Integrand size = 10, antiderivative size = 296 \[ \int e^x \tanh ^2(4 x) \, dx=e^x+\frac {e^x}{2 \left (1+e^{8 x}\right )}+\frac {1}{16} \sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {2-\sqrt {2}}-2 e^x}{\sqrt {2+\sqrt {2}}}\right )+\frac {1}{16} \sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {2+\sqrt {2}}-2 e^x}{\sqrt {2-\sqrt {2}}}\right )-\frac {1}{16} \sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {2-\sqrt {2}}+2 e^x}{\sqrt {2+\sqrt {2}}}\right )-\frac {1}{16} \sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {2+\sqrt {2}}+2 e^x}{\sqrt {2-\sqrt {2}}}\right )-\frac {1}{16} \sqrt {2-\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2-\sqrt {2}} e^x}{1+e^{2 x}}\right )-\frac {1}{16} \sqrt {2+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {2}} e^x}{1+e^{2 x}}\right ) \] Output:
exp(x)+exp(x)/(2+2*exp(8*x))+1/16*(2+2^(1/2))^(1/2)*arctan(((2-2^(1/2))^(1 /2)-2*exp(x))/(2+2^(1/2))^(1/2))+1/16*(2-2^(1/2))^(1/2)*arctan(((2+2^(1/2) )^(1/2)-2*exp(x))/(2-2^(1/2))^(1/2))-1/16*(2+2^(1/2))^(1/2)*arctan(((2-2^( 1/2))^(1/2)+2*exp(x))/(2+2^(1/2))^(1/2))-1/16*(2-2^(1/2))^(1/2)*arctan(((2 +2^(1/2))^(1/2)+2*exp(x))/(2-2^(1/2))^(1/2))-1/16*(2-2^(1/2))^(1/2)*arctan h((2-2^(1/2))^(1/2)*exp(x)/(1+exp(2*x)))-1/16*(2+2^(1/2))^(1/2)*arctanh((2 +2^(1/2))^(1/2)*exp(x)/(1+exp(2*x)))
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.05 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.17 \[ \int e^x \tanh ^2(4 x) \, dx=e^x+\frac {e^x}{2 \left (1+e^{8 x}\right )}+\frac {1}{16} \text {RootSum}\left [1+\text {$\#$1}^8\&,\frac {x-\log \left (e^x-\text {$\#$1}\right )}{\text {$\#$1}^7}\&\right ] \] Input:
Integrate[E^x*Tanh[4*x]^2,x]
Output:
E^x + E^x/(2*(1 + E^(8*x))) + RootSum[1 + #1^8 & , (x - Log[E^x - #1])/#1^ 7 & ]/16
Time = 1.12 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.29, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2720, 915, 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.
| \(\displaystyle \int e^x \tanh ^2(4 x) \, dx\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \int \frac {\left (1-e^{8 x}\right )^2}{\left (e^{8 x}+1\right )^2}de^x\) |
| \(\Big \downarrow \) 915 |
