Integrand size = 17, antiderivative size = 81 \[ \int \text {csch}^5(a+b x) \text {sech}^4(a+b x) \, dx=-\frac {35 \text {arctanh}(\cosh (a+b x))}{8 b}+\frac {13 \coth (a+b x) \text {csch}(a+b x)}{8 b}-\frac {\coth ^3(a+b x) \text {csch}(a+b x)}{4 b}+\frac {3 \text {sech}(a+b x)}{b}+\frac {\text {sech}^3(a+b x)}{3 b} \] Output:
-35/8*arctanh(cosh(b*x+a))/b+13/8*coth(b*x+a)*csch(b*x+a)/b-1/4*coth(b*x+a )^3*csch(b*x+a)/b+3*sech(b*x+a)/b+1/3*sech(b*x+a)^3/b
Time = 0.13 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.72 \[ \int \text {csch}^5(a+b x) \text {sech}^4(a+b x) \, dx=\frac {11 \text {csch}^2\left (\frac {1}{2} (a+b x)\right )}{32 b}-\frac {\text {csch}^4\left (\frac {1}{2} (a+b x)\right )}{64 b}-\frac {35 \log \left (\cosh \left (\frac {1}{2} (a+b x)\right )\right )}{8 b}+\frac {35 \log \left (\sinh \left (\frac {1}{2} (a+b x)\right )\right )}{8 b}+\frac {11 \text {sech}^2\left (\frac {1}{2} (a+b x)\right )}{32 b}+\frac {\text {sech}^4\left (\frac {1}{2} (a+b x)\right )}{64 b}+\frac {3 \text {sech}(a+b x)}{b}+\frac {\text {sech}^3(a+b x)}{3 b} \] Input:
Integrate[Csch[a + b*x]^5*Sech[a + b*x]^4,x]
Output:
(11*Csch[(a + b*x)/2]^2)/(32*b) - Csch[(a + b*x)/2]^4/(64*b) - (35*Log[Cos h[(a + b*x)/2]])/(8*b) + (35*Log[Sinh[(a + b*x)/2]])/(8*b) + (11*Sech[(a + b*x)/2]^2)/(32*b) + Sech[(a + b*x)/2]^4/(64*b) + (3*Sech[a + b*x])/b + Se ch[a + b*x]^3/(3*b)
Time = 0.27 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.16, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used = {3042, 26, 3102, 25, 252, 252, 254, 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.
| \(\displaystyle \int \text {csch}^5(a+b x) \text {sech}^4(a+b x) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int i \csc (i a+i b x)^5 \sec (i a+i b x)^4dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \csc (i a+i b x)^5 \sec (i a+i b x)^4dx\) |
| \(\Big \downarrow \) 3102 |
| \(\displaystyle \frac {\int -\frac {\text {sech}^8(a+b x)}{\left (1-\text {sech}^2(a+b x)\right )^3}d\text {sech}(a+b x)}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {\text {sech}^8(a+b x)}{\left (1-\text {sech}^2(a+b x)\right )^3}d\text {sech}(a+b x)}{b}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {\frac {7}{4} \int \frac {\text {sech}^6(a+b x)}{\left (1-\text {sech}^2(a+b x)\right )^2}d\text {sech}(a+b x)-\frac {\text {sech}^7(a+b x)}{4 \left (1-\text {sech}^2(a+b x)\right )^2}}{b}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {\frac {7}{4} \left (\frac {\text {sech}^5(a+b x)}{2 \left (1-\text {sech}^2(a+b x)\right )}-\frac {5}{2} \int \frac {\text {sech}^4(a+b x)}{1-\text {sech}^2(a+b x)}d\text {sech}(a+b x)\right )-\frac {\text {sech}^7(a+b x)}{4 \left (1-\text {sech}^2(a+b x)\right )^2}}{b}\) |
| \(\Big \downarrow \) 254 |
