Integrand size = 13, antiderivative size = 121 \[ \int x^3 \coth ^{-1}(\tanh (a+b x))^n \, dx=\frac {x^3 \coth ^{-1}(\tanh (a+b x))^{1+n}}{b (1+n)}-\frac {3 x^2 \coth ^{-1}(\tanh (a+b x))^{2+n}}{b^2 (1+n) (2+n)}+\frac {6 x \coth ^{-1}(\tanh (a+b x))^{3+n}}{b^3 (1+n) (2+n) (3+n)}-\frac {6 \coth ^{-1}(\tanh (a+b x))^{4+n}}{b^4 (1+n) (2+n) (3+n) (4+n)} \]
x^3*arccoth(tanh(b*x+a))^(1+n)/b/(1+n)-3*x^2*arccoth(tanh(b*x+a))^(2+n)/b^ 2/(1+n)/(2+n)+6*x*arccoth(tanh(b*x+a))^(3+n)/b^3/(3+n)/(n^2+3*n+2)-6*arcco th(tanh(b*x+a))^(4+n)/b^4/(n^2+5*n+4)/(n^2+5*n+6)
Time = 0.05 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.88 \[ \int x^3 \coth ^{-1}(\tanh (a+b x))^n \, dx=\frac {\coth ^{-1}(\tanh (a+b x))^{1+n} \left (b^3 \left (24+26 n+9 n^2+n^3\right ) x^3-3 b^2 \left (12+7 n+n^2\right ) x^2 \coth ^{-1}(\tanh (a+b x))+6 b (4+n) x \coth ^{-1}(\tanh (a+b x))^2-6 \coth ^{-1}(\tanh (a+b x))^3\right )}{b^4 (1+n) (2+n) (3+n) (4+n)} \]
(ArcCoth[Tanh[a + b*x]]^(1 + n)*(b^3*(24 + 26*n + 9*n^2 + n^3)*x^3 - 3*b^2 *(12 + 7*n + n^2)*x^2*ArcCoth[Tanh[a + b*x]] + 6*b*(4 + n)*x*ArcCoth[Tanh[ a + b*x]]^2 - 6*ArcCoth[Tanh[a + b*x]]^3))/(b^4*(1 + n)*(2 + n)*(3 + n)*(4 + n))
Time = 0.35 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.96, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2599, 2599, 2599, 2588, 15}
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.
| \(\displaystyle \int x^3 \coth ^{-1}(\tanh (a+b x))^n \, dx\) |
| \(\Big \downarrow \) 2599 |
| \(\displaystyle \frac {x^3 \coth ^{-1}(\tanh (a+b x))^{n+1}}{b (n+1)}-\frac {3 \int x^2 \coth ^{-1}(\tanh (a+b x))^{n+1}dx}{b (n+1)}\) |
| \(\Big \downarrow \) 2599 |
| \(\displaystyle \frac {x^3 \coth ^{-1}(\tanh (a+b x))^{n+1}}{b (n+1)}-\frac {3 \left (\frac {x^2 \coth ^{-1}(\tanh (a+b x))^{n+2}}{b (n+2)}-\frac {2 \int x \coth ^{-1}(\tanh (a+b x))^{n+2}dx}{b (n+2)}\right )}{b (n+1)}\) |
| \(\Big \downarrow \) 2599 |
| \(\displaystyle \frac {x^3 \coth ^{-1}(\tanh (a+b x))^{n+1}}{b (n+1)}-\frac {3 \left (\frac {x^2 \coth ^{-1}(\tanh (a+b x))^{n+2}}{b (n+2)}-\frac {2 \left (\frac {x \coth ^{-1}(\tanh (a+b x))^{n+3}}{b (n+3)}-\frac {\int \coth ^{-1}(\tanh (a+b x))^{n+3}dx}{b (n+3)}\right )}{b (n+2)}\right )}{b (n+1)}\) |
| \(\Big \downarrow \) 2588 |
| \(\displaystyle \frac {x^3 \coth ^{-1}(\tanh (a+b x))^{n+1}}{b (n+1)}-\frac {3 \left (\frac {x^2 \coth ^{-1}(\tanh (a+b x))^{n+2}}{b (n+2)}-\frac {2 \left (\frac {x \coth ^{-1}(\tanh (a+b x))^{n+3}}{b (n+3)}-\frac {\int \coth ^{-1}(\tanh (a+b x))^{n+3}d\coth ^{-1}(\tanh (a+b x))}{b^2 (n+3)}\right )}{b (n+2)}\right )}{b (n+1)}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {x^3 \coth ^{-1}(\tanh (a+b x))^{n+1}}{b (n+1)}-\frac {3 \left (\frac {x^2 \coth ^{-1}(\tanh (a+b x))^{n+2}}{b (n+2)}-\frac {2 \left (\frac {x \coth ^{-1}(\tanh (a+b x))^{n+3}}{b (n+3)}-\frac {\coth ^{-1}(\tanh (a+b x))^{n+4}}{b^2 (n+3) (n+4)}\right )}{b (n+2)}\right )}{b (n+1)}\) |