| \(\displaystyle \int \left (1-\frac {4 e^{8 x}}{\left (e^{8 x}+1\right )^2}\right )de^x\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\arctan \left (\frac {\sqrt {2-\sqrt {2}}-2 e^x}{\sqrt {2+\sqrt {2}}}\right )}{8 \sqrt {2 \left (2-\sqrt {2}\right )}}+\frac {\arctan \left (\frac {\sqrt {2+\sqrt {2}}-2 e^x}{\sqrt {2-\sqrt {2}}}\right )}{8 \sqrt {2 \left (2+\sqrt {2}\right )}}-\frac {\arctan \left (\frac {2 e^x+\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )}{8 \sqrt {2 \left (2-\sqrt {2}\right )}}-\frac {\arctan \left (\frac {2 e^x+\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )}{8 \sqrt {2 \left (2+\sqrt {2}\right )}}+e^x+\frac {e^x}{2 \left (e^{8 x}+1\right )}+\frac {1}{32} \sqrt {2-\sqrt {2}} \log \left (-\sqrt {2-\sqrt {2}} e^x+e^{2 x}+1\right )-\frac {1}{32} \sqrt {2-\sqrt {2}} \log \left (\sqrt {2-\sqrt {2}} e^x+e^{2 x}+1\right )+\frac {1}{32} \sqrt {2+\sqrt {2}} \log \left (-\sqrt {2+\sqrt {2}} e^x+e^{2 x}+1\right )-\frac {1}{32} \sqrt {2+\sqrt {2}} \log \left (\sqrt {2+\sqrt {2}} e^x+e^{2 x}+1\right )\) |
Input:
Int[E^x*Tanh[4*x]^2,x]
Output:
E^x + E^x/(2*(1 + E^(8*x))) + ArcTan[(Sqrt[2 - Sqrt[2]] - 2*E^x)/Sqrt[2 + Sqrt[2]]]/(8*Sqrt[2*(2 - Sqrt[2])]) + ArcTan[(Sqrt[2 + Sqrt[2]] - 2*E^x)/S qrt[2 - Sqrt[2]]]/(8*Sqrt[2*(2 + Sqrt[2])]) - ArcTan[(Sqrt[2 - Sqrt[2]] + 2*E^x)/Sqrt[2 + Sqrt[2]]]/(8*Sqrt[2*(2 - Sqrt[2])]) - ArcTan[(Sqrt[2 + Sqr t[2]] + 2*E^x)/Sqrt[2 - Sqrt[2]]]/(8*Sqrt[2*(2 + Sqrt[2])]) + (Sqrt[2 - Sq rt[2]]*Log[1 - Sqrt[2 - Sqrt[2]]*E^x + E^(2*x)])/32 - (Sqrt[2 - Sqrt[2]]*L og[1 + Sqrt[2 - Sqrt[2]]*E^x + E^(2*x)])/32 + (Sqrt[2 + Sqrt[2]]*Log[1 - S qrt[2 + Sqrt[2]]*E^x + E^(2*x)])/32 - (Sqrt[2 + Sqrt[2]]*Log[1 + Sqrt[2 + Sqrt[2]]*E^x + E^(2*x)])/32
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a , b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILtQ[q, 0] && GeQ[p, -q]
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 higher order function than in optimal. Order 9 vs. order 3.
Time = 0.69 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.12
| method | result | size |
| risch | \({\mathrm e}^{x}+\frac {{\mathrm e}^{x}}{2+2 \,{\mathrm e}^{8 x}}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (4294967296 \textit {\_Z}^{8}+1\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{x}-16 \textit {\_R} \right )\right )\) | \(36\) |
Input:
int(exp(x)*tanh(4*x)^2,x,method=_RETURNVERBOSE)
Output:
exp(x)+1/2*exp(x)/(1+exp(8*x))+sum(_R*ln(exp(x)-16*_R),_R=RootOf(429496729 6*_Z^8+1))
Result contains complex when optimal does not.
