Integrand size = 15, antiderivative size = 201 \[ \int \frac {1}{x^2 \left (4+\left (1+x^2\right )^4\right )} \, dx=-\frac {1}{5 x}+\frac {1}{80} \sqrt {\frac {1}{5} \left (11+5 \sqrt {5}\right )} \arctan \left (\frac {\sqrt {2 \left (-2+\sqrt {5}\right )}-2 x}{\sqrt {2 \left (2+\sqrt {5}\right )}}\right )-\frac {1}{80} \sqrt {\frac {1}{5} \left (11+5 \sqrt {5}\right )} \arctan \left (\frac {\sqrt {2 \left (-2+\sqrt {5}\right )}+2 x}{\sqrt {2 \left (2+\sqrt {5}\right )}}\right )+\frac {\arctan \left (1-\sqrt {2} x\right )}{8 \sqrt {2}}-\frac {\arctan \left (1+\sqrt {2} x\right )}{8 \sqrt {2}}-\frac {1}{80} \sqrt {\frac {1}{5} \left (-11+5 \sqrt {5}\right )} \text {arctanh}\left (\frac {\sqrt {2 \left (-2+\sqrt {5}\right )} x}{\sqrt {5}+x^2}\right ) \] Output:
-1/5/x+1/400*(55+25*5^(1/2))^(1/2)*arctan(((-4+2*5^(1/2))^(1/2)-2*x)/(4+2* 5^(1/2))^(1/2))-1/400*(55+25*5^(1/2))^(1/2)*arctan(((-4+2*5^(1/2))^(1/2)+2 *x)/(4+2*5^(1/2))^(1/2))-1/16*arctan(-1+x*2^(1/2))*2^(1/2)-1/16*arctan(1+x *2^(1/2))*2^(1/2)-1/400*(-55+25*5^(1/2))^(1/2)*arctanh((-4+2*5^(1/2))^(1/2 )*x/(5^(1/2)+x^2))
Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.45 \[ \int \frac {1}{x^2 \left (4+\left (1+x^2\right )^4\right )} \, dx=-\frac {80+(7+i) \sqrt {2-i} x \arctan \left (\frac {x}{\sqrt {2-i}}\right )+(7-i) \sqrt {2+i} x \arctan \left (\frac {x}{\sqrt {2+i}}\right )-25 \sqrt {2} x \arctan \left (1-\sqrt {2} x\right )+25 \sqrt {2} x \arctan \left (1+\sqrt {2} x\right )}{400 x} \] Input:
Integrate[1/(x^2*(4 + (1 + x^2)^4)),x]
Output:
-1/400*(80 + (7 + I)*Sqrt[2 - I]*x*ArcTan[x/Sqrt[2 - I]] + (7 - I)*Sqrt[2 + I]*x*ArcTan[x/Sqrt[2 + I]] - 25*Sqrt[2]*x*ArcTan[1 - Sqrt[2]*x] + 25*Sqr t[2]*x*ArcTan[1 + Sqrt[2]*x])/x
Time = 0.74 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.22, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2460, 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.
