Integrand size = 10, antiderivative size = 55 \[ \int \frac {1}{\left (1-\sinh ^2(x)\right )^3} \, dx=\frac {19 \text {arctanh}\left (\sqrt {2} \tanh (x)\right )}{32 \sqrt {2}}+\frac {\cosh (x) \sinh (x)}{8 \left (1-\sinh ^2(x)\right )^2}+\frac {9 \cosh (x) \sinh (x)}{32 \left (1-\sinh ^2(x)\right )} \] Output:
19/64*arctanh(2^(1/2)*tanh(x))*2^(1/2)+1/8*cosh(x)*sinh(x)/(1-sinh(x)^2)^2 +9*cosh(x)*sinh(x)/(32-32*sinh(x)^2)
Time = 0.31 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.93 \[ \int \frac {1}{\left (1-\sinh ^2(x)\right )^3} \, dx=\frac {19 \text {arctanh}\left (\sqrt {2} \tanh (x)\right )}{32 \sqrt {2}}+\frac {\sinh (2 x)}{4 (-3+\cosh (2 x))^2}-\frac {9 \sinh (2 x)}{32 (-3+\cosh (2 x))} \] Input:
Integrate[(1 - Sinh[x]^2)^(-3),x]
Output:
(19*ArcTanh[Sqrt[2]*Tanh[x]])/(32*Sqrt[2]) + Sinh[2*x]/(4*(-3 + Cosh[2*x]) ^2) - (9*Sinh[2*x])/(32*(-3 + Cosh[2*x]))
Time = 0.36 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.09, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {3042, 3663, 25, 3042, 3652, 27, 3042, 3660, 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.
| \(\displaystyle \int \frac {1}{\left (1-\sinh ^2(x)\right )^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (1+\sin (i x)^2\right )^3}dx\) |
| \(\Big \downarrow \) 3663 |
| \(\displaystyle \frac {\sinh (x) \cosh (x)}{8 \left (1-\sinh ^2(x)\right )^2}-\frac {1}{8} \int -\frac {2 \sinh ^2(x)+7}{\left (1-\sinh ^2(x)\right )^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{8} \int \frac {2 \sinh ^2(x)+7}{\left (1-\sinh ^2(x)\right )^2}dx+\frac {\sinh (x) \cosh (x)}{8 \left (1-\sinh ^2(x)\right )^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sinh (x) \cosh (x)}{8 \left (1-\sinh ^2(x)\right )^2}+\frac {1}{8} \int \frac {7-2 \sin (i x)^2}{\left (\sin (i x)^2+1\right )^2}dx\) |
| \(\Big \downarrow \) 3652 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{4} \int \frac {19}{1-\sinh ^2(x)}dx+\frac {9 \sinh (x) \cosh (x)}{4 \left (1-\sinh ^2(x)\right )}\right )+\frac {\sinh (x) \cosh (x)}{8 \left (1-\sinh ^2(x)\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \left (\frac {19}{4} \int \frac {1}{1-\sinh ^2(x)}dx+\frac {9 \sinh (x) \cosh (x)}{4 \left (1-\sinh ^2(x)\right )}\right )+\frac {\sinh (x) \cosh (x)}{8 \left (1-\sinh ^2(x)\right )^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sinh (x) \cosh (x)}{8 \left (1-\sinh ^2(x)\right )^2}+\frac {1}{8} \left (\frac {9 \sinh (x) \cosh (x)}{4 \left (1-\sinh ^2(x)\right )}+\frac {19}{4} \int \frac {1}{\sin (i x)^2+1}dx\right )\) |
| \(\Big \downarrow \) 3660 |
| \(\displaystyle \frac {1}{8} \left (\frac {19}{4} \int \frac {1}{1-2 \tanh ^2(x)}d\tanh (x)+\frac {9 \sinh (x) \cosh (x)}{4 \left (1-\sinh ^2(x)\right )}\right )+\frac {\sinh (x) \cosh (x)}{8 \left (1-\sinh ^2(x)\right )^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{8} \left (\frac {19 \text {arctanh}\left (\sqrt {2} \tanh (x)\right )}{4 \sqrt {2}}+\frac {9 \sinh (x) \cosh (x)}{4 \left (1-\sinh ^2(x)\right )}\right )+\frac {\sinh (x) \cosh (x)}{8 \left (1-\sinh ^2(x)\right )^2}\) |
