Integrand size = 17, antiderivative size = 212 \[ \int \frac {x^8}{4-\left (1+x^2\right )^4} \, dx=-x-\frac {1}{16} \sqrt {-43+27 \sqrt {3}} \arctan \left (\frac {\sqrt {2 \left (-1+\sqrt {3}\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )+\frac {\arctan \left (\sqrt {-1+\sqrt {2}} x\right )}{8 \sqrt {2 \left (-239+169 \sqrt {2}\right )}}+\frac {1}{16} \sqrt {-43+27 \sqrt {3}} \arctan \left (\frac {\sqrt {2 \left (-1+\sqrt {3}\right )}+2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )+\frac {\text {arctanh}\left (\sqrt {1+\sqrt {2}} x\right )}{8 \sqrt {2 \left (239+169 \sqrt {2}\right )}}-\frac {1}{16} \sqrt {43+27 \sqrt {3}} \text {arctanh}\left (\frac {\sqrt {2 \left (-1+\sqrt {3}\right )} x}{\sqrt {3}+x^2}\right ) \] Output:
-x-1/16*(-43+27*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)/(-478+338*2^(1/2))^(1/2)+1/16* (-43+27*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)/(478+338*2^(1/2))^(1/2)-1/16*(43+27*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 = 0.26 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.85 \[ \int \frac {x^8}{4-\left (1+x^2\right )^4} \, dx=-x+\frac {\left (7 i+4 \sqrt {2}\right ) \arctan \left (\frac {x}{\sqrt {1-i \sqrt {2}}}\right )}{8 \sqrt {2-2 i \sqrt {2}}}+\frac {\left (-7 i+4 \sqrt {2}\right ) \arctan \left (\frac {x}{\sqrt {1+i \sqrt {2}}}\right )}{8 \sqrt {2+2 i \sqrt {2}}}+\frac {\left (17+12 \sqrt {2}\right ) \arctan \left (\frac {x}{\sqrt {1+\sqrt {2}}}\right )}{8 \sqrt {2 \left (1+\sqrt {2}\right )}}-\frac {\left (-17+12 \sqrt {2}\right ) \text {arctanh}\left (\frac {x}{\sqrt {-1+\sqrt {2}}}\right )}{8 \sqrt {2 \left (-1+\sqrt {2}\right )}} \] Input:
Integrate[x^8/(4 - (1 + x^2)^4),x]
Output:
-x + ((7*I + 4*Sqrt[2])*ArcTan[x/Sqrt[1 - I*Sqrt[2]]])/(8*Sqrt[2 - (2*I)*S qrt[2]]) + ((-7*I + 4*Sqrt[2])*ArcTan[x/Sqrt[1 + I*Sqrt[2]]])/(8*Sqrt[2 + (2*I)*Sqrt[2]]) + ((17 + 12*Sqrt[2])*ArcTan[x/Sqrt[1 + Sqrt[2]]])/(8*Sqrt[ 2*(1 + Sqrt[2])]) - ((-17 + 12*Sqrt[2])*ArcTanh[x/Sqrt[-1 + Sqrt[2]]])/(8* Sqrt[2*(-1 + Sqrt[2])])
Time = 0.80 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.21, 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.
