Integrand size = 15, antiderivative size = 176 \[ \int \frac {x^2}{4+\left (1+x^2\right )^4} \, dx=\frac {1}{16} \sqrt {-1+\sqrt {5}} \arctan \left (\frac {\sqrt {2 \left (-2+\sqrt {5}\right )}-2 x}{\sqrt {2 \left (2+\sqrt {5}\right )}}\right )-\frac {1}{16} \sqrt {-1+\sqrt {5}} \arctan \left (\frac {\sqrt {2 \left (-2+\sqrt {5}\right )}+2 x}{\sqrt {2 \left (2+\sqrt {5}\right )}}\right )-\frac {\arctan \left (1-\sqrt {2} x\right )}{8 \sqrt {2}}+\frac {\arctan \left (1+\sqrt {2} x\right )}{8 \sqrt {2}}-\frac {1}{16} \sqrt {1+\sqrt {5}} \text {arctanh}\left (\frac {\sqrt {2 \left (-2+\sqrt {5}\right )} x}{\sqrt {5}+x^2}\right ) \] Output:
1/16*(5^(1/2)-1)^(1/2)*arctan(((-4+2*5^(1/2))^(1/2)-2*x)/(4+2*5^(1/2))^(1/ 2))-1/16*(5^(1/2)-1)^(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) -1/16*(5^(1/2)+1)^(1/2)*arctanh((-4+2*5^(1/2))^(1/2)*x/(5^(1/2)+x^2))
Result contains complex when optimal does not.
Time = 0.06 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.44 \[ \int \frac {x^2}{4+\left (1+x^2\right )^4} \, dx=\frac {1}{16} \left ((-1+i) \sqrt {2-i} \arctan \left (\frac {x}{\sqrt {2-i}}\right )-(1+i) \sqrt {2+i} \arctan \left (\frac {x}{\sqrt {2+i}}\right )+\sqrt {2} \left (-\arctan \left (1-\sqrt {2} x\right )+\arctan \left (1+\sqrt {2} x\right )\right )\right ) \] Input:
Integrate[x^2/(4 + (1 + x^2)^4),x]
Output:
((-1 + I)*Sqrt[2 - I]*ArcTan[x/Sqrt[2 - I]] - (1 + I)*Sqrt[2 + I]*ArcTan[x /Sqrt[2 + I]] + Sqrt[2]*(-ArcTan[1 - Sqrt[2]*x] + ArcTan[1 + Sqrt[2]*x]))/ 16
Time = 0.70 (sec) , antiderivative size = 215, 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.
| \(\displaystyle \int \frac {x^2}{\left (x^2+1\right )^4+4} \, dx\) |
| \(\Big \downarrow \) 2460 |
| \(\displaystyle \int \left (\frac {-x^2-5}{8 \left (x^4+4 x^2+5\right )}+\frac {x^2+1}{8 \left (x^4+1\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{16} \sqrt {\sqrt {5}-1} \arctan \left (\frac {\sqrt {2 \left (\sqrt {5}-2\right )}-2 x}{\sqrt {2 \left (2+\sqrt {5}\right )}}\right )-\frac {1}{16} \sqrt {\sqrt {5}-1} \arctan \left (\frac {2 x+\sqrt {2 \left (\sqrt {5}-2\right )}}{\sqrt {2 \left (2+\sqrt {5}\right )}}\right )-\frac {\arctan \left (1-\sqrt {2} x\right )}{8 \sqrt {2}}+\frac {\arctan \left (\sqrt {2} x+1\right )}{8 \sqrt {2}}+\frac {1}{32} \sqrt {1+\sqrt {5}} \log \left (x^2-\sqrt {2 \left (\sqrt {5}-2\right )} x+\sqrt {5}\right )-\frac {1}{32} \sqrt {1+\sqrt {5}} \log \left (x^2+\sqrt {2 \left (\sqrt {5}-2\right )} x+\sqrt {5}\right )\) |