Input:
Int[(1 - Sinh[x]^2)^(-3),x]
Output:
(Cosh[x]*Sinh[x])/(8*(1 - Sinh[x]^2)^2) + ((19*ArcTanh[Sqrt[2]*Tanh[x]])/( 4*Sqrt[2]) + (9*Cosh[x]*Sinh[x])/(4*(1 - Sinh[x]^2)))/8
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]))* 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_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b - a*B))*Cos[e + f*x]*Sin[e + f*x ]*((a + b*Sin[e + f*x]^2)^(p + 1)/(2*a*f*(a + b)*(p + 1))), x] - Simp[1/(2* a*(a + b)*(p + 1)) Int[(a + b*Sin[e + f*x]^2)^(p + 1)*Simp[a*B - A*(2*a*( p + 1) + b*(2*p + 3)) + 2*(A*b - a*B)*(p + 2)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B}, x] && LtQ[p, -1] && NeQ[a + b, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(-1), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff/f Subst[Int[1/(a + (a + b)*ff^2*x^ 2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Simp[(-b)*C os[e + f*x]*Sin[e + f*x]*((a + b*Sin[e + f*x]^2)^(p + 1)/(2*a*f*(p + 1)*(a + b))), x] + Simp[1/(2*a*(p + 1)*(a + b)) Int[(a + b*Sin[e + f*x]^2)^(p + 1)*Simp[2*a*(p + 1) + b*(2*p + 3) - 2*b*(p + 2)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a + b, 0] && LtQ[p, -1]
Time = 0.48 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.31
| method | result | size |
| risch | \(-\frac {19 \,{\mathrm e}^{6 x}-171 \,{\mathrm e}^{4 x}+89 \,{\mathrm e}^{2 x}-9}{16 \left ({\mathrm e}^{4 x}-6 \,{\mathrm e}^{2 x}+1\right )^{2}}+\frac {19 \sqrt {2}\, \ln \left ({\mathrm e}^{2 x}-3+2 \sqrt {2}\right )}{128}-\frac {19 \sqrt {2}\, \ln \left ({\mathrm e}^{2 x}-3-2 \sqrt {2}\right )}{128}\) | \(72\) |
| default | \(-\frac {-\frac {13 \tanh \left (\frac {x}{2}\right )^{3}}{8}-\frac {11 \tanh \left (\frac {x}{2}\right )^{2}}{8}+\frac {31 \tanh \left (\frac {x}{2}\right )}{8}-\frac {11}{8}}{4 \left (\tanh \left (\frac {x}{2}\right )^{2}+2 \tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {19 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2 \tanh \left (\frac {x}{2}\right )+2\right ) \sqrt {2}}{4}\right )}{64}-\frac {-\frac {13 \tanh \left (\frac {x}{2}\right )^{3}}{8}+\frac {11 \tanh \left (\frac {x}{2}\right )^{2}}{8}+\frac {31 \tanh \left (\frac {x}{2}\right )}{8}+\frac {11}{8}}{4 \left (\tanh \left (\frac {x}{2}\right )^{2}-2 \tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {19 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2 \tanh \left (\frac {x}{2}\right )-2\right ) \sqrt {2}}{4}\right )}{64}\) | \(124\) |
Input:
int(1/(1-sinh(x)^2)^3,x,method=_RETURNVERBOSE)
Output:
-1/16*(19*exp(6*x)-171*exp(4*x)+89*exp(2*x)-9)/(exp(4*x)-6*exp(2*x)+1)^2+1 9/128*2^(1/2)*ln(exp(2*x)-3+2*2^(1/2))-19/128*2^(1/2)*ln(exp(2*x)-3-2*2^(1 /2))
Leaf count of result is larger than twice the leaf count of optimal. 575 vs. \(2 (41) = 82\).
