Integrand size = 13, antiderivative size = 83 \[ \int \frac {\text {csch}^6(x)}{a+b \coth (x)} \, dx=\frac {a \left (a^2-2 b^2\right ) \coth (x)}{b^4}-\frac {\left (a^2-2 b^2\right ) \coth ^2(x)}{2 b^3}+\frac {a \coth ^3(x)}{3 b^2}-\frac {\coth ^4(x)}{4 b}-\frac {\left (a^2-b^2\right )^2 \log (a+b \coth (x))}{b^5} \] Output:
a*(a^2-2*b^2)*coth(x)/b^4-1/2*(a^2-2*b^2)*coth(x)^2/b^3+1/3*a*coth(x)^3/b^ 2-1/4*coth(x)^4/b-(a^2-b^2)^2*ln(a+b*coth(x))/b^5
Time = 4.49 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.07 \[ \int \frac {\text {csch}^6(x)}{a+b \coth (x)} \, dx=\frac {6 b^2 \left (-a^2+b^2\right ) \text {csch}^2(x)-3 b^4 \text {csch}^4(x)+4 a b \coth (x) \left (3 a^2-5 b^2+b^2 \text {csch}^2(x)\right )+12 \left (a^2-b^2\right )^2 (\log (\sinh (x))-\log (b \cosh (x)+a \sinh (x)))}{12 b^5} \] Input:
Integrate[Csch[x]^6/(a + b*Coth[x]),x]
Output:
(6*b^2*(-a^2 + b^2)*Csch[x]^2 - 3*b^4*Csch[x]^4 + 4*a*b*Coth[x]*(3*a^2 - 5 *b^2 + b^2*Csch[x]^2) + 12*(a^2 - b^2)^2*(Log[Sinh[x]] - Log[b*Cosh[x] + a *Sinh[x]]))/(12*b^5)
Time = 0.51 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {3042, 25, 3987, 27, 476, 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.
| \(\displaystyle \int \frac {\text {csch}^6(x)}{a+b \coth (x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\sec \left (-\frac {\pi }{2}+i x\right )^6}{a-i b \tan \left (-\frac {\pi }{2}+i x\right )}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\sec \left (i x-\frac {\pi }{2}\right )^6}{a-i b \tan \left (i x-\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 3987 |
| \(\displaystyle -\frac {\int \frac {\left (b^2-b^2 \coth ^2(x)\right )^2}{b^4 (a+b \coth (x))}d(b \coth (x))}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {\left (b^2-b^2 \coth ^2(x)\right )^2}{a+b \coth (x)}d(b \coth (x))}{b^5}\) |
| \(\Big \downarrow \) 476 |
| \(\displaystyle -\frac {\int \left (-\left (\left (1-\frac {2 b^2}{a^2}\right ) a^3\right )-b^2 \coth ^2(x) a+b^3 \coth ^3(x)+b \left (a^2-2 b^2\right ) \coth (x)+\frac {\left (a^2-b^2\right )^2}{a+b \coth (x)}\right )d(b \coth (x))}{b^5}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\frac {1}{2} b^2 \left (a^2-2 b^2\right ) \coth ^2(x)-a b \left (a^2-2 b^2\right ) \coth (x)+\left (a^2-b^2\right )^2 \log (a+b \coth (x))-\frac {1}{3} a b^3 \coth ^3(x)+\frac {1}{4} b^4 \coth ^4(x)}{b^5}\) |
Input:
Int[Csch[x]^6/(a + b*Coth[x]),x]
Output:
-((-(a*b*(a^2 - 2*b^2)*Coth[x]) + (b^2*(a^2 - 2*b^2)*Coth[x]^2)/2 - (a*b^3 *Coth[x]^3)/3 + (b^4*Coth[x]^4)/4 + (a^2 - b^2)^2*Log[a + b*Coth[x]])/b^5)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ ExpandIntegrand[(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[p, 0]
Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_ ), x_Symbol] :> Simp[1/(b*f) Subst[Int[(a + x)^n*(1 + x^2/b^2)^(m/2 - 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x] && NeQ[a^2 + b^2, 0] && IntegerQ[m/2]
Leaf count of result is larger than twice the leaf count of optimal. \(211\) vs. \(2(77)=154\).
