Integrand size = 13, antiderivative size = 121 \[ \int \frac {\tanh ^7(x)}{a+b \text {sech}(x)} \, dx=\frac {\log (\cosh (x))}{a}-\frac {\left (a^2-b^2\right )^3 \log (a+b \text {sech}(x))}{a b^6}+\frac {\left (a^4-3 a^2 b^2+3 b^4\right ) \text {sech}(x)}{b^5}-\frac {a \left (a^2-3 b^2\right ) \text {sech}^2(x)}{2 b^4}+\frac {\left (a^2-3 b^2\right ) \text {sech}^3(x)}{3 b^3}-\frac {a \text {sech}^4(x)}{4 b^2}+\frac {\text {sech}^5(x)}{5 b} \]
ln(cosh(x))/a-(a^2-b^2)^3*ln(a+b*sech(x))/a/b^6+(a^4-3*a^2*b^2+3*b^4)*sech (x)/b^5-1/2*a*(a^2-3*b^2)*sech(x)^2/b^4+1/3*(a^2-3*b^2)*sech(x)^3/b^3-1/4* a*sech(x)^4/b^2+1/5*sech(x)^5/b
Time = 0.31 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00 \[ \int \frac {\tanh ^7(x)}{a+b \text {sech}(x)} \, dx=\frac {\log (\cosh (x))}{a}-\frac {\left (a^2-b^2\right )^3 \log (a+b \text {sech}(x))}{a b^6}+\frac {\left (a^4-3 a^2 b^2+3 b^4\right ) \text {sech}(x)}{b^5}-\frac {a \left (a^2-3 b^2\right ) \text {sech}^2(x)}{2 b^4}+\frac {\left (a^2-3 b^2\right ) \text {sech}^3(x)}{3 b^3}-\frac {a \text {sech}^4(x)}{4 b^2}+\frac {\text {sech}^5(x)}{5 b} \]
Log[Cosh[x]]/a - ((a^2 - b^2)^3*Log[a + b*Sech[x]])/(a*b^6) + ((a^4 - 3*a^ 2*b^2 + 3*b^4)*Sech[x])/b^5 - (a*(a^2 - 3*b^2)*Sech[x]^2)/(2*b^4) + ((a^2 - 3*b^2)*Sech[x]^3)/(3*b^3) - (a*Sech[x]^4)/(4*b^2) + Sech[x]^5/(5*b)
Time = 0.35 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.04, 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.
| \(\displaystyle \int \frac {\tanh ^7(x)}{a+b \text {sech}(x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i \cot \left (\frac {\pi }{2}+i x\right )^7}{a+b \csc \left (\frac {\pi }{2}+i x\right )}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {\cot \left (i x+\frac {\pi }{2}\right )^7}{a+b \csc \left (i x+\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 4373 |
| \(\displaystyle -\frac {\int \frac {\cosh (x) \left (b^2-b^2 \text {sech}^2(x)\right )^3}{b (a+b \text {sech}(x))}d(b \text {sech}(x))}{b^6}\) |
| \(\Big \downarrow \) 522 |
| \(\displaystyle -\frac {\int \left (\frac {\cosh (x) b^5}{a}-\text {sech}^4(x) b^4+a \text {sech}^3(x) b^3-\left (a^2-3 b^2\right ) \text {sech}^2(x) b^2+a \left (a^2-3 b^2\right ) \text {sech}(x) b-a^4 \left (\frac {3 \left (b^2-a^2\right ) b^2}{a^4}+1\right )+\frac {\left (a^2-b^2\right )^3}{a (a+b \text {sech}(x))}\right )d(b \text {sech}(x))}{b^6}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\frac {1}{2} a b^2 \left (a^2-3 b^2\right ) \text {sech}^2(x)+\frac {\left (a^2-b^2\right )^3 \log (a+b \text {sech}(x))}{a}-\frac {1}{3} b^3 \left (a^2-3 b^2\right ) \text {sech}^3(x)-b \left (a^4-3 a^2 b^2+3 b^4\right ) \text {sech}(x)+\frac {b^6 \log (b \text {sech}(x))}{a}+\frac {1}{4} a b^4 \text {sech}^4(x)-\frac {1}{5} b^5 \text {sech}^5(x)}{b^6}\) |
-(((b^6*Log[b*Sech[x]])/a + ((a^2 - b^2)^3*Log[a + b*Sech[x]])/a - b*(a^4 - 3*a^2*b^2 + 3*b^4)*Sech[x] + (a*b^2*(a^2 - 3*b^2)*Sech[x]^2)/2 - (b^3*(a ^2 - 3*b^2)*Sech[x]^3)/3 + (a*b^4*Sech[x]^4)/4 - (b^5*Sech[x]^5)/5)/b^6)
3.2.13.3.1 Defintions of rubi rules used
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. \(245\) vs. \(2(113)=226\).
