Integrand size = 15, antiderivative size = 182 \[ \int \frac {x^6}{4+\left (1+x^2\right )^4} \, dx=-\frac {1}{16} \sqrt {-41+25 \sqrt {5}} \arctan \left (\frac {\sqrt {2 \left (-2+\sqrt {5}\right )}-2 x}{\sqrt {2 \left (2+\sqrt {5}\right )}}\right )+\frac {1}{16} \sqrt {-41+25 \sqrt {5}} \arctan \left (\frac {\sqrt {2 \left (-2+\sqrt {5}\right )}+2 x}{\sqrt {2 \left (2+\sqrt {5}\right )}}\right )+\frac {\arctan \left (1-\sqrt {2} x\right )}{8 \sqrt {2}}-\frac {\arctan \left (1+\sqrt {2} x\right )}{8 \sqrt {2}}-\frac {1}{16} \sqrt {41+25 \sqrt {5}} \text {arctanh}\left (\frac {\sqrt {2 \left (-2+\sqrt {5}\right )} x}{\sqrt {5}+x^2}\right ) \] Output:
-1/16*(-41+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*(-41+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)-1/16*(41+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 = 0.07 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.42 \[ \int \frac {x^6}{4+\left (1+x^2\right )^4} \, dx=\frac {1}{16} \left ((1+7 i) \sqrt {2-i} \arctan \left (\frac {x}{\sqrt {2-i}}\right )+(1-7 i) \sqrt {2+i} \arctan \left (\frac {x}{\sqrt {2+i}}\right )+\sqrt {2} \left (\arctan \left (1-\sqrt {2} x\right )-\arctan \left (1+\sqrt {2} x\right )\right )\right ) \] Input:
Integrate[x^6/(4 + (1 + x^2)^4),x]
Output:
((1 + 7*I)*Sqrt[2 - I]*ArcTan[x/Sqrt[2 - I]] + (1 - 7*I)*Sqrt[2 + I]*ArcTa n[x/Sqrt[2 + I]] + Sqrt[2]*(ArcTan[1 - Sqrt[2]*x] - ArcTan[1 + Sqrt[2]*x]) )/16
Time = 0.79 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.23, 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.
| \(\displaystyle \int \frac {x^6}{\left (x^2+1\right )^4+4} \, dx\) |
| \(\Big \downarrow \) 2460 |
| \(\displaystyle \int \left (\frac {-x^2-1}{8 \left (x^4+1\right )}+\frac {9 x^2+5}{8 \left (x^4+4 x^2+5\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {1}{16} \sqrt {25 \sqrt {5}-41} \arctan \left (\frac {\sqrt {2 \left (\sqrt {5}-2\right )}-2 x}{\sqrt {2 \left (2+\sqrt {5}\right )}}\right )+\frac {1}{16} \sqrt {25 \sqrt {5}-41} \arctan \left (\frac {2 x+\sqrt {2 \left (\sqrt {5}-2\right )}}{\sqrt {2 \left (2+\sqrt {5}\right )}}\right )+\frac {\arctan \left (1-\sqrt {2} x\right )}{8 \sqrt {2}}-\frac {\arctan \left (\sqrt {2} x+1\right )}{8 \sqrt {2}}+\frac {1}{32} \sqrt {41+25 \sqrt {5}} \log \left (x^2-\sqrt {2 \left (\sqrt {5}-2\right )} x+\sqrt {5}\right )-\frac {1}{32} \sqrt {41+25 \sqrt {5}} \log \left (x^2+\sqrt {2 \left (\sqrt {5}-2\right )} x+\sqrt {5}\right )\) |
Input:
Int[x^6/(4 + (1 + x^2)^4),x]
Output:
-1/16*(Sqrt[-41 + 25*Sqrt[5]]*ArcTan[(Sqrt[2*(-2 + Sqrt[5])] - 2*x)/Sqrt[2 *(2 + Sqrt[5])]]) + (Sqrt[-41 + 25*Sqrt[5]]*ArcTan[(Sqrt[2*(-2 + Sqrt[5])] + 2*x)/Sqrt[2*(2 + Sqrt[5])]])/16 + ArcTan[1 - Sqrt[2]*x]/(8*Sqrt[2]) - A rcTan[1 + Sqrt[2]*x]/(8*Sqrt[2]) + (Sqrt[41 + 25*Sqrt[5]]*Log[Sqrt[5] - Sq rt[2*(-2 + Sqrt[5])]*x + x^2])/32 - (Sqrt[41 + 25*Sqrt[5]]*Log[Sqrt[5] + S qrt[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 = 0.28 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.37