| \(\displaystyle \frac {\frac {7}{4} \left (\frac {\text {sech}^5(a+b x)}{2 \left (1-\text {sech}^2(a+b x)\right )}-\frac {5}{2} \int \left (-\text {sech}^2(a+b x)+\frac {1}{1-\text {sech}^2(a+b x)}-1\right )d\text {sech}(a+b x)\right )-\frac {\text {sech}^7(a+b x)}{4 \left (1-\text {sech}^2(a+b x)\right )^2}}{b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {7}{4} \left (\frac {\text {sech}^5(a+b x)}{2 \left (1-\text {sech}^2(a+b x)\right )}-\frac {5}{2} \left (\text {arctanh}(\text {sech}(a+b x))-\frac {1}{3} \text {sech}^3(a+b x)-\text {sech}(a+b x)\right )\right )-\frac {\text {sech}^7(a+b x)}{4 \left (1-\text {sech}^2(a+b x)\right )^2}}{b}\) |
Input:
Int[Csch[a + b*x]^5*Sech[a + b*x]^4,x]
Output:
(-1/4*Sech[a + b*x]^7/(1 - Sech[a + b*x]^2)^2 + (7*(Sech[a + b*x]^5/(2*(1 - Sech[a + b*x]^2)) - (5*(ArcTanh[Sech[a + b*x]] - Sech[a + b*x] - Sech[a + b*x]^3/3))/2))/4)/b
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_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x )^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* (p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c }, x] && LtQ[p, -1] && GtQ[m, 1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi alQ[a, b, c, 2, m, p, x]
Int[(x_)^(m_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^2, x], x] /; FreeQ[{a, b}, x] && IGtQ[m, 3]
Int[csc[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_), x_S ymbol] :> Simp[1/(f*a^n) Subst[Int[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/ 2), x], x, a*Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n + 1 )/2] && !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])
Time = 47.12 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.88
| method | result | size |
| derivativedivides | \(\frac {-\frac {1}{4 \sinh \left (b x +a \right )^{4} \cosh \left (b x +a \right )^{3}}+\frac {7}{8 \sinh \left (b x +a \right )^{2} \cosh \left (b x +a \right )^{3}}+\frac {35}{24 \cosh \left (b x +a \right )^{3}}+\frac {35}{8 \cosh \left (b x +a \right )}-\frac {35 \,\operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )}{4}}{b}\) | \(71\) |
| default | \(\frac {-\frac {1}{4 \sinh \left (b x +a \right )^{4} \cosh \left (b x +a \right )^{3}}+\frac {7}{8 \sinh \left (b x +a \right )^{2} \cosh \left (b x +a \right )^{3}}+\frac {35}{24 \cosh \left (b x +a \right )^{3}}+\frac {35}{8 \cosh \left (b x +a \right )}-\frac {35 \,\operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )}{4}}{b}\) | \(71\) |
| risch | \(\frac {{\mathrm e}^{b x +a} \left (105 \,{\mathrm e}^{12 b x +12 a}-70 \,{\mathrm e}^{10 b x +10 a}-329 \,{\mathrm e}^{8 b x +8 a}+204 \,{\mathrm e}^{6 b x +6 a}-329 \,{\mathrm e}^{4 b x +4 a}-70 \,{\mathrm e}^{2 b x +2 a}+105\right )}{12 b \left ({\mathrm e}^{2 b x +2 a}+1\right )^{3} \left ({\mathrm e}^{2 b x +2 a}-1\right )^{4}}-\frac {35 \ln \left ({\mathrm e}^{b x +a}+1\right )}{8 b}+\frac {35 \ln \left ({\mathrm e}^{b x +a}-1\right )}{8 b}\) | \(135\) |
Input:
int(csch(b*x+a)^5*sech(b*x+a)^4,x,method=_RETURNVERBOSE)
Output:
1/b*(-1/4/sinh(b*x+a)^4/cosh(b*x+a)^3+7/8/sinh(b*x+a)^2/cosh(b*x+a)^3+35/2 4/cosh(b*x+a)^3+35/8/cosh(b*x+a)-35/4*arctanh(exp(b*x+a)))
Leaf count of result is larger than twice the leaf count of optimal. 2802 vs. \(2 (73) = 146\).