Time = 2.31 (sec) , antiderivative size = 246, normalized size of antiderivative = 2.03
| method | result | size |
| default | \(-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{a}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{a}-\frac {\left (a -b \right )^{3} \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )^{2} a -\tanh \left (\frac {x}{2}\right )^{2} b +a +b \right )}{a \,b^{6}}+\frac {\frac {8 b^{3} \left (a^{2}+3 a b +3 b^{2}\right )}{3 \left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )^{3}}+a \left (a^{4}-3 a^{2} b^{2}+3 b^{4}\right ) \ln \left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )+\frac {32 b^{5}}{5 \left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )^{5}}-\frac {2 b^{2} \left (a^{3}+2 a^{2} b -2 b^{3}\right )}{\left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )^{2}}+\frac {2 b \left (a^{4}+a^{3} b -2 a^{2} b^{2}-2 a \,b^{3}+b^{4}\right )}{1+\tanh \left (\frac {x}{2}\right )^{2}}-\frac {4 b^{4} \left (a +4 b \right )}{\left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )^{4}}}{b^{6}}\) | \(246\) |
| risch | \(-\frac {x}{a}+\frac {2 \,{\mathrm e}^{x} \left (15 a^{4} {\mathrm e}^{8 x}-45 a^{2} b^{2} {\mathrm e}^{8 x}+45 b^{4} {\mathrm e}^{8 x}-15 a^{3} b \,{\mathrm e}^{7 x}+45 a \,b^{3} {\mathrm e}^{7 x}+60 a^{4} {\mathrm e}^{6 x}-160 a^{2} b^{2} {\mathrm e}^{6 x}+120 b^{4} {\mathrm e}^{6 x}-45 a^{3} b \,{\mathrm e}^{5 x}+105 a \,b^{3} {\mathrm e}^{5 x}+90 a^{4} {\mathrm e}^{4 x}-230 a^{2} b^{2} {\mathrm e}^{4 x}+198 b^{4} {\mathrm e}^{4 x}-45 a^{3} b \,{\mathrm e}^{3 x}+105 a \,b^{3} {\mathrm e}^{3 x}+60 a^{4} {\mathrm e}^{2 x}-160 a^{2} b^{2} {\mathrm e}^{2 x}+120 b^{4} {\mathrm e}^{2 x}-15 a^{3} b \,{\mathrm e}^{x}+45 a \,b^{3} {\mathrm e}^{x}+15 a^{4}-45 a^{2} b^{2}+45 b^{4}\right )}{15 b^{5} \left (1+{\mathrm e}^{2 x}\right )^{5}}+\frac {a^{5} \ln \left (1+{\mathrm e}^{2 x}\right )}{b^{6}}-\frac {3 a^{3} \ln \left (1+{\mathrm e}^{2 x}\right )}{b^{4}}+\frac {3 a \ln \left (1+{\mathrm e}^{2 x}\right )}{b^{2}}-\frac {a^{5} \ln \left ({\mathrm e}^{2 x}+\frac {2 b \,{\mathrm e}^{x}}{a}+1\right )}{b^{6}}+\frac {3 a^{3} \ln \left ({\mathrm e}^{2 x}+\frac {2 b \,{\mathrm e}^{x}}{a}+1\right )}{b^{4}}-\frac {3 a \ln \left ({\mathrm e}^{2 x}+\frac {2 b \,{\mathrm e}^{x}}{a}+1\right )}{b^{2}}+\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 b \,{\mathrm e}^{x}}{a}+1\right )}{a}\) | \(366\) |
-1/a*ln(tanh(1/2*x)+1)-1/a*ln(tanh(1/2*x)-1)-(a-b)^3*(a^3+3*a^2*b+3*a*b^2+ b^3)/a/b^6*ln(tanh(1/2*x)^2*a-tanh(1/2*x)^2*b+a+b)+1/b^6*(8/3*b^3*(a^2+3*a *b+3*b^2)/(1+tanh(1/2*x)^2)^3+a*(a^4-3*a^2*b^2+3*b^4)*ln(1+tanh(1/2*x)^2)+ 32/5*b^5/(1+tanh(1/2*x)^2)^5-2*b^2*(a^3+2*a^2*b-2*b^3)/(1+tanh(1/2*x)^2)^2 +2*b*(a^4+a^3*b-2*a^2*b^2-2*a*b^3+b^4)/(1+tanh(1/2*x)^2)-4*b^4*(a+4*b)/(1+ tanh(1/2*x)^2)^4)
Leaf count of result is larger than twice the leaf count of optimal. 4077 vs. \(2 (113) = 226\).