(x^3*ArcCoth[Tanh[a + b*x]]^(1 + n))/(b*(1 + n)) - (3*((x^2*ArcCoth[Tanh[a + b*x]]^(2 + n))/(b*(2 + n)) - (2*((x*ArcCoth[Tanh[a + b*x]]^(3 + n))/(b* (3 + n)) - ArcCoth[Tanh[a + b*x]]^(4 + n)/(b^2*(3 + n)*(4 + n))))/(b*(2 + n))))/(b*(1 + n))
3.2.86.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[(u_)^(m_.), x_Symbol] :> With[{c = Simplify[D[u, x]]}, Simp[1/c Subst [Int[x^m, x], x, u], x]] /; FreeQ[m, x] && PiecewiseLinearQ[u, x]
Int[(u_)^(m_)*(v_)^(n_.), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Sim plify[D[v, x]]}, Simp[u^(m + 1)*(v^n/(a*(m + 1))), x] - Simp[b*(n/(a*(m + 1 ))) Int[u^(m + 1)*v^(n - 1), x], x] /; NeQ[b*u - a*v, 0]] /; FreeQ[{m, n} , x] && PiecewiseLinearQ[u, v, x] && NeQ[m, -1] && ((LtQ[m, -1] && GtQ[n, 0 ] && !(ILtQ[m + n, -2] && (FractionQ[m] || GeQ[2*n + m + 1, 0]))) || (IGtQ [n, 0] && IGtQ[m, 0] && LeQ[n, m]) || (IGtQ[n, 0] && !IntegerQ[m]) || (ILt Q[m, 0] && !IntegerQ[n]))
Leaf count of result is larger than twice the leaf count of optimal. \(276\) vs. \(2(121)=242\).
Time = 10.86 (sec) , antiderivative size = 277, normalized size of antiderivative = 2.29
| method | result | size |
| parallelrisch | \(-\frac {-24 \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{n} \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{3} x b -x^{3} \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right ) \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{n} b^{3} n^{3}-9 x^{3} \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right ) \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{n} b^{3} n^{2}-26 x^{3} \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right ) \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{n} b^{3} n +3 x^{2} \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{2} \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{n} b^{2} n^{2}+21 x^{2} \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{2} \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{n} b^{2} n -6 x \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{3} \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{n} b n -24 \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{n} \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right ) x^{3} b^{3}+36 \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{n} \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{2} x^{2} b^{2}+6 \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{n} \operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{4}}{\left (n^{3}+6 n^{2}+11 n +6\right ) \left (4+n \right ) b^{4}}\) | \(277\) |
