Integrand size = 17, antiderivative size = 209 \[ \int \frac {x^6}{4-\left (1+x^2\right )^4} \, dx=-\frac {1}{16} \sqrt {-1+9 \sqrt {3}} \arctan \left (\frac {\sqrt {2 \left (-1+\sqrt {3}\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )-\frac {\arctan \left (\sqrt {-1+\sqrt {2}} x\right )}{8 \sqrt {2 \left (-41+29 \sqrt {2}\right )}}+\frac {1}{16} \sqrt {-1+9 \sqrt {3}} \arctan \left (\frac {\sqrt {2 \left (-1+\sqrt {3}\right )}+2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )+\frac {\text {arctanh}\left (\sqrt {1+\sqrt {2}} x\right )}{8 \sqrt {2 \left (41+29 \sqrt {2}\right )}}+\frac {1}{16} \sqrt {1+9 \sqrt {3}} \text {arctanh}\left (\frac {\sqrt {2 \left (-1+\sqrt {3}\right )} x}{\sqrt {3}+x^2}\right ) \] Output:
-1/16*(-1+9*3^(1/2))^(1/2)*arctan(((-2+2*3^(1/2))^(1/2)-2*x)/(2+2*3^(1/2)) ^(1/2))-1/8*arctan((2^(1/2)-1)^(1/2)*x)/(-82+58*2^(1/2))^(1/2)+1/16*(-1+9* 3^(1/2))^(1/2)*arctan(((-2+2*3^(1/2))^(1/2)+2*x)/(2+2*3^(1/2))^(1/2))+1/8* arctanh((1+2^(1/2))^(1/2)*x)/(82+58*2^(1/2))^(1/2)+1/16*(1+9*3^(1/2))^(1/2 )*arctanh((-2+2*3^(1/2))^(1/2)*x/(3^(1/2)+x^2))
Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.80 \[ \int \frac {x^6}{4-\left (1+x^2\right )^4} \, dx=\frac {\frac {\left (-5 i+\sqrt {2}\right ) \arctan \left (\frac {x}{\sqrt {1-i \sqrt {2}}}\right )}{\sqrt {1-i \sqrt {2}}}+\frac {\left (5 i+\sqrt {2}\right ) \arctan \left (\frac {x}{\sqrt {1+i \sqrt {2}}}\right )}{\sqrt {1+i \sqrt {2}}}-\frac {\left (7+5 \sqrt {2}\right ) \arctan \left (\frac {x}{\sqrt {1+\sqrt {2}}}\right )}{\sqrt {1+\sqrt {2}}}+\frac {\left (-7+5 \sqrt {2}\right ) \text {arctanh}\left (\frac {x}{\sqrt {-1+\sqrt {2}}}\right )}{\sqrt {-1+\sqrt {2}}}}{8 \sqrt {2}} \] Input:
Integrate[x^6/(4 - (1 + x^2)^4),x]
Output:
(((-5*I + Sqrt[2])*ArcTan[x/Sqrt[1 - I*Sqrt[2]]])/Sqrt[1 - I*Sqrt[2]] + (( 5*I + Sqrt[2])*ArcTan[x/Sqrt[1 + I*Sqrt[2]]])/Sqrt[1 + I*Sqrt[2]] - ((7 + 5*Sqrt[2])*ArcTan[x/Sqrt[1 + Sqrt[2]]])/Sqrt[1 + Sqrt[2]] + ((-7 + 5*Sqrt[ 2])*ArcTanh[x/Sqrt[-1 + Sqrt[2]]])/Sqrt[-1 + Sqrt[2]])/(8*Sqrt[2])
Time = 0.78 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.22, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, 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.