Time = 0.31 (sec) , antiderivative size = 4077, normalized size of antiderivative = 33.69 \[ \int \frac {\tanh ^7(x)}{a+b \text {sech}(x)} \, dx=\text {Too large to display} \]
-1/15*(15*b^6*x*cosh(x)^10 + 15*b^6*x*sinh(x)^10 - 30*(a^5*b - 3*a^3*b^3 + 3*a*b^5)*cosh(x)^9 + 30*(5*b^6*x*cosh(x) - a^5*b + 3*a^3*b^3 - 3*a*b^5)*s inh(x)^9 + 15*(5*b^6*x + 2*a^4*b^2 - 6*a^2*b^4)*cosh(x)^8 + 15*(45*b^6*x*c osh(x)^2 + 5*b^6*x + 2*a^4*b^2 - 6*a^2*b^4 - 18*(a^5*b - 3*a^3*b^3 + 3*a*b ^5)*cosh(x))*sinh(x)^8 - 40*(3*a^5*b - 8*a^3*b^3 + 6*a*b^5)*cosh(x)^7 + 40 *(45*b^6*x*cosh(x)^3 - 3*a^5*b + 8*a^3*b^3 - 6*a*b^5 - 27*(a^5*b - 3*a^3*b ^3 + 3*a*b^5)*cosh(x)^2 + 3*(5*b^6*x + 2*a^4*b^2 - 6*a^2*b^4)*cosh(x))*sin h(x)^7 + 15*b^6*x + 30*(5*b^6*x + 3*a^4*b^2 - 7*a^2*b^4)*cosh(x)^6 + 10*(3 15*b^6*x*cosh(x)^4 + 15*b^6*x + 9*a^4*b^2 - 21*a^2*b^4 - 252*(a^5*b - 3*a^ 3*b^3 + 3*a*b^5)*cosh(x)^3 + 42*(5*b^6*x + 2*a^4*b^2 - 6*a^2*b^4)*cosh(x)^ 2 - 28*(3*a^5*b - 8*a^3*b^3 + 6*a*b^5)*cosh(x))*sinh(x)^6 - 4*(45*a^5*b - 115*a^3*b^3 + 99*a*b^5)*cosh(x)^5 + 4*(945*b^6*x*cosh(x)^5 - 45*a^5*b + 11 5*a^3*b^3 - 99*a*b^5 - 945*(a^5*b - 3*a^3*b^3 + 3*a*b^5)*cosh(x)^4 + 210*( 5*b^6*x + 2*a^4*b^2 - 6*a^2*b^4)*cosh(x)^3 - 210*(3*a^5*b - 8*a^3*b^3 + 6* a*b^5)*cosh(x)^2 + 45*(5*b^6*x + 3*a^4*b^2 - 7*a^2*b^4)*cosh(x))*sinh(x)^5 + 30*(5*b^6*x + 3*a^4*b^2 - 7*a^2*b^4)*cosh(x)^4 + 10*(315*b^6*x*cosh(x)^ 6 + 15*b^6*x + 9*a^4*b^2 - 21*a^2*b^4 - 378*(a^5*b - 3*a^3*b^3 + 3*a*b^5)* cosh(x)^5 + 105*(5*b^6*x + 2*a^4*b^2 - 6*a^2*b^4)*cosh(x)^4 - 140*(3*a^5*b - 8*a^3*b^3 + 6*a*b^5)*cosh(x)^3 + 45*(5*b^6*x + 3*a^4*b^2 - 7*a^2*b^4)*c osh(x)^2 - 2*(45*a^5*b - 115*a^3*b^3 + 99*a*b^5)*cosh(x))*sinh(x)^4 - 4...
\[ \int \frac {\tanh ^7(x)}{a+b \text {sech}(x)} \, dx=\int \frac {\tanh ^{7}{\left (x \right )}}{a + b \operatorname {sech}{\left (x \right )}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 332 vs. \(2 (113) = 226\).
