-x) + 1)/a - 1/8*log(e^(-x) - 1)/a
Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (38) = 76\).
Time = 0.11 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.96 \[ \int \frac {\text {csch}^3(x)}{a+a \text {sech}(x)} \, dx=\frac {\log \left (e^{\left (-x\right )} + e^{x} + 2\right )}{16 \, a} - \frac {\log \left (e^{\left (-x\right )} + e^{x} - 2\right )}{16 \, a} + \frac {e^{\left (-x\right )} + e^{x} - 6}{16 \, a {\left (e^{\left (-x\right )} + e^{x} - 2\right )}} - \frac {3 \, {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} + 12 \, e^{\left (-x\right )} + 12 \, e^{x} - 4}{32 \, a {\left (e^{\left (-x\right )} + e^{x} + 2\right )}^{2}} \] Input:
integrate(csch(x)^3/(a+a*sech(x)),x, algorithm="giac")
Output:
1/16*log(e^(-x) + e^x + 2)/a - 1/16*log(e^(-x) + e^x - 2)/a + 1/16*(e^(-x) + e^x - 6)/(a*(e^(-x) + e^x - 2)) - 1/32*(3*(e^(-x) + e^x)^2 + 12*e^(-x) + 12*e^x - 4)/(a*(e^(-x) + e^x + 2)^2)
Time = 2.25 (sec) , antiderivative size = 121, normalized size of antiderivative = 2.63 \[ \int \frac {\text {csch}^3(x)}{a+a \text {sech}(x)} \, dx=\frac {1}{2\,a\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^x+1\right )}-\frac {1}{4\,a\,\left ({\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^x+1\right )}+\frac {1}{2\,a\,\left (6\,{\mathrm {e}}^{2\,x}+4\,{\mathrm {e}}^{3\,x}+{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^x+1\right )}-\frac {1}{4\,a\,\left ({\mathrm {e}}^x-1\right )}+\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^x\,\sqrt {-a^2}}{a}\right )}{4\,\sqrt {-a^2}}-\frac {1}{a\,\left (3\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{3\,x}+3\,{\mathrm {e}}^x+1\right )} \] Input:
int(1/(sinh(x)^3*(a + a/cosh(x))),x)
Output:
1/(2*a*(exp(2*x) + 2*exp(x) + 1)) - 1/(4*a*(exp(2*x) - 2*exp(x) + 1)) + 1/ (2*a*(6*exp(2*x) + 4*exp(3*x) + exp(4*x) + 4*exp(x) + 1)) - 1/(4*a*(exp(x) - 1)) + atan((exp(x)*(-a^2)^(1/2))/a)/(4*(-a^2)^(1/2)) - 1/(a*(3*exp(2*x) + exp(3*x) + 3*exp(x) + 1))
Time = 0.23 (sec) , antiderivative size = 238, normalized size of antiderivative = 5.17 \[ \int \frac {\text {csch}^3(x)}{a+a \text {sech}(x)} \, dx=\frac {-e^{6 x} \mathrm {log}\left (e^{x}-1\right )+e^{6 x} \mathrm {log}\left (e^{x}+1\right )+e^{6 x}-2 e^{5 x} \mathrm {log}\left (e^{x}-1\right )+2 e^{5 x} \mathrm {log}\left (e^{x}+1\right )+e^{4 x} \mathrm {log}\left (e^{x}-1\right )-e^{4 x} \mathrm {log}\left (e^{x}+1\right )-5 e^{4 x}+4 e^{3 x} \mathrm {log}\left (e^{x}-1\right )-4 e^{3 x} \mathrm {log}\left (e^{x}+1\right )-24 e^{3 x}+e^{2 x} \mathrm {log}\left (e^{x}-1\right )-e^{2 x} \mathrm {log}\left (e^{x}+1\right )-5 e^{2 x}-2 e^{x} \mathrm {log}\left (e^{x}-1\right )+2 e^{x} \mathrm {log}\left (e^{x}+1\right )-\mathrm {log}\left (e^{x}-1\right )+\mathrm {log}\left (e^{x}+1\right )+1}{8 a \left (e^{6 x}+2 e^{5 x}-e^{4 x}-4 e^{3 x}-e^{2 x}+2 e^{x}+1\right )} \] Input:
int(csch(x)^3/(a+a*sech(x)),x)
Output:
( - e**(6*x)*log(e**x - 1) + e**(6*x)*log(e**x + 1) + e**(6*x) - 2*e**(5*x )*log(e**x - 1) + 2*e**(5*x)*log(e**x + 1) + e**(4*x)*log(e**x - 1) - e**( 4*x)*log(e**x + 1) - 5*e**(4*x) + 4*e**(3*x)*log(e**x - 1) - 4*e**(3*x)*lo g(e**x + 1) - 24*e**(3*x) + e**(2*x)*log(e**x - 1) - e**(2*x)*log(e**x + 1 ) - 5*e**(2*x) - 2*e**x*log(e**x - 1) + 2*e**x*log(e**x + 1) - log(e**x - 1) + log(e**x + 1) + 1)/(8*a*(e**(6*x) + 2*e**(5*x) - e**(4*x) - 4*e**(3*x ) - e**(2*x) + 2*e**x + 1))