Integrand size = 11, antiderivative size = 98 \[ \int \frac {1}{(a \coth (x)+b \text {csch}(x))^5} \, dx=-\frac {\left (a^2-b^2\right )^2}{4 a^5 (b+a \cosh (x))^4}-\frac {4 b \left (a^2-b^2\right )}{3 a^5 (b+a \cosh (x))^3}+\frac {a^2-3 b^2}{a^5 (b+a \cosh (x))^2}+\frac {4 b}{a^5 (b+a \cosh (x))}+\frac {\log (b+a \cosh (x))}{a^5} \] Output:
-1/4*(a^2-b^2)^2/a^5/(b+a*cosh(x))^4-4/3*b*(a^2-b^2)/a^5/(b+a*cosh(x))^3+( a^2-3*b^2)/a^5/(b+a*cosh(x))^2+4*b/a^5/(b+a*cosh(x))+ln(b+a*cosh(x))/a^5
Time = 0.30 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.41 \[ \int \frac {1}{(a \coth (x)+b \text {csch}(x))^5} \, dx=\frac {-3 a^4+2 a^2 b^2+25 b^4+12 b^4 \log (b+a \cosh (x))+12 a^4 \cosh ^4(x) \log (b+a \cosh (x))+48 a^3 b \cosh ^3(x) (1+\log (b+a \cosh (x)))+12 a^2 \cosh ^2(x) \left (a^2+9 b^2+6 b^2 \log (b+a \cosh (x))\right )+8 a b \cosh (x) \left (a^2+11 b^2+6 b^2 \log (b+a \cosh (x))\right )}{12 a^5 (b+a \cosh (x))^4} \] Input:
Integrate[(a*Coth[x] + b*Csch[x])^(-5),x]
Output:
(-3*a^4 + 2*a^2*b^2 + 25*b^4 + 12*b^4*Log[b + a*Cosh[x]] + 12*a^4*Cosh[x]^ 4*Log[b + a*Cosh[x]] + 48*a^3*b*Cosh[x]^3*(1 + Log[b + a*Cosh[x]]) + 12*a^ 2*Cosh[x]^2*(a^2 + 9*b^2 + 6*b^2*Log[b + a*Cosh[x]]) + 8*a*b*Cosh[x]*(a^2 + 11*b^2 + 6*b^2*Log[b + a*Cosh[x]]))/(12*a^5*(b + a*Cosh[x])^4)
Time = 0.67 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.88, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.818, Rules used = {3042, 4892, 26, 26, 3042, 26, 3147, 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.
| \(\displaystyle \int \frac {1}{(a \coth (x)+b \text {csch}(x))^5} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(i a \cot (i x)+i b \csc (i x))^5}dx\) |
| \(\Big \downarrow \) 4892 |
| \(\displaystyle \int \frac {i \sinh ^5(x)}{(i a \cosh (x)+i b)^5}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int -\frac {i \sinh ^5(x)}{(b+a \cosh (x))^5}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \int \frac {\sinh ^5(x)}{(a \cosh (x)+b)^5}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i \cos \left (-\frac {\pi }{2}+i x\right )^5}{\left (b-a \sin \left (-\frac {\pi }{2}+i x\right )\right )^5}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {\cos \left (i x-\frac {\pi }{2}\right )^5}{\left (b-a \sin \left (i x-\frac {\pi }{2}\right )\right )^5}dx\) |
| \(\Big \downarrow \) 3147 |
| \(\displaystyle \frac {\int \frac {\left (a^2-a^2 \cosh ^2(x)\right )^2}{(b+a \cosh (x))^5}d(a \cosh (x))}{a^5}\) |
| \(\Big \downarrow \) 476 |
| \(\displaystyle \frac {\int \left (\frac {\left (a^2-b^2\right )^2}{(b+a \cosh (x))^5}+\frac {1}{b+a \cosh (x)}-\frac {4 b}{(b+a \cosh (x))^2}-\frac {2 \left (a^2-3 b^2\right )}{(b+a \cosh (x))^3}-\frac {4 b \left (b^2-a^2\right )}{(b+a \cosh (x))^4}\right )d(a \cosh (x))}{a^5}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {\left (a^2-b^2\right )^2}{4 (a \cosh (x)+b)^4}-\frac {4 b \left (a^2-b^2\right )}{3 (a \cosh (x)+b)^3}+\frac {a^2-3 b^2}{(a \cosh (x)+b)^2}+\frac {4 b}{a \cosh (x)+b}+\log (a \cosh (x)+b)}{a^5}\) |
