Integrand size = 16, antiderivative size = 71 \[ \int e^x \coth ^2(2 x) \text {csch}^2(2 x) \, dx=\frac {4 e^x}{3 \left (1-e^{4 x}\right )^3}-\frac {13 e^x}{6 \left (1-e^{4 x}\right )^2}+\frac {29 e^x}{24 \left (1-e^{4 x}\right )}-\frac {3 \arctan \left (e^x\right )}{16}-\frac {3 \text {arctanh}\left (e^x\right )}{16} \] Output:
4/3*exp(x)/(1-exp(4*x))^3-13/6*exp(x)/(1-exp(4*x))^2+29*exp(x)/(24-24*exp( 4*x))-3/16*arctan(exp(x))-3/16*arctanh(exp(x))
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 4.31 (sec) , antiderivative size = 310, normalized size of antiderivative = 4.37 \[ \int e^x \coth ^2(2 x) \text {csch}^2(2 x) \, dx=\frac {e^{-7 x} \left (-1070609085-946471617 e^{4 x}+369641285 e^{8 x}+351173641 e^{12 x}-23818496 e^{16 x}+1070609085 \operatorname {Hypergeometric2F1}\left (\frac {1}{4},1,\frac {5}{4},e^{4 x}\right )+732349800 e^{4 x} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},1,\frac {5}{4},e^{4 x}\right )-635067810 e^{8 x} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},1,\frac {5}{4},e^{4 x}\right )-384831720 e^{12 x} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},1,\frac {5}{4},e^{4 x}\right )+60913125 e^{16 x} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},1,\frac {5}{4},e^{4 x}\right )+1280 e^{16 x} \left (821+1346 e^{4 x}+557 e^{8 x}\right ) \, _4F_3\left (2,2,2,\frac {9}{4};1,1,\frac {21}{4};e^{4 x}\right )+10240 e^{16 x} \left (23+42 e^{4 x}+19 e^{8 x}\right ) \, _5F_4\left (2,2,2,2,\frac {9}{4};1,1,1,\frac {21}{4};e^{4 x}\right )+20480 e^{16 x} \, _6F_5\left (2,2,2,2,2,\frac {9}{4};1,1,1,1,\frac {21}{4};e^{4 x}\right )+40960 e^{20 x} \, _6F_5\left (2,2,2,2,2,\frac {9}{4};1,1,1,1,\frac {21}{4};e^{4 x}\right )+20480 e^{24 x} \, _6F_5\left (2,2,2,2,2,\frac {9}{4};1,1,1,1,\frac {21}{4};e^{4 x}\right )\right )}{3818880} \] Input:
Integrate[E^x*Coth[2*x]^2*Csch[2*x]^2,x]
Output:
(-1070609085 - 946471617*E^(4*x) + 369641285*E^(8*x) + 351173641*E^(12*x) - 23818496*E^(16*x) + 1070609085*Hypergeometric2F1[1/4, 1, 5/4, E^(4*x)] + 732349800*E^(4*x)*Hypergeometric2F1[1/4, 1, 5/4, E^(4*x)] - 635067810*E^( 8*x)*Hypergeometric2F1[1/4, 1, 5/4, E^(4*x)] - 384831720*E^(12*x)*Hypergeo metric2F1[1/4, 1, 5/4, E^(4*x)] + 60913125*E^(16*x)*Hypergeometric2F1[1/4, 1, 5/4, E^(4*x)] + 1280*E^(16*x)*(821 + 1346*E^(4*x) + 557*E^(8*x))*Hyper geometricPFQ[{2, 2, 2, 9/4}, {1, 1, 21/4}, E^(4*x)] + 10240*E^(16*x)*(23 + 42*E^(4*x) + 19*E^(8*x))*HypergeometricPFQ[{2, 2, 2, 2, 9/4}, {1, 1, 1, 2 1/4}, E^(4*x)] + 20480*E^(16*x)*HypergeometricPFQ[{2, 2, 2, 2, 2, 9/4}, {1 , 1, 1, 1, 21/4}, E^(4*x)] + 40960*E^(20*x)*HypergeometricPFQ[{2, 2, 2, 2, 2, 9/4}, {1, 1, 1, 1, 21/4}, E^(4*x)] + 20480*E^(24*x)*HypergeometricPFQ[ {2, 2, 2, 2, 2, 9/4}, {1, 1, 1, 1, 21/4}, E^(4*x)])/(3818880*E^(7*x))
Time = 0.26 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.30, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {2720, 27, 963, 27, 957, 817, 756, 216, 219}
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.