Time = 6.71 (sec) , antiderivative size = 212, normalized size of antiderivative = 2.55
| method | result | size |
| default | \(\frac {-\frac {\tanh \left (\frac {x}{2}\right )^{4} b^{3}}{4}+\frac {2 a \tanh \left (\frac {x}{2}\right )^{3} b^{2}}{3}-2 a^{2} b \tanh \left (\frac {x}{2}\right )^{2}+3 b^{3} \tanh \left (\frac {x}{2}\right )^{2}+8 \tanh \left (\frac {x}{2}\right ) a^{3}-14 a \,b^{2} \tanh \left (\frac {x}{2}\right )}{16 b^{4}}+\frac {\left (-16 a^{4}+32 a^{2} b^{2}-16 b^{4}\right ) \ln \left (b \tanh \left (\frac {x}{2}\right )^{2}+2 a \tanh \left (\frac {x}{2}\right )+b \right )}{16 b^{5}}-\frac {1}{64 b \tanh \left (\frac {x}{2}\right )^{4}}-\frac {4 a^{2}-6 b^{2}}{32 b^{3} \tanh \left (\frac {x}{2}\right )^{2}}+\frac {\left (16 a^{4}-32 a^{2} b^{2}+16 b^{4}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{16 b^{5}}+\frac {a}{24 b^{2} \tanh \left (\frac {x}{2}\right )^{3}}+\frac {a \left (4 a^{2}-7 b^{2}\right )}{8 b^{4} \tanh \left (\frac {x}{2}\right )}\) | \(212\) |
| risch | \(\frac {2 a^{3} {\mathrm e}^{6 x}-2 a^{2} b \,{\mathrm e}^{6 x}-2 a \,b^{2} {\mathrm e}^{6 x}+2 b^{3} {\mathrm e}^{6 x}-6 a^{3} {\mathrm e}^{4 x}+4 a^{2} b \,{\mathrm e}^{4 x}+10 a \,b^{2} {\mathrm e}^{4 x}-8 b^{3} {\mathrm e}^{4 x}+6 a^{3} {\mathrm e}^{2 x}-2 a^{2} b \,{\mathrm e}^{2 x}-\frac {34 a \,b^{2} {\mathrm e}^{2 x}}{3}+2 b^{3} {\mathrm e}^{2 x}-2 a^{3}+\frac {10 a \,b^{2}}{3}}{b^{4} \left ({\mathrm e}^{2 x}-1\right )^{4}}+\frac {\ln \left ({\mathrm e}^{2 x}-1\right ) a^{4}}{b^{5}}-\frac {2 \ln \left ({\mathrm e}^{2 x}-1\right ) a^{2}}{b^{3}}+\frac {\ln \left ({\mathrm e}^{2 x}-1\right )}{b}-\frac {\ln \left ({\mathrm e}^{2 x}-\frac {a -b}{a +b}\right ) a^{4}}{b^{5}}+\frac {2 \ln \left ({\mathrm e}^{2 x}-\frac {a -b}{a +b}\right ) a^{2}}{b^{3}}-\frac {\ln \left ({\mathrm e}^{2 x}-\frac {a -b}{a +b}\right )}{b}\) | \(256\) |
Input:
int(csch(x)^6/(a+b*coth(x)),x,method=_RETURNVERBOSE)
Output:
1/16/b^4*(-1/4*tanh(1/2*x)^4*b^3+2/3*a*tanh(1/2*x)^3*b^2-2*a^2*b*tanh(1/2* x)^2+3*b^3*tanh(1/2*x)^2+8*tanh(1/2*x)*a^3-14*a*b^2*tanh(1/2*x))+1/16/b^5* (-16*a^4+32*a^2*b^2-16*b^4)*ln(b*tanh(1/2*x)^2+2*a*tanh(1/2*x)+b)-1/64/b/t anh(1/2*x)^4-1/32*(4*a^2-6*b^2)/b^3/tanh(1/2*x)^2+1/16/b^5*(16*a^4-32*a^2* b^2+16*b^4)*ln(tanh(1/2*x))+1/24/b^2*a/tanh(1/2*x)^3+1/8*a*(4*a^2-7*b^2)/b ^4/tanh(1/2*x)
Leaf count of result is larger than twice the leaf count of optimal. 1843 vs. \(2 (77) = 154\).