| method | result | size |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (4 \textit {\_Z}^{4}-164 \textit {\_Z}^{2}+3125\right )}{\sum }\textit {\_R} \ln \left (14 \textit {\_R}^{3}-249 \textit {\_R} +950 x \right )\right )}{16}-\frac {\sqrt {2}\, \arctan \left (\frac {x \sqrt {2}}{2}\right )}{16}-\frac {\sqrt {2}\, \arctan \left (\frac {x^{3} \sqrt {2}}{2}+\frac {x \sqrt {2}}{2}\right )}{16}\) | \(68\) |
| default | \(-\frac {\sqrt {2}\, \left (\ln \left (\frac {x^{2}+x \sqrt {2}+1}{x^{2}-x \sqrt {2}+1}\right )+2 \arctan \left (x \sqrt {2}+1\right )+2 \arctan \left (x \sqrt {2}-1\right )\right )}{64}-\frac {\sqrt {2}\, \left (\ln \left (\frac {x^{2}-x \sqrt {2}+1}{x^{2}+x \sqrt {2}+1}\right )+2 \arctan \left (x \sqrt {2}+1\right )+2 \arctan \left (x \sqrt {2}-1\right )\right )}{64}-\frac {\left (-7 \sqrt {-4+2 \sqrt {5}}\, \sqrt {5}-13 \sqrt {-4+2 \sqrt {5}}\right ) \ln \left (x^{2}-x \sqrt {-4+2 \sqrt {5}}+\sqrt {5}\right )}{64}-\frac {\left (-2 \sqrt {5}+\frac {\left (-7 \sqrt {-4+2 \sqrt {5}}\, \sqrt {5}-13 \sqrt {-4+2 \sqrt {5}}\right ) \sqrt {-4+2 \sqrt {5}}}{2}\right ) \arctan \left (\frac {2 x -\sqrt {-4+2 \sqrt {5}}}{\sqrt {4+2 \sqrt {5}}}\right )}{16 \sqrt {4+2 \sqrt {5}}}+\frac {\left (-7 \sqrt {-4+2 \sqrt {5}}\, \sqrt {5}-13 \sqrt {-4+2 \sqrt {5}}\right ) \ln \left (x^{2}+x \sqrt {-4+2 \sqrt {5}}+\sqrt {5}\right )}{64}+\frac {\left (2 \sqrt {5}-\frac {\left (-7 \sqrt {-4+2 \sqrt {5}}\, \sqrt {5}-13 \sqrt {-4+2 \sqrt {5}}\right ) \sqrt {-4+2 \sqrt {5}}}{2}\right ) \arctan \left (\frac {\sqrt {-4+2 \sqrt {5}}+2 x}{\sqrt {4+2 \sqrt {5}}}\right )}{16 \sqrt {4+2 \sqrt {5}}}\) | \(357\) |
Input:
int(x^6/(4+(x^2+1)^4),x,method=_RETURNVERBOSE)
Output:
1/16*sum(_R*ln(14*_R^3-249*_R+950*x),_R=RootOf(4*_Z^4-164*_Z^2+3125))-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 = 232, normalized size of antiderivative = 1.27 \[ \int \frac {x^6}{4+\left (1+x^2\right )^4} \, dx=-\frac {1}{16} \, \sqrt {25 \, \sqrt {5} - 41} \arctan \left (\frac {1}{38} \, \sqrt {25 \, \sqrt {5} + 41} \sqrt {25 \, \sqrt {5} - 41} {\left (\sqrt {5} - 2\right )} + \frac {1}{38} \, {\left (\sqrt {5} x - 9 \, x\right )} \sqrt {25 \, \sqrt {5} - 41}\right ) + \frac {1}{16} \, \sqrt {25 \, \sqrt {5} - 41} \arctan \left (\frac {1}{38} \, \sqrt {25 \, \sqrt {5} + 41} \sqrt {25 \, \sqrt {5} - 41} {\left (\sqrt {5} - 2\right )} - \frac {1}{38} \, {\left (\sqrt {5} x - 9 \, x\right )} \sqrt {25 \, \sqrt {5} - 41}\right ) - \frac {1}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (x^{3} + x\right )}\right ) - \frac {1}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) - \frac {1}{32} \, \sqrt {25 \, \sqrt {5} + 41} \log \left (38 \, x^{2} + {\left (7 \, \sqrt {5} x - 13 \, x\right )} \sqrt {25 \, \sqrt {5} + 41} + 38 \, \sqrt {5}\right ) + \frac {1}{32} \, \sqrt {25 \, \sqrt {5} + 41} \log \left (38 \, x^{2} - {\left (7 \, \sqrt {5} x - 13 \, x\right )} \sqrt {25 \, \sqrt {5} + 41} + 38 \, \sqrt {5}\right ) \] Input:
integrate(x^6/(4+(x^2+1)^4),x, algorithm="fricas")
Output:
-1/16*sqrt(25*sqrt(5) - 41)*arctan(1/38*sqrt(25*sqrt(5) + 41)*sqrt(25*sqrt (5) - 41)*(sqrt(5) - 2) + 1/38*(sqrt(5)*x - 9*x)*sqrt(25*sqrt(5) - 41)) + 1/16*sqrt(25*sqrt(5) - 41)*arctan(1/38*sqrt(25*sqrt(5) + 41)*sqrt(25*sqrt( 5) - 41)*(sqrt(5) - 2) - 1/38*(sqrt(5)*x - 9*x)*sqrt(25*sqrt(5) - 41)) - 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(25*sqrt(5) + 41)*log(38*x^2 + (7*sqrt(5)*x - 13*x)*sqrt(2 5*sqrt(5) + 41) + 38*sqrt(5)) + 1/32*sqrt(25*sqrt(5) + 41)*log(38*x^2 - (7 *sqrt(5)*x - 13*x)*sqrt(25*sqrt(5) + 41) + 38*sqrt(5))
Leaf count of result is larger than twice the leaf count of optimal. 1187 vs. \(2 (155) = 310\).
Time = 0.67 (sec) , antiderivative size = 1187, normalized size of antiderivative = 6.52 \[ \int \frac {x^6}{4+\left (1+x^2\right )^4} \, dx=\text {Too large to display} \] Input:
integrate(x**6/(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(41/1024 + 25*sqrt(5)/1024)*log(x**2 + x*(-14*sqrt(5)*sqrt(41 + 25*sq rt(5))/19 - 268*sqrt(41 + 25*sqrt(5))/475 + 21*sqrt(2)*sqrt(41 + 25*sqrt(5 ))*sqrt(1025*sqrt(5) + 2403)/950) - 631257*sqrt(2)*sqrt(1025*sqrt(5) + 240 3)/451250 - 4823*sqrt(10)*sqrt(1025*sqrt(5) + 2403)/9025 + 56025287/451250 + 1044793*sqrt(5)/18050) - sqrt(41/1024 + 25*sqrt(5)/1024)*log(x**2 + x*( -21*sqrt(2)*sqrt(41 + 25*sqrt(5))*sqrt(1025*sqrt(5) + 2403)/950 + 268*sqrt (41 + 25*sqrt(5))/475 + 14*sqrt(5)*sqrt(41 + 25*sqrt(5))/19) - 631257*sqrt (2)*sqrt(1025*sqrt(5) + 2403)/451250 - 4823*sqrt(10)*sqrt(1025*sqrt(5) + 2 403)/9025 + 56025287/451250 + 1044793*sqrt(5)/18050) + 2*sqrt(-sqrt(2)*sqr t(1025*sqrt(5) + 2403)/512 + 41/1024 + 75*sqrt(5)/1024)*atan(1900*x/(38*sq rt(-2*sqrt(2)*sqrt(1025*sqrt(5) + 2403) + 41 + 75*sqrt(5)) + 7*sqrt(2)*sqr t(1025*sqrt(5) + 2403)*sqrt(-2*sqrt(2)*sqrt(1025*sqrt(5) + 2403) + 41 + 75 *sqrt(5))) - 700*sqrt(5)*sqrt(41 + 25*sqrt(5))/(38*sqrt(-2*sqrt(2)*sqrt(10 25*sqrt(5) + 2403) + 41 + 75*sqrt(5)) + 7*sqrt(2)*sqrt(1025*sqrt(5) + 2403 )*sqrt(-2*sqrt(2)*sqrt(1025*sqrt(5) + 2403) + 41 + 75*sqrt(5))) - 536*sqrt (41 + 25*sqrt(5))/(38*sqrt(-2*sqrt(2)*sqrt(1025*sqrt(5) + 2403) + 41 + 75* sqrt(5)) + 7*sqrt(2)*sqrt(1025*sqrt(5) + 2403)*sqrt(-2*sqrt(2)*sqrt(1025*s qrt(5) + 2403) + 41 + 75*sqrt(5))) + 21*sqrt(2)*sqrt(41 + 25*sqrt(5))*sqrt (1025*sqrt(5) + 2403)/(38*sqrt(-2*sqrt(2)*sqrt(1025*sqrt(5) + 2403) + 4...