| \(\displaystyle \int e^x \coth ^2(2 x) \text {csch}^2(2 x) \, dx\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \int \frac {4 e^{4 x} \left (e^{4 x}+1\right )^2}{\left (1-e^{4 x}\right )^4}de^x\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 4 \int \frac {e^{4 x} \left (1+e^{4 x}\right )^2}{\left (1-e^{4 x}\right )^4}de^x\) |
| \(\Big \downarrow \) 963 |
| \(\displaystyle 4 \left (\frac {e^{5 x}}{3 \left (1-e^{4 x}\right )^3}-\frac {1}{12} \int \frac {4 e^{4 x} \left (2+3 e^{4 x}\right )}{\left (1-e^{4 x}\right )^3}de^x\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 4 \left (\frac {e^{5 x}}{3 \left (1-e^{4 x}\right )^3}-\frac {1}{3} \int \frac {e^{4 x} \left (2+3 e^{4 x}\right )}{\left (1-e^{4 x}\right )^3}de^x\right )\) |
| \(\Big \downarrow \) 957 |
| \(\displaystyle 4 \left (\frac {1}{3} \left (\frac {9}{8} \int \frac {e^{4 x}}{\left (1-e^{4 x}\right )^2}de^x-\frac {5 e^{5 x}}{8 \left (1-e^{4 x}\right )^2}\right )+\frac {e^{5 x}}{3 \left (1-e^{4 x}\right )^3}\right )\) |
| \(\Big \downarrow \) 817 |
| \(\displaystyle 4 \left (\frac {1}{3} \left (\frac {9}{8} \left (\frac {e^x}{4 \left (1-e^{4 x}\right )}-\frac {1}{4} \int \frac {1}{1-e^{4 x}}de^x\right )-\frac {5 e^{5 x}}{8 \left (1-e^{4 x}\right )^2}\right )+\frac {e^{5 x}}{3 \left (1-e^{4 x}\right )^3}\right )\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle 4 \left (\frac {1}{3} \left (\frac {9}{8} \left (\frac {1}{4} \left (-\frac {1}{2} \int \frac {1}{1-e^{2 x}}de^x-\frac {1}{2} \int \frac {1}{1+e^{2 x}}de^x\right )+\frac {e^x}{4 \left (1-e^{4 x}\right )}\right )-\frac {5 e^{5 x}}{8 \left (1-e^{4 x}\right )^2}\right )+\frac {e^{5 x}}{3 \left (1-e^{4 x}\right )^3}\right )\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle 4 \left (\frac {1}{3} \left (\frac {9}{8} \left (\frac {1}{4} \left (-\frac {1}{2} \int \frac {1}{1-e^{2 x}}de^x-\frac {1}{2} \arctan \left (e^x\right )\right )+\frac {e^x}{4 \left (1-e^{4 x}\right )}\right )-\frac {5 e^{5 x}}{8 \left (1-e^{4 x}\right )^2}\right )+\frac {e^{5 x}}{3 \left (1-e^{4 x}\right )^3}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle 4 \left (\frac {1}{3} \left (\frac {9}{8} \left (\frac {1}{4} \left (-\frac {1}{2} \arctan \left (e^x\right )-\frac {\text {arctanh}\left (e^x\right )}{2}\right )+\frac {e^x}{4 \left (1-e^{4 x}\right )}\right )-\frac {5 e^{5 x}}{8 \left (1-e^{4 x}\right )^2}\right )+\frac {e^{5 x}}{3 \left (1-e^{4 x}\right )^3}\right )\) |
Input:
Int[E^x*Coth[2*x]^2*Csch[2*x]^2,x]
Output:
4*(E^(5*x)/(3*(1 - E^(4*x))^3) + ((-5*E^(5*x))/(8*(1 - E^(4*x))^2) + (9*(E ^x/(4*(1 - E^(4*x))) + (-1/2*ArcTan[E^x] - ArcTanh[E^x]/2)/4))/8)/3)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^( n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*n*(p + 1))), x] - Simp[c^n *((m - n + 1)/(b*n*(p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x], x ] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] && ! ILtQ[(m + n*(p + 1) + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _)), x_Symbol] :> Simp[(-(b*c - a*d))*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a *b*e*n*(p + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*b*n* (p + 1)) Int[(e*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && (( !IntegerQ[p + 1/2] && N eQ[p, -5/4]) || !RationalQ[m] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0] && LeQ[-1 , m, (-n)*(p + 1)]))
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^2, x_Symbol] :> Simp[(-(b*c - a*d)^2)*(e*x)^(m + 1)*((a + b*x^n)^(p + 1) /(a*b^2*e*n*(p + 1))), x] + Simp[1/(a*b^2*n*(p + 1)) Int[(e*x)^m*(a + b*x ^n)^(p + 1)*Simp[(b*c - a*d)^2*(m + 1) + b^2*c^2*n*(p + 1) + a*b*d^2*n*(p + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Result contains complex when optimal does not.
