Integrand size = 13, antiderivative size = 70 \[ \int \frac {\coth ^5(x)}{a+b \text {csch}(x)} \, dx=-\frac {\left (a^2+2 b^2\right ) \text {csch}(x)}{b^3}+\frac {a \text {csch}^2(x)}{2 b^2}-\frac {\text {csch}^3(x)}{3 b}+\frac {\left (a^2+b^2\right )^2 \log (a+b \text {csch}(x))}{a b^4}+\frac {\log (\sinh (x))}{a} \] Output:
-(a^2+2*b^2)*csch(x)/b^3+1/2*a*csch(x)^2/b^2-1/3*csch(x)^3/b+(a^2+b^2)^2*l n(a+b*csch(x))/a/b^4+ln(sinh(x))/a
Time = 0.09 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.19 \[ \int \frac {\coth ^5(x)}{a+b \text {csch}(x)} \, dx=\frac {-6 a b \left (a^2+2 b^2\right ) \text {csch}(x)+3 a^2 b^2 \text {csch}^2(x)-2 a b^3 \text {csch}^3(x)-6 a^2 \left (a^2+2 b^2\right ) \log (\sinh (x))+6 \left (a^2+b^2\right )^2 \log (b+a \sinh (x))}{6 a b^4} \] Input:
Integrate[Coth[x]^5/(a + b*Csch[x]),x]
Output:
(-6*a*b*(a^2 + 2*b^2)*Csch[x] + 3*a^2*b^2*Csch[x]^2 - 2*a*b^3*Csch[x]^3 - 6*a^2*(a^2 + 2*b^2)*Log[Sinh[x]] + 6*(a^2 + b^2)^2*Log[b + a*Sinh[x]])/(6* a*b^4)
Time = 0.30 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.07, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3042, 26, 4373, 522, 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.
| \(\displaystyle \int \frac {\coth ^5(x)}{a+b \text {csch}(x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {i \cot (i x)^5}{a+i b \csc (i x)}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {\cot (i x)^5}{a+i b \csc (i x)}dx\) |
| \(\Big \downarrow \) 4373 |
| \(\displaystyle -\frac {\int \frac {\left (\text {csch}^2(x) b^2+b^2\right )^2 \sinh (x)}{b (a+b \text {csch}(x))}d(b \text {csch}(x))}{b^4}\) |
| \(\Big \downarrow \) 522 |
| \(\displaystyle -\frac {\int \left (\frac {\sinh (x) b^3}{a}+\text {csch}^2(x) b^2-a \text {csch}(x) b+a^2 \left (\frac {2 b^2}{a^2}+1\right )-\frac {\left (a^2+b^2\right )^2}{a (a+b \text {csch}(x))}\right )d(b \text {csch}(x))}{b^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {b \left (a^2+2 b^2\right ) \text {csch}(x)-\frac {\left (a^2+b^2\right )^2 \log (a+b \text {csch}(x))}{a}+\frac {b^4 \log (b \text {csch}(x))}{a}-\frac {1}{2} a b^2 \text {csch}^2(x)+\frac {1}{3} b^3 \text {csch}^3(x)}{b^4}\) |
Input:
Int[Coth[x]^5/(a + b*Csch[x]),x]
Output:
-((b*(a^2 + 2*b^2)*Csch[x] - (a*b^2*Csch[x]^2)/2 + (b^3*Csch[x]^3)/3 + (b^ 4*Log[b*Csch[x]])/a - ((a^2 + b^2)^2*Log[a + b*Csch[x]])/a)/b^4)
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[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. ), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n _), x_Symbol] :> Simp[-(-1)^((m - 1)/2)/(d*b^(m - 1)) Subst[Int[(b^2 - x^ 2)^((m - 1)/2)*((a + x)^n/x), x], x, b*Csc[c + d*x]], x] /; FreeQ[{a, b, c, d, n}, x] && IntegerQ[(m - 1)/2] && NeQ[a^2 - b^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(173\) vs. \(2(66)=132\).