Input:
Int[x^2/(4 + (1 + x^2)^4),x]
Output:
(Sqrt[-1 + Sqrt[5]]*ArcTan[(Sqrt[2*(-2 + Sqrt[5])] - 2*x)/Sqrt[2*(2 + Sqrt [5])]])/16 - (Sqrt[-1 + Sqrt[5]]*ArcTan[(Sqrt[2*(-2 + Sqrt[5])] + 2*x)/Sqr t[2*(2 + Sqrt[5])]])/16 - ArcTan[1 - Sqrt[2]*x]/(8*Sqrt[2]) + ArcTan[1 + S qrt[2]*x]/(8*Sqrt[2]) + (Sqrt[1 + Sqrt[5]]*Log[Sqrt[5] - Sqrt[2*(-2 + Sqrt [5])]*x + x^2])/32 - (Sqrt[1 + Sqrt[5]]*Log[Sqrt[5] + Sqrt[2*(-2 + Sqrt[5] )]*x + x^2])/32
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 = 0.16 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.39
| method | result | size |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (4 \textit {\_Z}^{4}-4 \textit {\_Z}^{2}+5\right )}{\sum }\textit {\_R} \ln \left (-2 \textit {\_R}^{3}-\textit {\_R} +2 x \right )\right )}{16}+\frac {\sqrt {2}\, \arctan \left (\frac {x \sqrt {2}}{2}\right )}{16}+\frac {\sqrt {2}\, \arctan \left (\frac {x^{3} \sqrt {2}}{2}+\frac {x \sqrt {2}}{2}\right )}{16}\) | \(68\) |
| default | \(\frac {\sqrt {2}\, \left (\ln \left (\frac {x^{2}+x \sqrt {2}+1}{x^{2}-x \sqrt {2}+1}\right )+2 \arctan \left (x \sqrt {2}+1\right )+2 \arctan \left (x \sqrt {2}-1\right )\right )}{64}+\frac {\sqrt {2}\, \left (\ln \left (\frac {x^{2}-x \sqrt {2}+1}{x^{2}+x \sqrt {2}+1}\right )+2 \arctan \left (x \sqrt {2}+1\right )+2 \arctan \left (x \sqrt {2}-1\right )\right )}{64}-\frac {\left (-\sqrt {-4+2 \sqrt {5}}\, \sqrt {5}-3 \sqrt {-4+2 \sqrt {5}}\right ) \ln \left (x^{2}-x \sqrt {-4+2 \sqrt {5}}+\sqrt {5}\right )}{64}-\frac {\left (2 \sqrt {5}+\frac {\left (-\sqrt {-4+2 \sqrt {5}}\, \sqrt {5}-3 \sqrt {-4+2 \sqrt {5}}\right ) \sqrt {-4+2 \sqrt {5}}}{2}\right ) \arctan \left (\frac {2 x -\sqrt {-4+2 \sqrt {5}}}{\sqrt {4+2 \sqrt {5}}}\right )}{16 \sqrt {4+2 \sqrt {5}}}+\frac {\left (-\sqrt {-4+2 \sqrt {5}}\, \sqrt {5}-3 \sqrt {-4+2 \sqrt {5}}\right ) \ln \left (x^{2}+x \sqrt {-4+2 \sqrt {5}}+\sqrt {5}\right )}{64}+\frac {\left (-2 \sqrt {5}-\frac {\left (-\sqrt {-4+2 \sqrt {5}}\, \sqrt {5}-3 \sqrt {-4+2 \sqrt {5}}\right ) \sqrt {-4+2 \sqrt {5}}}{2}\right ) \arctan \left (\frac {\sqrt {-4+2 \sqrt {5}}+2 x}{\sqrt {4+2 \sqrt {5}}}\right )}{16 \sqrt {4+2 \sqrt {5}}}\) | \(357\) |
Input:
int(x^2/(4+(x^2+1)^4),x,method=_RETURNVERBOSE)
Output:
1/16*sum(_R*ln(-2*_R^3-_R+2*x),_R=RootOf(4*_Z^4-4*_Z^2+5))+1/16*2^(1/2)*ar ctan(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 = 206, normalized size of antiderivative = 1.17 \[ \int \frac {x^2}{4+\left (1+x^2\right )^4} \, dx=-\frac {1}{16} \, \sqrt {\sqrt {5} - 1} \arctan \left (\frac {1}{2} \, \sqrt {\sqrt {5} + 1} \sqrt {\sqrt {5} - 1} {\left (\sqrt {5} - 2\right )} + \frac {1}{2} \, {\left (\sqrt {5} x - x\right )} \sqrt {\sqrt {5} - 1}\right ) + \frac {1}{16} \, \sqrt {\sqrt {5} - 1} \arctan \left (\frac {1}{2} \, \sqrt {\sqrt {5} + 1} \sqrt {\sqrt {5} - 1} {\left (\sqrt {5} - 2\right )} - \frac {1}{2} \, {\left (\sqrt {5} x - x\right )} \sqrt {\sqrt {5} - 1}\right ) + \frac {1}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (x^{3} + x\right )}\right ) + \frac {1}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) + \frac {1}{32} \, \sqrt {\sqrt {5} + 1} \log \left (2 \, x^{2} + {\left (\sqrt {5} x - 3 \, x\right )} \sqrt {\sqrt {5} + 1} + 2 \, \sqrt {5}\right ) - \frac {1}{32} \, \sqrt {\sqrt {5} + 1} \log \left (2 \, x^{2} - {\left (\sqrt {5} x - 3 \, x\right )} \sqrt {\sqrt {5} + 1} + 2 \, \sqrt {5}\right ) \] Input:
integrate(x^2/(4+(x^2+1)^4),x, algorithm="fricas")
Output:
-1/16*sqrt(sqrt(5) - 1)*arctan(1/2*sqrt(sqrt(5) + 1)*sqrt(sqrt(5) - 1)*(sq rt(5) - 2) + 1/2*(sqrt(5)*x - x)*sqrt(sqrt(5) - 1)) + 1/16*sqrt(sqrt(5) - 1)*arctan(1/2*sqrt(sqrt(5) + 1)*sqrt(sqrt(5) - 1)*(sqrt(5) - 2) - 1/2*(sqr t(5)*x - x)*sqrt(sqrt(5) - 1)) + 1/16*sqrt(2)*arctan(1/2*sqrt(2)*(x^3 + x) ) + 1/16*sqrt(2)*arctan(1/2*sqrt(2)*x) + 1/32*sqrt(sqrt(5) + 1)*log(2*x^2 + (sqrt(5)*x - 3*x)*sqrt(sqrt(5) + 1) + 2*sqrt(5)) - 1/32*sqrt(sqrt(5) + 1 )*log(2*x^2 - (sqrt(5)*x - 3*x)*sqrt(sqrt(5) + 1) + 2*sqrt(5))
Leaf count of result is larger than twice the leaf count of optimal. 879 vs. \(2 (150) = 300\).
Time = 0.65 (sec) , antiderivative size = 879, normalized size of antiderivative = 4.99 \[ \int \frac {x^2}{4+\left (1+x^2\right )^4} \, dx =\text {Too large to display} \] Input:
integrate(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 - sqrt(1/1024 + sqrt(5)/1024)*log(x**2 + x*(-2*sqrt(5)*sqrt(1 + sqrt(5)) + 3 *sqrt(2)*sqrt(1 + sqrt(5))*sqrt(sqrt(5) + 3)/2) - 17*sqrt(2)*sqrt(sqrt(5) + 3)/2 - sqrt(10)*sqrt(sqrt(5) + 3) + 27/2 + 21*sqrt(5)/2) + sqrt(1/1024 + sqrt(5)/1024)*log(x**2 + x*(-3*sqrt(2)*sqrt(1 + sqrt(5))*sqrt(sqrt(5) + 3 )/2 + 2*sqrt(5)*sqrt(1 + sqrt(5))) - 17*sqrt(2)*sqrt(sqrt(5) + 3)/2 - sqrt (10)*sqrt(sqrt(5) + 3) + 27/2 + 21*sqrt(5)/2) - 2*sqrt(-sqrt(2)*sqrt(sqrt( 5) + 3)/512 + 1/1024 + 3*sqrt(5)/1024)*atan(4*x/(2*sqrt(-2*sqrt(2)*sqrt(sq rt(5) + 3) + 1 + 3*sqrt(5)) + sqrt(2)*sqrt(sqrt(5) + 3)*sqrt(-2*sqrt(2)*sq rt(sqrt(5) + 3) + 1 + 3*sqrt(5))) - 4*sqrt(5)*sqrt(1 + sqrt(5))/(2*sqrt(-2 *sqrt(2)*sqrt(sqrt(5) + 3) + 1 + 3*sqrt(5)) + sqrt(2)*sqrt(sqrt(5) + 3)*sq rt(-2*sqrt(2)*sqrt(sqrt(5) + 3) + 1 + 3*sqrt(5))) + 3*sqrt(2)*sqrt(1 + sqr t(5))*sqrt(sqrt(5) + 3)/(2*sqrt(-2*sqrt(2)*sqrt(sqrt(5) + 3) + 1 + 3*sqrt( 5)) + sqrt(2)*sqrt(sqrt(5) + 3)*sqrt(-2*sqrt(2)*sqrt(sqrt(5) + 3) + 1 + 3* sqrt(5)))) - 2*sqrt(-sqrt(2)*sqrt(sqrt(5) + 3)/512 + 1/1024 + 3*sqrt(5)/10 24)*atan(4*x/(2*sqrt(-2*sqrt(2)*sqrt(sqrt(5) + 3) + 1 + 3*sqrt(5)) + sqrt( 2)*sqrt(sqrt(5) + 3)*sqrt(-2*sqrt(2)*sqrt(sqrt(5) + 3) + 1 + 3*sqrt(5))) - 3*sqrt(2)*sqrt(1 + sqrt(5))*sqrt(sqrt(5) + 3)/(2*sqrt(-2*sqrt(2)*sqrt(sqr t(5) + 3) + 1 + 3*sqrt(5)) + sqrt(2)*sqrt(sqrt(5) + 3)*sqrt(-2*sqrt(2)*sqr t(sqrt(5) + 3) + 1 + 3*sqrt(5))) + 4*sqrt(5)*sqrt(1 + sqrt(5))/(2*sqrt(...
\[ \int \frac {x^2}{4+\left (1+x^2\right )^4} \, dx=\int { \frac {x^{2}}{{\left (x^{2} + 1\right )}^{4} + 4} \,d x } \] Input:
integrate(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))) - 1/8*integrate((x^2 + 5)/(x^4 + 4*x^2 + 5), x)
Time = 0.33 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.08 \[ \int \frac {x^2}{4+\left (1+x^2\right )^4} \, dx=\frac {1}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + \sqrt {2}\right )}\right ) + \frac {1}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - \sqrt {2}\right )}\right ) - \frac {1}{80} \, \sqrt {25 \, \sqrt {5} - 41} \arctan \left (\frac {5^{\frac {3}{4}} {\left (x + 5^{\frac {1}{4}} \sqrt {-\frac {1}{5} \, \sqrt {5} + \frac {1}{2}}\right )}}{5 \, \sqrt {\frac {1}{5} \, \sqrt {5} + \frac {1}{2}}}\right ) - \frac {1}{80} \, \sqrt {25 \, \sqrt {5} - 41} \arctan \left (\frac {5^{\frac {3}{4}} {\left (x - 5^{\frac {1}{4}} \sqrt {-\frac {1}{5} \, \sqrt {5} + \frac {1}{2}}\right )}}{5 \, \sqrt {\frac {1}{5} \, \sqrt {5} + \frac {1}{2}}}\right ) + \frac {1}{160} \, \sqrt {25 \, \sqrt {5} + 41} \log \left (x^{2} + 2 \cdot 5^{\frac {1}{4}} x \sqrt {-\frac {1}{5} \, \sqrt {5} + \frac {1}{2}} + \sqrt {5}\right ) - \frac {1}{160} \, \sqrt {25 \, \sqrt {5} + 41} \log \left (x^{2} - 2 \cdot 5^{\frac {1}{4}} x \sqrt {-\frac {1}{5} \, \sqrt {5} + \frac {1}{2}} + \sqrt {5}\right ) \] Input:
integrate(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(25*sqrt(5) - 41)*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*sqr t(25*sqrt(5) - 41)*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(25*sqrt(5) + 41)*log(x^2 + 2*5^(1 /4)*x*sqrt(-1/5*sqrt(5) + 1/2) + sqrt(5)) - 1/160*sqrt(25*sqrt(5) + 41)*lo g(x^2 - 2*5^(1/4)*x*sqrt(-1/5*sqrt(5) + 1/2) + sqrt(5))
Time = 0.13 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.38 \[ \int \frac {x^2}{4+\left (1+x^2\right )^4} \, dx=\frac {\sqrt {2}\,\left (2\,\mathrm {atan}\left (\frac {\sqrt {2}\,x^3}{2}+\frac {\sqrt {2}\,x}{2}\right )+2\,\mathrm {atan}\left (\frac {\sqrt {2}\,x}{2}\right )\right )}{32}-\mathrm {atanh}\left (\frac {128\,\sqrt {5}\,x\,\sqrt {\frac {1}{1024}-\frac {\sqrt {5}}{1024}}}{10240\,\sqrt {\frac {1}{1024}-\frac {\sqrt {5}}{1024}}\,\sqrt {\frac {\sqrt {5}}{1024}+\frac {1}{1024}}-20}+\frac {128\,\sqrt {5}\,x\,\sqrt {\frac {\sqrt {5}}{1024}+\frac {1}{1024}}}{10240\,\sqrt {\frac {1}{1024}-\frac {\sqrt {5}}{1024}}\,\sqrt {\frac {\sqrt {5}}{1024}+\frac {1}{1024}}-20}\right )\,\left (2\,\sqrt {\frac {1}{1024}-\frac {\sqrt {5}}{1024}}-2\,\sqrt {\frac {\sqrt {5}}{1024}+\frac {1}{1024}}\right )+\mathrm {atanh}\left (\frac {128\,\sqrt {5}\,x\,\sqrt {\frac {1}{1024}-\frac {\sqrt {5}}{1024}}}{10240\,\sqrt {\frac {1}{1024}-\frac {\sqrt {5}}{1024}}\,\sqrt {\frac {\sqrt {5}}{1024}+\frac {1}{1024}}+20}-\frac {128\,\sqrt {5}\,x\,\sqrt {\frac {\sqrt {5}}{1024}+\frac {1}{1024}}}{10240\,\sqrt {\frac {1}{1024}-\frac {\sqrt {5}}{1024}}\,\sqrt {\frac {\sqrt {5}}{1024}+\frac {1}{1024}}+20}\right )\,\left (2\,\sqrt {\frac {1}{1024}-\frac {\sqrt {5}}{1024}}+2\,\sqrt {\frac {\sqrt {5}}{1024}+\frac {1}{1024}}\right ) \] Input:
int(x^2/((x^2 + 1)^4 + 4),x)
Output:
(2^(1/2)*(2*atan((2^(1/2)*x)/2 + (2^(1/2)*x^3)/2) + 2*atan((2^(1/2)*x)/2)) )/32 - atanh((128*5^(1/2)*x*(1/1024 - 5^(1/2)/1024)^(1/2))/(10240*(1/1024 - 5^(1/2)/1024)^(1/2)*(5^(1/2)/1024 + 1/1024)^(1/2) - 20) + (128*5^(1/2)*x *(5^(1/2)/1024 + 1/1024)^(1/2))/(10240*(1/1024 - 5^(1/2)/1024)^(1/2)*(5^(1 /2)/1024 + 1/1024)^(1/2) - 20))*(2*(1/1024 - 5^(1/2)/1024)^(1/2) - 2*(5^(1 /2)/1024 + 1/1024)^(1/2)) + atanh((128*5^(1/2)*x*(1/1024 - 5^(1/2)/1024)^( 1/2))/(10240*(1/1024 - 5^(1/2)/1024)^(1/2)*(5^(1/2)/1024 + 1/1024)^(1/2) + 20) - (128*5^(1/2)*x*(5^(1/2)/1024 + 1/1024)^(1/2))/(10240*(1/1024 - 5^(1 /2)/1024)^(1/2)*(5^(1/2)/1024 + 1/1024)^(1/2) + 20))*(2*(1/1024 - 5^(1/2)/ 1024)^(1/2) + 2*(5^(1/2)/1024 + 1/1024)^(1/2))