Time = 0.65 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.85
| method | result | size |
| risch | \(-\frac {{\mathrm e}^{x} \left (29 \,{\mathrm e}^{8 x}-6 \,{\mathrm e}^{4 x}+9\right )}{24 \left ({\mathrm e}^{4 x}-1\right )^{3}}+\frac {3 \ln \left ({\mathrm e}^{x}-1\right )}{32}-\frac {3 \ln \left (1+{\mathrm e}^{x}\right )}{32}+\frac {3 i \ln \left ({\mathrm e}^{x}-i\right )}{32}-\frac {3 i \ln \left ({\mathrm e}^{x}+i\right )}{32}\) | \(60\) |
Input:
int(exp(x)*coth(2*x)^2*csch(2*x)^2,x,method=_RETURNVERBOSE)
Output:
-1/24*exp(x)*(29*exp(8*x)-6*exp(4*x)+9)/(exp(4*x)-1)^3+3/32*ln(exp(x)-1)-3 /32*ln(1+exp(x))+3/32*I*ln(exp(x)-I)-3/32*I*ln(exp(x)+I)
Leaf count of result is larger than twice the leaf count of optimal. 992 vs. \(2 (47) = 94\).
Time = 0.09 (sec) , antiderivative size = 992, normalized size of antiderivative = 13.97 \[ \int e^x \coth ^2(2 x) \text {csch}^2(2 x) \, dx=\text {Too large to display} \] Input:
integrate(exp(x)*coth(2*x)^2*csch(2*x)^2,x, algorithm="fricas")
Output:
-1/96*(116*cosh(x)^9 + 9744*cosh(x)^3*sinh(x)^6 + 4176*cosh(x)^2*sinh(x)^7 + 1044*cosh(x)*sinh(x)^8 + 116*sinh(x)^9 + 24*(609*cosh(x)^4 - 1)*sinh(x) ^5 - 24*cosh(x)^5 + 24*(609*cosh(x)^5 - 5*cosh(x))*sinh(x)^4 + 48*(203*cos h(x)^6 - 5*cosh(x)^2)*sinh(x)^3 + 48*(87*cosh(x)^7 - 5*cosh(x)^3)*sinh(x)^ 2 + 18*(cosh(x)^12 + 220*cosh(x)^3*sinh(x)^9 + 66*cosh(x)^2*sinh(x)^10 + 1 2*cosh(x)*sinh(x)^11 + sinh(x)^12 + 3*(165*cosh(x)^4 - 1)*sinh(x)^8 - 3*co sh(x)^8 + 24*(33*cosh(x)^5 - cosh(x))*sinh(x)^7 + 84*(11*cosh(x)^6 - cosh( x)^2)*sinh(x)^6 + 24*(33*cosh(x)^7 - 7*cosh(x)^3)*sinh(x)^5 + 3*(165*cosh( x)^8 - 70*cosh(x)^4 + 1)*sinh(x)^4 + 3*cosh(x)^4 + 4*(55*cosh(x)^9 - 42*co sh(x)^5 + 3*cosh(x))*sinh(x)^3 + 6*(11*cosh(x)^10 - 14*cosh(x)^6 + 3*cosh( x)^2)*sinh(x)^2 + 12*(cosh(x)^11 - 2*cosh(x)^7 + cosh(x)^3)*sinh(x) - 1)*a rctan(cosh(x) + sinh(x)) + 9*(cosh(x)^12 + 220*cosh(x)^3*sinh(x)^9 + 66*co sh(x)^2*sinh(x)^10 + 12*cosh(x)*sinh(x)^11 + sinh(x)^12 + 3*(165*cosh(x)^4 - 1)*sinh(x)^8 - 3*cosh(x)^8 + 24*(33*cosh(x)^5 - cosh(x))*sinh(x)^7 + 84 *(11*cosh(x)^6 - cosh(x)^2)*sinh(x)^6 + 24*(33*cosh(x)^7 - 7*cosh(x)^3)*si nh(x)^5 + 3*(165*cosh(x)^8 - 70*cosh(x)^4 + 1)*sinh(x)^4 + 3*cosh(x)^4 + 4 *(55*cosh(x)^9 - 42*cosh(x)^5 + 3*cosh(x))*sinh(x)^3 + 6*(11*cosh(x)^10 - 14*cosh(x)^6 + 3*cosh(x)^2)*sinh(x)^2 + 12*(cosh(x)^11 - 2*cosh(x)^7 + cos h(x)^3)*sinh(x) - 1)*log(cosh(x) + sinh(x) + 1) - 9*(cosh(x)^12 + 220*cosh (x)^3*sinh(x)^9 + 66*cosh(x)^2*sinh(x)^10 + 12*cosh(x)*sinh(x)^11 + sin...
\[ \int e^x \coth ^2(2 x) \text {csch}^2(2 x) \, dx=\int e^{x} \coth ^{2}{\left (2 x \right )} \operatorname {csch}^{2}{\left (2 x \right )}\, dx \] Input:
integrate(exp(x)*coth(2*x)**2*csch(2*x)**2,x)
Output:
Integral(exp(x)*coth(2*x)**2*csch(2*x)**2, x)
Time = 0.12 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.83 \[ \int e^x \coth ^2(2 x) \text {csch}^2(2 x) \, dx=-\frac {29 \, e^{\left (9 \, x\right )} - 6 \, e^{\left (5 \, x\right )} + 9 \, e^{x}}{24 \, {\left (e^{\left (12 \, x\right )} - 3 \, e^{\left (8 \, x\right )} + 3 \, e^{\left (4 \, x\right )} - 1\right )}} - \frac {3}{16} \, \arctan \left (e^{x}\right ) - \frac {3}{32} \, \log \left (e^{x} + 1\right ) + \frac {3}{32} \, \log \left (e^{x} - 1\right ) \] Input:
integrate(exp(x)*coth(2*x)^2*csch(2*x)^2,x, algorithm="maxima")
Output:
-1/24*(29*e^(9*x) - 6*e^(5*x) + 9*e^x)/(e^(12*x) - 3*e^(8*x) + 3*e^(4*x) - 1) - 3/16*arctan(e^x) - 3/32*log(e^x + 1) + 3/32*log(e^x - 1)
Time = 0.11 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.68 \[ \int e^x \coth ^2(2 x) \text {csch}^2(2 x) \, dx=-\frac {29 \, e^{\left (9 \, x\right )} - 6 \, e^{\left (5 \, x\right )} + 9 \, e^{x}}{24 \, {\left (e^{\left (4 \, x\right )} - 1\right )}^{3}} - \frac {3}{16} \, \arctan \left (e^{x}\right ) - \frac {3}{32} \, \log \left (e^{x} + 1\right ) + \frac {3}{32} \, \log \left ({\left | e^{x} - 1 \right |}\right ) \] Input:
integrate(exp(x)*coth(2*x)^2*csch(2*x)^2,x, algorithm="giac")
Output:
-1/24*(29*e^(9*x) - 6*e^(5*x) + 9*e^x)/(e^(4*x) - 1)^3 - 3/16*arctan(e^x) - 3/32*log(e^x + 1) + 3/32*log(abs(e^x - 1))