Time = 1.60 (sec) , antiderivative size = 174, normalized size of antiderivative = 2.49
| method | result | size |
| risch | \(-\frac {x}{a}-\frac {2 \,{\mathrm e}^{x} \left (3 \,{\mathrm e}^{4 x} a^{2}+6 b^{2} {\mathrm e}^{4 x}-3 \,{\mathrm e}^{3 x} a b -6 \,{\mathrm e}^{2 x} a^{2}-8 b^{2} {\mathrm e}^{2 x}+3 a \,{\mathrm e}^{x} b +3 a^{2}+6 b^{2}\right )}{3 b^{3} \left ({\mathrm e}^{2 x}-1\right )^{3}}-\frac {a^{3} \ln \left ({\mathrm e}^{2 x}-1\right )}{b^{4}}-\frac {2 a \ln \left ({\mathrm e}^{2 x}-1\right )}{b^{2}}+\frac {a^{3} \ln \left ({\mathrm e}^{2 x}+\frac {2 b \,{\mathrm e}^{x}}{a}-1\right )}{b^{4}}+\frac {2 a \ln \left ({\mathrm e}^{2 x}+\frac {2 b \,{\mathrm e}^{x}}{a}-1\right )}{b^{2}}+\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 b \,{\mathrm e}^{x}}{a}-1\right )}{a}\) | \(174\) |
| default | \(\frac {\frac {\tanh \left (\frac {x}{2}\right )^{3} b^{2}}{3}+\tanh \left (\frac {x}{2}\right )^{2} a b +4 \tanh \left (\frac {x}{2}\right ) a^{2}+7 \tanh \left (\frac {x}{2}\right ) b^{2}}{8 b^{3}}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{a}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{a}-\frac {1}{24 b \tanh \left (\frac {x}{2}\right )^{3}}-\frac {4 a^{2}+7 b^{2}}{8 b^{3} \tanh \left (\frac {x}{2}\right )}+\frac {a}{8 b^{2} \tanh \left (\frac {x}{2}\right )^{2}}-\frac {a \left (a^{2}+2 b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{b^{4}}+\frac {\left (8 a^{4}+16 a^{2} b^{2}+8 b^{4}\right ) \ln \left (-b \tanh \left (\frac {x}{2}\right )^{2}+2 a \tanh \left (\frac {x}{2}\right )+b \right )}{8 b^{4} a}\) | \(181\) |
Input:
int(coth(x)^5/(a+b*csch(x)),x,method=_RETURNVERBOSE)
Output:
-x/a-2/3*exp(x)*(3*exp(4*x)*a^2+6*b^2*exp(4*x)-3*exp(3*x)*a*b-6*exp(2*x)*a ^2-8*b^2*exp(2*x)+3*a*exp(x)*b+3*a^2+6*b^2)/b^3/(exp(2*x)-1)^3-1/b^4*a^3*l n(exp(2*x)-1)-2/b^2*a*ln(exp(2*x)-1)+1/b^4*a^3*ln(exp(2*x)+2*b/a*exp(x)-1) +2*a/b^2*ln(exp(2*x)+2*b/a*exp(x)-1)+1/a*ln(exp(2*x)+2*b/a*exp(x)-1)
Leaf count of result is larger than twice the leaf count of optimal. 1288 vs. \(2 (66) = 132\).