Time = 0.29 (sec) , antiderivative size = 332, normalized size of antiderivative = 2.74 \[ \int \frac {\tanh ^7(x)}{a+b \text {sech}(x)} \, dx=\frac {2 \, {\left (15 \, {\left (a^{4} - 3 \, a^{2} b^{2} + 3 \, b^{4}\right )} e^{\left (-x\right )} - 15 \, {\left (a^{3} b - 3 \, a b^{3}\right )} e^{\left (-2 \, x\right )} + 20 \, {\left (3 \, a^{4} - 8 \, a^{2} b^{2} + 6 \, b^{4}\right )} e^{\left (-3 \, x\right )} - 15 \, {\left (3 \, a^{3} b - 7 \, a b^{3}\right )} e^{\left (-4 \, x\right )} + 2 \, {\left (45 \, a^{4} - 115 \, a^{2} b^{2} + 99 \, b^{4}\right )} e^{\left (-5 \, x\right )} - 15 \, {\left (3 \, a^{3} b - 7 \, a b^{3}\right )} e^{\left (-6 \, x\right )} + 20 \, {\left (3 \, a^{4} - 8 \, a^{2} b^{2} + 6 \, b^{4}\right )} e^{\left (-7 \, x\right )} - 15 \, {\left (a^{3} b - 3 \, a b^{3}\right )} e^{\left (-8 \, x\right )} + 15 \, {\left (a^{4} - 3 \, a^{2} b^{2} + 3 \, b^{4}\right )} e^{\left (-9 \, x\right )}\right )}}{15 \, {\left (5 \, b^{5} e^{\left (-2 \, x\right )} + 10 \, b^{5} e^{\left (-4 \, x\right )} + 10 \, b^{5} e^{\left (-6 \, x\right )} + 5 \, b^{5} e^{\left (-8 \, x\right )} + b^{5} e^{\left (-10 \, x\right )} + b^{5}\right )}} + \frac {x}{a} + \frac {{\left (a^{5} - 3 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \log \left (e^{\left (-2 \, x\right )} + 1\right )}{b^{6}} - \frac {{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \log \left (2 \, b e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} + a\right )}{a b^{6}} \]
2/15*(15*(a^4 - 3*a^2*b^2 + 3*b^4)*e^(-x) - 15*(a^3*b - 3*a*b^3)*e^(-2*x) + 20*(3*a^4 - 8*a^2*b^2 + 6*b^4)*e^(-3*x) - 15*(3*a^3*b - 7*a*b^3)*e^(-4*x ) + 2*(45*a^4 - 115*a^2*b^2 + 99*b^4)*e^(-5*x) - 15*(3*a^3*b - 7*a*b^3)*e^ (-6*x) + 20*(3*a^4 - 8*a^2*b^2 + 6*b^4)*e^(-7*x) - 15*(a^3*b - 3*a*b^3)*e^ (-8*x) + 15*(a^4 - 3*a^2*b^2 + 3*b^4)*e^(-9*x))/(5*b^5*e^(-2*x) + 10*b^5*e ^(-4*x) + 10*b^5*e^(-6*x) + 5*b^5*e^(-8*x) + b^5*e^(-10*x) + b^5) + x/a + (a^5 - 3*a^3*b^2 + 3*a*b^4)*log(e^(-2*x) + 1)/b^6 - (a^6 - 3*a^4*b^2 + 3*a ^2*b^4 - b^6)*log(2*b*e^(-x) + a*e^(-2*x) + a)/(a*b^6)
Leaf count of result is larger than twice the leaf count of optimal. 267 vs. \(2 (113) = 226\).