Time = 0.29 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.49 \[ \int \frac {x^2}{4+\left (1+x^2\right )^4} \, dx=\frac {\sqrt {2}\, \left (-4 \mathit {atan} \left (\frac {\sqrt {2}-2 x}{\sqrt {2}}\right )+4 \mathit {atan} \left (\frac {\sqrt {2}+2 x}{\sqrt {2}}\right )-2 \sqrt {\sqrt {5}+2}\, \sqrt {5}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {5}-2}\, \sqrt {2}-2 x}{\sqrt {\sqrt {5}+2}\, \sqrt {2}}\right )+6 \sqrt {\sqrt {5}+2}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {5}-2}\, \sqrt {2}-2 x}{\sqrt {\sqrt {5}+2}\, \sqrt {2}}\right )+2 \sqrt {\sqrt {5}+2}\, \sqrt {5}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {5}-2}\, \sqrt {2}+2 x}{\sqrt {\sqrt {5}+2}\, \sqrt {2}}\right )-6 \sqrt {\sqrt {5}+2}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {5}-2}\, \sqrt {2}+2 x}{\sqrt {\sqrt {5}+2}\, \sqrt {2}}\right )+\sqrt {\sqrt {5}-2}\, \sqrt {5}\, \mathrm {log}\left (-\sqrt {\sqrt {5}-2}\, \sqrt {2}\, x +\sqrt {5}+x^{2}\right )-\sqrt {\sqrt {5}-2}\, \sqrt {5}\, \mathrm {log}\left (\sqrt {\sqrt {5}-2}\, \sqrt {2}\, x +\sqrt {5}+x^{2}\right )+3 \sqrt {\sqrt {5}-2}\, \mathrm {log}\left (-\sqrt {\sqrt {5}-2}\, \sqrt {2}\, x +\sqrt {5}+x^{2}\right )-3 \sqrt {\sqrt {5}-2}\, \mathrm {log}\left (\sqrt {\sqrt {5}-2}\, \sqrt {2}\, x +\sqrt {5}+x^{2}\right )\right )}{64} \] Input:
int(x^2/(4+(x^2+1)^4),x)
Output:
(sqrt(2)*( - 4*atan((sqrt(2) - 2*x)/sqrt(2)) + 4*atan((sqrt(2) + 2*x)/sqrt (2)) - 2*sqrt(sqrt(5) + 2)*sqrt(5)*atan((sqrt(sqrt(5) - 2)*sqrt(2) - 2*x)/ (sqrt(sqrt(5) + 2)*sqrt(2))) + 6*sqrt(sqrt(5) + 2)*atan((sqrt(sqrt(5) - 2) *sqrt(2) - 2*x)/(sqrt(sqrt(5) + 2)*sqrt(2))) + 2*sqrt(sqrt(5) + 2)*sqrt(5) *atan((sqrt(sqrt(5) - 2)*sqrt(2) + 2*x)/(sqrt(sqrt(5) + 2)*sqrt(2))) - 6*s qrt(sqrt(5) + 2)*atan((sqrt(sqrt(5) - 2)*sqrt(2) + 2*x)/(sqrt(sqrt(5) + 2) *sqrt(2))) + sqrt(sqrt(5) - 2)*sqrt(5)*log( - sqrt(sqrt(5) - 2)*sqrt(2)*x + sqrt(5) + x**2) - sqrt(sqrt(5) - 2)*sqrt(5)*log(sqrt(sqrt(5) - 2)*sqrt(2 )*x + sqrt(5) + x**2) + 3*sqrt(sqrt(5) - 2)*log( - sqrt(sqrt(5) - 2)*sqrt( 2)*x + sqrt(5) + x**2) - 3*sqrt(sqrt(5) - 2)*log(sqrt(sqrt(5) - 2)*sqrt(2) *x + sqrt(5) + x**2)))/64