| \(\displaystyle \int \frac {1}{x^2 \left (\left (x^2+1\right )^4+4\right )} \, dx\) |
| \(\Big \downarrow \) 2460 |
| \(\displaystyle \int \left (\frac {1}{5 x^2}+\frac {-3 x^2-7}{40 \left (x^4+4 x^2+5\right )}+\frac {-x^2-1}{8 \left (x^4+1\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{80} \sqrt {\frac {1}{5} \left (11+5 \sqrt {5}\right )} \arctan \left (\frac {\sqrt {2 \left (\sqrt {5}-2\right )}-2 x}{\sqrt {2 \left (2+\sqrt {5}\right )}}\right )-\frac {1}{80} \sqrt {\frac {1}{5} \left (11+5 \sqrt {5}\right )} \arctan \left (\frac {2 x+\sqrt {2 \left (\sqrt {5}-2\right )}}{\sqrt {2 \left (2+\sqrt {5}\right )}}\right )+\frac {\arctan \left (1-\sqrt {2} x\right )}{8 \sqrt {2}}-\frac {\arctan \left (\sqrt {2} x+1\right )}{8 \sqrt {2}}+\frac {1}{160} \sqrt {\frac {1}{5} \left (5 \sqrt {5}-11\right )} \log \left (x^2-\sqrt {2 \left (\sqrt {5}-2\right )} x+\sqrt {5}\right )-\frac {1}{160} \sqrt {\frac {1}{5} \left (5 \sqrt {5}-11\right )} \log \left (x^2+\sqrt {2 \left (\sqrt {5}-2\right )} x+\sqrt {5}\right )-\frac {1}{5 x}\) |
Input:
Int[1/(x^2*(4 + (1 + x^2)^4)),x]
Output:
-1/5*1/x + (Sqrt[(11 + 5*Sqrt[5])/5]*ArcTan[(Sqrt[2*(-2 + Sqrt[5])] - 2*x) /Sqrt[2*(2 + Sqrt[5])]])/80 - (Sqrt[(11 + 5*Sqrt[5])/5]*ArcTan[(Sqrt[2*(-2 + Sqrt[5])] + 2*x)/Sqrt[2*(2 + Sqrt[5])]])/80 + ArcTan[1 - Sqrt[2]*x]/(8* Sqrt[2]) - ArcTan[1 + Sqrt[2]*x]/(8*Sqrt[2]) + (Sqrt[(-11 + 5*Sqrt[5])/5]* Log[Sqrt[5] - Sqrt[2*(-2 + Sqrt[5])]*x + x^2])/160 - (Sqrt[(-11 + 5*Sqrt[5 ])/5]*Log[Sqrt[5] + Sqrt[2*(-2 + Sqrt[5])]*x + x^2])/160
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px /. x -> Sqrt[x]]}, Int[ExpandIntegrand[u*(Qx /. x -> x^2)^p, x], x] /; !SumQ[NonfreeFactors[Q x, x]]] /; PolyQ[Px, x^2] && GtQ[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0] && RationalFunctionQ[u, x]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.19 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.36
| method | result | size |
| risch | \(-\frac {1}{5 x}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (20 \textit {\_Z}^{4}+44 \textit {\_Z}^{2}+25\right )}{\sum }\textit {\_R} \ln \left (10 \textit {\_R}^{3}-3 \textit {\_R} +10 x \right )\right )}{80}-\frac {\sqrt {2}\, \arctan \left (\frac {x \sqrt {2}}{2}\right )}{16}-\frac {\sqrt {2}\, \arctan \left (\frac {x^{3} \sqrt {2}}{2}+\frac {x \sqrt {2}}{2}\right )}{16}\) | \(73\) |
