Integrand size = 13, antiderivative size = 140 \[ \int \frac {\text {sech}^8(x)}{a+b \tanh (x)} \, dx=-\frac {\left (a^2-b^2\right )^3 \log (a+b \tanh (x))}{b^7}+\frac {a \left (a^4-3 a^2 b^2+3 b^4\right ) \tanh (x)}{b^6}-\frac {\left (a^4-3 a^2 b^2+3 b^4\right ) \tanh ^2(x)}{2 b^5}+\frac {a \left (a^2-3 b^2\right ) \tanh ^3(x)}{3 b^4}-\frac {\left (a^2-3 b^2\right ) \tanh ^4(x)}{4 b^3}+\frac {a \tanh ^5(x)}{5 b^2}-\frac {\tanh ^6(x)}{6 b} \]
-(a^2-b^2)^3*ln(a+b*tanh(x))/b^7+a*(a^4-3*a^2*b^2+3*b^4)*tanh(x)/b^6-1/2*( a^4-3*a^2*b^2+3*b^4)*tanh(x)^2/b^5+1/3*a*(a^2-3*b^2)*tanh(x)^3/b^4-1/4*(a^ 2-3*b^2)*tanh(x)^4/b^3+1/5*a*tanh(x)^5/b^2-1/6*tanh(x)^6/b
Time = 0.44 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.91 \[ \int \frac {\text {sech}^8(x)}{a+b \tanh (x)} \, dx=\frac {-60 \left (a^2-b^2\right )^3 \log (a+b \tanh (x))+15 b^4 \left (-a^2+b^2\right ) \text {sech}^4(x)+10 b^6 \text {sech}^6(x)+60 a b \left (a^4-3 a^2 b^2+3 b^4\right ) \tanh (x)-30 b^2 \left (a^2-b^2\right )^2 \tanh ^2(x)+20 a b^3 \left (a^2-3 b^2\right ) \tanh ^3(x)+12 a b^5 \tanh ^5(x)}{60 b^7} \]
(-60*(a^2 - b^2)^3*Log[a + b*Tanh[x]] + 15*b^4*(-a^2 + b^2)*Sech[x]^4 + 10 *b^6*Sech[x]^6 + 60*a*b*(a^4 - 3*a^2*b^2 + 3*b^4)*Tanh[x] - 30*b^2*(a^2 - b^2)^2*Tanh[x]^2 + 20*a*b^3*(a^2 - 3*b^2)*Tanh[x]^3 + 12*a*b^5*Tanh[x]^5)/ (60*b^7)
Time = 0.37 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.99, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3042, 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.
| \(\displaystyle \int \frac {\text {sech}^8(x)}{a+b \tanh (x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sec (i x)^8}{a-i b \tan (i x)}dx\) |
| \(\Big \downarrow \) 3987 |
| \(\displaystyle \frac {\int \frac {\left (b^2-b^2 \tanh ^2(x)\right )^3}{b^6 (a+b \tanh (x))}d(b \tanh (x))}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\left (b^2-b^2 \tanh ^2(x)\right )^3}{a+b \tanh (x)}d(b \tanh (x))}{b^7}\) |
| \(\Big \downarrow \) 476 |
| \(\displaystyle \frac {\int \left (\left (\frac {3 \left (b^2-a^2\right ) b^2}{a^4}+1\right ) a^5+b^4 \tanh ^4(x) a+b^2 \left (a^2-3 b^2\right ) \tanh ^2(x) a-b^5 \tanh ^5(x)-b^3 \left (a^2-3 b^2\right ) \tanh ^3(x)-b \left (a^4-3 b^2 a^2+3 b^4\right ) \tanh (x)-\frac {\left (a^2-b^2\right )^3}{a+b \tanh (x)}\right )d(b \tanh (x))}{b^7}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\left (a^2-b^2\right )^3 \log (a+b \tanh (x))-\frac {1}{4} b^4 \left (a^2-3 b^2\right ) \tanh ^4(x)+\frac {1}{3} a b^3 \left (a^2-3 b^2\right ) \tanh ^3(x)-\frac {1}{2} b^2 \left (a^4-3 a^2 b^2+3 b^4\right ) \tanh ^2(x)+a b \left (a^4-3 a^2 b^2+3 b^4\right ) \tanh (x)+\frac {1}{5} a b^5 \tanh ^5(x)-\frac {1}{6} b^6 \tanh ^6(x)}{b^7}\) |
(-((a^2 - b^2)^3*Log[a + b*Tanh[x]]) + a*b*(a^4 - 3*a^2*b^2 + 3*b^4)*Tanh[ x] - (b^2*(a^4 - 3*a^2*b^2 + 3*b^4)*Tanh[x]^2)/2 + (a*b^3*(a^2 - 3*b^2)*Ta nh[x]^3)/3 - (b^4*(a^2 - 3*b^2)*Tanh[x]^4)/4 + (a*b^5*Tanh[x]^5)/5 - (b^6* Tanh[x]^6)/6)/b^7
3.2.2.3.1 Defintions of rubi rules used
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. \(412\) vs. \(2(130)=260\).