Time = 0.13 (sec) , antiderivative size = 1843, normalized size of antiderivative = 22.20 \[ \int \frac {\text {csch}^6(x)}{a+b \coth (x)} \, dx=\text {Too large to display} \] Input:
integrate(csch(x)^6/(a+b*coth(x)),x, algorithm="fricas")
Output:
1/3*(6*(a^3*b - a^2*b^2 - a*b^3 + b^4)*cosh(x)^6 + 36*(a^3*b - a^2*b^2 - a *b^3 + b^4)*cosh(x)*sinh(x)^5 + 6*(a^3*b - a^2*b^2 - a*b^3 + b^4)*sinh(x)^ 6 - 6*(3*a^3*b - 2*a^2*b^2 - 5*a*b^3 + 4*b^4)*cosh(x)^4 - 6*(3*a^3*b - 2*a ^2*b^2 - 5*a*b^3 + 4*b^4 - 15*(a^3*b - a^2*b^2 - a*b^3 + b^4)*cosh(x)^2)*s inh(x)^4 - 6*a^3*b + 10*a*b^3 + 24*(5*(a^3*b - a^2*b^2 - a*b^3 + b^4)*cosh (x)^3 - (3*a^3*b - 2*a^2*b^2 - 5*a*b^3 + 4*b^4)*cosh(x))*sinh(x)^3 + 2*(9* a^3*b - 3*a^2*b^2 - 17*a*b^3 + 3*b^4)*cosh(x)^2 + 2*(45*(a^3*b - a^2*b^2 - a*b^3 + b^4)*cosh(x)^4 + 9*a^3*b - 3*a^2*b^2 - 17*a*b^3 + 3*b^4 - 18*(3*a ^3*b - 2*a^2*b^2 - 5*a*b^3 + 4*b^4)*cosh(x)^2)*sinh(x)^2 - 3*((a^4 - 2*a^2 *b^2 + b^4)*cosh(x)^8 + 8*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)*sinh(x)^7 + (a^4 - 2*a^2*b^2 + b^4)*sinh(x)^8 - 4*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^6 - 4*(a ^4 - 2*a^2*b^2 + b^4 - 7*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^2)*sinh(x)^6 + 8* (7*(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)^5 + 6*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^4 + 2*(35*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^4 + 3*a^4 - 6*a^2*b^2 + 3*b^4 - 30*(a^4 - 2*a^2*b^2 + b^4)*c osh(x)^2)*sinh(x)^4 + a^4 - 2*a^2*b^2 + b^4 + 8*(7*(a^4 - 2*a^2*b^2 + b^4) *cosh(x)^5 - 10*(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 - 4*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^2 + 4*(7*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^6 - 15*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^4 - a^4 + 2*a^2*b^2 - b^4 + 9*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^2)*sinh(x)^2 + 8*(...
\[ \int \frac {\text {csch}^6(x)}{a+b \coth (x)} \, dx=\int \frac {\operatorname {csch}^{6}{\left (x \right )}}{a + b \coth {\left (x \right )}}\, dx \] Input:
integrate(csch(x)**6/(a+b*coth(x)),x)
Output:
Integral(csch(x)**6/(a + b*coth(x)), x)
Leaf count of result is larger than twice the leaf count of optimal. 230 vs. \(2 (77) = 154\).
Time = 0.04 (sec) , antiderivative size = 230, normalized size of antiderivative = 2.77 \[ \int \frac {\text {csch}^6(x)}{a+b \coth (x)} \, dx=-\frac {2 \, {\left (3 \, a^{3} - 5 \, a b^{2} - {\left (9 \, a^{3} + 3 \, a^{2} b - 17 \, a b^{2} - 3 \, b^{3}\right )} e^{\left (-2 \, x\right )} + 3 \, {\left (3 \, a^{3} + 2 \, a^{2} b - 5 \, a b^{2} - 4 \, b^{3}\right )} e^{\left (-4 \, x\right )} - 3 \, {\left (a^{3} + a^{2} b - a b^{2} - b^{3}\right )} e^{\left (-6 \, x\right )}\right )}}{3 \, {\left (4 \, b^{4} e^{\left (-2 \, x\right )} - 6 \, b^{4} e^{\left (-4 \, x\right )} + 4 \, b^{4} e^{\left (-6 \, x\right )} - b^{4} e^{\left (-8 \, x\right )} - b^{4}\right )}} - \frac {{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \log \left (-{\left (a - b\right )} e^{\left (-2 \, x\right )} + a + b\right )}{b^{5}} + \frac {{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \log \left (e^{\left (-x\right )} + 1\right )}{b^{5}} + \frac {{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \log \left (e^{\left (-x\right )} - 1\right )}{b^{5}} \] Input:
integrate(csch(x)^6/(a+b*coth(x)),x, algorithm="maxima")
Output:
-2/3*(3*a^3 - 5*a*b^2 - (9*a^3 + 3*a^2*b - 17*a*b^2 - 3*b^3)*e^(-2*x) + 3* (3*a^3 + 2*a^2*b - 5*a*b^2 - 4*b^3)*e^(-4*x) - 3*(a^3 + a^2*b - a*b^2 - b^ 3)*e^(-6*x))/(4*b^4*e^(-2*x) - 6*b^4*e^(-4*x) + 4*b^4*e^(-6*x) - b^4*e^(-8 *x) - b^4) - (a^4 - 2*a^2*b^2 + b^4)*log(-(a - b)*e^(-2*x) + a + b)/b^5 + (a^4 - 2*a^2*b^2 + b^4)*log(e^(-x) + 1)/b^5 + (a^4 - 2*a^2*b^2 + b^4)*log( e^(-x) - 1)/b^5
Leaf count of result is larger than twice the leaf count of optimal. 317 vs. \(2 (77) = 154\).