Time = 0.10 (sec) , antiderivative size = 2802, normalized size of antiderivative = 34.59 \[ \int \text {csch}^5(a+b x) \text {sech}^4(a+b x) \, dx=\text {Too large to display} \] Input:
integrate(csch(b*x+a)^5*sech(b*x+a)^4,x, algorithm="fricas")
Output:
1/24*(210*cosh(b*x + a)^13 + 2730*cosh(b*x + a)*sinh(b*x + a)^12 + 210*sin h(b*x + a)^13 + 140*(117*cosh(b*x + a)^2 - 1)*sinh(b*x + a)^11 - 140*cosh( b*x + a)^11 + 1540*(39*cosh(b*x + a)^3 - cosh(b*x + a))*sinh(b*x + a)^10 + 14*(10725*cosh(b*x + a)^4 - 550*cosh(b*x + a)^2 - 47)*sinh(b*x + a)^9 - 6 58*cosh(b*x + a)^9 + 42*(6435*cosh(b*x + a)^5 - 550*cosh(b*x + a)^3 - 141* cosh(b*x + a))*sinh(b*x + a)^8 + 24*(15015*cosh(b*x + a)^6 - 1925*cosh(b*x + a)^4 - 987*cosh(b*x + a)^2 + 17)*sinh(b*x + a)^7 + 408*cosh(b*x + a)^7 + 168*(2145*cosh(b*x + a)^7 - 385*cosh(b*x + a)^5 - 329*cosh(b*x + a)^3 + 17*cosh(b*x + a))*sinh(b*x + a)^6 + 14*(19305*cosh(b*x + a)^8 - 4620*cosh( b*x + a)^6 - 5922*cosh(b*x + a)^4 + 612*cosh(b*x + a)^2 - 47)*sinh(b*x + a )^5 - 658*cosh(b*x + a)^5 + 14*(10725*cosh(b*x + a)^9 - 3300*cosh(b*x + a) ^7 - 5922*cosh(b*x + a)^5 + 1020*cosh(b*x + a)^3 - 235*cosh(b*x + a))*sinh (b*x + a)^4 + 28*(2145*cosh(b*x + a)^10 - 825*cosh(b*x + a)^8 - 1974*cosh( b*x + a)^6 + 510*cosh(b*x + a)^4 - 235*cosh(b*x + a)^2 - 5)*sinh(b*x + a)^ 3 - 140*cosh(b*x + a)^3 + 28*(585*cosh(b*x + a)^11 - 275*cosh(b*x + a)^9 - 846*cosh(b*x + a)^7 + 306*cosh(b*x + a)^5 - 235*cosh(b*x + a)^3 - 15*cosh (b*x + a))*sinh(b*x + a)^2 - 105*(cosh(b*x + a)^14 + 14*cosh(b*x + a)*sinh (b*x + a)^13 + sinh(b*x + a)^14 + (91*cosh(b*x + a)^2 - 1)*sinh(b*x + a)^1 2 - cosh(b*x + a)^12 + 4*(91*cosh(b*x + a)^3 - 3*cosh(b*x + a))*sinh(b*x + a)^11 + (1001*cosh(b*x + a)^4 - 66*cosh(b*x + a)^2 - 3)*sinh(b*x + a)^...
\[ \int \text {csch}^5(a+b x) \text {sech}^4(a+b x) \, dx=\int \operatorname {csch}^{5}{\left (a + b x \right )} \operatorname {sech}^{4}{\left (a + b x \right )}\, dx \] Input:
integrate(csch(b*x+a)**5*sech(b*x+a)**4,x)
Output:
Integral(csch(a + b*x)**5*sech(a + b*x)**4, x)
Leaf count of result is larger than twice the leaf count of optimal. 195 vs. \(2 (73) = 146\).