| \(\displaystyle \int \frac {x^6}{4-\left (x^2+1\right )^4} \, dx\) |
| \(\Big \downarrow \) 2460 |
| \(\displaystyle \int \left (\frac {2-5 x^2}{4 \left (x^4+2 x^2-1\right )}+\frac {x^2+6}{4 \left (x^4+2 x^2+3\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {1}{16} \sqrt {9 \sqrt {3}-1} \arctan \left (\frac {\sqrt {2 \left (\sqrt {3}-1\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )-\frac {1}{8} \sqrt {\frac {1}{2} \left (41+29 \sqrt {2}\right )} \arctan \left (\frac {x}{\sqrt {1+\sqrt {2}}}\right )+\frac {1}{16} \sqrt {9 \sqrt {3}-1} \arctan \left (\frac {2 x+\sqrt {2 \left (\sqrt {3}-1\right )}}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )+\frac {1}{8} \sqrt {\frac {1}{2} \left (29 \sqrt {2}-41\right )} \text {arctanh}\left (\frac {x}{\sqrt {\sqrt {2}-1}}\right )-\frac {1}{32} \sqrt {1+9 \sqrt {3}} \log \left (x^2-\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )+\frac {1}{32} \sqrt {1+9 \sqrt {3}} \log \left (x^2+\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )\) |
Input:
Int[x^6/(4 - (1 + x^2)^4),x]
Output:
-1/16*(Sqrt[-1 + 9*Sqrt[3]]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] - 2*x)/Sqrt[2*( 1 + Sqrt[3])]]) - (Sqrt[(41 + 29*Sqrt[2])/2]*ArcTan[x/Sqrt[1 + Sqrt[2]]])/ 8 + (Sqrt[-1 + 9*Sqrt[3]]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] + 2*x)/Sqrt[2*(1 + Sqrt[3])]])/16 + (Sqrt[(-41 + 29*Sqrt[2])/2]*ArcTanh[x/Sqrt[-1 + Sqrt[2] ]])/8 - (Sqrt[1 + 9*Sqrt[3]]*Log[Sqrt[3] - Sqrt[2*(-1 + Sqrt[3])]*x + x^2] )/32 + (Sqrt[1 + 9*Sqrt[3]]*Log[Sqrt[3] + Sqrt[2*(-1 + Sqrt[3])]*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 = 0.10 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.33
| method | result | size |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (4 \textit {\_Z}^{4}-4 \textit {\_Z}^{2}+243\right )}{\sum }\textit {\_R} \ln \left (2 \textit {\_R}^{3}+43 \textit {\_R} +99 x \right )\right )}{16}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (4 \textit {\_Z}^{4}+164 \textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (6 \textit {\_R}^{3}+239 \textit {\_R} +29 x \right )\right )}{16}\) | \(68\) |
| default | \(-\frac {\left (\sqrt {-2+2 \sqrt {3}}\, \sqrt {3}+5 \sqrt {-2+2 \sqrt {3}}\right ) \ln \left (x^{2}-x \sqrt {-2+2 \sqrt {3}}+\sqrt {3}\right )}{64}-\frac {\left (-8 \sqrt {3}+\frac {\left (\sqrt {-2+2 \sqrt {3}}\, \sqrt {3}+5 \sqrt {-2+2 \sqrt {3}}\right ) \sqrt {-2+2 \sqrt {3}}}{2}\right ) \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{16 \sqrt {2+2 \sqrt {3}}}+\frac {\left (\sqrt {-2+2 \sqrt {3}}\, \sqrt {3}+5 \sqrt {-2+2 \sqrt {3}}\right ) \ln \left (x^{2}+x \sqrt {-2+2 \sqrt {3}}+\sqrt {3}\right )}{64}+\frac {\left (8 \sqrt {3}-\frac {\left (\sqrt {-2+2 \sqrt {3}}\, \sqrt {3}+5 \sqrt {-2+2 \sqrt {3}}\right ) \sqrt {-2+2 \sqrt {3}}}{2}\right ) \arctan \left (\frac {\sqrt {-2+2 \sqrt {3}}+2 x}{\sqrt {2+2 \sqrt {3}}}\right )}{16 \sqrt {2+2 \sqrt {3}}}+\frac {\left (-7+5 \sqrt {2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {x}{\sqrt {\sqrt {2}-1}}\right )}{16 \sqrt {\sqrt {2}-1}}-\frac {\left (7+5 \sqrt {2}\right ) \sqrt {2}\, \arctan \left (\frac {x}{\sqrt {1+\sqrt {2}}}\right )}{16 \sqrt {1+\sqrt {2}}}\) | \(309\) |
Input:
int(x^6/(4-(x^2+1)^4),x,method=_RETURNVERBOSE)
Output:
1/16*sum(_R*ln(2*_R^3+43*_R+99*x),_R=RootOf(4*_Z^4-4*_Z^2+243))+1/16*sum(_ R*ln(6*_R^3+239*_R+29*x),_R=RootOf(4*_Z^4+164*_Z^2-1))
Leaf count of result is larger than twice the leaf count of optimal. 299 vs. \(2 (148) = 296\).