| default | \(-\frac {\sqrt {2}\, \left (\ln \left (\frac {x^{2}+x \sqrt {2}+1}{x^{2}-x \sqrt {2}+1}\right )+2 \arctan \left (x \sqrt {2}+1\right )+2 \arctan \left (x \sqrt {2}-1\right )\right )}{64}-\frac {\sqrt {2}\, \left (\ln \left (\frac {x^{2}-x \sqrt {2}+1}{x^{2}+x \sqrt {2}+1}\right )+2 \arctan \left (x \sqrt {2}+1\right )+2 \arctan \left (x \sqrt {2}-1\right )\right )}{64}-\frac {1}{5 x}-\frac {\left (\sqrt {-4+2 \sqrt {5}}\, \sqrt {5}-5 \sqrt {-4+2 \sqrt {5}}\right ) \ln \left (x^{2}-x \sqrt {-4+2 \sqrt {5}}+\sqrt {5}\right )}{1600}-\frac {\left (14 \sqrt {5}+\frac {\left (\sqrt {-4+2 \sqrt {5}}\, \sqrt {5}-5 \sqrt {-4+2 \sqrt {5}}\right ) \sqrt {-4+2 \sqrt {5}}}{2}\right ) \arctan \left (\frac {2 x -\sqrt {-4+2 \sqrt {5}}}{\sqrt {4+2 \sqrt {5}}}\right )}{400 \sqrt {4+2 \sqrt {5}}}+\frac {\left (\sqrt {-4+2 \sqrt {5}}\, \sqrt {5}-5 \sqrt {-4+2 \sqrt {5}}\right ) \ln \left (x^{2}+x \sqrt {-4+2 \sqrt {5}}+\sqrt {5}\right )}{1600}+\frac {\left (-14 \sqrt {5}-\frac {\left (\sqrt {-4+2 \sqrt {5}}\, \sqrt {5}-5 \sqrt {-4+2 \sqrt {5}}\right ) \sqrt {-4+2 \sqrt {5}}}{2}\right ) \arctan \left (\frac {\sqrt {-4+2 \sqrt {5}}+2 x}{\sqrt {4+2 \sqrt {5}}}\right )}{400 \sqrt {4+2 \sqrt {5}}}\) | \(358\) |
Input:
int(1/x^2/(4+(x^2+1)^4),x,method=_RETURNVERBOSE)
Output:
-1/5/x+1/80*sum(_R*ln(10*_R^3-3*_R+10*x),_R=RootOf(20*_Z^4+44*_Z^2+25))-1/ 16*2^(1/2)*arctan(1/2*x*2^(1/2))-1/16*2^(1/2)*arctan(1/2*x^3*2^(1/2)+1/2*x *2^(1/2))
Time = 0.08 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.01 \[ \int \frac {1}{x^2 \left (4+\left (1+x^2\right )^4\right )} \, dx=-\frac {10 \, \sqrt {2} x \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (x^{3} + x\right )}\right ) + 10 \, \sqrt {2} x \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) + 2 \, x \sqrt {\sqrt {5} + \frac {11}{5}} \arctan \left (\frac {1}{2} \, {\left (7 \, \sqrt {5} x + 5 \, {\left (\sqrt {5} - 2\right )} \sqrt {\sqrt {5} - \frac {11}{5}} - 15 \, x\right )} \sqrt {\sqrt {5} + \frac {11}{5}}\right ) - 2 \, x \sqrt {\sqrt {5} + \frac {11}{5}} \arctan \left (-\frac {1}{2} \, {\left (7 \, \sqrt {5} x - 5 \, {\left (\sqrt {5} - 2\right )} \sqrt {\sqrt {5} - \frac {11}{5}} - 15 \, x\right )} \sqrt {\sqrt {5} + \frac {11}{5}}\right ) + x \sqrt {\sqrt {5} - \frac {11}{5}} \log \left (2 \, x^{2} + {\left (\sqrt {5} x + 5 \, x\right )} \sqrt {\sqrt {5} - \frac {11}{5}} + 2 \, \sqrt {5}\right ) - x \sqrt {\sqrt {5} - \frac {11}{5}} \log \left (2 \, x^{2} - {\left (\sqrt {5} x + 5 \, x\right )} \sqrt {\sqrt {5} - \frac {11}{5}} + 2 \, \sqrt {5}\right ) + 32}{160 \, x} \] Input:
integrate(1/x^2/(4+(x^2+1)^4),x, algorithm="fricas")
Output:
-1/160*(10*sqrt(2)*x*arctan(1/2*sqrt(2)*(x^3 + x)) + 10*sqrt(2)*x*arctan(1 /2*sqrt(2)*x) + 2*x*sqrt(sqrt(5) + 11/5)*arctan(1/2*(7*sqrt(5)*x + 5*(sqrt (5) - 2)*sqrt(sqrt(5) - 11/5) - 15*x)*sqrt(sqrt(5) + 11/5)) - 2*x*sqrt(sqr t(5) + 11/5)*arctan(-1/2*(7*sqrt(5)*x - 5*(sqrt(5) - 2)*sqrt(sqrt(5) - 11/ 5) - 15*x)*sqrt(sqrt(5) + 11/5)) + x*sqrt(sqrt(5) - 11/5)*log(2*x^2 + (sqr t(5)*x + 5*x)*sqrt(sqrt(5) - 11/5) + 2*sqrt(5)) - x*sqrt(sqrt(5) - 11/5)*l og(2*x^2 - (sqrt(5)*x + 5*x)*sqrt(sqrt(5) - 11/5) + 2*sqrt(5)) + 32)/x
Time = 0.42 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.35 \[ \int \frac {1}{x^2 \left (4+\left (1+x^2\right )^4\right )} \, dx=\frac {\sqrt {2} \left (- 2 \operatorname {atan}{\left (\frac {\sqrt {2} x}{2} \right )} - 2 \operatorname {atan}{\left (\frac {\sqrt {2} x^{3}}{2} + \frac {\sqrt {2} x}{2} \right )}\right )}{32} + \operatorname {RootSum} {\left (32768000 t^{4} + 11264 t^{2} + 1, \left ( t \mapsto t \log {\left (512000 t^{3} - 24 t + x \right )} \right )\right )} - \frac {1}{5 x} \] Input:
integrate(1/x**2/(4+(x**2+1)**4),x)
Output:
sqrt(2)*(-2*atan(sqrt(2)*x/2) - 2*atan(sqrt(2)*x**3/2 + sqrt(2)*x/2))/32 + RootSum(32768000*_t**4 + 11264*_t**2 + 1, Lambda(_t, _t*log(512000*_t**3 - 24*_t + x))) - 1/(5*x)
\[ \int \frac {1}{x^2 \left (4+\left (1+x^2\right )^4\right )} \, dx=\int { \frac {1}{{\left ({\left (x^{2} + 1\right )}^{4} + 4\right )} x^{2}} \,d x } \] Input:
integrate(1/x^2/(4+(x^2+1)^4),x, algorithm="maxima")
Output:
-1/16*sqrt(2)*arctan(1/2*sqrt(2)*(2*x + sqrt(2))) - 1/16*sqrt(2)*arctan(1/ 2*sqrt(2)*(2*x - sqrt(2))) - 1/5/x - 1/40*integrate((3*x^2 + 7)/(x^4 + 4*x ^2 + 5), x)
Time = 0.37 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.93 \[ \int \frac {1}{x^2 \left (4+\left (1+x^2\right )^4\right )} \, dx=-\frac {1}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + \sqrt {2}\right )}\right ) - \frac {1}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - \sqrt {2}\right )}\right ) + \frac {1}{80} \, \sqrt {\sqrt {5} - 1} \arctan \left (\frac {5^{\frac {3}{4}} {\left (x + 5^{\frac {1}{4}} \sqrt {-\frac {1}{5} \, \sqrt {5} + \frac {1}{2}}\right )}}{5 \, \sqrt {\frac {1}{5} \, \sqrt {5} + \frac {1}{2}}}\right ) + \frac {1}{80} \, \sqrt {\sqrt {5} - 1} \arctan \left (\frac {5^{\frac {3}{4}} {\left (x - 5^{\frac {1}{4}} \sqrt {-\frac {1}{5} \, \sqrt {5} + \frac {1}{2}}\right )}}{5 \, \sqrt {\frac {1}{5} \, \sqrt {5} + \frac {1}{2}}}\right ) + \frac {1}{160} \, \sqrt {\sqrt {5} + 1} \log \left (x^{2} + 2 \cdot 5^{\frac {1}{4}} x \sqrt {-\frac {1}{5} \, \sqrt {5} + \frac {1}{2}} + \sqrt {5}\right ) - \frac {1}{160} \, \sqrt {\sqrt {5} + 1} \log \left (x^{2} - 2 \cdot 5^{\frac {1}{4}} x \sqrt {-\frac {1}{5} \, \sqrt {5} + \frac {1}{2}} + \sqrt {5}\right ) - \frac {1}{5 \, x} \] Input:
integrate(1/x^2/(4+(x^2+1)^4),x, algorithm="giac")
Output:
-1/16*sqrt(2)*arctan(1/2*sqrt(2)*(2*x + sqrt(2))) - 1/16*sqrt(2)*arctan(1/ 2*sqrt(2)*(2*x - sqrt(2))) + 1/80*sqrt(sqrt(5) - 1)*arctan(1/5*5^(3/4)*(x + 5^(1/4)*sqrt(-1/5*sqrt(5) + 1/2))/sqrt(1/5*sqrt(5) + 1/2)) + 1/80*sqrt(s qrt(5) - 1)*arctan(1/5*5^(3/4)*(x - 5^(1/4)*sqrt(-1/5*sqrt(5) + 1/2))/sqrt (1/5*sqrt(5) + 1/2)) + 1/160*sqrt(sqrt(5) + 1)*log(x^2 + 2*5^(1/4)*x*sqrt( -1/5*sqrt(5) + 1/2) + sqrt(5)) - 1/160*sqrt(sqrt(5) + 1)*log(x^2 - 2*5^(1/ 4)*x*sqrt(-1/5*sqrt(5) + 1/2) + sqrt(5)) - 1/5/x
Time = 10.56 (sec) , antiderivative size = 1457, normalized size of antiderivative = 7.25 \[ \int \frac {1}{x^2 \left (4+\left (1+x^2\right )^4\right )} \, dx=\text {Too large to display} \] Input:
int(1/(x^2*((x^2 + 1)^4 + 4)),x)
Output:
atan((((8*x)/25 + (((16384*x)/25 - (((98566144*x)/25 + (2147483648*x*((- 5 ^(1/2)/25600 - 11/128000)^(1/2) + (5^(1/2)/25600 - 11/128000)^(1/2)) - 402 653184/5)*((- 5^(1/2)/25600 - 11/128000)^(1/2) + (5^(1/2)/25600 - 11/12800 0)^(1/2)))*((- 5^(1/2)/25600 - 11/128000)^(1/2) + (5^(1/2)/25600 - 11/1280 00)^(1/2)) - 4456448/25)*((- 5^(1/2)/25600 - 11/128000)^(1/2) + (5^(1/2)/2 5600 - 11/128000)^(1/2)))*((- 5^(1/2)/25600 - 11/128000)^(1/2) + (5^(1/2)/ 25600 - 11/128000)^(1/2)) + 1024/25)*((- 5^(1/2)/25600 - 11/128000)^(1/2) + (5^(1/2)/25600 - 11/128000)^(1/2)))*((- 5^(1/2)/25600 - 11/128000)^(1/2) + (5^(1/2)/25600 - 11/128000)^(1/2))*1i + ((8*x)/25 + (((16384*x)/25 - (( (98566144*x)/25 + (2147483648*x*((- 5^(1/2)/25600 - 11/128000)^(1/2) + (5^ (1/2)/25600 - 11/128000)^(1/2)) + 402653184/5)*((- 5^(1/2)/25600 - 11/1280 00)^(1/2) + (5^(1/2)/25600 - 11/128000)^(1/2)))*((- 5^(1/2)/25600 - 11/128 000)^(1/2) + (5^(1/2)/25600 - 11/128000)^(1/2)) + 4456448/25)*((- 5^(1/2)/ 25600 - 11/128000)^(1/2) + (5^(1/2)/25600 - 11/128000)^(1/2)))*((- 5^(1/2) /25600 - 11/128000)^(1/2) + (5^(1/2)/25600 - 11/128000)^(1/2)) - 1024/25)* ((- 5^(1/2)/25600 - 11/128000)^(1/2) + (5^(1/2)/25600 - 11/128000)^(1/2))) *((- 5^(1/2)/25600 - 11/128000)^(1/2) + (5^(1/2)/25600 - 11/128000)^(1/2)) *1i)/(((8*x)/25 + (((16384*x)/25 - (((98566144*x)/25 + (2147483648*x*((- 5 ^(1/2)/25600 - 11/128000)^(1/2) + (5^(1/2)/25600 - 11/128000)^(1/2)) - 402 653184/5)*((- 5^(1/2)/25600 - 11/128000)^(1/2) + (5^(1/2)/25600 - 11/12...