Time = 235.72 (sec) , antiderivative size = 413, normalized size of antiderivative = 2.95
| method | result | size |
| default | \(\frac {\frac {2 \left (\left (a^{5} b -3 a^{3} b^{3}+3 a \,b^{5}\right ) \tanh \left (\frac {x}{2}\right )^{11}+\left (-a^{4} b^{2}+3 a^{2} b^{4}-3 b^{6}\right ) \tanh \left (\frac {x}{2}\right )^{10}+\left (5 a^{5} b -\frac {41}{3} a^{3} b^{3}+11 a \,b^{5}\right ) \tanh \left (\frac {x}{2}\right )^{9}+\left (-4 a^{4} b^{2}+10 a^{2} b^{4}-6 b^{6}\right ) \tanh \left (\frac {x}{2}\right )^{8}+\left (10 a^{5} b -26 a^{3} b^{3}+\frac {106}{5} a \,b^{5}\right ) \tanh \left (\frac {x}{2}\right )^{7}+\left (-6 a^{4} b^{2}+14 a^{2} b^{4}-\frac {34}{3} b^{6}\right ) \tanh \left (\frac {x}{2}\right )^{6}+\left (10 a^{5} b -26 a^{3} b^{3}+\frac {106}{5} a \,b^{5}\right ) \tanh \left (\frac {x}{2}\right )^{5}+\left (-4 a^{4} b^{2}+10 a^{2} b^{4}-6 b^{6}\right ) \tanh \left (\frac {x}{2}\right )^{4}+\left (5 a^{5} b -\frac {41}{3} a^{3} b^{3}+11 a \,b^{5}\right ) \tanh \left (\frac {x}{2}\right )^{3}+\left (-a^{4} b^{2}+3 a^{2} b^{4}-3 b^{6}\right ) \tanh \left (\frac {x}{2}\right )^{2}+\left (a^{5} b -3 a^{3} b^{3}+3 a \,b^{5}\right ) \tanh \left (\frac {x}{2}\right )\right )}{\left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )^{6}}+\left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \ln \left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )}{b^{7}}-\frac {\left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )^{2} a +2 b \tanh \left (\frac {x}{2}\right )+a \right )}{b^{7}}\) | \(413\) |
| risch | \(-\frac {2 \left (330 a \,b^{4} {\mathrm e}^{6 x}+150 a^{5} {\mathrm e}^{4 x}+75 a^{5} {\mathrm e}^{2 x}-400 a^{3} b^{2} {\mathrm e}^{6 x}+105 a \,b^{4} {\mathrm e}^{8 x}+30 a^{2} b^{3} {\mathrm e}^{2 x}-15 a^{4} b \,{\mathrm e}^{10 x}-30 a^{3} b^{2} {\mathrm e}^{10 x}+30 a^{2} b^{3} {\mathrm e}^{10 x}+15 a \,b^{4} {\mathrm e}^{10 x}-60 a^{4} b \,{\mathrm e}^{8 x}+15 a^{5}-420 a^{3} b^{2} {\mathrm e}^{4 x}+390 a \,b^{4} {\mathrm e}^{4 x}-210 a^{3} b^{2} {\mathrm e}^{2 x}+183 a \,b^{4} {\mathrm e}^{2 x}-40 a^{3} b^{2}+33 a \,b^{4}-15 a^{4} b \,{\mathrm e}^{2 x}+15 a^{5} {\mathrm