Time = 0.14 (sec) , antiderivative size = 1303, normalized size of antiderivative = 4.40 \[ \int e^x \tanh ^2(4 x) \, dx=\text {Too large to display} \] Input:
integrate(exp(x)*tanh(4*x)^2,x, algorithm="fricas")
Output:
1/32*(32*cosh(x)^9 + 1152*cosh(x)^7*sinh(x)^2 + 2688*cosh(x)^6*sinh(x)^3 + 4032*cosh(x)^5*sinh(x)^4 + 4032*cosh(x)^4*sinh(x)^5 + 2688*cosh(x)^3*sinh (x)^6 + 1152*cosh(x)^2*sinh(x)^7 + 288*cosh(x)*sinh(x)^8 + 32*sinh(x)^9 + (-(I + 1)*sqrt(2)*(-1)^(1/8)*cosh(x)^8 - (8*I + 8)*sqrt(2)*(-1)^(1/8)*cosh (x)^7*sinh(x) - (28*I + 28)*sqrt(2)*(-1)^(1/8)*cosh(x)^6*sinh(x)^2 - (56*I + 56)*sqrt(2)*(-1)^(1/8)*cosh(x)^5*sinh(x)^3 - (70*I + 70)*sqrt(2)*(-1)^( 1/8)*cosh(x)^4*sinh(x)^4 - (56*I + 56)*sqrt(2)*(-1)^(1/8)*cosh(x)^3*sinh(x )^5 - (28*I + 28)*sqrt(2)*(-1)^(1/8)*cosh(x)^2*sinh(x)^6 - (8*I + 8)*sqrt( 2)*(-1)^(1/8)*cosh(x)*sinh(x)^7 - (I + 1)*sqrt(2)*(-1)^(1/8)*sinh(x)^8 - ( I + 1)*sqrt(2)*(-1)^(1/8))*log((I + 1)*sqrt(2)*(-1)^(1/8) + 2*cosh(x) + 2* sinh(x)) + ((I - 1)*sqrt(2)*(-1)^(1/8)*cosh(x)^8 + (8*I - 8)*sqrt(2)*(-1)^ (1/8)*cosh(x)^7*sinh(x) + (28*I - 28)*sqrt(2)*(-1)^(1/8)*cosh(x)^6*sinh(x) ^2 + (56*I - 56)*sqrt(2)*(-1)^(1/8)*cosh(x)^5*sinh(x)^3 + (70*I - 70)*sqrt (2)*(-1)^(1/8)*cosh(x)^4*sinh(x)^4 + (56*I - 56)*sqrt(2)*(-1)^(1/8)*cosh(x )^3*sinh(x)^5 + (28*I - 28)*sqrt(2)*(-1)^(1/8)*cosh(x)^2*sinh(x)^6 + (8*I - 8)*sqrt(2)*(-1)^(1/8)*cosh(x)*sinh(x)^7 + (I - 1)*sqrt(2)*(-1)^(1/8)*sin h(x)^8 + (I - 1)*sqrt(2)*(-1)^(1/8))*log(-(I - 1)*sqrt(2)*(-1)^(1/8) + 2*c osh(x) + 2*sinh(x)) + (-(I - 1)*sqrt(2)*(-1)^(1/8)*cosh(x)^8 - (8*I - 8)*s qrt(2)*(-1)^(1/8)*cosh(x)^7*sinh(x) - (28*I - 28)*sqrt(2)*(-1)^(1/8)*cosh( x)^6*sinh(x)^2 - (56*I - 56)*sqrt(2)*(-1)^(1/8)*cosh(x)^5*sinh(x)^3 - (...
\[ \int e^x \tanh ^2(4 x) \, dx=\int e^{x} \tanh ^{2}{\left (4 x \right )}\, dx \] Input:
integrate(exp(x)*tanh(4*x)**2,x)
Output:
Integral(exp(x)*tanh(4*x)**2, x)
\[ \int e^x \tanh ^2(4 x) \, dx=\int { e^{x} \tanh \left (4 \, x\right )^{2} \,d x } \] Input:
integrate(exp(x)*tanh(4*x)^2,x, algorithm="maxima")
Output:
1/2*(2*e^(9*x) + 3*e^x)/(e^(8*x) + 1) - integrate(1/2*e^x/(e^(8*x) + 1), x )