Time = 0.09 (sec) , antiderivative size = 575, normalized size of antiderivative = 10.45 \[ \int \frac {1}{\left (1-\sinh ^2(x)\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(1/(1-sinh(x)^2)^3,x, algorithm="fricas")
Output:
-1/128*(152*cosh(x)^6 + 912*cosh(x)*sinh(x)^5 + 152*sinh(x)^6 + 456*(5*cos h(x)^2 - 3)*sinh(x)^4 - 1368*cosh(x)^4 + 608*(5*cosh(x)^3 - 9*cosh(x))*sin h(x)^3 + 8*(285*cosh(x)^4 - 1026*cosh(x)^2 + 89)*sinh(x)^2 + 712*cosh(x)^2 - 19*(sqrt(2)*cosh(x)^8 + 8*sqrt(2)*cosh(x)*sinh(x)^7 + sqrt(2)*sinh(x)^8 + 4*(7*sqrt(2)*cosh(x)^2 - 3*sqrt(2))*sinh(x)^6 - 12*sqrt(2)*cosh(x)^6 + 8*(7*sqrt(2)*cosh(x)^3 - 9*sqrt(2)*cosh(x))*sinh(x)^5 + 2*(35*sqrt(2)*cosh (x)^4 - 90*sqrt(2)*cosh(x)^2 + 19*sqrt(2))*sinh(x)^4 + 38*sqrt(2)*cosh(x)^ 4 + 8*(7*sqrt(2)*cosh(x)^5 - 30*sqrt(2)*cosh(x)^3 + 19*sqrt(2)*cosh(x))*si nh(x)^3 + 4*(7*sqrt(2)*cosh(x)^6 - 45*sqrt(2)*cosh(x)^4 + 57*sqrt(2)*cosh( x)^2 - 3*sqrt(2))*sinh(x)^2 - 12*sqrt(2)*cosh(x)^2 + 8*(sqrt(2)*cosh(x)^7 - 9*sqrt(2)*cosh(x)^5 + 19*sqrt(2)*cosh(x)^3 - 3*sqrt(2)*cosh(x))*sinh(x) + sqrt(2))*log(-(3*(2*sqrt(2) - 3)*cosh(x)^2 - 4*(3*sqrt(2) - 4)*cosh(x)*s inh(x) + 3*(2*sqrt(2) - 3)*sinh(x)^2 - 2*sqrt(2) + 3)/(cosh(x)^2 + sinh(x) ^2 - 3)) + 16*(57*cosh(x)^5 - 342*cosh(x)^3 + 89*cosh(x))*sinh(x) - 72)/(c osh(x)^8 + 8*cosh(x)*sinh(x)^7 + sinh(x)^8 + 4*(7*cosh(x)^2 - 3)*sinh(x)^6 - 12*cosh(x)^6 + 8*(7*cosh(x)^3 - 9*cosh(x))*sinh(x)^5 + 2*(35*cosh(x)^4 - 90*cosh(x)^2 + 19)*sinh(x)^4 + 38*cosh(x)^4 + 8*(7*cosh(x)^5 - 30*cosh(x )^3 + 19*cosh(x))*sinh(x)^3 + 4*(7*cosh(x)^6 - 45*cosh(x)^4 + 57*cosh(x)^2 - 3)*sinh(x)^2 - 12*cosh(x)^2 + 8*(cosh(x)^7 - 9*cosh(x)^5 + 19*cosh(x)^3 - 3*cosh(x))*sinh(x) + 1)
Leaf count of result is larger than twice the leaf count of optimal. 5666 vs. \(2 (51) = 102\).