Input:
Int[(a*Coth[x] + b*Csch[x])^(-5),x]
Output:
(-1/4*(a^2 - b^2)^2/(b + a*Cosh[x])^4 - (4*b*(a^2 - b^2))/(3*(b + a*Cosh[x ])^3) + (a^2 - 3*b^2)/(b + a*Cosh[x])^2 + (4*b)/(b + a*Cosh[x]) + Log[b + a*Cosh[x]])/a^5
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[((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[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m _.), x_Symbol] :> Simp[1/(b^p*f) Subst[Int[(a + x)^m*(b^2 - x^2)^((p - 1) /2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && IntegerQ[(p - 1)/2] && NeQ[a^2 - b^2, 0]
Int[(cot[(c_.) + (d_.)*(x_)]^(n_.)*(a_.) + csc[(c_.) + (d_.)*(x_)]^(n_.)*(b _.))^(p_)*(u_.), x_Symbol] :> Int[ActivateTrig[u]*Csc[c + d*x]^(n*p)*(b + a *Cos[c + d*x]^n)^p, x] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p]
Time = 147.51 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.78
| method | result | size |
| risch | \(-\frac {x}{a^{5}}+\frac {4 \left (6 a^{3} b \,{\mathrm e}^{6 x}+3 a^{4} {\mathrm e}^{5 x}+27 a^{2} b^{2} {\mathrm e}^{5 x}+22 a^{3} b \,{\mathrm e}^{4 x}+44 a \,b^{3} {\mathrm e}^{4 x}+3 a^{4} {\mathrm e}^{3 x}+56 a^{2} b^{2} {\mathrm e}^{3 x}+25 b^{4} {\mathrm e}^{3 x}+22 a^{3} b \,{\mathrm e}^{2 x}+44 a \,b^{3} {\mathrm e}^{2 x}+3 a^{4} {\mathrm e}^{x}+27 a^{2} b^{2} {\mathrm e}^{x}+6 a^{3} b \right ) {\mathrm e}^{x}}{3 a^{5} \left ({\mathrm e}^{2 x} a +2 b \,{\mathrm e}^{x}+a \right )^{4}}+\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 b \,{\mathrm e}^{x}}{a}+1\right )}{a^{5}}\) | \(174\) |
| default | \(-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{a^{5}}+\frac {\frac {8 a^{3} \left (3 a^{2}+2 a b -b^{2}\right )}{3 \left (a -b \right )^{2} \left (a \tanh \left (\frac {x}{2}\right )^{2}-b \tanh \left (\frac {x}{2}\right )^{2}+a +b \right )^{3}}-\frac {2 a}{a \tanh \left (\frac {x}{2}\right )^{2}-b \tanh \left (\frac {x}{2}\right )^{2}+a +b}+\ln \left (a \tanh \left (\frac {x}{2}\right )^{2}-b \tanh \left (\frac {x}{2}\right )^{2}+a +b \right )-\frac {4 a^{4} \left (a^{2}+2 a b +b^{2}\right )}{\left (a -b \right )^{2} \left (a \tanh \left (\frac {x}{2}\right )^{2}-b \tanh \left (\frac {x}{2}\right )^{2}+a +b \right )^{4}}-\frac {2 a^{2}}{\left (a \tanh \left (\frac {x}{2}\right )^{2}-b \tanh \left (\frac {x}{2}\right )^{2}+a +b \right )^{2}}}{a^{5}}-\frac {\ln \left (1+\tanh \left (\frac {x}{2}\right )\right )}{a^{5}}\) | \(198\) |
Input:
int(1/(a*coth(x)+b*csch(x))^5,x,method=_RETURNVERBOSE)
Output:
-1/a^5*x+4/3*(6*a^3*b*exp(6*x)+3*a^4*exp(5*x)+27*a^2*b^2*exp(5*x)+22*a^3*b *exp(4*x)+44*a*b^3*exp(4*x)+3*a^4*exp(3*x)+56*a^2*b^2*exp(3*x)+25*b^4*exp( 3*x)+22*a^3*b*exp(2*x)+44*a*b^3*exp(2*x)+3*a^4*exp(x)+27*a^2*b^2*exp(x)+6* a^3*b)/a^5*exp(x)/(exp(2*x)*a+2*b*exp(x)+a)^4+1/a^5*ln(exp(2*x)+2*b/a*exp( x)+1)
Leaf count of result is larger than twice the leaf count of optimal. 2564 vs. \(2 (94) = 188\).