| risch | \(\text {Expression too large to display}\) | \(129477\) |
-(-24*arccoth(tanh(b*x+a))^n*arccoth(tanh(b*x+a))^3*x*b-x^3*arccoth(tanh(b *x+a))*arccoth(tanh(b*x+a))^n*b^3*n^3-9*x^3*arccoth(tanh(b*x+a))*arccoth(t anh(b*x+a))^n*b^3*n^2-26*x^3*arccoth(tanh(b*x+a))*arccoth(tanh(b*x+a))^n*b ^3*n+3*x^2*arccoth(tanh(b*x+a))^2*arccoth(tanh(b*x+a))^n*b^2*n^2+21*x^2*ar ccoth(tanh(b*x+a))^2*arccoth(tanh(b*x+a))^n*b^2*n-6*x*arccoth(tanh(b*x+a)) ^3*arccoth(tanh(b*x+a))^n*b*n-24*arccoth(tanh(b*x+a))^n*arccoth(tanh(b*x+a ))*x^3*b^3+36*arccoth(tanh(b*x+a))^n*arccoth(tanh(b*x+a))^2*x^2*b^2+6*arcc oth(tanh(b*x+a))^n*arccoth(tanh(b*x+a))^4)/(n^3+6*n^2+11*n+6)/(4+n)/b^4
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 498, normalized size of antiderivative = 4.12 \[ \int x^3 \coth ^{-1}(\tanh (a+b x))^n \, dx=\frac {{\left (48 \, a^{3} b n x + 8 \, {\left (b^{4} n^{3} + 6 \, b^{4} n^{2} + 11 \, b^{4} n + 6 \, b^{4}\right )} x^{4} - 3 \, \pi ^{4} - 6 i \, \pi ^{3} {\left (b n x - 4 \, a\right )} - 48 \, a^{4} + 8 \, {\left (a b^{3} n^{3} + 3 \, a b^{3} n^{2} + 2 \, a b^{3} n\right )} x^{3} - 6 \, \pi ^{2} {\left (6 \, a b n x - {\left (b^{2} n^{2} + b^{2} n\right )} x^{2} - 12 \, a^{2}\right )} - 24 \, {\left (a^{2} b^{2} n^{2} + a^{2} b^{2} n\right )} x^{2} + 4 i \, \pi {\left (18 \, a^{2} b n x + {\left (b^{3} n^{3} + 3 \, b^{3} n^{2} + 2 \, b^{3} n\right )} x^{3} - 24 \, a^{3} - 6 \, {\left (a b^{2} n^{2} + a b^{2} n\right )} x^{2}\right )}\right )} \cosh \left (n \log \left (\frac {1}{2} i \, \pi + b x + a\right )\right ) + {\left (48 \, a^{3} b n x + 8 \, {\left (b^{4} n^{3} + 6 \, b^{4} n^{2} + 11 \, b^{4} n + 6 \, b^{4}\right )} x^{4} - 3 \, \pi ^{4} - 6 i \, \pi ^{3} {\left (b n x - 4 \, a\right )} - 48 \, a^{4} + 8 \, {\left (a b^{3} n^{3} + 3 \, a b^{3} n^{2} + 2 \, a b^{3} n\right )} x^{3} - 6 \, \pi ^{2} {\left (6 \, a b n x - {\left (b^{2} n^{2} + b^{2} n\right )} x^{2} - 12 \, a^{2}\right )} - 24 \, {\left (a^{2} b^{2} n^{2} + a^{2} b^{2} n\right )} x^{2} + 4 i \, \pi {\left (18 \, a^{2} b n x + {\left (b^{3} n^{3} + 3 \, b^{3} n^{2} + 2 \, b^{3} n\right )} x^{3} - 24 \, a^{3} - 6 \, {\left (a b^{2} n^{2} + a b^{2} n\right )} x^{2}\right )}\right )} \sinh \left (n \log \left (\frac {1}{2} i \, \pi + b x + a\right )\right )}{8 \, {\left (b^{4} n^{4} + 10 \, b^{4} n^{3} + 35 \, b^{4} n^{2} + 50 \, b^{4} n + 24 \, b^{4}\right )}} \]