Time = 11.67 (sec) , antiderivative size = 5666, normalized size of antiderivative = 103.02 \[ \int \frac {1}{\left (1-\sinh ^2(x)\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(1/(1-sinh(x)**2)**3,x)
Output:
10001001174720*log(tanh(x/2) - 1 + sqrt(2))*tanh(x/2)**8/(33687582904320*s qrt(2)*tanh(x/2)**8 + 47641436627072*tanh(x/2)**8 - 571697239524864*tanh(x /2)**6 - 404250994851840*sqrt(2)*tanh(x/2)**6 + 1280128150364160*sqrt(2)*t anh(x/2)**4 + 1810374591828736*tanh(x/2)**4 - 571697239524864*tanh(x/2)**2 - 404250994851840*sqrt(2)*tanh(x/2)**2 + 33687582904320*sqrt(2) + 4764143 6627072) + 7071775749331*sqrt(2)*log(tanh(x/2) - 1 + sqrt(2))*tanh(x/2)**8 /(33687582904320*sqrt(2)*tanh(x/2)**8 + 47641436627072*tanh(x/2)**8 - 5716 97239524864*tanh(x/2)**6 - 404250994851840*sqrt(2)*tanh(x/2)**6 + 12801281 50364160*sqrt(2)*tanh(x/2)**4 + 1810374591828736*tanh(x/2)**4 - 5716972395 24864*tanh(x/2)**2 - 404250994851840*sqrt(2)*tanh(x/2)**2 + 33687582904320 *sqrt(2) + 47641436627072) - 84861308991972*sqrt(2)*log(tanh(x/2) - 1 + sq rt(2))*tanh(x/2)**6/(33687582904320*sqrt(2)*tanh(x/2)**8 + 47641436627072* tanh(x/2)**8 - 571697239524864*tanh(x/2)**6 - 404250994851840*sqrt(2)*tanh (x/2)**6 + 1280128150364160*sqrt(2)*tanh(x/2)**4 + 1810374591828736*tanh(x /2)**4 - 571697239524864*tanh(x/2)**2 - 404250994851840*sqrt(2)*tanh(x/2)* *2 + 33687582904320*sqrt(2) + 47641436627072) - 120012014096640*log(tanh(x /2) - 1 + sqrt(2))*tanh(x/2)**6/(33687582904320*sqrt(2)*tanh(x/2)**8 + 476 41436627072*tanh(x/2)**8 - 571697239524864*tanh(x/2)**6 - 404250994851840* sqrt(2)*tanh(x/2)**6 + 1280128150364160*sqrt(2)*tanh(x/2)**4 + 18103745918 28736*tanh(x/2)**4 - 571697239524864*tanh(x/2)**2 - 404250994851840*sqr...
Leaf count of result is larger than twice the leaf count of optimal. 111 vs. \(2 (41) = 82\).
Time = 0.11 (sec) , antiderivative size = 111, normalized size of antiderivative = 2.02 \[ \int \frac {1}{\left (1-\sinh ^2(x)\right )^3} \, dx=\frac {19}{128} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - e^{\left (-x\right )} + 1}{\sqrt {2} + e^{\left (-x\right )} - 1}\right ) - \frac {19}{128} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - e^{\left (-x\right )} - 1}{\sqrt {2} + e^{\left (-x\right )} + 1}\right ) - \frac {89 \, e^{\left (-2 \, x\right )} - 171 \, e^{\left (-4 \, x\right )} + 19 \, e^{\left (-6 \, x\right )} - 9}{16 \, {\left (12 \, e^{\left (-2 \, x\right )} - 38 \, e^{\left (-4 \, x\right )} + 12 \, e^{\left (-6 \, x\right )} - e^{\left (-8 \, x\right )} - 1\right )}} \] Input:
integrate(1/(1-sinh(x)^2)^3,x, algorithm="maxima")
Output:
19/128*sqrt(2)*log(-(sqrt(2) - e^(-x) + 1)/(sqrt(2) + e^(-x) - 1)) - 19/12 8*sqrt(2)*log(-(sqrt(2) - e^(-x) - 1)/(sqrt(2) + e^(-x) + 1)) - 1/16*(89*e ^(-2*x) - 171*e^(-4*x) + 19*e^(-6*x) - 9)/(12*e^(-2*x) - 38*e^(-4*x) + 12* e^(-6*x) - e^(-8*x) - 1)
Time = 0.12 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.35 \[ \int \frac {1}{\left (1-\sinh ^2(x)\right )^3} \, dx=-\frac {19}{128} \, \sqrt {2} \log \left (\frac {{\left | -4 \, \sqrt {2} + 2 \, e^{\left (2 \, x\right )} - 6 \right |}}{{\left | 4 \, \sqrt {2} + 2 \, e^{\left (2 \, x\right )} - 6 \right |}}\right ) - \frac {19 \, e^{\left (6 \, x\right )} - 171 \, e^{\left (4 \, x\right )} + 89 \, e^{\left (2 \, x\right )} - 9}{16 \, {\left (e^{\left (4 \, x\right )} - 6 \, e^{\left (2 \, x\right )} + 1\right )}^{2}} \] Input:
integrate(1/(1-sinh(x)^2)^3,x, algorithm="giac")
Output:
-19/128*sqrt(2)*log(abs(-4*sqrt(2) + 2*e^(2*x) - 6)/abs(4*sqrt(2) + 2*e^(2 *x) - 6)) - 1/16*(19*e^(6*x) - 171*e^(4*x) + 89*e^(2*x) - 9)/(e^(4*x) - 6* e^(2*x) + 1)^2
Time = 1.74 (sec) , antiderivative size = 112, normalized size of antiderivative = 2.04 \[ \int \frac {1}{\left (1-\sinh ^2(x)\right )^3} \, dx=\frac {17\,{\mathrm {e}}^{2\,x}-3}{38\,{\mathrm {e}}^{4\,x}-12\,{\mathrm {e}}^{2\,x}-12\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1}-\frac {19\,\sqrt {2}\,\ln \left (\frac {19\,{\mathrm {e}}^{2\,x}}{8}-\frac {19\,\sqrt {2}\,\left (12\,{\mathrm {e}}^{2\,x}-4\right )}{128}\right )}{128}+\frac {19\,\sqrt {2}\,\ln \left (\frac {19\,{\mathrm {e}}^{2\,x}}{8}+\frac {19\,\sqrt {2}\,\left (12\,{\mathrm {e}}^{2\,x}-4\right )}{128}\right )}{128}-\frac {\frac {19\,{\mathrm {e}}^{2\,x}}{16}-\frac {57}{16}}{{\mathrm {e}}^{4\,x}-6\,{\mathrm {e}}^{2\,x}+1} \] Input:
int(-1/(sinh(x)^2 - 1)^3,x)
Output:
(17*exp(2*x) - 3)/(38*exp(4*x) - 12*exp(2*x) - 12*exp(6*x) + exp(8*x) + 1) - (19*2^(1/2)*log((19*exp(2*x))/8 - (19*2^(1/2)*(12*exp(2*x) - 4))/128))/ 128 + (19*2^(1/2)*log((19*exp(2*x))/8 + (19*2^(1/2)*(12*exp(2*x) - 4))/128 ))/128 - ((19*exp(2*x))/16 - 57/16)/(exp(4*x) - 6*exp(2*x) + 1)