Time = 0.04 (sec) , antiderivative size = 195, normalized size of antiderivative = 2.41 \[ \int \text {csch}^5(a+b x) \text {sech}^4(a+b x) \, dx=-\frac {35 \, \log \left (e^{\left (-b x - a\right )} + 1\right )}{8 \, b} + \frac {35 \, \log \left (e^{\left (-b x - a\right )} - 1\right )}{8 \, b} - \frac {105 \, e^{\left (-b x - a\right )} - 70 \, e^{\left (-3 \, b x - 3 \, a\right )} - 329 \, e^{\left (-5 \, b x - 5 \, a\right )} + 204 \, e^{\left (-7 \, b x - 7 \, a\right )} - 329 \, e^{\left (-9 \, b x - 9 \, a\right )} - 70 \, e^{\left (-11 \, b x - 11 \, a\right )} + 105 \, e^{\left (-13 \, b x - 13 \, a\right )}}{12 \, b {\left (e^{\left (-2 \, b x - 2 \, a\right )} + 3 \, e^{\left (-4 \, b x - 4 \, a\right )} - 3 \, e^{\left (-6 \, b x - 6 \, a\right )} - 3 \, e^{\left (-8 \, b x - 8 \, a\right )} + 3 \, e^{\left (-10 \, b x - 10 \, a\right )} + e^{\left (-12 \, b x - 12 \, a\right )} - e^{\left (-14 \, b x - 14 \, a\right )} - 1\right )}} \] Input:
integrate(csch(b*x+a)^5*sech(b*x+a)^4,x, algorithm="maxima")
Output:
-35/8*log(e^(-b*x - a) + 1)/b + 35/8*log(e^(-b*x - a) - 1)/b - 1/12*(105*e ^(-b*x - a) - 70*e^(-3*b*x - 3*a) - 329*e^(-5*b*x - 5*a) + 204*e^(-7*b*x - 7*a) - 329*e^(-9*b*x - 9*a) - 70*e^(-11*b*x - 11*a) + 105*e^(-13*b*x - 13 *a))/(b*(e^(-2*b*x - 2*a) + 3*e^(-4*b*x - 4*a) - 3*e^(-6*b*x - 6*a) - 3*e^ (-8*b*x - 8*a) + 3*e^(-10*b*x - 10*a) + e^(-12*b*x - 12*a) - e^(-14*b*x - 14*a) - 1))
Leaf count of result is larger than twice the leaf count of optimal. 152 vs. \(2 (73) = 146\).