| \(\displaystyle \int \frac {x^8}{4-\left (x^2+1\right )^4} \, dx\) |
| \(\Big \downarrow \) 2460 |
| \(\displaystyle \int \left (\frac {4 x^2-3}{4 \left (x^4+2 x^2+3\right )}+\frac {12 x^2-5}{4 \left (x^4+2 x^2-1\right )}-1\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {1}{16} \sqrt {27 \sqrt {3}-43} \arctan \left (\frac {\sqrt {2 \left (\sqrt {3}-1\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )+\frac {1}{8} \sqrt {\frac {1}{2} \left (239+169 \sqrt {2}\right )} \arctan \left (\frac {x}{\sqrt {1+\sqrt {2}}}\right )+\frac {1}{16} \sqrt {27 \sqrt {3}-43} \arctan \left (\frac {2 x+\sqrt {2 \left (\sqrt {3}-1\right )}}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )+\frac {1}{8} \sqrt {\frac {1}{2} \left (169 \sqrt {2}-239\right )} \text {arctanh}\left (\frac {x}{\sqrt {\sqrt {2}-1}}\right )+\frac {1}{32} \sqrt {43+27 \sqrt {3}} \log \left (x^2-\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )-\frac {1}{32} \sqrt {43+27 \sqrt {3}} \log \left (x^2+\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )-x\) |
Input:
Int[x^8/(4 - (1 + x^2)^4),x]
Output:
-x - (Sqrt[-43 + 27*Sqrt[3]]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] - 2*x)/Sqrt[2* (1 + Sqrt[3])]])/16 + (Sqrt[(239 + 169*Sqrt[2])/2]*ArcTan[x/Sqrt[1 + Sqrt[ 2]]])/8 + (Sqrt[-43 + 27*Sqrt[3]]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] + 2*x)/Sq rt[2*(1 + Sqrt[3])]])/16 + (Sqrt[(-239 + 169*Sqrt[2])/2]*ArcTanh[x/Sqrt[-1 + Sqrt[2]]])/8 + (Sqrt[43 + 27*Sqrt[3]]*Log[Sqrt[3] - Sqrt[2*(-1 + Sqrt[3 ])]*x + x^2])/32 - (Sqrt[43 + 27*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 = 0.11 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.33
| method | result | size |
| risch | \(-x +\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (4 \textit {\_Z}^{4}-172 \textit {\_Z}^{2}+2187\right )}{\sum }\textit {\_R} \ln \left (10 \textit {\_R}^{3}-241 \textit {\_R} +351 x \right )\right )}{16}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (4 \textit {\_Z}^{4}+956 \textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (14 \textit {\_R}^{3}+3363 \textit {\_R} +169 x \right )\right )}{16}\) | \(71\) |
| default | \(-x -\frac {\left (-5 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}-7 \sqrt {-2+2 \sqrt {3}}\right ) \ln \left (x^{2}-x \sqrt {-2+2 \sqrt {3}}+\sqrt {3}\right )}{64}-\frac {\left (4 \sqrt {3}+\frac {\left (-5 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}-7 \sqrt {-2+2 \sqrt {3}}\right ) \sqrt {-2+2 \sqrt {3}}}{2}\right ) \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{16 \sqrt {2+2 \sqrt {3}}}+\frac {\left (-5 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}-7 \sqrt {-2+2 \sqrt {3}}\right ) \ln \left (x^{2}+x \sqrt {-2+2 \sqrt {3}}+\sqrt {3}\right )}{64}+\frac {\left (-4 \sqrt {3}-\frac {\left (-5 \sqrt {-2+2 \sqrt {3}}\, \sqrt {3}-7 \sqrt {-2+2 \sqrt {3}}\right ) \sqrt {-2+2 \sqrt {3}}}{2}\right ) \arctan \left (\frac {\sqrt {-2+2 \sqrt {3}}+2 x}{\sqrt {2+2 \sqrt {3}}}\right )}{16 \sqrt {2+2 \sqrt {3}}}-\frac {\left (-17+12 \sqrt {2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {x}{\sqrt {\sqrt {2}-1}}\right )}{16 \sqrt {\sqrt {2}-1}}+\frac {\left (17+12 \sqrt {2}\right ) \sqrt {2}\, \arctan \left (\frac {x}{\sqrt {1+\sqrt {2}}}\right )}{16 \sqrt {1+\sqrt {2}}}\) | \(316\) |