Time = 0.15 (sec) , antiderivative size = 396, normalized size of antiderivative = 7.20 \[ \int \frac {1}{\left (1-\sinh ^2(x)\right )^3} \, dx=\frac {-57 e^{8 x} \sqrt {2}\, \mathrm {log}\left (e^{x}-\sqrt {2}-1\right )+57 e^{8 x} \sqrt {2}\, \mathrm {log}\left (e^{x}-\sqrt {2}+1\right )+57 e^{8 x} \sqrt {2}\, \mathrm {log}\left (e^{x}+\sqrt {2}-1\right )-57 e^{8 x} \sqrt {2}\, \mathrm {log}\left (e^{x}+\sqrt {2}+1\right )-38 e^{8 x}+684 e^{6 x} \sqrt {2}\, \mathrm {log}\left (e^{x}-\sqrt {2}-1\right )-684 e^{6 x} \sqrt {2}\, \mathrm {log}\left (e^{x}-\sqrt {2}+1\right )-684 e^{6 x} \sqrt {2}\, \mathrm {log}\left (e^{x}+\sqrt {2}-1\right )+684 e^{6 x} \sqrt {2}\, \mathrm {log}\left (e^{x}+\sqrt {2}+1\right )-2166 e^{4 x} \sqrt {2}\, \mathrm {log}\left (e^{x}-\sqrt {2}-1\right )+2166 e^{4 x} \sqrt {2}\, \mathrm {log}\left (e^{x}-\sqrt {2}+1\right )+2166 e^{4 x} \sqrt {2}\, \mathrm {log}\left (e^{x}+\sqrt {2}-1\right )-2166 e^{4 x} \sqrt {2}\, \mathrm {log}\left (e^{x}+\sqrt {2}+1\right )+2660 e^{4 x}+684 e^{2 x} \sqrt {2}\, \mathrm {log}\left (e^{x}-\sqrt {2}-1\right )-684 e^{2 x} \sqrt {2}\, \mathrm {log}\left (e^{x}-\sqrt {2}+1\right )-684 e^{2 x} \sqrt {2}\, \mathrm {log}\left (e^{x}+\sqrt {2}-1\right )+684 e^{2 x} \sqrt {2}\, \mathrm {log}\left (e^{x}+\sqrt {2}+1\right )-1680 e^{2 x}-57 \sqrt {2}\, \mathrm {log}\left (e^{x}-\sqrt {2}-1\right )+57 \sqrt {2}\, \mathrm {log}\left (e^{x}-\sqrt {2}+1\right )+57 \sqrt {2}\, \mathrm {log}\left (e^{x}+\sqrt {2}-1\right )-57 \sqrt {2}\, \mathrm {log}\left (e^{x}+\sqrt {2}+1\right )+178}{384 e^{8 x}-4608 e^{6 x}+14592 e^{4 x}-4608 e^{2 x}+384} \] Input:
int(1/(1-sinh(x)^2)^3,x)
Output:
( - 57*e**(8*x)*sqrt(2)*log(e**x - sqrt(2) - 1) + 57*e**(8*x)*sqrt(2)*log( e**x - sqrt(2) + 1) + 57*e**(8*x)*sqrt(2)*log(e**x + sqrt(2) - 1) - 57*e** (8*x)*sqrt(2)*log(e**x + sqrt(2) + 1) - 38*e**(8*x) + 684*e**(6*x)*sqrt(2) *log(e**x - sqrt(2) - 1) - 684*e**(6*x)*sqrt(2)*log(e**x - sqrt(2) + 1) - 684*e**(6*x)*sqrt(2)*log(e**x + sqrt(2) - 1) + 684*e**(6*x)*sqrt(2)*log(e* *x + sqrt(2) + 1) - 2166*e**(4*x)*sqrt(2)*log(e**x - sqrt(2) - 1) + 2166*e **(4*x)*sqrt(2)*log(e**x - sqrt(2) + 1) + 2166*e**(4*x)*sqrt(2)*log(e**x + sqrt(2) - 1) - 2166*e**(4*x)*sqrt(2)*log(e**x + sqrt(2) + 1) + 2660*e**(4 *x) + 684*e**(2*x)*sqrt(2)*log(e**x - sqrt(2) - 1) - 684*e**(2*x)*sqrt(2)* log(e**x - sqrt(2) + 1) - 684*e**(2*x)*sqrt(2)*log(e**x + sqrt(2) - 1) + 6 84*e**(2*x)*sqrt(2)*log(e**x + sqrt(2) + 1) - 1680*e**(2*x) - 57*sqrt(2)*l og(e**x - sqrt(2) - 1) + 57*sqrt(2)*log(e**x - sqrt(2) + 1) + 57*sqrt(2)*l og(e**x + sqrt(2) - 1) - 57*sqrt(2)*log(e**x + sqrt(2) + 1) + 178)/(384*(e **(8*x) - 12*e**(6*x) + 38*e**(4*x) - 12*e**(2*x) + 1))