Time = 0.13 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.88 \[ \int \text {csch}^5(a+b x) \text {sech}^4(a+b x) \, dx=\frac {\frac {12 \, {\left (11 \, {\left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )}\right )}^{3} - 52 \, e^{\left (b x + a\right )} - 52 \, e^{\left (-b x - a\right )}\right )}}{{\left ({\left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )}\right )}^{2} - 4\right )}^{2}} + \frac {32 \, {\left (9 \, {\left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )}\right )}^{2} + 4\right )}}{{\left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )}\right )}^{3}} - 105 \, \log \left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )} + 2\right ) + 105 \, \log \left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )} - 2\right )}{48 \, b} \] Input:
integrate(csch(b*x+a)^5*sech(b*x+a)^4,x, algorithm="giac")
Output:
1/48*(12*(11*(e^(b*x + a) + e^(-b*x - a))^3 - 52*e^(b*x + a) - 52*e^(-b*x - a))/((e^(b*x + a) + e^(-b*x - a))^2 - 4)^2 + 32*(9*(e^(b*x + a) + e^(-b* x - a))^2 + 4)/(e^(b*x + a) + e^(-b*x - a))^3 - 105*log(e^(b*x + a) + e^(- b*x - a) + 2) + 105*log(e^(b*x + a) + e^(-b*x - a) - 2))/b
Time = 0.97 (sec) , antiderivative size = 295, normalized size of antiderivative = 3.64 \[ \int \text {csch}^5(a+b x) \text {sech}^4(a+b x) \, dx=\frac {7\,{\mathrm {e}}^{a+b\,x}}{2\,b\,\left ({\mathrm {e}}^{4\,a+4\,b\,x}-2\,{\mathrm {e}}^{2\,a+2\,b\,x}+1\right )}-\frac {35\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {-b^2}}{b}\right )}{4\,\sqrt {-b^2}}+\frac {8\,{\mathrm {e}}^{a+b\,x}}{3\,b\,\left (2\,{\mathrm {e}}^{2\,a+2\,b\,x}+{\mathrm {e}}^{4\,a+4\,b\,x}+1\right )}-\frac {6\,{\mathrm {e}}^{a+b\,x}}{b\,\left (3\,{\mathrm {e}}^{2\,a+2\,b\,x}-3\,{\mathrm {e}}^{4\,a+4\,b\,x}+{\mathrm {e}}^{6\,a+6\,b\,x}-1\right )}-\frac {8\,{\mathrm {e}}^{a+b\,x}}{3\,b\,\left (3\,{\mathrm {e}}^{2\,a+2\,b\,x}+3\,{\mathrm {e}}^{4\,a+4\,b\,x}+{\mathrm {e}}^{6\,a+6\,b\,x}+1\right )}-\frac {4\,{\mathrm {e}}^{a+b\,x}}{b\,\left (6\,{\mathrm {e}}^{4\,a+4\,b\,x}-4\,{\mathrm {e}}^{2\,a+2\,b\,x}-4\,{\mathrm {e}}^{6\,a+6\,b\,x}+{\mathrm {e}}^{8\,a+8\,b\,x}+1\right )}+\frac {11\,{\mathrm {e}}^{a+b\,x}}{4\,b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )}+\frac {6\,{\mathrm {e}}^{a+b\,x}}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}+1\right )} \] Input:
int(1/(cosh(a + b*x)^4*sinh(a + b*x)^5),x)
Output:
(7*exp(a + b*x))/(2*b*(exp(4*a + 4*b*x) - 2*exp(2*a + 2*b*x) + 1)) - (35*a tan((exp(b*x)*exp(a)*(-b^2)^(1/2))/b))/(4*(-b^2)^(1/2)) + (8*exp(a + b*x)) /(3*b*(2*exp(2*a + 2*b*x) + exp(4*a + 4*b*x) + 1)) - (6*exp(a + b*x))/(b*( 3*exp(2*a + 2*b*x) - 3*exp(4*a + 4*b*x) + exp(6*a + 6*b*x) - 1)) - (8*exp( a + b*x))/(3*b*(3*exp(2*a + 2*b*x) + 3*exp(4*a + 4*b*x) + exp(6*a + 6*b*x) + 1)) - (4*exp(a + b*x))/(b*(6*exp(4*a + 4*b*x) - 4*exp(2*a + 2*b*x) - 4* exp(6*a + 6*b*x) + exp(8*a + 8*b*x) + 1)) + (11*exp(a + b*x))/(4*b*(exp(2* a + 2*b*x) - 1)) + (6*exp(a + b*x))/(b*(exp(2*a + 2*b*x) + 1))