Time = 0.13 (sec) , antiderivative size = 2564, normalized size of antiderivative = 26.16 \[ \int \frac {1}{(a \coth (x)+b \text {csch}(x))^5} \, dx=\text {Too large to display} \] Input:
integrate(1/(a*coth(x)+b*csch(x))^5,x, algorithm="fricas")
Output:
-1/3*(3*a^4*x*cosh(x)^8 + 3*a^4*x*sinh(x)^8 + 24*(a^3*b*x - a^3*b)*cosh(x) ^7 + 24*(a^4*x*cosh(x) + a^3*b*x - a^3*b)*sinh(x)^7 - 12*(a^4 + 9*a^2*b^2 - (a^4 + 6*a^2*b^2)*x)*cosh(x)^6 + 12*(7*a^4*x*cosh(x)^2 - a^4 - 9*a^2*b^2 + (a^4 + 6*a^2*b^2)*x + 14*(a^3*b*x - a^3*b)*cosh(x))*sinh(x)^6 - 8*(11*a ^3*b + 22*a*b^3 - 3*(3*a^3*b + 4*a*b^3)*x)*cosh(x)^5 + 8*(21*a^4*x*cosh(x) ^3 - 11*a^3*b - 22*a*b^3 + 63*(a^3*b*x - a^3*b)*cosh(x)^2 + 3*(3*a^3*b + 4 *a*b^3)*x - 9*(a^4 + 9*a^2*b^2 - (a^4 + 6*a^2*b^2)*x)*cosh(x))*sinh(x)^5 + 3*a^4*x - 2*(6*a^4 + 112*a^2*b^2 + 50*b^4 - 3*(3*a^4 + 24*a^2*b^2 + 8*b^4 )*x)*cosh(x)^4 + 2*(105*a^4*x*cosh(x)^4 - 6*a^4 - 112*a^2*b^2 - 50*b^4 + 4 20*(a^3*b*x - a^3*b)*cosh(x)^3 - 90*(a^4 + 9*a^2*b^2 - (a^4 + 6*a^2*b^2)*x )*cosh(x)^2 + 3*(3*a^4 + 24*a^2*b^2 + 8*b^4)*x - 20*(11*a^3*b + 22*a*b^3 - 3*(3*a^3*b + 4*a*b^3)*x)*cosh(x))*sinh(x)^4 - 8*(11*a^3*b + 22*a*b^3 - 3* (3*a^3*b + 4*a*b^3)*x)*cosh(x)^3 + 8*(21*a^4*x*cosh(x)^5 + 105*(a^3*b*x - a^3*b)*cosh(x)^4 - 11*a^3*b - 22*a*b^3 - 30*(a^4 + 9*a^2*b^2 - (a^4 + 6*a^ 2*b^2)*x)*cosh(x)^3 - 10*(11*a^3*b + 22*a*b^3 - 3*(3*a^3*b + 4*a*b^3)*x)*c osh(x)^2 + 3*(3*a^3*b + 4*a*b^3)*x - (6*a^4 + 112*a^2*b^2 + 50*b^4 - 3*(3* a^4 + 24*a^2*b^2 + 8*b^4)*x)*cosh(x))*sinh(x)^3 - 12*(a^4 + 9*a^2*b^2 - (a ^4 + 6*a^2*b^2)*x)*cosh(x)^2 + 4*(21*a^4*x*cosh(x)^6 + 126*(a^3*b*x - a^3* b)*cosh(x)^5 - 45*(a^4 + 9*a^2*b^2 - (a^4 + 6*a^2*b^2)*x)*cosh(x)^4 - 3*a^ 4 - 27*a^2*b^2 - 20*(11*a^3*b + 22*a*b^3 - 3*(3*a^3*b + 4*a*b^3)*x)*cos...
\[ \int \frac {1}{(a \coth (x)+b \text {csch}(x))^5} \, dx=\int \frac {1}{\left (a \coth {\left (x \right )} + b \operatorname {csch}{\left (x \right )}\right )^{5}}\, dx \] Input:
integrate(1/(a*coth(x)+b*csch(x))**5,x)
Output:
Integral((a*coth(x) + b*csch(x))**(-5), x)
Leaf count of result is larger than twice the leaf count of optimal. 285 vs. \(2 (94) = 188\).