e}^{10 x}-15 b^{5} {\mathrm e}^{10 x}+75 a^{5} {\mathrm e}^{8 x}-90 b^{5} {\mathrm e}^{8 x}-180 a^{3} b^{2} {\mathrm e}^{8 x}+150 a^{2} b^{3} {\mathrm e}^{8 x}-90 a^{4} b \,{\mathrm e}^{6 x}+240 a^{2} b^{3} {\mathrm e}^{6 x}-60 a^{4} b \,{\mathrm e}^{4 x}+150 a^{2} b^{3} {\mathrm e}^{4 x}+150 a^{5} {\mathrm e}^{6 x}-230 b^{5} {\mathrm e}^{6 x}-90 b^{5} {\mathrm e}^{4 x}-15 b^{5} {\mathrm e}^{2 x}\right )}{15 b^{6} \left (1+{\mathrm e}^{2 x}\right )^{6}}+\frac {\ln \left (1+{\mathrm e}^{2 x}\right ) a^{6}}{b^{7}}-\frac {3 \ln \left (1+{\mathrm e}^{2 x}\right ) a^{4}}{b^{5}}+\frac {3 \ln \left (1+{\mathrm e}^{2 x}\right ) a^{2}}{b^{3}}-\frac {\ln \left (1+{\mathrm e}^{2 x}\right )}{b}-\frac {\ln \left ({\mathrm e}^{2 x}+\frac {a -b}{a +b}\right ) a^{6}}{b^{7}}+\frac {3 \ln \left ({\mathrm e}^{2 x}+\frac {a -b}{a +b}\right ) a^{4}}{b^{5}}-\frac {3 \ln \left ({\mathrm e}^{2 x}+\frac {a -b}{a +b}\right ) a^{2}}{b^{3}}+\frac {\ln \left ({\mathrm e}^{2 x}+\frac {a -b}{a +b}\right )}{b}\) | \(497\) |
2/b^7*(((a^5*b-3*a^3*b^3+3*a*b^5)*tanh(1/2*x)^11+(-a^4*b^2+3*a^2*b^4-3*b^6 )*tanh(1/2*x)^10+(5*a^5*b-41/3*a^3*b^3+11*a*b^5)*tanh(1/2*x)^9+(-4*a^4*b^2 +10*a^2*b^4-6*b^6)*tanh(1/2*x)^8+(10*a^5*b-26*a^3*b^3+106/5*a*b^5)*tanh(1/ 2*x)^7+(-6*a^4*b^2+14*a^2*b^4-34/3*b^6)*tanh(1/2*x)^6+(10*a^5*b-26*a^3*b^3 +106/5*a*b^5)*tanh(1/2*x)^5+(-4*a^4*b^2+10*a^2*b^4-6*b^6)*tanh(1/2*x)^4+(5 *a^5*b-41/3*a^3*b^3+11*a*b^5)*tanh(1/2*x)^3+(-a^4*b^2+3*a^2*b^4-3*b^6)*tan h(1/2*x)^2+(a^5*b-3*a^3*b^3+3*a*b^5)*tanh(1/2*x))/(1+tanh(1/2*x)^2)^6+1/2* (a^6-3*a^4*b^2+3*a^2*b^4-b^6)*ln(1+tanh(1/2*x)^2))-(a^6-3*a^4*b^2+3*a^2*b^ 4-b^6)/b^7*ln(tanh(1/2*x)^2*a+2*b*tanh(1/2*x)+a)
Leaf count of result is larger than twice the leaf count of optimal. 5275 vs. \(2 (130) = 260\).
Time = 0.35 (sec) , antiderivative size = 5275, normalized size of antiderivative = 37.68 \[ \int \frac {\text {sech}^8(x)}{a+b \tanh (x)} \, dx=\text {Too large to display} \]
\[ \int \frac {\text {sech}^8(x)}{a+b \tanh (x)} \, dx=\int \frac {\operatorname {sech}^{8}{\left (x \right )}}{a + b \tanh {\left (x \right )}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 386 vs. \(2 (130) = 260\).