Time = 0.10 (sec) , antiderivative size = 1288, normalized size of antiderivative = 18.40 \[ \int \frac {\coth ^5(x)}{a+b \text {csch}(x)} \, dx=\text {Too large to display} \] Input:
integrate(coth(x)^5/(a+b*csch(x)),x, algorithm="fricas")
Output:
-1/3*(3*b^4*x*cosh(x)^6 + 3*b^4*x*sinh(x)^6 + 6*(a^3*b + 2*a*b^3)*cosh(x)^ 5 + 6*(3*b^4*x*cosh(x) + a^3*b + 2*a*b^3)*sinh(x)^5 - 3*b^4*x - 3*(3*b^4*x + 2*a^2*b^2)*cosh(x)^4 + 3*(15*b^4*x*cosh(x)^2 - 3*b^4*x - 2*a^2*b^2 + 10 *(a^3*b + 2*a*b^3)*cosh(x))*sinh(x)^4 - 4*(3*a^3*b + 4*a*b^3)*cosh(x)^3 + 4*(15*b^4*x*cosh(x)^3 - 3*a^3*b - 4*a*b^3 + 15*(a^3*b + 2*a*b^3)*cosh(x)^2 - 3*(3*b^4*x + 2*a^2*b^2)*cosh(x))*sinh(x)^3 + 3*(3*b^4*x + 2*a^2*b^2)*co sh(x)^2 + 3*(15*b^4*x*cosh(x)^4 + 3*b^4*x + 2*a^2*b^2 + 20*(a^3*b + 2*a*b^ 3)*cosh(x)^3 - 6*(3*b^4*x + 2*a^2*b^2)*cosh(x)^2 - 4*(3*a^3*b + 4*a*b^3)*c osh(x))*sinh(x)^2 + 6*(a^3*b + 2*a*b^3)*cosh(x) - 3*((a^4 + 2*a^2*b^2 + b^ 4)*cosh(x)^6 + 6*(a^4 + 2*a^2*b^2 + b^4)*cosh(x)*sinh(x)^5 + (a^4 + 2*a^2* b^2 + b^4)*sinh(x)^6 - 3*(a^4 + 2*a^2*b^2 + b^4)*cosh(x)^4 - 3*(a^4 + 2*a^ 2*b^2 + b^4 - 5*(a^4 + 2*a^2*b^2 + b^4)*cosh(x)^2)*sinh(x)^4 - a^4 - 2*a^2 *b^2 - b^4 + 4*(5*(a^4 + 2*a^2*b^2 + b^4)*cosh(x)^3 - 3*(a^4 + 2*a^2*b^2 + b^4)*cosh(x))*sinh(x)^3 + 3*(a^4 + 2*a^2*b^2 + b^4)*cosh(x)^2 + 3*(5*(a^4 + 2*a^2*b^2 + b^4)*cosh(x)^4 + a^4 + 2*a^2*b^2 + b^4 - 6*(a^4 + 2*a^2*b^2 + b^4)*cosh(x)^2)*sinh(x)^2 + 6*((a^4 + 2*a^2*b^2 + b^4)*cosh(x)^5 - 2*(a ^4 + 2*a^2*b^2 + b^4)*cosh(x)^3 + (a^4 + 2*a^2*b^2 + b^4)*cosh(x))*sinh(x) )*log(2*(a*sinh(x) + b)/(cosh(x) - sinh(x))) + 3*((a^4 + 2*a^2*b^2)*cosh(x )^6 + 6*(a^4 + 2*a^2*b^2)*cosh(x)*sinh(x)^5 + (a^4 + 2*a^2*b^2)*sinh(x)^6 - 3*(a^4 + 2*a^2*b^2)*cosh(x)^4 - 3*(a^4 + 2*a^2*b^2 - 5*(a^4 + 2*a^2*b...
\[ \int \frac {\coth ^5(x)}{a+b \text {csch}(x)} \, dx=\int \frac {\coth ^{5}{\left (x \right )}}{a + b \operatorname {csch}{\left (x \right )}}\, dx \] Input:
integrate(coth(x)**5/(a+b*csch(x)),x)
Output:
Integral(coth(x)**5/(a + b*csch(x)), x)
Leaf count of result is larger than twice the leaf count of optimal. 190 vs. \(2 (66) = 132\).