Time = 0.07 (sec) , antiderivative size = 285, normalized size of antiderivative = 2.91 \[ \int \frac {1}{(a \coth (x)+b \text {csch}(x))^5} \, dx=\frac {4 \, {\left (6 \, a^{3} b e^{\left (-x\right )} + 6 \, a^{3} b e^{\left (-7 \, x\right )} + 3 \, {\left (a^{4} + 9 \, a^{2} b^{2}\right )} e^{\left (-2 \, x\right )} + 22 \, {\left (a^{3} b + 2 \, a b^{3}\right )} e^{\left (-3 \, x\right )} + {\left (3 \, a^{4} + 56 \, a^{2} b^{2} + 25 \, b^{4}\right )} e^{\left (-4 \, x\right )} + 22 \, {\left (a^{3} b + 2 \, a b^{3}\right )} e^{\left (-5 \, x\right )} + 3 \, {\left (a^{4} + 9 \, a^{2} b^{2}\right )} e^{\left (-6 \, x\right )}\right )}}{3 \, {\left (8 \, a^{8} b e^{\left (-x\right )} + 8 \, a^{8} b e^{\left (-7 \, x\right )} + a^{9} e^{\left (-8 \, x\right )} + a^{9} + 4 \, {\left (a^{9} + 6 \, a^{7} b^{2}\right )} e^{\left (-2 \, x\right )} + 8 \, {\left (3 \, a^{8} b + 4 \, a^{6} b^{3}\right )} e^{\left (-3 \, x\right )} + 2 \, {\left (3 \, a^{9} + 24 \, a^{7} b^{2} + 8 \, a^{5} b^{4}\right )} e^{\left (-4 \, x\right )} + 8 \, {\left (3 \, a^{8} b + 4 \, a^{6} b^{3}\right )} e^{\left (-5 \, x\right )} + 4 \, {\left (a^{9} + 6 \, a^{7} b^{2}\right )} e^{\left (-6 \, x\right )}\right )}} + \frac {x}{a^{5}} + \frac {\log \left (2 \, b e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} + a\right )}{a^{5}} \] Input:
integrate(1/(a*coth(x)+b*csch(x))^5,x, algorithm="maxima")
Output:
4/3*(6*a^3*b*e^(-x) + 6*a^3*b*e^(-7*x) + 3*(a^4 + 9*a^2*b^2)*e^(-2*x) + 22 *(a^3*b + 2*a*b^3)*e^(-3*x) + (3*a^4 + 56*a^2*b^2 + 25*b^4)*e^(-4*x) + 22* (a^3*b + 2*a*b^3)*e^(-5*x) + 3*(a^4 + 9*a^2*b^2)*e^(-6*x))/(8*a^8*b*e^(-x) + 8*a^8*b*e^(-7*x) + a^9*e^(-8*x) + a^9 + 4*(a^9 + 6*a^7*b^2)*e^(-2*x) + 8*(3*a^8*b + 4*a^6*b^3)*e^(-3*x) + 2*(3*a^9 + 24*a^7*b^2 + 8*a^5*b^4)*e^(- 4*x) + 8*(3*a^8*b + 4*a^6*b^3)*e^(-5*x) + 4*(a^9 + 6*a^7*b^2)*e^(-6*x)) + x/a^5 + log(2*b*e^(-x) + a*e^(-2*x) + a)/a^5
Time = 0.14 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.38 \[ \int \frac {1}{(a \coth (x)+b \text {csch}(x))^5} \, dx=\frac {\log \left ({\left | a {\left (e^{\left (-x\right )} + e^{x}\right )} + 2 \, b \right |}\right )}{a^{5}} - \frac {25 \, a^{3} {\left (e^{\left (-x\right )} + e^{x}\right )}^{4} + 104 \, a^{2} b {\left (e^{\left (-x\right )} + e^{x}\right )}^{3} - 48 \, a^{3} {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} + 168 \, a b^{2} {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 64 \, a^{2} b {\left (e^{\left (-x\right )} + e^{x}\right )} + 96 \, b^{3} {\left (e^{\left (-x\right )} + e^{x}\right )} + 48 \, a^{3} - 32 \, a b^{2}}{12 \, {\left (a {\left (e^{\left (-x\right )} + e^{x}\right )} + 2 \, b\right )}^{4} a^{4}} \] Input:
integrate(1/(a*coth(x)+b*csch(x))^5,x, algorithm="giac")
Output:
log(abs(a*(e^(-x) + e^x) + 2*b))/a^5 - 1/12*(25*a^3*(e^(-x) + e^x)^4 + 104 *a^2*b*(e^(-x) + e^x)^3 - 48*a^3*(e^(-x) + e^x)^2 + 168*a*b^2*(e^(-x) + e^ x)^2 - 64*a^2*b*(e^(-x) + e^x) + 96*b^3*(e^(-x) + e^x) + 48*a^3 - 32*a*b^2 )/((a*(e^(-x) + e^x) + 2*b)^4*a^4)
Timed out. \[ \int \frac {1}{(a \coth (x)+b \text {csch}(x))^5} \, dx=\int \frac {1}{{\left (\frac {b}{\mathrm {sinh}\left (x\right )}+a\,\mathrm {coth}\left (x\right )\right )}^5} \,d x \] Input:
int(1/(b/sinh(x) + a*coth(x))^5,x)
Output:
int(1/(b/sinh(x) + a*coth(x))^5, x)
Time = 0.22 (sec) , antiderivative size = 853, normalized size of antiderivative = 8.70 \[ \int \frac {1}{(a \coth (x)+b \text {csch}(x))^5} \, dx=\frac {36 e^{6 x} a^{2} b^{2}+80 e^{5 x} a \,b^{3}-48 e^{4 x} b^{4} x +16 e^{3 x} a^{3} b +80 e^{3 x} a \,b^{3}+36 e^{2 x} a^{2} b^{2}-3 a^{4}+24 e^{7 x} \mathrm {log}\left (e^{2 x} a +2 e^{x} b +a \right ) a^{3} b -24 e^{7 x} a^{3} b x +72 e^{6 x} \mathrm {log}\left (e^{2 x} a +2 e^{x} b +a \right ) a^{2} b^{2}-72 e^{6 x} a^{2} b^{2} x +72 e^{5 x} \mathrm {log}\left (e^{2 x} a +2 e^{x} b +a \right ) a^{3} b +96 e^{5 x} \mathrm {log}\left (e^{2 x} a +2 e^{x} b +a \right ) a \,b^{3}-72 e^{5 x} a^{3} b x -96 e^{5 x} a \,b^{3} x +144 e^{4 x} \mathrm {log}\left (e^{2 x} a +2 e^{x} b +a \right ) a^{2} b^{2}-144 e^{4 x} a^{2} b^{2} x +72 e^{3 x} \mathrm {log}\left (e^{2 x} a +2 e^{x} b +a \right ) a^{3} b +96 e^{3 x} \mathrm {log}\left (e^{2 x} a +2 e^{x} b +a \right ) a \,b^{3}-72 e^{3 x} a^{3} b x -96 e^{3 x} a \,b^{3} x +72 e^{2 x} \mathrm {log}\left (e^{2 x} a +2 e^{x} b +a \right ) a^{2} b^{2}-72 e^{2 x} a^{2} b^{2} x +24 e^{x} \mathrm {log}\left (e^{2 x} a +2 e^{x} b +a \right ) a^{3} b -24 e^{x} a^{3} b x -12 e^{6 x} a^{4} x +16 e^{5 x} a^{3} b -18 e^{4 x} a^{4} x -12 e^{2 x} a^{4} x -3 e^{8 x} a^{4}-6 e^{4 x} a^{4}+3 \,\mathrm {log}\left (e^{2 