Input:
int(x^8/(4-(x^2+1)^4),x,method=_RETURNVERBOSE)
Output:
-x+1/16*sum(_R*ln(10*_R^3-241*_R+351*x),_R=RootOf(4*_Z^4-172*_Z^2+2187))+1 /16*sum(_R*ln(14*_R^3+3363*_R+169*x),_R=RootOf(4*_Z^4+956*_Z^2-1))
Time = 0.09 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.42 \[ \int \frac {x^8}{4-\left (1+x^2\right )^4} \, dx=\frac {1}{16} \, \sqrt {27 \, \sqrt {3} - 43} \arctan \left (\frac {1}{26} \, \sqrt {27 \, \sqrt {3} + 43} \sqrt {27 \, \sqrt {3} - 43} {\left (\sqrt {3} - 1\right )} + \frac {1}{13} \, {\left (\sqrt {3} x + 4 \, x\right )} \sqrt {27 \, \sqrt {3} - 43}\right ) - \frac {1}{16} \, \sqrt {27 \, \sqrt {3} - 43} \arctan \left (\frac {1}{26} \, \sqrt {27 \, \sqrt {3} + 43} \sqrt {27 \, \sqrt {3} - 43} {\left (\sqrt {3} - 1\right )} - \frac {1}{13} \, {\left (\sqrt {3} x + 4 \, x\right )} \sqrt {27 \, \sqrt {3} - 43}\right ) + \frac {1}{8} \, \sqrt {\frac {169}{2} \, \sqrt {2} + \frac {239}{2}} \arctan \left ({\left (17 \, \sqrt {2} x - 24 \, x\right )} \sqrt {\frac {169}{2} \, \sqrt {2} + \frac {239}{2}}\right ) - \frac {1}{32} \, \sqrt {27 \, \sqrt {3} + 43} \log \left (13 \, x^{2} + {\left (5 \, \sqrt {3} x - 7 \, x\right )} \sqrt {27 \, \sqrt {3} + 43} + 13 \, \sqrt {3}\right ) + \frac {1}{32} \, \sqrt {27 \, \sqrt {3} + 43} \log \left (13 \, x^{2} - {\left (5 \, \sqrt {3} x - 7 \, x\right )} \sqrt {27 \, \sqrt {3} + 43} + 13 \, \sqrt {3}\right ) + \frac {1}{16} \, \sqrt {\frac {169}{2} \, \sqrt {2} - \frac {239}{2}} \log \left (\sqrt {\frac {169}{2} \, \sqrt {2} - \frac {239}{2}} {\left (7 \, \sqrt {2} + 10\right )} + x\right ) - \frac {1}{16} \, \sqrt {\frac {169}{2} \, \sqrt {2} - \frac {239}{2}} \log \left (-\sqrt {\frac {169}{2} \, \sqrt {2} - \frac {239}{2}} {\left (7 \, \sqrt {2} + 10\right )} + x\right ) - x \] Input:
integrate(x^8/(4-(x^2+1)^4),x, algorithm="fricas")
Output:
1/16*sqrt(27*sqrt(3) - 43)*arctan(1/26*sqrt(27*sqrt(3) + 43)*sqrt(27*sqrt( 3) - 43)*(sqrt(3) - 1) + 1/13*(sqrt(3)*x + 4*x)*sqrt(27*sqrt(3) - 43)) - 1 /16*sqrt(27*sqrt(3) - 43)*arctan(1/26*sqrt(27*sqrt(3) + 43)*sqrt(27*sqrt(3 ) - 43)*(sqrt(3) - 1) - 1/13*(sqrt(3)*x + 4*x)*sqrt(27*sqrt(3) - 43)) + 1/ 8*sqrt(169/2*sqrt(2) + 239/2)*arctan((17*sqrt(2)*x - 24*x)*sqrt(169/2*sqrt (2) + 239/2)) - 1/32*sqrt(27*sqrt(3) + 43)*log(13*x^2 + (5*sqrt(3)*x - 7*x )*sqrt(27*sqrt(3) + 43) + 13*sqrt(3)) + 1/32*sqrt(27*sqrt(3) + 43)*log(13* x^2 - (5*sqrt(3)*x - 7*x)*sqrt(27*sqrt(3) + 43) + 13*sqrt(3)) + 1/16*sqrt( 169/2*sqrt(2) - 239/2)*log(sqrt(169/2*sqrt(2) - 239/2)*(7*sqrt(2) + 10) + x) - 1/16*sqrt(169/2*sqrt(2) - 239/2)*log(-sqrt(169/2*sqrt(2) - 239/2)*(7* sqrt(2) + 10) + x) - x