Time = 0.23 (sec) , antiderivative size = 505, normalized size of antiderivative = 6.23 \[ \int \text {csch}^5(a+b x) \text {sech}^4(a+b x) \, dx=\frac {105 e^{14 b x +14 a} \mathrm {log}\left (e^{b x +a}-1\right )-105 e^{14 b x +14 a} \mathrm {log}\left (e^{b x +a}+1\right )+210 e^{13 b x +13 a}-105 e^{12 b x +12 a} \mathrm {log}\left (e^{b x +a}-1\right )+105 e^{12 b x +12 a} \mathrm {log}\left (e^{b x +a}+1\right )-140 e^{11 b x +11 a}-315 e^{10 b x +10 a} \mathrm {log}\left (e^{b x +a}-1\right )+315 e^{10 b x +10 a} \mathrm {log}\left (e^{b x +a}+1\right )-658 e^{9 b x +9 a}+315 e^{8 b x +8 a} \mathrm {log}\left (e^{b x +a}-1\right )-315 e^{8 b x +8 a} \mathrm {log}\left (e^{b x +a}+1\right )+408 e^{7 b x +7 a}+315 e^{6 b x +6 a} \mathrm {log}\left (e^{b x +a}-1\right )-315 e^{6 b x +6 a} \mathrm {log}\left (e^{b x +a}+1\right )-658 e^{5 b x +5 a}-315 e^{4 b x +4 a} \mathrm {log}\left (e^{b x +a}-1\right )+315 e^{4 b x +4 a} \mathrm {log}\left (e^{b x +a}+1\right )-140 e^{3 b x +3 a}-105 e^{2 b x +2 a} \mathrm {log}\left (e^{b x +a}-1\right )+105 e^{2 b x +2 a} \mathrm {log}\left (e^{b x +a}+1\right )+210 e^{b x +a}+105 \,\mathrm {log}\left (e^{b x +a}-1\right )-105 \,\mathrm {log}\left (e^{b x +a}+1\right )}{24 b \left (e^{14 b x +14 a}-e^{12 b x +12 a}-3 e^{10 b x +10 a}+3 e^{8 b x +8 a}+3 e^{6 b x +6 a}-3 e^{4 b x +4 a}-e^{2 b x +2 a}+1\right )} \] Input:
int(csch(b*x+a)^5*sech(b*x+a)^4,x)
Output:
(105*e**(14*a + 14*b*x)*log(e**(a + b*x) - 1) - 105*e**(14*a + 14*b*x)*log (e**(a + b*x) + 1) + 210*e**(13*a + 13*b*x) - 105*e**(12*a + 12*b*x)*log(e **(a + b*x) - 1) + 105*e**(12*a + 12*b*x)*log(e**(a + b*x) + 1) - 140*e**( 11*a + 11*b*x) - 315*e**(10*a + 10*b*x)*log(e**(a + b*x) - 1) + 315*e**(10 *a + 10*b*x)*log(e**(a + b*x) + 1) - 658*e**(9*a + 9*b*x) + 315*e**(8*a + 8*b*x)*log(e**(a + b*x) - 1) - 315*e**(8*a + 8*b*x)*log(e**(a + b*x) + 1) + 408*e**(7*a + 7*b*x) + 315*e**(6*a + 6*b*x)*log(e**(a + b*x) - 1) - 315* e**(6*a + 6*b*x)*log(e**(a + b*x) + 1) - 658*e**(5*a + 5*b*x) - 315*e**(4* a + 4*b*x)*log(e**(a + b*x) - 1) + 315*e**(4*a + 4*b*x)*log(e**(a + b*x) + 1) - 140*e**(3*a + 3*b*x) - 105*e**(2*a + 2*b*x)*log(e**(a + b*x) - 1) + 105*e**(2*a + 2*b*x)*log(e**(a + b*x) + 1) + 210*e**(a + b*x) + 105*log(e* *(a + b*x) - 1) - 105*log(e**(a + b*x) + 1))/(24*b*(e**(14*a + 14*b*x) - e **(12*a + 12*b*x) - 3*e**(10*a + 10*b*x) + 3*e**(8*a + 8*b*x) + 3*e**(6*a + 6*b*x) - 3*e**(4*a + 4*b*x) - e**(2*a + 2*b*x) + 1))