Time = 0.04 (sec) , antiderivative size = 190, normalized size of antiderivative = 2.71 \[ \int \frac {\coth ^5(x)}{a+b \text {csch}(x)} \, dx=-\frac {2 \, {\left (3 \, a b e^{\left (-2 \, x\right )} - 3 \, a b e^{\left (-4 \, x\right )} - 3 \, {\left (a^{2} + 2 \, b^{2}\right )} e^{\left (-x\right )} + 2 \, {\left (3 \, a^{2} + 4 \, b^{2}\right )} e^{\left (-3 \, x\right )} - 3 \, {\left (a^{2} + 2 \, b^{2}\right )} e^{\left (-5 \, x\right )}\right )}}{3 \, {\left (3 \, b^{3} e^{\left (-2 \, x\right )} - 3 \, b^{3} e^{\left (-4 \, x\right )} + b^{3} e^{\left (-6 \, x\right )} - b^{3}\right )}} + \frac {x}{a} - \frac {{\left (a^{3} + 2 \, a b^{2}\right )} \log \left (e^{\left (-x\right )} + 1\right )}{b^{4}} - \frac {{\left (a^{3} + 2 \, a b^{2}\right )} \log \left (e^{\left (-x\right )} - 1\right )}{b^{4}} + \frac {{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \log \left (-2 \, b e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} - a\right )}{a b^{4}} \] Input:
integrate(coth(x)^5/(a+b*csch(x)),x, algorithm="maxima")
Output:
-2/3*(3*a*b*e^(-2*x) - 3*a*b*e^(-4*x) - 3*(a^2 + 2*b^2)*e^(-x) + 2*(3*a^2 + 4*b^2)*e^(-3*x) - 3*(a^2 + 2*b^2)*e^(-5*x))/(3*b^3*e^(-2*x) - 3*b^3*e^(- 4*x) + b^3*e^(-6*x) - b^3) + x/a - (a^3 + 2*a*b^2)*log(e^(-x) + 1)/b^4 - ( a^3 + 2*a*b^2)*log(e^(-x) - 1)/b^4 + (a^4 + 2*a^2*b^2 + b^4)*log(-2*b*e^(- x) + a*e^(-2*x) - a)/(a*b^4)
Leaf count of result is larger than twice the leaf count of optimal. 170 vs. \(2 (66) = 132\).
Time = 0.13 (sec) , antiderivative size = 170, normalized size of antiderivative = 2.43 \[ \int \frac {\coth ^5(x)}{a+b \text {csch}(x)} \, dx=-\frac {{\left (a^{3} + 2 \, a b^{2}\right )} \log \left ({\left | -e^{\left (-x\right )} + e^{x} \right |}\right )}{b^{4}} + \frac {{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \log \left ({\left | -a {\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, b \right |}\right )}{a b^{4}} + \frac {11 \, a^{3} {\left (e^{\left (-x\right )} - e^{x}\right )}^{3} + 22 \, a b^{2} {\left (e^{\left (-x\right )} - e^{x}\right )}^{3} + 12 \, a^{2} b {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 24 \, b^{3} {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 12 \, a b^{2} {\left (e^{\left (-x\right )} - e^{x}\right )} + 16 \, b^{3}}{6 \, b^{4} {\left (e^{\left (-x\right )} - e^{x}\right )}^{3}} \] Input:
integrate(coth(x)^5/(a+b*csch(x)),x, algorithm="giac")
Output:
-(a^3 + 2*a*b^2)*log(abs(-e^(-x) + e^x))/b^4 + (a^4 + 2*a^2*b^2 + b^4)*log (abs(-a*(e^(-x) - e^x) + 2*b))/(a*b^4) + 1/6*(11*a^3*(e^(-x) - e^x)^3 + 22 *a*b^2*(e^(-x) - e^x)^3 + 12*a^2*b*(e^(-x) - e^x)^2 + 24*b^3*(e^(-x) - e^x )^2 + 12*a*b^2*(e^(-x) - e^x) + 16*b^3)/(b^4*(e^(-x) - e^x)^3)
Time = 3.13 (sec) , antiderivative size = 155, normalized size of antiderivative = 2.21 \[ \int \frac {\coth ^5(x)}{a+b \text {csch}(x)} \, dx=\frac {\frac {2\,a}{b^2}-\frac {2\,{\mathrm {e}}^x\,\left (a^2+2\,b^2\right )}{b^3}}{{\mathrm {e}}^{2\,x}-1}-\frac {x}{a}+\frac {\frac {2\,a}{b^2}-\frac {8\,{\mathrm {e}}^x}{3\,b}}{{\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1}-\frac {8\,{\mathrm {e}}^x}{3\,b\,\left (3\,{\mathrm {e}}^{2\,x}-3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1\right )}-\frac {\ln \left ({\mathrm {e}}^{2\,x}-1\right )\,\left (a^3+2\,a\,b^2\right )}{b^4}+\frac {\ln \left (2\,b\,{\mathrm {e}}^x-a+a\,{\mathrm {e}}^{2\,x}\right )\,\left (a^4+2\,a^2\,b^2+b^4\right )}{a\,b^4} \] Input:
int(coth(x)^5/(a + b/sinh(x)),x)
Output:
((2*a)/b^2 - (2*exp(x)*(a^2 + 2*b^2))/b^3)/(exp(2*x) - 1) - x/a + ((2*a)/b ^2 - (8*exp(x))/(3*b))/(exp(4*x) - 2*exp(2*x) + 1) - (8*exp(x))/(3*b*(3*ex p(2*x) - 3*exp(4*x) + exp(6*x) - 1)) - (log(exp(2*x) - 1)*(2*a*b^2 + a^3)) /b^4 + (log(2*b*exp(x) - a + a*exp(2*x))*(a^4 + b^4 + 2*a^2*b^2))/(a*b^4)
Time = 0.23 (sec) , antiderivative size = 747, normalized size of antiderivative = 10.67 \[ \int \frac {\coth ^5(x)}{a+b \text {csch}(x)} \, dx=\frac {12 e^{3 x} a^{3} b +16 e^{3 x} a \,b^{3}-6 e^{x} a^{3} b +2 e^{6 x} a^{2} b^{2}-6 e^{5 x} a^{3} b -12 e^{5 x} a \,b^{3}+9 e^{4 x} b^{4} x -9 e^{4 x} \mathrm {log}\left (e^{2 x} a +2 e^{x} b -a \right ) b^{4}-9 e^{2 x} b^{4} x -12 e^{x} a \,b^{3}+9 e^{2 x} \mathrm {log}\left (e^{2 x} a +2 e^{x} b -a \right ) b^{4}-6 e^{6 x} \mathrm {log}\left (e^{x}-1\right ) a^{2} b^{2}-6 e^{6 x} \mathrm {log}\left (e^{x}+1\right ) a^{2} b^{2}+6 e^{6 x} \mathrm {log}\left (e^{2 x} a +2 e^{x} b -a \right ) a^{2} b^{2}-18 e^{4 x} \mathrm {log}\left (e^{2 x} a +2 e^{x} b -a \right ) a^{2} b^{2}+18 e^{2 x} \mathrm {log}\left (e^{2 x} a +2 e^{x} b -a \right ) a^{2} b^{2}-2 a^{2} b^{2}+18 e^{4 x} \mathrm {log}\left (e^{x}-1\right ) a^{2} b^{2}+18 e^{4 x} \mathrm {log}\left (e^{x}+1\right ) a^{2} b^{2}+6 \,\mathrm {log}\left (e^{x}-1\right ) a^{2} b^{2}+6 \,\mathrm {log}\left (e^{x}+1\right ) a^{2} b^{2}-3 \,\mathrm {log}\left (e^{2 x} a +2 e^{x} b -a \right ) b^{4}+3 b^{4} x +9 e^{4 