Time = 1.04 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.41 \[ \int \frac {x^8}{4-\left (1+x^2\right )^4} \, dx=- x - \operatorname {RootSum} {\left (262144 t^{4} - 44032 t^{2} + 2187, \left ( t \mapsto t \log {\left (- \frac {3573552913580032 t^{7}}{84748676571} - \frac {2745951176359936 t^{5}}{84748676571} + \frac {174117810636800 t^{3}}{28249558857} - \frac {26985428064460 t}{84748676571} + x \right )} \right )\right )} - \operatorname {RootSum} {\left (262144 t^{4} + 244736 t^{2} - 1, \left ( t \mapsto t \log {\left (- \frac {3573552913580032 t^{7}}{84748676571} - \frac {2745951176359936 t^{5}}{84748676571} + \frac {174117810636800 t^{3}}{28249558857} - \frac {26985428064460 t}{84748676571} + x \right )} \right )\right )} \] Input:
integrate(x**8/(4-(x**2+1)**4),x)
Output:
-x - RootSum(262144*_t**4 - 44032*_t**2 + 2187, Lambda(_t, _t*log(-3573552 913580032*_t**7/84748676571 - 2745951176359936*_t**5/84748676571 + 1741178 10636800*_t**3/28249558857 - 26985428064460*_t/84748676571 + x))) - RootSu m(262144*_t**4 + 244736*_t**2 - 1, Lambda(_t, _t*log(-3573552913580032*_t* *7/84748676571 - 2745951176359936*_t**5/84748676571 + 174117810636800*_t** 3/28249558857 - 26985428064460*_t/84748676571 + x)))
\[ \int \frac {x^8}{4-\left (1+x^2\right )^4} \, dx=\int { -\frac {x^{8}}{{\left (x^{2} + 1\right )}^{4} - 4} \,d x } \] Input:
integrate(x^8/(4-(x^2+1)^4),x, algorithm="maxima")
Output:
-x + 1/4*integrate((12*x^2 - 5)/(x^4 + 2*x^2 - 1), x) + 1/4*integrate((4*x ^2 - 3)/(x^4 + 2*x^2 + 3), x)
Leaf count of result is larger than twice the leaf count of optimal. 612 vs. \(2 (151) = 302\).
Time = 0.49 (sec) , antiderivative size = 612, normalized size of antiderivative = 2.89 \[ \int \frac {x^8}{4-\left (1+x^2\right )^4} \, dx =\text {Too large to display} \] Input:
integrate(x^8/(4-(x^2+1)^4),x, algorithm="giac")
Output:
-1/2592*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*s qrt(3) + 18) - 3^(3/4)*(-6*sqrt(3) + 18)^(3/2) + 27*3^(1/4)*sqrt(2)*sqrt(6 *sqrt(3) + 18) + 27*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)) - 1/2592*sqrt( 2)*(3^(3/4)*sqrt(2)*(6*sqrt(3) + 18)^(3/2) + 18*3^(3/4)*sqrt(2)*sqrt(6*sqr t(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) + 27*3^(1/4)*sqrt(2)*sqrt(6*sqrt(3) + 18 ) + 27*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)) - 1/5184*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) + 27*3^(1/4)*sqrt(2)*sqrt(-6*sqrt(3) + 18) - 27*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)) + 1/5184*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) + 1 8)^(3/2) - 18*3^(3/4)*sqrt(6*sqrt(3) + 18)*(sqrt(3) - 3) + 27*3^(1/4)*sqrt (2)*sqrt(-6*sqrt(3) + 18) - 27*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)) + 1/16*sqrt(338*sqrt(2) + 478 )*arctan(x/sqrt(sqrt(2) + 1)) + 1/32*sqrt(338*sqrt(2) - 478)*log(abs(x ...