x} a +2 e^{x} b +a \right ) a^{4}+3 e^{8 x} \mathrm {log}\left (e^{2 x} a +2 e^{x} b +a \right ) a^{4}-3 e^{8 x} a^{4} x +12 e^{6 x} \mathrm {log}\left (e^{2 x} a +2 e^{x} b +a \right ) a^{4}+18 e^{4 x} \mathrm {log}\left (e^{2 x} a +2 e^{x} b +a \right ) a^{4}+48 e^{4 x} \mathrm {log}\left (e^{2 x} a +2 e^{x} b +a \right ) b^{4}+80 e^{4 x} a^{2} b^{2}+12 e^{2 x} \mathrm {log}\left (e^{2 x} a +2 e^{x} b +a \right ) a^{4}+52 e^{4 x} b^{4}-3 a^{4} x}{3 a^{5} \left (e^{8 x} a^{4}+8 e^{7 x} a^{3} b +4 e^{6 x} a^{4}+24 e^{6 x} a^{2} b^{2}+24 e^{5 x} a^{3} b +32 e^{5 x} a \,b^{3}+6 e^{4 x} a^{4}+48 e^{4 x} a^{2} b^{2}+16 e^{4 x} b^{4}+24 e^{3 x} a^{3} b +32 e^{3 x} a \,b^{3}+4 e^{2 x} a^{4}+24 e^{2 x} a^{2} b^{2}+8 e^{x} a^{3} b +a^{4}\right )} \] Input:
int(1/(a*coth(x)+b*csch(x))^5,x)
Output:
(3*e**(8*x)*log(e**(2*x)*a + 2*e**x*b + a)*a**4 - 3*e**(8*x)*a**4*x - 3*e* *(8*x)*a**4 + 24*e**(7*x)*log(e**(2*x)*a + 2*e**x*b + a)*a**3*b - 24*e**(7 *x)*a**3*b*x + 12*e**(6*x)*log(e**(2*x)*a + 2*e**x*b + a)*a**4 + 72*e**(6* x)*log(e**(2*x)*a + 2*e**x*b + a)*a**2*b**2 - 12*e**(6*x)*a**4*x - 72*e**( 6*x)*a**2*b**2*x + 36*e**(6*x)*a**2*b**2 + 72*e**(5*x)*log(e**(2*x)*a + 2* e**x*b + a)*a**3*b + 96*e**(5*x)*log(e**(2*x)*a + 2*e**x*b + a)*a*b**3 - 7 2*e**(5*x)*a**3*b*x + 16*e**(5*x)*a**3*b - 96*e**(5*x)*a*b**3*x + 80*e**(5 *x)*a*b**3 + 18*e**(4*x)*log(e**(2*x)*a + 2*e**x*b + a)*a**4 + 144*e**(4*x )*log(e**(2*x)*a + 2*e**x*b + a)*a**2*b**2 + 48*e**(4*x)*log(e**(2*x)*a + 2*e**x*b + a)*b**4 - 18*e**(4*x)*a**4*x - 6*e**(4*x)*a**4 - 144*e**(4*x)*a **2*b**2*x + 80*e**(4*x)*a**2*b**2 - 48*e**(4*x)*b**4*x + 52*e**(4*x)*b**4 + 72*e**(3*x)*log(e**(2*x)*a + 2*e**x*b + a)*a**3*b + 96*e**(3*x)*log(e** (2*x)*a + 2*e**x*b + a)*a*b**3 - 72*e**(3*x)*a**3*b*x + 16*e**(3*x)*a**3*b - 96*e**(3*x)*a*b**3*x + 80*e**(3*x)*a*b**3 + 12*e**(2*x)*log(e**(2*x)*a + 2*e**x*b + a)*a**4 + 72*e**(2*x)*log(e**(2*x)*a + 2*e**x*b + a)*a**2*b** 2 - 12*e**(2*x)*a**4*x - 72*e**(2*x)*a**2*b**2*x + 36*e**(2*x)*a**2*b**2 + 24*e**x*log(e**(2*x)*a + 2*e**x*b + a)*a**3*b - 24*e**x*a**3*b*x + 3*log( e**(2*x)*a + 2*e**x*b + a)*a**4 - 3*a**4*x - 3*a**4)/(3*a**5*(e**(8*x)*a** 4 + 8*e**(7*x)*a**3*b + 4*e**(6*x)*a**4 + 24*e**(6*x)*a**2*b**2 + 24*e**(5 *x)*a**3*b + 32*e**(5*x)*a*b**3 + 6*e**(4*x)*a**4 + 48*e**(4*x)*a**2*b*...