1/8*((48*a^3*b*n*x + 8*(b^4*n^3 + 6*b^4*n^2 + 11*b^4*n + 6*b^4)*x^4 - 3*pi ^4 - 6*I*pi^3*(b*n*x - 4*a) - 48*a^4 + 8*(a*b^3*n^3 + 3*a*b^3*n^2 + 2*a*b^ 3*n)*x^3 - 6*pi^2*(6*a*b*n*x - (b^2*n^2 + b^2*n)*x^2 - 12*a^2) - 24*(a^2*b ^2*n^2 + a^2*b^2*n)*x^2 + 4*I*pi*(18*a^2*b*n*x + (b^3*n^3 + 3*b^3*n^2 + 2* b^3*n)*x^3 - 24*a^3 - 6*(a*b^2*n^2 + a*b^2*n)*x^2))*cosh(n*log(1/2*I*pi + b*x + a)) + (48*a^3*b*n*x + 8*(b^4*n^3 + 6*b^4*n^2 + 11*b^4*n + 6*b^4)*x^4 - 3*pi^4 - 6*I*pi^3*(b*n*x - 4*a) - 48*a^4 + 8*(a*b^3*n^3 + 3*a*b^3*n^2 + 2*a*b^3*n)*x^3 - 6*pi^2*(6*a*b*n*x - (b^2*n^2 + b^2*n)*x^2 - 12*a^2) - 24 *(a^2*b^2*n^2 + a^2*b^2*n)*x^2 + 4*I*pi*(18*a^2*b*n*x + (b^3*n^3 + 3*b^3*n ^2 + 2*b^3*n)*x^3 - 24*a^3 - 6*(a*b^2*n^2 + a*b^2*n)*x^2))*sinh(n*log(1/2* I*pi + b*x + a)))/(b^4*n^4 + 10*b^4*n^3 + 35*b^4*n^2 + 50*b^4*n + 24*b^4)
\[ \int x^3 \coth ^{-1}(\tanh (a+b x))^n \, dx=\text {Too large to display} \]
Piecewise((x**4*acoth(tanh(a))**n/4, Eq(b, 0)), (-x**3/(3*b*acoth(tanh(a + b*x))**3) - x**2/(2*b**2*acoth(tanh(a + b*x))**2) - x/(b**3*acoth(tanh(a + b*x))) + log(acoth(tanh(a + b*x)))/b**4, Eq(n, -4)), (Integral(x**3/acot h(tanh(a + b*x))**3, x), Eq(n, -3)), (Integral(x**3/acoth(tanh(a + b*x))** 2, x), Eq(n, -2)), (Integral(x**3/acoth(tanh(a + b*x)), x), Eq(n, -1)), (b **3*n**3*x**3*acoth(tanh(a + b*x))*acoth(tanh(a + b*x))**n/(b**4*n**4 + 10 *b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 9*b**3*n**2*x**3*acoth( tanh(a + b*x))*acoth(tanh(a + b*x))**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4 *n**2 + 50*b**4*n + 24*b**4) + 26*b**3*n*x**3*acoth(tanh(a + b*x))*acoth(t anh(a + b*x))**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24 *b**4) + 24*b**3*x**3*acoth(tanh(a + b*x))*acoth(tanh(a + b*x))**n/(b**4*n **4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) - 3*b**2*n**2*x** 2*acoth(tanh(a + b*x))**2*acoth(tanh(a + b*x))**n/(b**4*n**4 + 10*b**4*n** 3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) - 21*b**2*n*x**2*acoth(tanh(a + b* x))**2*acoth(tanh(a + b*x))**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) - 36*b**2*x**2*acoth(tanh(a + b*x))**2*acoth(tanh(a + b*x))**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 6*b*n*x*acoth(tanh(a + b*x))**3*acoth(tanh(a + b*x))**n/(b**4*n**4 + 10* b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 24*b*x*acoth(tanh(a + b* x))**3*acoth(tanh(a + b*x))**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2...
Result contains complex when optimal does not.