Time = 0.14 (sec) , antiderivative size = 317, normalized size of antiderivative = 3.82 \[ \int \frac {\text {csch}^6(x)}{a+b \coth (x)} \, dx=-\frac {{\left (a^{5} + a^{4} b - 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} + b^{5}\right )} \log \left ({\left | a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} - a + b \right |}\right )}{a b^{5} + b^{6}} + \frac {{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right )}{b^{5}} - \frac {25 \, a^{4} e^{\left (8 \, x\right )} - 50 \, a^{2} b^{2} e^{\left (8 \, x\right )} + 25 \, b^{4} e^{\left (8 \, x\right )} - 100 \, a^{4} e^{\left (6 \, x\right )} - 24 \, a^{3} b e^{\left (6 \, x\right )} + 224 \, a^{2} b^{2} e^{\left (6 \, x\right )} + 24 \, a b^{3} e^{\left (6 \, x\right )} - 124 \, b^{4} e^{\left (6 \, x\right )} + 150 \, a^{4} e^{\left (4 \, x\right )} + 72 \, a^{3} b e^{\left (4 \, x\right )} - 348 \, a^{2} b^{2} e^{\left (4 \, x\right )} - 120 \, a b^{3} e^{\left (4 \, x\right )} + 246 \, b^{4} e^{\left (4 \, x\right )} - 100 \, a^{4} e^{\left (2 \, x\right )} - 72 \, a^{3} b e^{\left (2 \, x\right )} + 224 \, a^{2} b^{2} e^{\left (2 \, x\right )} + 136 \, a b^{3} e^{\left (2 \, x\right )} - 124 \, b^{4} e^{\left (2 \, x\right )} + 25 \, a^{4} + 24 \, a^{3} b - 50 \, a^{2} b^{2} - 40 \, a b^{3} + 25 \, b^{4}}{12 \, b^{5} {\left (e^{\left (2 \, x\right )} - 1\right )}^{4}} \] Input:
integrate(csch(x)^6/(a+b*coth(x)),x, algorithm="giac")
Output:
-(a^5 + a^4*b - 2*a^3*b^2 - 2*a^2*b^3 + a*b^4 + b^5)*log(abs(a*e^(2*x) + b *e^(2*x) - a + b))/(a*b^5 + b^6) + (a^4 - 2*a^2*b^2 + b^4)*log(abs(e^(2*x) - 1))/b^5 - 1/12*(25*a^4*e^(8*x) - 50*a^2*b^2*e^(8*x) + 25*b^4*e^(8*x) - 100*a^4*e^(6*x) - 24*a^3*b*e^(6*x) + 224*a^2*b^2*e^(6*x) + 24*a*b^3*e^(6*x ) - 124*b^4*e^(6*x) + 150*a^4*e^(4*x) + 72*a^3*b*e^(4*x) - 348*a^2*b^2*e^( 4*x) - 120*a*b^3*e^(4*x) + 246*b^4*e^(4*x) - 100*a^4*e^(2*x) - 72*a^3*b*e^ (2*x) + 224*a^2*b^2*e^(2*x) + 136*a*b^3*e^(2*x) - 124*b^4*e^(2*x) + 25*a^4 + 24*a^3*b - 50*a^2*b^2 - 40*a*b^3 + 25*b^4)/(b^5*(e^(2*x) - 1)^4)
Time = 2.48 (sec) , antiderivative size = 169, normalized size of antiderivative = 2.04 \[ \int \frac {\text {csch}^6(x)}{a+b \coth (x)} \, dx=\frac {8\,\left (a-3\,b\right )}{3\,b^2\,\left (3\,{\mathrm {e}}^{2\,x}-3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1\right )}-\frac {2\,{\left (a-b\right )}^2}{b^3\,\left ({\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1\right )}-\frac {4}{b\,\left (6\,{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^{2\,x}-4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}+\frac {2\,\left (a+b\right )\,{\left (a-b\right )}^2}{b^4\,\left ({\mathrm {e}}^{2\,x}-1\right )}-\frac {\ln \left (b-a+a\,{\mathrm {e}}^{2\,x}+b\,{\mathrm {e}}^{2\,x}\right )\,{\left (a+b\right )}^2\,{\left (a-b\right )}^2}{b^5}+\frac {\ln \left ({\mathrm {e}}^{2\,x}-1\right )\,{\left (a+b\right )}^2\,{\left (a-b\right )}^2}{b^5} \] Input:
int(1/(sinh(x)^6*(a + b*coth(x))),x)
Output:
(8*(a - 3*b))/(3*b^2*(3*exp(2*x) - 3*exp(4*x) + exp(6*x) - 1)) - (2*(a - b )^2)/(b^3*(exp(4*x) - 2*exp(2*x) + 1)) - 4/(b*(6*exp(4*x) - 4*exp(2*x) - 4 *exp(6*x) + exp(8*x) + 1)) + (2*(a + b)*(a - b)^2)/(b^4*(exp(2*x) - 1)) - (log(b - a + a*exp(2*x) + b*exp(2*x))*(a + b)^2*(a - b)^2)/b^5 + (log(exp( 2*x) - 1)*(a + b)^2*(a - b)^2)/b^5
Time = 0.29 (sec) , antiderivative size = 1103, normalized size of antiderivative = 13.29 \[ \int \frac {\text {csch}^6(x)}{a+b \coth (x)} \, dx =\text {Too large to display} \] Input:
int(csch(x)^6/(a+b*coth(x)),x)
Output:
(6*e**(8*x)*log(e**x - 1)*a**4 - 12*e**(8*x)*log(e**x - 1)*a**2*b**2 + 6*e **(8*x)*log(e**x - 1)*b**4 + 6*e**(8*x)*log(e**x + 1)*a**4 - 12*e**(8*x)*l og(e**x + 1)*a**2*b**2 + 6*e**(8*x)*log(e**x + 1)*b**4 - 6*e**(8*x)*log(e* *(2*x)*a + e**(2*x)*b - a + b)*a**4 + 12*e**(8*x)*log(e**(2*x)*a + e**(2*x )*b - a + b)*a**2*b**2 - 6*e**(8*x)*log(e**(2*x)*a + e**(2*x)*b - a + b)*b **4 + 3*e**(8*x)*a**3*b - 3*e**(8*x)*a**2*b**2 - 3*e**(8*x)*a*b**3 + 3*e** (8*x)*b**4 - 24*e**(6*x)*log(e**x - 1)*a**4 + 48*e**(6*x)*log(e**x - 1)*a* *2*b**2 - 24*e**(6*x)*log(e**x - 1)*b**4 - 24*e**(6*x)*log(e**x + 1)*a**4 + 48*e**(6*x)*log(e**x + 1)*a**2*b**2 - 24*e**(6*x)*log(e**x + 1)*b**4 + 2 4*e**(6*x)*log(e**(2*x)*a + e**(2*x)*b - a + b)*a**4 - 48*e**(6*x)*log(e** (2*x)*a + e**(2*x)*b - a + b)*a**2*b**2 + 24*e**(6*x)*log(e**(2*x)*a + e** (2*x)*b - a + b)*b**4 + 36*e**(4*x)*log(e**x - 1)*a**4 - 72*e**(4*x)*log(e **x - 1)*a**2*b**2 + 36*e**(4*x)*log(e**x - 1)*b**4 + 36*e**(4*x)*log(e**x + 1)*a**4 - 72*e**(4*x)*log(e**x + 1)*a**2*b**2 + 36*e**(4*x)*log(e**x + 1)*b**4 - 36*e**(4*x)*log(e**(2*x)*a + e**(2*x)*b - a + b)*a**4 + 72*e**(4 *x)*log(e**(2*x)*a + e**(2*x)*b - a + b)*a**2*b**2 - 36*e**(4*x)*log(e**(2 *x)*a + e**(2*x)*b - a + b)*b**4 - 18*e**(4*x)*a**3*b + 6*e**(4*x)*a**2*b* *2 + 42*e**(4*x)*a*b**3 - 30*e**(4*x)*b**4 - 24*e**(2*x)*log(e**x - 1)*a** 4 + 48*e**(2*x)*log(e**x - 1)*a**2*b**2 - 24*e**(2*x)*log(e**x - 1)*b**4 - 24*e**(2*x)*log(e**x + 1)*a**4 + 48*e**(2*x)*log(e**x + 1)*a**2*b**2 -...