Time = 0.31 (sec) , antiderivative size = 386, normalized size of antiderivative = 2.76 \[ \int \frac {\text {sech}^8(x)}{a+b \tanh (x)} \, dx=\frac {2 \, {\left (15 \, a^{5} - 40 \, a^{3} b^{2} + 33 \, a b^{4} + 3 \, {\left (25 \, a^{5} + 5 \, a^{4} b - 70 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 61 \, a b^{4} + 5 \, b^{5}\right )} e^{\left (-2 \, x\right )} + 30 \, {\left (5 \, a^{5} + 2 \, a^{4} b - 14 \, a^{3} b^{2} - 5 \, a^{2} b^{3} + 13 \, a b^{4} + 3 \, b^{5}\right )} e^{\left (-4 \, x\right )} + 10 \, {\left (15 \, a^{5} + 9 \, a^{4} b - 40 \, a^{3} b^{2} - 24 \, a^{2} b^{3} + 33 \, a b^{4} + 23 \, b^{5}\right )} e^{\left (-6 \, x\right )} + 15 \, {\left (5 \, a^{5} + 4 \, a^{4} b - 12 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 7 \, a b^{4} + 6 \, b^{5}\right )} e^{\left (-8 \, x\right )} + 15 \, {\left (a^{5} + a^{4} b - 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} + b^{5}\right )} e^{\left (-10 \, x\right )}\right )}}{15 \, {\left (6 \, b^{6} e^{\left (-2 \, x\right )} + 15 \, b^{6} e^{\left (-4 \, x\right )} + 20 \, b^{6} e^{\left (-6 \, x\right )} + 15 \, b^{6} e^{\left (-8 \, x\right )} + 6 \, b^{6} e^{\left (-10 \, x\right )} + b^{6} e^{\left (-12 \, x\right )} + b^{6}\right )}} - \frac {{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \log \left (-{\left (a - b\right )} e^{\left (-2 \, x\right )} - a - b\right )}{b^{7}} + \frac {{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \log \left (e^{\left (-2 \, x\right )} + 1\right )}{b^{7}} \]
2/15*(15*a^5 - 40*a^3*b^2 + 33*a*b^4 + 3*(25*a^5 + 5*a^4*b - 70*a^3*b^2 - 10*a^2*b^3 + 61*a*b^4 + 5*b^5)*e^(-2*x) + 30*(5*a^5 + 2*a^4*b - 14*a^3*b^2 - 5*a^2*b^3 + 13*a*b^4 + 3*b^5)*e^(-4*x) + 10*(15*a^5 + 9*a^4*b - 40*a^3* b^2 - 24*a^2*b^3 + 33*a*b^4 + 23*b^5)*e^(-6*x) + 15*(5*a^5 + 4*a^4*b - 12* a^3*b^2 - 10*a^2*b^3 + 7*a*b^4 + 6*b^5)*e^(-8*x) + 15*(a^5 + a^4*b - 2*a^3 *b^2 - 2*a^2*b^3 + a*b^4 + b^5)*e^(-10*x))/(6*b^6*e^(-2*x) + 15*b^6*e^(-4* x) + 20*b^6*e^(-6*x) + 15*b^6*e^(-8*x) + 6*b^6*e^(-10*x) + b^6*e^(-12*x) + b^6) - (a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*log(-(a - b)*e^(-2*x) - a - b) /b^7 + (a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*log(e^(-2*x) + 1)/b^7
Leaf count of result is larger than twice the leaf count of optimal. 593 vs. \(2 (130) = 260\).