Time = 2.93 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.61 \[ \int e^x \coth ^2(2 x) \text {csch}^2(2 x) \, dx=\frac {3\,\ln \left (\frac {3}{8}-\frac {3\,{\mathrm {e}}^x}{8}\right )}{32}-\frac {3\,\ln \left (-\frac {3\,{\mathrm {e}}^x}{8}-\frac {3}{8}\right )}{32}-\frac {7\,{\mathrm {e}}^x}{8\,\left ({\mathrm {e}}^{4\,x}-1\right )}-\frac {\frac {2\,{\mathrm {e}}^{5\,x}}{3}+\frac {{\mathrm {e}}^{9\,x}}{3}+\frac {{\mathrm {e}}^x}{3}}{3\,{\mathrm {e}}^{4\,x}-3\,{\mathrm {e}}^{8\,x}+{\mathrm {e}}^{12\,x}-1}-\frac {5\,{\mathrm {e}}^x}{6\,\left ({\mathrm {e}}^{8\,x}-2\,{\mathrm {e}}^{4\,x}+1\right )}-\frac {\ln \left (-\frac {3\,{\mathrm {e}}^x}{8}-\frac {3}{8}{}\mathrm {i}\right )\,3{}\mathrm {i}}{32}+\frac {\ln \left (-\frac {3\,{\mathrm {e}}^x}{8}+\frac {3}{8}{}\mathrm {i}\right )\,3{}\mathrm {i}}{32} \] Input:
int((coth(2*x)^2*exp(x))/sinh(2*x)^2,x)
Output:
(3*log(3/8 - (3*exp(x))/8))/32 - (3*log(- (3*exp(x))/8 - 3/8))/32 - (log(- (3*exp(x))/8 - 3i/8)*3i)/32 + (log(3i/8 - (3*exp(x))/8)*3i)/32 - (7*exp(x ))/(8*(exp(4*x) - 1)) - ((2*exp(5*x))/3 + exp(9*x)/3 + exp(x)/3)/(3*exp(4* x) - 3*exp(8*x) + exp(12*x) - 1) - (5*exp(x))/(6*(exp(8*x) - 2*exp(4*x) + 1))
Time = 0.21 (sec) , antiderivative size = 179, normalized size of antiderivative = 2.52 \[ \int e^x \coth ^2(2 x) \text {csch}^2(2 x) \, dx=\frac {-18 e^{12 x} \mathit {atan} \left (e^{x}\right )+54 e^{8 x} \mathit {atan} \left (e^{x}\right )-54 e^{4 x} \mathit {atan} \left (e^{x}\right )+18 \mathit {atan} \left (e^{x}\right )+9 e^{12 x} \mathrm {log}\left (e^{x}-1\right )-9 e^{12 x} \mathrm {log}\left (e^{x}+1\right )-116 e^{9 x}-27 e^{8 x} \mathrm {log}\left (e^{x}-1\right )+27 e^{8 x} \mathrm {log}\left (e^{x}+1\right )+24 e^{5 x}+27 e^{4 x} \mathrm {log}\left (e^{x}-1\right )-27 e^{4 x} \mathrm {log}\left (e^{x}+1\right )-36 e^{x}-9 \,\mathrm {log}\left (e^{x}-1\right )+9 \,\mathrm {log}\left (e^{x}+1\right )}{96 e^{12 x}-288 e^{8 x}+288 e^{4 x}-96} \] Input:
int(exp(x)*coth(2*x)^2*csch(2*x)^2,x)
Output:
( - 18*e**(12*x)*atan(e**x) + 54*e**(8*x)*atan(e**x) - 54*e**(4*x)*atan(e* *x) + 18*atan(e**x) + 9*e**(12*x)*log(e**x - 1) - 9*e**(12*x)*log(e**x + 1 ) - 116*e**(9*x) - 27*e**(8*x)*log(e**x - 1) + 27*e**(8*x)*log(e**x + 1) + 24*e**(5*x) + 27*e**(4*x)*log(e**x - 1) - 27*e**(4*x)*log(e**x + 1) - 36* e**x - 9*log(e**x - 1) + 9*log(e**x + 1))/(96*(e**(12*x) - 3*e**(8*x) + 3* e**(4*x) - 1))