x} \mathrm {log}\left (e^{x}-1\right ) a^{4}+9 e^{4 x} \mathrm {log}\left (e^{x}+1\right ) a^{4}-9 e^{2 x} \mathrm {log}\left (e^{x}-1\right ) a^{4}-9 e^{2 x} \mathrm {log}\left (e^{x}+1\right ) a^{4}+3 \,\mathrm {log}\left (e^{x}-1\right ) a^{4}+3 \,\mathrm {log}\left (e^{x}+1\right ) a^{4}-3 \,\mathrm {log}\left (e^{2 x} a +2 e^{x} b -a \right ) a^{4}-18 e^{2 x} \mathrm {log}\left (e^{x}-1\right ) a^{2} b^{2}-18 e^{2 x} \mathrm {log}\left (e^{x}+1\right ) a^{2} b^{2}-3 e^{6 x} \mathrm {log}\left (e^{x}-1\right ) a^{4}-3 e^{6 x} \mathrm {log}\left (e^{x}+1\right ) a^{4}+3 e^{6 x} \mathrm {log}\left (e^{2 x} a +2 e^{x} b -a \right ) a^{4}+3 e^{6 x} \mathrm {log}\left (e^{2 x} a +2 e^{x} b -a \right ) b^{4}-3 e^{6 x} b^{4} x -9 e^{4 x} \mathrm {log}\left (e^{2 x} a +2 e^{x} b -a \right ) a^{4}+9 e^{2 x} \mathrm {log}\left (e^{2 x} a +2 e^{x} b -a \right ) a^{4}-6 \,\mathrm {log}\left (e^{2 x} a +2 e^{x} b -a \right ) a^{2} b^{2}}{3 a \,b^{4} \left (e^{6 x}-3 e^{4 x}+3 e^{2 x}-1\right )} \] Input:
int(coth(x)^5/(a+b*csch(x)),x)
Output:
( - 3*e**(6*x)*log(e**x - 1)*a**4 - 6*e**(6*x)*log(e**x - 1)*a**2*b**2 - 3 *e**(6*x)*log(e**x + 1)*a**4 - 6*e**(6*x)*log(e**x + 1)*a**2*b**2 + 3*e**( 6*x)*log(e**(2*x)*a + 2*e**x*b - a)*a**4 + 6*e**(6*x)*log(e**(2*x)*a + 2*e **x*b - a)*a**2*b**2 + 3*e**(6*x)*log(e**(2*x)*a + 2*e**x*b - a)*b**4 + 2* e**(6*x)*a**2*b**2 - 3*e**(6*x)*b**4*x - 6*e**(5*x)*a**3*b - 12*e**(5*x)*a *b**3 + 9*e**(4*x)*log(e**x - 1)*a**4 + 18*e**(4*x)*log(e**x - 1)*a**2*b** 2 + 9*e**(4*x)*log(e**x + 1)*a**4 + 18*e**(4*x)*log(e**x + 1)*a**2*b**2 - 9*e**(4*x)*log(e**(2*x)*a + 2*e**x*b - a)*a**4 - 18*e**(4*x)*log(e**(2*x)* a + 2*e**x*b - a)*a**2*b**2 - 9*e**(4*x)*log(e**(2*x)*a + 2*e**x*b - a)*b* *4 + 9*e**(4*x)*b**4*x + 12*e**(3*x)*a**3*b + 16*e**(3*x)*a*b**3 - 9*e**(2 *x)*log(e**x - 1)*a**4 - 18*e**(2*x)*log(e**x - 1)*a**2*b**2 - 9*e**(2*x)* log(e**x + 1)*a**4 - 18*e**(2*x)*log(e**x + 1)*a**2*b**2 + 9*e**(2*x)*log( e**(2*x)*a + 2*e**x*b - a)*a**4 + 18*e**(2*x)*log(e**(2*x)*a + 2*e**x*b - a)*a**2*b**2 + 9*e**(2*x)*log(e**(2*x)*a + 2*e**x*b - a)*b**4 - 9*e**(2*x) *b**4*x - 6*e**x*a**3*b - 12*e**x*a*b**3 + 3*log(e**x - 1)*a**4 + 6*log(e* *x - 1)*a**2*b**2 + 3*log(e**x + 1)*a**4 + 6*log(e**x + 1)*a**2*b**2 - 3*l og(e**(2*x)*a + 2*e**x*b - a)*a**4 - 6*log(e**(2*x)*a + 2*e**x*b - a)*a**2 *b**2 - 3*log(e**(2*x)*a + 2*e**x*b - a)*b**4 - 2*a**2*b**2 + 3*b**4*x)/(3 *a*b**4*(e**(6*x) - 3*e**(4*x) + 3*e**(2*x) - 1))