\[ \int \frac {x^6}{4+\left (1+x^2\right )^4} \, dx=\int { \frac {x^{6}}{{\left (x^{2} + 1\right )}^{4} + 4} \,d x } \] Input:
integrate(x^6/(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((9*x^2 + 5)/(x^4 + 4*x^2 + 5), x)
Time = 0.36 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.04 \[ \int \frac {x^6}{4+\left (1+x^2\right )^4} \, dx=-\frac {1}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + \sqrt {2}\right )}\right ) - \frac {1}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - \sqrt {2}\right )}\right ) - \frac {1}{80} \, \sqrt {625 \, \sqrt {5} + 1199} \arctan \left (\frac {5^{\frac {3}{4}} {\left (x + 5^{\frac {1}{4}} \sqrt {-\frac {1}{5} \, \sqrt {5} + \frac {1}{2}}\right )}}{5 \, \sqrt {\frac {1}{5} \, \sqrt {5} + \frac {1}{2}}}\right ) - \frac {1}{80} \, \sqrt {625 \, \sqrt {5} + 1199} \arctan \left (\frac {5^{\frac {3}{4}} {\left (x - 5^{\frac {1}{4}} \sqrt {-\frac {1}{5} \, \sqrt {5} + \frac {1}{2}}\right )}}{5 \, \sqrt {\frac {1}{5} \, \sqrt {5} + \frac {1}{2}}}\right ) + \frac {1}{160} \, \sqrt {625 \, \sqrt {5} - 1199} \log \left (x^{2} + 2 \cdot 5^{\frac {1}{4}} x \sqrt {-\frac {1}{5} \, \sqrt {5} + \frac {1}{2}} + \sqrt {5}\right ) - \frac {1}{160} \, \sqrt {625 \, \sqrt {5} - 1199} \log \left (x^{2} - 2 \cdot 5^{\frac {1}{4}} x \sqrt {-\frac {1}{5} \, \sqrt {5} + \frac {1}{2}} + \sqrt {5}\right ) \] Input:
integrate(x^6/(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(625*sqrt(5) + 1199)*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(625*sqrt(5) + 1199)*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(625*sqrt(5) - 1199)*log(x^ 2 + 2*5^(1/4)*x*sqrt(-1/5*sqrt(5) + 1/2) + sqrt(5)) - 1/160*sqrt(625*sqrt( 5) - 1199)*log(x^2 - 2*5^(1/4)*x*sqrt(-1/5*sqrt(5) + 1/2) + sqrt(5))
Time = 10.62 (sec) , antiderivative size = 1287, normalized size of antiderivative = 7.07 \[ \int \frac {x^6}{4+\left (1+x^2\right )^4} \, dx=\text {Too large to display} \] Input:
int(x^6/((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 - atan(((62968*x + (((41/1024 - (25*5^(1/2))/1024)^(1/2) - ((25*5^( 1/2))/1024 + 41/1024)^(1/2))*(31637504*x - (((41/1024 - (25*5^(1/2))/1024) ^(1/2) - ((25*5^(1/2))/1024 + 41/1024)^(1/2))*(304087040*x - (2147483648*x *((41/1024 - (25*5^(1/2))/1024)^(1/2) - ((25*5^(1/2))/1024 + 41/1024)^(1/2 )) - 671088640)*((41/1024 - (25*5^(1/2))/1024)^(1/2) - ((25*5^(1/2))/1024 + 41/1024)^(1/2))) - 64225280)*((41/1024 - (25*5^(1/2))/1024)^(1/2) - ((25 *5^(1/2))/1024 + 41/1024)^(1/2))) - 768000)*((41/1024 - (25*5^(1/2))/1024) ^(1/2) - ((25*5^(1/2))/1024 + 41/1024)^(1/2)))*((41/1024 - (25*5^(1/2))/10 24)^(1/2) - ((25*5^(1/2))/1024 + 41/1024)^(1/2))*1i + (62968*x + (((41/102 4 - (25*5^(1/2))/1024)^(1/2) - ((25*5^(1/2))/1024 + 41/1024)^(1/2))*(31637 504*x - (((41/1024 - (25*5^(1/2))/1024)^(1/2) - ((25*5^(1/2))/1024 + 41/10 24)^(1/2))*(304087040*x - (2147483648*x*((41/1024 - (25*5^(1/2))/1024)^(1/ 2) - ((25*5^(1/2))/1024 + 41/1024)^(1/2)) + 671088640)*((41/1024 - (25*5^( 1/2))/1024)^(1/2) - ((25*5^(1/2))/1024 + 41/1024)^(1/2))) + 64225280)*((41 /1024 - (25*5^(1/2))/1024)^(1/2) - ((25*5^(1/2))/1024 + 41/1024)^(1/2))) + 768000)*((41/1024 - (25*5^(1/2))/1024)^(1/2) - ((25*5^(1/2))/1024 + 41/10 24)^(1/2)))*((41/1024 - (25*5^(1/2))/1024)^(1/2) - ((25*5^(1/2))/1024 + 41 /1024)^(1/2))*1i)/((62968*x + (((41/1024 - (25*5^(1/2))/1024)^(1/2) - ((25 *5^(1/2))/1024 + 41/1024)^(1/2))*(31637504*x - (((41/1024 - (25*5^(1/2)...