Time = 0.12 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.89 \[ \int e^x \tanh ^2(4 x) \, dx=-\frac {1}{16} \, \sqrt {-\sqrt {2} + 2} \arctan \left (\frac {\sqrt {\sqrt {2} + 2} + 2 \, e^{x}}{\sqrt {-\sqrt {2} + 2}}\right ) - \frac {1}{16} \, \sqrt {-\sqrt {2} + 2} \arctan \left (-\frac {\sqrt {\sqrt {2} + 2} - 2 \, e^{x}}{\sqrt {-\sqrt {2} + 2}}\right ) - \frac {1}{16} \, \sqrt {\sqrt {2} + 2} \arctan \left (\frac {\sqrt {-\sqrt {2} + 2} + 2 \, e^{x}}{\sqrt {\sqrt {2} + 2}}\right ) - \frac {1}{16} \, \sqrt {\sqrt {2} + 2} \arctan \left (-\frac {\sqrt {-\sqrt {2} + 2} - 2 \, e^{x}}{\sqrt {\sqrt {2} + 2}}\right ) - \frac {1}{32} \, \sqrt {\sqrt {2} + 2} \log \left (\sqrt {\sqrt {2} + 2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac {1}{32} \, \sqrt {\sqrt {2} + 2} \log \left (-\sqrt {\sqrt {2} + 2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) - \frac {1}{32} \, \sqrt {-\sqrt {2} + 2} \log \left (\sqrt {-\sqrt {2} + 2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac {1}{32} \, \sqrt {-\sqrt {2} + 2} \log \left (-\sqrt {-\sqrt {2} + 2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac {e^{x}}{2 \, {\left (e^{\left (8 \, x\right )} + 1\right )}} + e^{x} \] Input:
integrate(exp(x)*tanh(4*x)^2,x, algorithm="giac")
Output:
-1/16*sqrt(-sqrt(2) + 2)*arctan((sqrt(sqrt(2) + 2) + 2*e^x)/sqrt(-sqrt(2) + 2)) - 1/16*sqrt(-sqrt(2) + 2)*arctan(-(sqrt(sqrt(2) + 2) - 2*e^x)/sqrt(- sqrt(2) + 2)) - 1/16*sqrt(sqrt(2) + 2)*arctan((sqrt(-sqrt(2) + 2) + 2*e^x) /sqrt(sqrt(2) + 2)) - 1/16*sqrt(sqrt(2) + 2)*arctan(-(sqrt(-sqrt(2) + 2) - 2*e^x)/sqrt(sqrt(2) + 2)) - 1/32*sqrt(sqrt(2) + 2)*log(sqrt(sqrt(2) + 2)* e^x + e^(2*x) + 1) + 1/32*sqrt(sqrt(2) + 2)*log(-sqrt(sqrt(2) + 2)*e^x + e ^(2*x) + 1) - 1/32*sqrt(-sqrt(2) + 2)*log(sqrt(-sqrt(2) + 2)*e^x + e^(2*x) + 1) + 1/32*sqrt(-sqrt(2) + 2)*log(-sqrt(-sqrt(2) + 2)*e^x + e^(2*x) + 1) + 1/2*e^x/(e^(8*x) + 1) + e^x
Time = 5.21 (sec) , antiderivative size = 474, normalized size of antiderivative = 1.60 \[ \int e^x \tanh ^2(4 x) \, dx =\text {Too large to display} \] Input:
int(tanh(4*x)^2*exp(x),x)
Output:
exp(x) + exp(x)/(2*(exp(8*x) + 1)) + log(exp(x)/2 - (2^(1/2) + 2)^(1/2)/4 - ((2 - 2^(1/2))^(1/2)*1i)/4)*((2^(1/2) + 2)^(1/2)/32 + ((2 - 2^(1/2))^(1/ 2)*1i)/32) - log(exp(x)/2 + (2^(1/2) + 2)^(1/2)/4 + ((2 - 2^(1/2))^(1/2)*1 i)/4)*((2^(1/2) + 2)^(1/2)/32 + ((2 - 2^(1/2))^(1/2)*1i)/32) + log(exp(x)/ 2 - ((2^(1/2) + 2)^(1/2)*1i)/4 + (2 - 2^(1/2))^(1/2)/4)*(((2^(1/2) + 2)^(1 /2)*1i)/32 - (2 - 2^(1/2))^(1/2)/32) - log(exp(x)/2 + ((2^(1/2) + 2)^(1/2) *1i)/4 - (2 - 2^(1/2))^(1/2)/4)*(((2^(1/2) + 2)^(1/2)*1i)/32 - (2 - 2^(1/2 ))^(1/2)/32) + 2^(1/2)*log(exp(x)/2 - 2^(1/2)*((2^(1/2) + 2)^(1/2)/32 + (( 2 - 2^(1/2))^(1/2)*1i)/32)*(4 + 4i))*((2^(1/2) + 2)^(1/2)/32 + ((2 - 2^(1/ 2))^(1/2)*1i)/32)*(1/2 + 1i/2) + 2^(1/2)*log(exp(x)/2 - 2^(1/2)*((2^(1/2) + 2)^(1/2)/32 + ((2 - 2^(1/2))^(1/2)*1i)/32)*(4 - 4i))*((2^(1/2) + 2)^(1/2 )/32 + ((2 - 2^(1/2))^(1/2)*1i)/32)*(1/2 - 1i/2) - 2^(1/2)*log(exp(x)/2 + 2^(1/2)*((2^(1/2) + 2)^(1/2)/32 + ((2 - 2^(1/2))^(1/2)*1i)/32)*(4 - 4i))*( (2^(1/2) + 2)^(1/2)/32 + ((2 - 2^(1/2))^(1/2)*1i)/32)*(1/2 - 1i/2) - 2^(1/ 2)*log(exp(x)/2 + 2^(1/2)*((2^(1/2) + 2)^(1/2)/32 + ((2 - 2^(1/2))^(1/2)*1 i)/32)*(4 + 4i))*((2^(1/2) + 2)^(1/2)/32 + ((2 - 2^(1/2))^(1/2)*1i)/32)*(1 /2 + 1i/2)
Time = 0.28 (sec) , antiderivative size = 513, normalized size of antiderivative = 1.73 \[ \int e^x \tanh ^2(4 x) \, dx=\frac {2 e^{8 x} \sqrt {\sqrt {2}+2}\, \mathit {atan} \left (\frac {\sqrt {-\sqrt {2}+2}-2 e^{x}}{\sqrt {\sqrt {2}+2}}\right )+2 \sqrt {\sqrt {2}+2}\, \mathit {atan} \left (\frac {\sqrt {-\sqrt {2}+2}-2 e^{x}}{\sqrt {\sqrt {2}+2}}\right )-2 e^{8 x} \sqrt {\sqrt {2}+2}\, \mathit {atan} \left (\frac {\sqrt {-\sqrt {2}+2}+2 e^{x}}{\sqrt {\sqrt {2}+2}}\right )-2 \sqrt {\sqrt {2}+2}\, \mathit {atan} \left (\frac {\sqrt {-\sqrt {2}+2}+2 e^{x}}{\sqrt {\sqrt {2}+2}}\right )+2 e^{8 x} \sqrt {-\sqrt {2}+2}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {2}+2}-2 e^{x}}{\sqrt {-\sqrt {2}+2}}\right )+2 \sqrt {-\sqrt {2}+2}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {2}+2}-2 e^{x}}{\sqrt {-\sqrt {2}+2}}\right )-2 e^{8 x} \sqrt {-\sqrt {2}+2}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {2}+2}+2 e^{x}}{\sqrt {-\sqrt {2}+2}}\right )-2 \sqrt {-\sqrt {2}+2}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {2}+2}+2 