Time = 0.23 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.42 \[ \int \frac {1}{x^2 \left (4+\left (1+x^2\right )^4\right )} \, dx=\frac {100 \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {2}-2 x}{\sqrt {2}}\right ) x -100 \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {2}+2 x}{\sqrt {2}}\right ) x +2 \sqrt {\sqrt {5}+2}\, \sqrt {10}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {5}-2}\, \sqrt {2}-2 x}{\sqrt {\sqrt {5}+2}\, \sqrt {2}}\right ) x +10 \sqrt {\sqrt {5}+2}\, \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {5}-2}\, \sqrt {2}-2 x}{\sqrt {\sqrt {5}+2}\, \sqrt {2}}\right ) x -2 \sqrt {\sqrt {5}+2}\, \sqrt {10}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {5}-2}\, \sqrt {2}+2 x}{\sqrt {\sqrt {5}+2}\, \sqrt {2}}\right ) x -10 \sqrt {\sqrt {5}+2}\, \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {5}-2}\, \sqrt {2}+2 x}{\sqrt {\sqrt {5}+2}\, \sqrt {2}}\right ) x -\sqrt {\sqrt {5}-2}\, \sqrt {10}\, \mathrm {log}\left (-\sqrt {\sqrt {5}-2}\, \sqrt {2}\, x +\sqrt {5}+x^{2}\right ) x +\sqrt {\sqrt {5}-2}\, \sqrt {10}\, \mathrm {log}\left (\sqrt {\sqrt {5}-2}\, \sqrt {2}\, x +\sqrt {5}+x^{2}\right ) x +5 \sqrt {\sqrt {5}-2}\, \sqrt {2}\, \mathrm {log}\left (-\sqrt {\sqrt {5}-2}\, \sqrt {2}\, x +\sqrt {5}+x^{2}\right ) x -5 \sqrt {\sqrt {5}-2}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {\sqrt {5}-2}\, \sqrt {2}\, x +\sqrt {5}+x^{2}\right ) x -320}{1600 x} \] Input:
int(1/x^2/(4+(x^2+1)^4),x)
Output:
(100*sqrt(2)*atan((sqrt(2) - 2*x)/sqrt(2))*x - 100*sqrt(2)*atan((sqrt(2) + 2*x)/sqrt(2))*x + 2*sqrt(sqrt(5) + 2)*sqrt(10)*atan((sqrt(sqrt(5) - 2)*sq rt(2) - 2*x)/(sqrt(sqrt(5) + 2)*sqrt(2)))*x + 10*sqrt(sqrt(5) + 2)*sqrt(2) *atan((sqrt(sqrt(5) - 2)*sqrt(2) - 2*x)/(sqrt(sqrt(5) + 2)*sqrt(2)))*x - 2 *sqrt(sqrt(5) + 2)*sqrt(10)*atan((sqrt(sqrt(5) - 2)*sqrt(2) + 2*x)/(sqrt(s qrt(5) + 2)*sqrt(2)))*x - 10*sqrt(sqrt(5) + 2)*sqrt(2)*atan((sqrt(sqrt(5) - 2)*sqrt(2) + 2*x)/(sqrt(sqrt(5) + 2)*sqrt(2)))*x - sqrt(sqrt(5) - 2)*sqr t(10)*log( - sqrt(sqrt(5) - 2)*sqrt(2)*x + sqrt(5) + x**2)*x + sqrt(sqrt(5 ) - 2)*sqrt(10)*log(sqrt(sqrt(5) - 2)*sqrt(2)*x + sqrt(5) + x**2)*x + 5*sq rt(sqrt(5) - 2)*sqrt(2)*log( - sqrt(sqrt(5) - 2)*sqrt(2)*x + sqrt(5) + x** 2)*x - 5*sqrt(sqrt(5) - 2)*sqrt(2)*log(sqrt(sqrt(5) - 2)*sqrt(2)*x + sqrt( 5) + x**2)*x - 320)/(1600*x)