Time = 0.23 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.21 \[ \int \frac {x^8}{4-\left (1+x^2\right )^4} \, dx=-x+\frac {\mathrm {atan}\left (\frac {x\,\sqrt {-338\,\sqrt {2}-478}\,10562838{}\mathrm {i}}{342765462\,\sqrt {2}+484704168}+\frac {\sqrt {2}\,x\,\sqrt {-338\,\sqrt {2}-478}\,6799884{}\mathrm {i}}{342765462\,\sqrt {2}+484704168}\right )\,\sqrt {-338\,\sqrt {2}-478}\,1{}\mathrm {i}}{16}-\frac {\mathrm {atan}\left (\frac {x\,\sqrt {338\,\sqrt {2}-478}\,10562838{}\mathrm {i}}{342765462\,\sqrt {2}-484704168}-\frac {\sqrt {2}\,x\,\sqrt {338\,\sqrt {2}-478}\,6799884{}\mathrm {i}}{342765462\,\sqrt {2}-484704168}\right )\,\sqrt {338\,\sqrt {2}-478}\,1{}\mathrm {i}}{16}-\frac {\mathrm {atan}\left (\frac {x\,\sqrt {86-\sqrt {2}\,26{}\mathrm {i}}\,95758{}\mathrm {i}}{-2437188+\sqrt {2}\,284622{}\mathrm {i}}+\frac {119366\,\sqrt {2}\,x\,\sqrt {86-\sqrt {2}\,26{}\mathrm {i}}}{-2437188+\sqrt {2}\,284622{}\mathrm {i}}\right )\,\sqrt {86-\sqrt {2}\,26{}\mathrm {i}}\,1{}\mathrm {i}}{16}+\frac {\mathrm {atan}\left (\frac {x\,\sqrt {86+\sqrt {2}\,26{}\mathrm {i}}\,95758{}\mathrm {i}}{2437188+\sqrt {2}\,284622{}\mathrm {i}}-\frac {119366\,\sqrt {2}\,x\,\sqrt {86+\sqrt {2}\,26{}\mathrm {i}}}{2437188+\sqrt {2}\,284622{}\mathrm {i}}\right )\,\sqrt {86+\sqrt {2}\,26{}\mathrm {i}}\,1{}\mathrm {i}}{16} \] Input:
int(-x^8/((x^2 + 1)^4 - 4),x)
Output:
(atan((x*(- 338*2^(1/2) - 478)^(1/2)*10562838i)/(342765462*2^(1/2) + 48470 4168) + (2^(1/2)*x*(- 338*2^(1/2) - 478)^(1/2)*6799884i)/(342765462*2^(1/2 ) + 484704168))*(- 338*2^(1/2) - 478)^(1/2)*1i)/16 - x - (atan((x*(338*2^( 1/2) - 478)^(1/2)*10562838i)/(342765462*2^(1/2) - 484704168) - (2^(1/2)*x* (338*2^(1/2) - 478)^(1/2)*6799884i)/(342765462*2^(1/2) - 484704168))*(338* 2^(1/2) - 478)^(1/2)*1i)/16 - (atan((x*(86 - 2^(1/2)*26i)^(1/2)*95758i)/(2 ^(1/2)*284622i - 2437188) + (119366*2^(1/2)*x*(86 - 2^(1/2)*26i)^(1/2))/(2 ^(1/2)*284622i - 2437188))*(86 - 2^(1/2)*26i)^(1/2)*1i)/16 + (atan((x*(2^( 1/2)*26i + 86)^(1/2)*95758i)/(2^(1/2)*284622i + 2437188) - (119366*2^(1/2) *x*(2^(1/2)*26i + 86)^(1/2))/(2^(1/2)*284622i + 2437188))*(2^(1/2)*26i + 8 6)^(1/2)*1i)/16
Time = 0.28 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.63 \[ \int \frac {x^8}{4-\left (1+x^2\right )^4} \, dx=-\frac {5 \sqrt {\sqrt {3}+1}\, \sqrt {6}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {3}-1}\, \sqrt {2}-2 x}{\sqrt {\sqrt {3}+1}\, \sqrt {2}}\right )}{32}+\frac {7 \sqrt {\sqrt {3}+1}\, \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {3}-1}\, \sqrt {2}-2 x}{\sqrt {\sqrt {3}+1}\, \sqrt {2}}\right )}{32}+\frac {5 \sqrt {\sqrt {3}+1}\, \sqrt {6}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {3}-1}\, \sqrt {2}+2 x}{\sqrt {\sqrt {3}+1}\, \sqrt {2}}\right )}{32}-\frac {7 \sqrt {\sqrt {3}+1}\, \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {3}-1}\, \sqrt {2}+2 x}{\sqrt {\sqrt {3}+1}\, \sqrt {2}}\right )}{32}+\frac {7 \sqrt {\sqrt {2}+1}\, \sqrt {2}\, \mathit {atan} \left (\frac {x}{\sqrt {\sqrt {2}+1}}\right )}{16}+\frac {5 \sqrt {\sqrt {2}+1}\, \mathit {atan} \left (\frac {x}{\sqrt {\sqrt {2}+1}}\right )}{8}+\frac {5 \sqrt {\sqrt {3}-1}\, \sqrt {6}\, \mathrm {log}\left (-\sqrt {\sqrt {3}-1}\, \sqrt {2}\, x +\sqrt {3}+x^{2}\right )}{64}-\frac {5 \sqrt {\sqrt {3}-1}\, \sqrt {6}\, \mathrm {log}\left (\sqrt {\sqrt {3}-1}\, \sqrt {2}\, x +\sqrt {3}+x^{2}\right )}{64}+\frac {7 \sqrt {\sqrt {3}-1}\, \sqrt {2}\, \mathrm {log}\left (-\sqrt {\sqrt {3}-1}\, \sqrt {2}\, x +\sqrt {3}+x^{2}\right )}{64}-\frac {7 \sqrt {\sqrt {3}-1}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {\sqrt {3}-1}\, \sqrt {2}\, x +\sqrt {3}+x^{2}\right )}{64}+\frac {7 \sqrt {\sqrt {2}-1}\, \sqrt {2}\, \mathrm {log}\left (-\sqrt {\sqrt {2}-1}+x \right )}{32}-\frac {7 \sqrt {\sqrt {2}-1}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {\sqrt {2}-1}+x \right )}{32}-\frac {5 \sqrt {\sqrt {2}-1}\, \mathrm {log}\left (-\sqrt {\sqrt {2}-1}+x \right )}{16}+\frac {5 \sqrt {\sqrt {2}-1}\, \mathrm {log}\left (\sqrt {\sqrt {2}-1}+x \right )}{16}-x \] Input:
int(x^8/(4-(x^2+1)^4),x)
Output:
( - 10*sqrt(sqrt(3) + 1)*sqrt(6)*atan((sqrt(sqrt(3) - 1)*sqrt(2) - 2*x)/(s qrt(sqrt(3) + 1)*sqrt(2))) + 14*sqrt(sqrt(3) + 1)*sqrt(2)*atan((sqrt(sqrt( 3) - 1)*sqrt(2) - 2*x)/(sqrt(sqrt(3) + 1)*sqrt(2))) + 10*sqrt(sqrt(3) + 1) *sqrt(6)*atan((sqrt(sqrt(3) - 1)*sqrt(2) + 2*x)/(sqrt(sqrt(3) + 1)*sqrt(2) )) - 14*sqrt(sqrt(3) + 1)*sqrt(2)*atan((sqrt(sqrt(3) - 1)*sqrt(2) + 2*x)/( sqrt(sqrt(3) + 1)*sqrt(2))) + 28*sqrt(sqrt(2) + 1)*sqrt(2)*atan(x/sqrt(sqr t(2) + 1)) + 40*sqrt(sqrt(2) + 1)*atan(x/sqrt(sqrt(2) + 1)) + 5*sqrt(sqrt( 3) - 1)*sqrt(6)*log( - sqrt(sqrt(3) - 1)*sqrt(2)*x + sqrt(3) + x**2) - 5*s qrt(sqrt(3) - 1)*sqrt(6)*log(sqrt(sqrt(3) - 1)*sqrt(2)*x + sqrt(3) + x**2) + 7*sqrt(sqrt(3) - 1)*sqrt(2)*log( - sqrt(sqrt(3) - 1)*sqrt(2)*x + sqrt(3 ) + x**2) - 7*sqrt(sqrt(3) - 1)*sqrt(2)*log(sqrt(sqrt(3) - 1)*sqrt(2)*x + sqrt(3) + x**2) + 14*sqrt(sqrt(2) - 1)*sqrt(2)*log( - sqrt(sqrt(2) - 1) + x) - 14*sqrt(sqrt(2) - 1)*sqrt(2)*log(sqrt(sqrt(2) - 1) + x) - 20*sqrt(sqr t(2) - 1)*log( - sqrt(sqrt(2) - 1) + x) + 20*sqrt(sqrt(2) - 1)*log(sqrt(sq rt(2) - 1) + x) - 64*x)/64