e^{x}}{\sqrt {-\sqrt {2}+2}}\right )+e^{8 x} \sqrt {-\sqrt {2}+2}\, \mathrm {log}\left (-e^{x} \sqrt {-\sqrt {2}+2}+e^{2 x}+1\right )-e^{8 x} \sqrt {-\sqrt {2}+2}\, \mathrm {log}\left (e^{x} \sqrt {-\sqrt {2}+2}+e^{2 x}+1\right )+\sqrt {-\sqrt {2}+2}\, \mathrm {log}\left (-e^{x} \sqrt {-\sqrt {2}+2}+e^{2 x}+1\right )-\sqrt {-\sqrt {2}+2}\, \mathrm {log}\left (e^{x} \sqrt {-\sqrt {2}+2}+e^{2 x}+1\right )+e^{8 x} \sqrt {\sqrt {2}+2}\, \mathrm {log}\left (-e^{x} \sqrt {\sqrt {2}+2}+e^{2 x}+1\right )-e^{8 x} \sqrt {\sqrt {2}+2}\, \mathrm {log}\left (e^{x} \sqrt {\sqrt {2}+2}+e^{2 x}+1\right )+\sqrt {\sqrt {2}+2}\, \mathrm {log}\left (-e^{x} \sqrt {\sqrt {2}+2}+e^{2 x}+1\right )-\sqrt {\sqrt {2}+2}\, \mathrm {log}\left (e^{x} \sqrt {\sqrt {2}+2}+e^{2 x}+1\right )+32 e^{9 x}+48 e^{x}}{32 e^{8 x}+32} \] Input:
int(exp(x)*tanh(4*x)^2,x)
Output:
(2*e**(8*x)*sqrt(sqrt(2) + 2)*atan((sqrt( - sqrt(2) + 2) - 2*e**x)/sqrt(sq rt(2) + 2)) + 2*sqrt(sqrt(2) + 2)*atan((sqrt( - sqrt(2) + 2) - 2*e**x)/sqr t(sqrt(2) + 2)) - 2*e**(8*x)*sqrt(sqrt(2) + 2)*atan((sqrt( - sqrt(2) + 2) + 2*e**x)/sqrt(sqrt(2) + 2)) - 2*sqrt(sqrt(2) + 2)*atan((sqrt( - sqrt(2) + 2) + 2*e**x)/sqrt(sqrt(2) + 2)) + 2*e**(8*x)*sqrt( - sqrt(2) + 2)*atan((s qrt(sqrt(2) + 2) - 2*e**x)/sqrt( - sqrt(2) + 2)) + 2*sqrt( - sqrt(2) + 2)* atan((sqrt(sqrt(2) + 2) - 2*e**x)/sqrt( - sqrt(2) + 2)) - 2*e**(8*x)*sqrt( - sqrt(2) + 2)*atan((sqrt(sqrt(2) + 2) + 2*e**x)/sqrt( - sqrt(2) + 2)) - 2*sqrt( - sqrt(2) + 2)*atan((sqrt(sqrt(2) + 2) + 2*e**x)/sqrt( - sqrt(2) + 2)) + e**(8*x)*sqrt( - sqrt(2) + 2)*log( - e**x*sqrt( - sqrt(2) + 2) + e* *(2*x) + 1) - e**(8*x)*sqrt( - sqrt(2) + 2)*log(e**x*sqrt( - sqrt(2) + 2) + e**(2*x) + 1) + sqrt( - sqrt(2) + 2)*log( - e**x*sqrt( - sqrt(2) + 2) + e**(2*x) + 1) - sqrt( - sqrt(2) + 2)*log(e**x*sqrt( - sqrt(2) + 2) + e**(2 *x) + 1) + e**(8*x)*sqrt(sqrt(2) + 2)*log( - e**x*sqrt(sqrt(2) + 2) + e**( 2*x) + 1) - e**(8*x)*sqrt(sqrt(2) + 2)*log(e**x*sqrt(sqrt(2) + 2) + e**(2* x) + 1) + sqrt(sqrt(2) + 2)*log( - e**x*sqrt(sqrt(2) + 2) + e**(2*x) + 1) - sqrt(sqrt(2) + 2)*log(e**x*sqrt(sqrt(2) + 2) + e**(2*x) + 1) + 32*e**(9* x) + 48*e**x)/(32*(e**(8*x) + 1))