Time = 0.28 (sec) , antiderivative size = 593, normalized size of antiderivative = 4.24 \[ \int \frac {\text {sech}^8(x)}{a+b \tanh (x)} \, dx=-\frac {{\left (a^{7} + a^{6} b - 3 \, a^{5} b^{2} - 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} + 3 \, a^{2} b^{5} - a b^{6} - b^{7}\right )} \log \left ({\left | a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b \right |}\right )}{a b^{7} + b^{8}} + \frac {{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \log \left (e^{\left (2 \, x\right )} + 1\right )}{b^{7}} - \frac {147 \, a^{6} e^{\left (12 \, x\right )} - 441 \, a^{4} b^{2} e^{\left (12 \, x\right )} + 441 \, a^{2} b^{4} e^{\left (12 \, x\right )} - 147 \, b^{6} e^{\left (12 \, x\right )} + 882 \, a^{6} e^{\left (10 \, x\right )} + 120 \, a^{5} b e^{\left (10 \, x\right )} - 2766 \, a^{4} b^{2} e^{\left (10 \, x\right )} - 240 \, a^{3} b^{3} e^{\left (10 \, x\right )} + 2886 \, a^{2} b^{4} e^{\left (10 \, x\right )} + 120 \, a b^{5} e^{\left (10 \, x\right )} - 1002 \, b^{6} e^{\left (10 \, x\right )} + 2205 \, a^{6} e^{\left (8 \, x\right )} + 600 \, a^{5} b e^{\left (8 \, x\right )} - 7095 \, a^{4} b^{2} e^{\left (8 \, x\right )} - 1440 \, a^{3} b^{3} e^{\left (8 \, x\right )} + 7815 \, a^{2} b^{4} e^{\left (8 \, x\right )} + 840 \, a b^{5} e^{\left (8 \, x\right )} - 2925 \, b^{6} e^{\left (8 \, x\right )} + 2940 \, a^{6} e^{\left (6 \, x\right )} + 1200 \, a^{5} b e^{\left (6 \, x\right )} - 9540 \, a^{4} b^{2} e^{\left (6 \, x\right )} - 3200 \, a^{3} b^{3} e^{\left (6 \, x\right )} + 10740 \, a^{2} b^{4} e^{\left (6 \, x\right )} + 2640 \, a b^{5} e^{\left (6 \, x\right )} - 4780 \, b^{6} e^{\left (6 \, x\right )} + 2205 \, a^{6} e^{\left (4 \, x\right )} + 1200 \, a^{5} b e^{\left (4 \, x\right )} - 7095 \, a^{4} b^{2} e^{\left (4 \, x\right )} - 3360 \, a^{3} b^{3} e^{\left (4 \, x\right )} + 7815 \, a^{2} b^{4} e^{\left (4 \, x\right )} + 3120 \, a b^{5} e^{\left (4 \, x\right )} - 2925 \, b^{6} e^{\left (4 \, x\right )} + 882 \, a^{6} e^{\left (2 \, x\right )} + 600 \, a^{5} b e^{\left (2 \, x\right )} - 2766 \, a^{4} b^{2} e^{\left (2 \, x\right )} - 1680 \, a^{3} b^{3} e^{\left (2 \, x\right )} + 2886 \, a^{2} b^{4} e^{\left (2 \, x\right )} + 1464 \, a b^{5} e^{\left (2 \, x\right )} - 1002 \, b^{6} e^{\left (2 \, x\right )} + 147 \, a^{6} + 120 \, a^{5} b - 441 \, a^{4} b^{2} - 320 \, a^{3} b^{3} + 441 \, a^{2} b^{4} + 264 \, a b^{5} - 147 \, b^{6}}{60 \, b^{7} {\left (e^{\left (2 \, x\right )} + 1\right )}^{6}} \]