Time = 0.25 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.45 \[ \int \frac {x^6}{4+\left (1+x^2\right )^4} \, dx=\frac {\sqrt {2}\, \left (4 \mathit {atan} \left (\frac {\sqrt {2}-2 x}{\sqrt {2}}\right )-4 \mathit {atan} \left (\frac {\sqrt {2}+2 x}{\sqrt {2}}\right )-14 \sqrt {\sqrt {5}+2}\, \sqrt {5}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {5}-2}\, \sqrt {2}-2 x}{\sqrt {\sqrt {5}+2}\, \sqrt {2}}\right )+26 \sqrt {\sqrt {5}+2}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {5}-2}\, \sqrt {2}-2 x}{\sqrt {\sqrt {5}+2}\, \sqrt {2}}\right )+14 \sqrt {\sqrt {5}+2}\, \sqrt {5}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {5}-2}\, \sqrt {2}+2 x}{\sqrt {\sqrt {5}+2}\, \sqrt {2}}\right )-26 \sqrt {\sqrt {5}+2}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {5}-2}\, \sqrt {2}+2 x}{\sqrt {\sqrt {5}+2}\, \sqrt {2}}\right )+7 \sqrt {\sqrt {5}-2}\, \sqrt {5}\, \mathrm {log}\left (-\sqrt {\sqrt {5}-2}\, \sqrt {2}\, x +\sqrt {5}+x^{2}\right )-7 \sqrt {\sqrt {5}-2}\, \sqrt {5}\, \mathrm {log}\left (\sqrt {\sqrt {5}-2}\, \sqrt {2}\, x +\sqrt {5}+x^{2}\right )+13 \sqrt {\sqrt {5}-2}\, \mathrm {log}\left (-\sqrt {\sqrt {5}-2}\, \sqrt {2}\, x +\sqrt {5}+x^{2}\right )-13 \sqrt {\sqrt {5}-2}\, \mathrm {log}\left (\sqrt {\sqrt {5}-2}\, \sqrt {2}\, x +\sqrt {5}+x^{2}\right )\right )}{64} \] Input:
int(x^6/(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) ) - 14*sqrt(sqrt(5) + 2)*sqrt(5)*atan((sqrt(sqrt(5) - 2)*sqrt(2) - 2*x)/(s qrt(sqrt(5) + 2)*sqrt(2))) + 26*sqrt(sqrt(5) + 2)*atan((sqrt(sqrt(5) - 2)* sqrt(2) - 2*x)/(sqrt(sqrt(5) + 2)*sqrt(2))) + 14*sqrt(sqrt(5) + 2)*sqrt(5) *atan((sqrt(sqrt(5) - 2)*sqrt(2) + 2*x)/(sqrt(sqrt(5) + 2)*sqrt(2))) - 26* sqrt(sqrt(5) + 2)*atan((sqrt(sqrt(5) - 2)*sqrt(2) + 2*x)/(sqrt(sqrt(5) + 2 )*sqrt(2))) + 7*sqrt(sqrt(5) - 2)*sqrt(5)*log( - sqrt(sqrt(5) - 2)*sqrt(2) *x + sqrt(5) + x**2) - 7*sqrt(sqrt(5) - 2)*sqrt(5)*log(sqrt(sqrt(5) - 2)*s qrt(2)*x + sqrt(5) + x**2) + 13*sqrt(sqrt(5) - 2)*log( - sqrt(sqrt(5) - 2) *sqrt(2)*x + sqrt(5) + x**2) - 13*sqrt(sqrt(5) - 2)*log(sqrt(sqrt(5) - 2)* sqrt(2)*x + sqrt(5) + x**2)))/64