Time = 0.12 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.43 \[ \int \frac {x^6}{4-\left (1+x^2\right )^4} \, dx=\frac {1}{16} \, \sqrt {9 \, \sqrt {3} - 1} \arctan \left (\frac {1}{22} \, \sqrt {9 \, \sqrt {3} + 1} \sqrt {9 \, \sqrt {3} - 1} {\left (\sqrt {3} - 1\right )} + \frac {1}{11} \, {\left (2 \, \sqrt {3} x - x\right )} \sqrt {9 \, \sqrt {3} - 1}\right ) - \frac {1}{16} \, \sqrt {9 \, \sqrt {3} - 1} \arctan \left (\frac {1}{22} \, \sqrt {9 \, \sqrt {3} + 1} \sqrt {9 \, \sqrt {3} - 1} {\left (\sqrt {3} - 1\right )} - \frac {1}{11} \, {\left (2 \, \sqrt {3} x - x\right )} \sqrt {9 \, \sqrt {3} - 1}\right ) + \frac {1}{8} \, \sqrt {\frac {29}{2} \, \sqrt {2} + \frac {41}{2}} \arctan \left ({\left (7 \, \sqrt {2} x - 10 \, x\right )} \sqrt {\frac {29}{2} \, \sqrt {2} + \frac {41}{2}}\right ) - \frac {1}{32} \, \sqrt {9 \, \sqrt {3} + 1} \log \left (11 \, x^{2} + {\left (\sqrt {3} x - 5 \, x\right )} \sqrt {9 \, \sqrt {3} + 1} + 11 \, \sqrt {3}\right ) + \frac {1}{32} \, \sqrt {9 \, \sqrt {3} + 1} \log \left (11 \, x^{2} - {\left (\sqrt {3} x - 5 \, x\right )} \sqrt {9 \, \sqrt {3} + 1} + 11 \, \sqrt {3}\right ) + \frac {1}{16} \, \sqrt {\frac {29}{2} \, \sqrt {2} - \frac {41}{2}} \log \left (\sqrt {\frac {29}{2} \, \sqrt {2} - \frac {41}{2}} {\left (3 \, \sqrt {2} + 4\right )} + x\right ) - \frac {1}{16} \, \sqrt {\frac {29}{2} \, \sqrt {2} - \frac {41}{2}} \log \left (-\sqrt {\frac {29}{2} \, \sqrt {2} - \frac {41}{2}} {\left (3 \, \sqrt {2} + 4\right )} + x\right ) \] Input:
integrate(x^6/(4-(x^2+1)^4),x, algorithm="fricas")
Output:
1/16*sqrt(9*sqrt(3) - 1)*arctan(1/22*sqrt(9*sqrt(3) + 1)*sqrt(9*sqrt(3) - 1)*(sqrt(3) - 1) + 1/11*(2*sqrt(3)*x - x)*sqrt(9*sqrt(3) - 1)) - 1/16*sqrt (9*sqrt(3) - 1)*arctan(1/22*sqrt(9*sqrt(3) + 1)*sqrt(9*sqrt(3) - 1)*(sqrt( 3) - 1) - 1/11*(2*sqrt(3)*x - x)*sqrt(9*sqrt(3) - 1)) + 1/8*sqrt(29/2*sqrt (2) + 41/2)*arctan((7*sqrt(2)*x - 10*x)*sqrt(29/2*sqrt(2) + 41/2)) - 1/32* sqrt(9*sqrt(3) + 1)*log(11*x^2 + (sqrt(3)*x - 5*x)*sqrt(9*sqrt(3) + 1) + 1 1*sqrt(3)) + 1/32*sqrt(9*sqrt(3) + 1)*log(11*x^2 - (sqrt(3)*x - 5*x)*sqrt( 9*sqrt(3) + 1) + 11*sqrt(3)) + 1/16*sqrt(29/2*sqrt(2) - 41/2)*log(sqrt(29/ 2*sqrt(2) - 41/2)*(3*sqrt(2) + 4) + x) - 1/16*sqrt(29/2*sqrt(2) - 41/2)*lo g(-sqrt(29/2*sqrt(2) - 41/2)*(3*sqrt(2) + 4) + x)
Time = 1.02 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.41 \[ \int \frac {x^6}{4-\left (1+x^2\right )^4} \, dx=- \operatorname {RootSum} {\left (262144 t^{4} - 1024 t^{2} + 243, \left ( t \mapsto t \log {\left (- \frac {130739341361152 t^{7}}{155496231} - \frac {20444980707328 t^{5}}{155496231} - \frac {17400141824 t^{3}}{51832077} - \frac {20505938212 t}{155496231} + x \right )} \right )\right )} - \operatorname {RootSum} {\left (262144 t^{4} + 41984 t^{2} - 1, \left ( t \mapsto t \log {\left (- \frac {130739341361152 t^{7}}{155496231} - \frac {20444980707328 t^{5}}{155496231} - \frac {17400141824 t^{3}}{51832077} - \frac {20505938212 t}{155496231} + x \right )} \right )\right )} \] Input:
integrate(x**6/(4-(x**2+1)**4),x)
Output:
-RootSum(262144*_t**4 - 1024*_t**2 + 243, Lambda(_t, _t*log(-1307393413611 52*_t**7/155496231 - 20444980707328*_t**5/155496231 - 17400141824*_t**3/51 832077 - 20505938212*_t/155496231 + x))) - RootSum(262144*_t**4 + 41984*_t **2 - 1, Lambda(_t, _t*log(-130739341361152*_t**7/155496231 - 204449807073 28*_t**5/155496231 - 17400141824*_t**3/51832077 - 20505938212*_t/155496231 + x)))
\[ \int \frac {x^6}{4-\left (1+x^2\right )^4} \, dx=\int { -\frac {x^{6}}{{\left (x^{2} + 1\right )}^{4} - 4} \,d x } \] Input:
integrate(x^6/(4-(x^2+1)^4),x, algorithm="maxima")
Output:
-integrate(x^6/((x^2 + 1)^4 - 4), x)
Leaf count of result is larger than twice the leaf count of optimal. 609 vs. \(2 (148) = 296\).