Time = 0.31 (sec) , antiderivative size = 267, normalized size of antiderivative = 2.21 \[ \int \frac {\tanh ^7(x)}{a+b \text {sech}(x)} \, dx=\frac {{\left (a^{5} - 3 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \log \left (e^{\left (-x\right )} + e^{x}\right )}{b^{6}} - \frac {{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \log \left ({\left | a {\left (e^{\left (-x\right )} + e^{x}\right )} + 2 \, b \right |}\right )}{a b^{6}} - \frac {137 \, a^{5} {\left (e^{\left (-x\right )} + e^{x}\right )}^{5} - 411 \, a^{3} b^{2} {\left (e^{\left (-x\right )} + e^{x}\right )}^{5} + 411 \, a b^{4} {\left (e^{\left (-x\right )} + e^{x}\right )}^{5} - 120 \, a^{4} b {\left (e^{\left (-x\right )} + e^{x}\right )}^{4} + 360 \, a^{2} b^{3} {\left (e^{\left (-x\right )} + e^{x}\right )}^{4} - 360 \, b^{5} {\left (e^{\left (-x\right )} + e^{x}\right )}^{4} + 120 \, a^{3} b^{2} {\left (e^{\left (-x\right )} + e^{x}\right )}^{3} - 360 \, a b^{4} {\left (e^{\left (-x\right )} + e^{x}\right )}^{3} - 160 \, a^{2} b^{3} {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} + 480 \, b^{5} {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} + 240 \, a b^{4} {\left (e^{\left (-x\right )} + e^{x}\right )} - 384 \, b^{5}}{60 \, b^{6} {\left (e^{\left (-x\right )} + e^{x}\right )}^{5}} \]
(a^5 - 3*a^3*b^2 + 3*a*b^4)*log(e^(-x) + e^x)/b^6 - (a^6 - 3*a^4*b^2 + 3*a ^2*b^4 - b^6)*log(abs(a*(e^(-x) + e^x) + 2*b))/(a*b^6) - 1/60*(137*a^5*(e^ (-x) + e^x)^5 - 411*a^3*b^2*(e^(-x) + e^x)^5 + 411*a*b^4*(e^(-x) + e^x)^5 - 120*a^4*b*(e^(-x) + e^x)^4 + 360*a^2*b^3*(e^(-x) + e^x)^4 - 360*b^5*(e^( -x) + e^x)^4 + 120*a^3*b^2*(e^(-x) + e^x)^3 - 360*a*b^4*(e^(-x) + e^x)^3 - 160*a^2*b^3*(e^(-x) + e^x)^2 + 480*b^5*(e^(-x) + e^x)^2 + 240*a*b^4*(e^(- x) + e^x) - 384*b^5)/(b^6*(e^(-x) + e^x)^5)
Time = 2.59 (sec) , antiderivative size = 316, normalized size of antiderivative = 2.61 \[ \int \frac {\tanh ^7(x)}{a+b \text {sech}(x)} \, dx=\frac {\frac {8\,a}{b^2}-\frac {8\,{\mathrm {e}}^x\,\left (5\,a^2-27\,b^2\right )}{15\,b^3}}{3\,{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}+1}-\frac {\frac {4\,a}{b^2}+\frac {64\,{\mathrm {e}}^x}{5\,b}}{4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1}+\frac {\frac {8\,{\mathrm {e}}^x\,\left (a^2-3\,b^2\right )}{3\,b^3}+\frac {2\,\left (a^4-5\,a^2\,b^2\right )}{a\,b^4}}{2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1}-\frac {x}{a}+\frac {\frac {2\,{\mathrm {e}}^x\,\left (a^4-3\,a^2\,b^2+3\,b^4\right )}{b^5}-\frac {2\,\left (a^4-3\,a^2\,b^2\right )}{a\,b^4}}{{\mathrm {e}}^{2\,x}+1}+\frac {32\,{\mathrm {e}}^x}{5\,b\,\left (5\,{\mathrm {e}}^{2\,x}+10\,{\mathrm {e}}^{4\,x}+10\,{\mathrm {e}}^{6\,x}+5\,{\mathrm {e}}^{8\,x}+{\mathrm {e}}^{10\,x}+1\right )}+\frac {\ln \left ({\mathrm {e}}^{2\,x}+1\right )\,\left (a^5-3\,a^3\,b^2+3\,a\,b^4\right )}{b^6}-\frac {\ln \left (a+2\,b\,{\mathrm {e}}^x+a\,{\mathrm {e}}^{2\,x}\right )\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}{a\,b^6} \]
((8*a)/b^2 - (8*exp(x)*(5*a^2 - 27*b^2))/(15*b^3))/(3*exp(2*x) + 3*exp(4*x ) + exp(6*x) + 1) - ((4*a)/b^2 + (64*exp(x))/(5*b))/(4*exp(2*x) + 6*exp(4* x) + 4*exp(6*x) + exp(8*x) + 1) + ((8*exp(x)*(a^2 - 3*b^2))/(3*b^3) + (2*( a^4 - 5*a^2*b^2))/(a*b^4))/(2*exp(2*x) + exp(4*x) + 1) - x/a + ((2*exp(x)* (a^4 + 3*b^4 - 3*a^2*b^2))/b^5 - (2*(a^4 - 3*a^2*b^2))/(a*b^4))/(exp(2*x) + 1) + (32*exp(x))/(5*b*(5*exp(2*x) + 10*exp(4*x) + 10*exp(6*x) + 5*exp(8* x) + exp(10*x) + 1)) + (log(exp(2*x) + 1)*(3*a*b^4 + a^5 - 3*a^3*b^2))/b^6 - (log(a + 2*b*exp(x) + a*exp(2*x))*(a^6 - b^6 + 3*a^2*b^4 - 3*a^4*b^2))/ (a*b^6)