Time = 0.38 (sec) , antiderivative size = 257, normalized size of antiderivative = 2.12 \[ \int x^3 \coth ^{-1}(\tanh (a+b x))^n \, dx=\frac {{\left (8 \, {\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{4} x^{4} - 3 \, \pi ^{4} - 24 i \, \pi ^{3} a + 72 \, \pi ^{2} a^{2} + 96 i \, \pi a^{3} - 48 \, a^{4} - 4 \, {\left (i \, \pi {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} b^{3} - 2 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a b^{3}\right )} x^{3} + 6 \, {\left (\pi ^{2} {\left (n^{2} + n\right )} b^{2} + 4 i \, \pi {\left (n^{2} + n\right )} a b^{2} - 4 \, {\left (n^{2} + n\right )} a^{2} b^{2}\right )} x^{2} - 6 \, {\left (-i \, \pi ^{3} b n + 6 \, \pi ^{2} a b n + 12 i \, \pi a^{2} b n - 8 \, a^{3} b n\right )} x\right )} {\left (\cosh \left (-n \log \left (-i \, \pi + 2 \, b x + 2 \, a\right )\right ) - \sinh \left (-n \log \left (-i \, \pi + 2 \, b x + 2 \, a\right )\right )\right )}}{{\left (2^{n + 3} n^{4} + 5 \cdot 2^{n + 4} n^{3} + 35 \cdot 2^{n + 3} n^{2} + 25 \cdot 2^{n + 4} n + 3 \cdot 2^{n + 6}\right )} b^{4}} \]
(8*(n^3 + 6*n^2 + 11*n + 6)*b^4*x^4 - 3*pi^4 - 24*I*pi^3*a + 72*pi^2*a^2 + 96*I*pi*a^3 - 48*a^4 - 4*(I*pi*(n^3 + 3*n^2 + 2*n)*b^3 - 2*(n^3 + 3*n^2 + 2*n)*a*b^3)*x^3 + 6*(pi^2*(n^2 + n)*b^2 + 4*I*pi*(n^2 + n)*a*b^2 - 4*(n^2 + n)*a^2*b^2)*x^2 - 6*(-I*pi^3*b*n + 6*pi^2*a*b*n + 12*I*pi*a^2*b*n - 8*a ^3*b*n)*x)*(cosh(-n*log(-I*pi + 2*b*x + 2*a)) - sinh(-n*log(-I*pi + 2*b*x + 2*a)))/((2^(n + 3)*n^4 + 5*2^(n + 4)*n^3 + 35*2^(n + 3)*n^2 + 25*2^(n + 4)*n + 3*2^(n + 6))*b^4)
\[ \int x^3 \coth ^{-1}(\tanh (a+b x))^n \, dx=\int { x^{3} \operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{n} \,d x } \]
Time = 4.20 (sec) , antiderivative size = 418, normalized size of antiderivative = 3.45 \[ \int x^3 \coth ^{-1}(\tanh (a+b x))^n \, dx=-{\left (\frac {\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )}{2}-\frac {\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )}{2}\right )}^n\,\left (\frac {3\,{\left (\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^4}{8\,b^4\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}-\frac {x^4\,\left (n^3+6\,n^2+11\,n+6\right )}{n^4+10\,n^3+35\,n^2+50\,n+24}+\frac {3\,n\,x\,{\left (\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^3}{4\,b^3\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {n\,x^3\,\left (\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )\,\left (n^2+3\,n+2\right )}{2\,b\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {3\,n\,x^2\,\left (n+1\right )\,{\left (\ln \left (-\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1}\right )+2\,b\,x\right )}^2}{4\,b^2\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}\right ) \]
-(log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) - 1))/2 - log(-2/(exp(2 *a)*exp(2*b*x) - 1))/2)^n*((3*(log(-2/(exp(2*a)*exp(2*b*x) - 1)) - log((2* exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) - 1)) + 2*b*x)^4)/(8*b^4*(50*n + 35*n^2 + 10*n^3 + n^4 + 24)) - (x^4*(11*n + 6*n^2 + n^3 + 6))/(50*n + 35* n^2 + 10*n^3 + n^4 + 24) + (3*n*x*(log(-2/(exp(2*a)*exp(2*b*x) - 1)) - log ((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) - 1)) + 2*b*x)^3)/(4*b^3*(50 *n + 35*n^2 + 10*n^3 + n^4 + 24)) + (n*x^3*(log(-2/(exp(2*a)*exp(2*b*x) - 1)) - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) - 1)) + 2*b*x)*(3*n + n^2 + 2))/(2*b*(50*n + 35*n^2 + 10*n^3 + n^4 + 24)) + (3*n*x^2*(n + 1)* (log(-2/(exp(2*a)*exp(2*b*x) - 1)) - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a) *exp(2*b*x) - 1)) + 2*b*x)^2)/(4*b^2*(50*n + 35*n^2 + 10*n^3 + n^4 + 24)))