Time = 0.47 (sec) , antiderivative size = 609, normalized size of antiderivative = 2.91 \[ \int \frac {x^6}{4-\left (1+x^2\right )^4} \, dx =\text {Too large to display} \] Input:
integrate(x^6/(4-(x^2+1)^4),x, algorithm="giac")
Output:
-1/10368*sqrt(2)*(3^(3/4)*sqrt(2)*(6*sqrt(3) + 18)^(3/2) + 18*3^(3/4)*sqrt (2)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) + 18*3^(3/4)*(sqrt(3) + 3)*sqrt(-6* sqrt(3) + 18) - 3^(3/4)*(-6*sqrt(3) + 18)^(3/2) - 216*3^(1/4)*sqrt(2)*sqrt (6*sqrt(3) + 18) - 216*3^(1/4)*sqrt(-6*sqrt(3) + 18))*arctan(1/3*3^(3/4)*( x + 3^(1/4)*sqrt(-1/6*sqrt(3) + 1/2))/sqrt(1/6*sqrt(3) + 1/2)) - 1/10368*s qrt(2)*(3^(3/4)*sqrt(2)*(6*sqrt(3) + 18)^(3/2) + 18*3^(3/4)*sqrt(2)*sqrt(6 *sqrt(3) + 18)*(sqrt(3) - 3) + 18*3^(3/4)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) - 3^(3/4)*(-6*sqrt(3) + 18)^(3/2) - 216*3^(1/4)*sqrt(2)*sqrt(6*sqrt(3) + 18) - 216*3^(1/4)*sqrt(-6*sqrt(3) + 18))*arctan(1/3*3^(3/4)*(x - 3^(1/4 )*sqrt(-1/6*sqrt(3) + 1/2))/sqrt(1/6*sqrt(3) + 1/2)) - 1/20736*sqrt(2)*(18 *3^(3/4)*sqrt(2)*(sqrt(3) + 3)*sqrt(-6*sqrt(3) + 18) - 3^(3/4)*sqrt(2)*(-6 *sqrt(3) + 18)^(3/2) - 3^(3/4)*(6*sqrt(3) + 18)^(3/2) - 18*3^(3/4)*sqrt(6* sqrt(3) + 18)*(sqrt(3) - 3) - 216*3^(1/4)*sqrt(2)*sqrt(-6*sqrt(3) + 18) + 216*3^(1/4)*sqrt(6*sqrt(3) + 18))*log(x^2 + 2*3^(1/4)*x*sqrt(-1/6*sqrt(3) + 1/2) + sqrt(3)) + 1/20736*sqrt(2)*(18*3^(3/4)*sqrt(2)*(sqrt(3) + 3)*sqrt (-6*sqrt(3) + 18) - 3^(3/4)*sqrt(2)*(-6*sqrt(3) + 18)^(3/2) - 3^(3/4)*(6*s qrt(3) + 18)^(3/2) - 18*3^(3/4)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) - 216*3 ^(1/4)*sqrt(2)*sqrt(-6*sqrt(3) + 18) + 216*3^(1/4)*sqrt(6*sqrt(3) + 18))*l og(x^2 - 2*3^(1/4)*x*sqrt(-1/6*sqrt(3) + 1/2) + sqrt(3)) - 1/16*sqrt(58*sq rt(2) + 82)*arctan(x/sqrt(sqrt(2) + 1)) + 1/32*sqrt(58*sqrt(2) - 82)*lo...