-(a^7 + a^6*b - 3*a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 + 3*a^2*b^5 - a*b^6 - b^ 7)*log(abs(a*e^(2*x) + b*e^(2*x) + a - b))/(a*b^7 + b^8) + (a^6 - 3*a^4*b^ 2 + 3*a^2*b^4 - b^6)*log(e^(2*x) + 1)/b^7 - 1/60*(147*a^6*e^(12*x) - 441*a ^4*b^2*e^(12*x) + 441*a^2*b^4*e^(12*x) - 147*b^6*e^(12*x) + 882*a^6*e^(10* x) + 120*a^5*b*e^(10*x) - 2766*a^4*b^2*e^(10*x) - 240*a^3*b^3*e^(10*x) + 2 886*a^2*b^4*e^(10*x) + 120*a*b^5*e^(10*x) - 1002*b^6*e^(10*x) + 2205*a^6*e ^(8*x) + 600*a^5*b*e^(8*x) - 7095*a^4*b^2*e^(8*x) - 1440*a^3*b^3*e^(8*x) + 7815*a^2*b^4*e^(8*x) + 840*a*b^5*e^(8*x) - 2925*b^6*e^(8*x) + 2940*a^6*e^ (6*x) + 1200*a^5*b*e^(6*x) - 9540*a^4*b^2*e^(6*x) - 3200*a^3*b^3*e^(6*x) + 10740*a^2*b^4*e^(6*x) + 2640*a*b^5*e^(6*x) - 4780*b^6*e^(6*x) + 2205*a^6* e^(4*x) + 1200*a^5*b*e^(4*x) - 7095*a^4*b^2*e^(4*x) - 3360*a^3*b^3*e^(4*x) + 7815*a^2*b^4*e^(4*x) + 3120*a*b^5*e^(4*x) - 2925*b^6*e^(4*x) + 882*a^6* e^(2*x) + 600*a^5*b*e^(2*x) - 2766*a^4*b^2*e^(2*x) - 1680*a^3*b^3*e^(2*x) + 2886*a^2*b^4*e^(2*x) + 1464*a*b^5*e^(2*x) - 1002*b^6*e^(2*x) + 147*a^6 + 120*a^5*b - 441*a^4*b^2 - 320*a^3*b^3 + 441*a^2*b^4 + 264*a*b^5 - 147*b^6 )/(b^7*(e^(2*x) + 1)^6)
Time = 2.04 (sec) , antiderivative size = 301, normalized size of antiderivative = 2.15 \[ \int \frac {\text {sech}^8(x)}{a+b \tanh (x)} \, dx=\frac {\ln \left ({\mathrm {e}}^{2\,x}+1\right )\,{\left (a+b\right )}^3\,{\left (a-b\right )}^3}{b^7}-\frac {32\,\left (a-5\,b\right )}{5\,b^2\,\left (5\,{\mathrm {e}}^{2\,x}+10\,{\mathrm {e}}^{4\,x}+10\,{\mathrm {e}}^{6\,x}+5\,{\mathrm {e}}^{8\,x}+{\mathrm {e}}^{10\,x}+1\right )}-\frac {4\,\left (a^2-4\,a\,b+7\,b^2\right )}{b^3\,\left (4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}-\frac {\ln \left (a-b+a\,{\mathrm {e}}^{2\,x}+b\,{\mathrm {e}}^{2\,x}\right )\,{\left (a+b\right )}^3\,{\left (a-b\right )}^3}{b^7}-\frac {32}{3\,b\,\left (6\,{\mathrm {e}}^{2\,x}+15\,{\mathrm {e}}^{4\,x}+20\,{\mathrm {e}}^{6\,x}+15\,{\mathrm {e}}^{8\,x}+6\,{\mathrm {e}}^{10\,x}+{\mathrm {e}}^{12\,x}+1\right )}-\frac {8\,\left (a-b\right )\,\left (a^2-2\,a\,b+b^2\right )}{3\,b^4\,\left (3\,{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}+1\right )}-\frac {2\,{\left (a+b\right )}^2\,\left (a-b\right )\,\left (a^2-2\,a\,b+b^2\right )}{b^6\,\left ({\mathrm {e}}^{2\,x}+1\right )}-\frac {2\,\left (a+b\right )\,\left (a-b\right )\,\left (a^2-2\,a\,b+b^2\right )}{b^5\,\left (2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1\right )} \]
(log(exp(2*x) + 1)*(a + b)^3*(a - b)^3)/b^7 - (32*(a - 5*b))/(5*b^2*(5*exp (2*x) + 10*exp(4*x) + 10*exp(6*x) + 5*exp(8*x) + exp(10*x) + 1)) - (4*(a^2 - 4*a*b + 7*b^2))/(b^3*(4*exp(2*x) + 6*exp(4*x) + 4*exp(6*x) + exp(8*x) + 1)) - (log(a - b + a*exp(2*x) + b*exp(2*x))*(a + b)^3*(a - b)^3)/b^7 - 32 /(3*b*(6*exp(2*x) + 15*exp(4*x) + 20*exp(6*x) + 15*exp(8*x) + 6*exp(10*x) + exp(12*x) + 1)) - (8*(a - b)*(a^2 - 2*a*b + b^2))/(3*b^4*(3*exp(2*x) + 3 *exp(4*x) + exp(6*x) + 1)) - (2*(a + b)^2*(a - b)*(a^2 - 2*a*b + b^2))/(b^ 6*(exp(2*x) + 1)) - (2*(a + b)*(a - b)*(a^2 - 2*a*b + b^2))/(b^5*(2*exp(2* x) + exp(4*x) + 1))