Time = 0.25 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.21 \[ \int \frac {x^6}{4-\left (1+x^2\right )^4} \, dx=\frac {\mathrm {atan}\left (\frac {x\,\sqrt {2-\sqrt {2}\,22{}\mathrm {i}}\,4202{}\mathrm {i}}{63624+\sqrt {2}\,9966{}\mathrm {i}}+\frac {5522\,\sqrt {2}\,x\,\sqrt {2-\sqrt {2}\,22{}\mathrm {i}}}{63624+\sqrt {2}\,9966{}\mathrm {i}}\right )\,\sqrt {2-\sqrt {2}\,22{}\mathrm {i}}\,1{}\mathrm {i}}{16}-\frac {\mathrm {atan}\left (\frac {x\,\sqrt {2+\sqrt {2}\,22{}\mathrm {i}}\,4202{}\mathrm {i}}{-63624+\sqrt {2}\,9966{}\mathrm {i}}-\frac {5522\,\sqrt {2}\,x\,\sqrt {2+\sqrt {2}\,22{}\mathrm {i}}}{-63624+\sqrt {2}\,9966{}\mathrm {i}}\right )\,\sqrt {2+\sqrt {2}\,22{}\mathrm {i}}\,1{}\mathrm {i}}{16}-\frac {\mathrm {atan}\left (\frac {x\,\sqrt {-58\,\sqrt {2}-82}\,54462{}\mathrm {i}}{638754\,\sqrt {2}+905148}+\frac {\sqrt {2}\,x\,\sqrt {-58\,\sqrt {2}-82}\,25752{}\mathrm {i}}{638754\,\sqrt {2}+905148}\right )\,\sqrt {-58\,\sqrt {2}-82}\,1{}\mathrm {i}}{16}+\frac {\mathrm {atan}\left (\frac {x\,\sqrt {58\,\sqrt {2}-82}\,54462{}\mathrm {i}}{638754\,\sqrt {2}-905148}-\frac {\sqrt {2}\,x\,\sqrt {58\,\sqrt {2}-82}\,25752{}\mathrm {i}}{638754\,\sqrt {2}-905148}\right )\,\sqrt {58\,\sqrt {2}-82}\,1{}\mathrm {i}}{16} \] Input:
int(-x^6/((x^2 + 1)^4 - 4),x)
Output:
(atan((x*(2 - 2^(1/2)*22i)^(1/2)*4202i)/(2^(1/2)*9966i + 63624) + (5522*2^ (1/2)*x*(2 - 2^(1/2)*22i)^(1/2))/(2^(1/2)*9966i + 63624))*(2 - 2^(1/2)*22i )^(1/2)*1i)/16 - (atan((x*(2^(1/2)*22i + 2)^(1/2)*4202i)/(2^(1/2)*9966i - 63624) - (5522*2^(1/2)*x*(2^(1/2)*22i + 2)^(1/2))/(2^(1/2)*9966i - 63624)) *(2^(1/2)*22i + 2)^(1/2)*1i)/16 - (atan((x*(- 58*2^(1/2) - 82)^(1/2)*54462 i)/(638754*2^(1/2) + 905148) + (2^(1/2)*x*(- 58*2^(1/2) - 82)^(1/2)*25752i )/(638754*2^(1/2) + 905148))*(- 58*2^(1/2) - 82)^(1/2)*1i)/16 + (atan((x*( 58*2^(1/2) - 82)^(1/2)*54462i)/(638754*2^(1/2) - 905148) - (2^(1/2)*x*(58* 2^(1/2) - 82)^(1/2)*25752i)/(638754*2^(1/2) - 905148))*(58*2^(1/2) - 82)^( 1/2)*1i)/16
Time = 0.26 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.64 \[ \int \frac {x^6}{4-\left (1+x^2\right )^4} \, dx=\frac {\sqrt {\sqrt {3}+1}\, \sqrt {6}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {3}-1}\, \sqrt {2}-2 x}{\sqrt {\sqrt {3}+1}\, \sqrt {2}}\right )}{32}-\frac {5 \sqrt {\sqrt {3}+1}\, \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {3}-1}\, \sqrt {2}-2 x}{\sqrt {\sqrt {3}+1}\, \sqrt {2}}\right )}{32}-\frac {\sqrt {\sqrt {3}+1}\, \sqrt {6}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {3}-1}\, \sqrt {2}+2 x}{\sqrt {\sqrt {3}+1}\, \sqrt {2}}\right )}{32}+\frac {5 \sqrt {\sqrt {3}+1}\, \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {3}-1}\, \sqrt {2}+2 x}{\sqrt {\sqrt {3}+1}\, \sqrt {2}}\right )}{32}-\frac {3 \sqrt {\sqrt {2}+1}\, \sqrt {2}\, \mathit {atan} \left (\frac {x}{\sqrt {\sqrt {2}+1}}\right )}{16}-\frac {\sqrt {\sqrt {2}+1}\, \mathit {atan} \left (\frac {x}{\sqrt {\sqrt {2}+1}}\right )}{4}-\frac {\sqrt {\sqrt {3}-1}\, \sqrt {6}\, \mathrm {log}\left (-\sqrt {\sqrt {3}-1}\, \sqrt {2}\, x +\sqrt {3}+x^{2}\right )}{64}+\frac {\sqrt {\sqrt {3}-1}\, \sqrt {6}\, \mathrm {log}\left (\sqrt {\sqrt {3}-1}\, \sqrt {2}\, x +\sqrt {3}+x^{2}\right )}{64}-\frac {5 \sqrt {\sqrt {3}-1}\, \sqrt {2}\, \mathrm {log}\left (-\sqrt {\sqrt {3}-1}\, \sqrt {2}\, x +\sqrt {3}+x^{2}\right )}{64}+\frac {5 \sqrt {\sqrt {3}-1}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {\sqrt {3}-1}\, \sqrt {2}\, x +\sqrt {3}+x^{2}\right )}{64}-\frac {3 \sqrt {\sqrt {2}-1}\, \sqrt {2}\, \mathrm {log}\left (-\sqrt {\sqrt {2}-1}+x \right )}{32}+\frac {3 \sqrt {\sqrt {2}-1}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {\sqrt {2}-1}+x \right )}{32}+\frac {\sqrt {\sqrt {2}-1}\, \mathrm {log}\left (-\sqrt {\sqrt {2}-1}+x \right )}{8}-\frac {\sqrt {\sqrt {2}-1}\, \mathrm {log}\left (\sqrt {\sqrt {2}-1}+x \right )}{8} \] Input:
int(x^6/(4-(x^2+1)^4),x)
Output:
(2*sqrt(sqrt(3) + 1)*sqrt(6)*atan((sqrt(sqrt(3) - 1)*sqrt(2) - 2*x)/(sqrt( sqrt(3) + 1)*sqrt(2))) - 10*sqrt(sqrt(3) + 1)*sqrt(2)*atan((sqrt(sqrt(3) - 1)*sqrt(2) - 2*x)/(sqrt(sqrt(3) + 1)*sqrt(2))) - 2*sqrt(sqrt(3) + 1)*sqrt (6)*atan((sqrt(sqrt(3) - 1)*sqrt(2) + 2*x)/(sqrt(sqrt(3) + 1)*sqrt(2))) + 10*sqrt(sqrt(3) + 1)*sqrt(2)*atan((sqrt(sqrt(3) - 1)*sqrt(2) + 2*x)/(sqrt( sqrt(3) + 1)*sqrt(2))) - 12*sqrt(sqrt(2) + 1)*sqrt(2)*atan(x/sqrt(sqrt(2) + 1)) - 16*sqrt(sqrt(2) + 1)*atan(x/sqrt(sqrt(2) + 1)) - sqrt(sqrt(3) - 1) *sqrt(6)*log( - sqrt(sqrt(3) - 1)*sqrt(2)*x + sqrt(3) + x**2) + sqrt(sqrt( 3) - 1)*sqrt(6)*log(sqrt(sqrt(3) - 1)*sqrt(2)*x + sqrt(3) + x**2) - 5*sqrt (sqrt(3) - 1)*sqrt(2)*log( - sqrt(sqrt(3) - 1)*sqrt(2)*x + sqrt(3) + x**2) + 5*sqrt(sqrt(3) - 1)*sqrt(2)*log(sqrt(sqrt(3) - 1)*sqrt(2)*x + sqrt(3) + x**2) - 6*sqrt(sqrt(2) - 1)*sqrt(2)*log( - sqrt(sqrt(2) - 1) + x) + 6*sqr t(sqrt(2) - 1)*sqrt(2)*log(sqrt(sqrt(2) - 1) + x) + 8*sqrt(sqrt(2) - 1)*lo g( - sqrt(sqrt(2) - 1) + x) - 8*sqrt(sqrt(2) - 1)*log(sqrt(sqrt(2) - 1) + x))/64