Integrand size = 13, antiderivative size = 209 \[ \int \frac {1}{4-\left (1+x^2\right )^4} \, dx=-\frac {1}{16} \sqrt {\frac {1}{3} \left (-1+\sqrt {3}\right )} \arctan \left (\frac {\sqrt {2 \left (-1+\sqrt {3}\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )+\frac {\arctan \left (\frac {x}{\sqrt {1+\sqrt {2}}}\right )}{8 \sqrt {2 \left (1+\sqrt {2}\right )}}+\frac {1}{16} \sqrt {\frac {1}{3} \left (-1+\sqrt {3}\right )} \arctan \left (\frac {\sqrt {2 \left (-1+\sqrt {3}\right )}+2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )+\frac {\text {arctanh}\left (\frac {x}{\sqrt {-1+\sqrt {2}}}\right )}{8 \sqrt {2 \left (-1+\sqrt {2}\right )}}+\frac {\text {arctanh}\left (\frac {\sqrt {2 \left (-1+\sqrt {3}\right )} x}{\sqrt {3}+x^2}\right )}{8 \sqrt {6 \left (-1+\sqrt {3}\right )}} \] Output:
-1/48*(-3+3*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(x/(1+2^(1/2))^(1/2))/(2+2*2^(1/2))^(1/2)+1/48*(-3+3*3^( 1/2))^(1/2)*arctan(((-2+2*3^(1/2))^(1/2)+2*x)/(2+2*3^(1/2))^(1/2))+1/8*arc tanh(x/(2^(1/2)-1)^(1/2))/(-2+2*2^(1/2))^(1/2)+1/8*arctanh((-2+2*3^(1/2))^ (1/2)*x/(3^(1/2)+x^2))/(-6+6*3^(1/2))^(1/2)
Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.67 \[ \int \frac {1}{4-\left (1+x^2\right )^4} \, dx=\frac {-i \sqrt {1+i \sqrt {2}} \arctan \left (\frac {x}{\sqrt {1-i \sqrt {2}}}\right )+i \sqrt {1-i \sqrt {2}} \arctan \left (\frac {x}{\sqrt {1+i \sqrt {2}}}\right )+\sqrt {3 \left (-1+\sqrt {2}\right )} \arctan \left (\frac {x}{\sqrt {1+\sqrt {2}}}\right )+\sqrt {3 \left (1+\sqrt {2}\right )} \text {arctanh}\left (\frac {x}{\sqrt {-1+\sqrt {2}}}\right )}{8 \sqrt {6}} \] Input:
Integrate[(4 - (1 + x^2)^4)^(-1),x]
Output:
((-I)*Sqrt[1 + I*Sqrt[2]]*ArcTan[x/Sqrt[1 - I*Sqrt[2]]] + I*Sqrt[1 - I*Sqr t[2]]*ArcTan[x/Sqrt[1 + I*Sqrt[2]]] + Sqrt[3*(-1 + Sqrt[2])]*ArcTan[x/Sqrt [1 + Sqrt[2]]] + Sqrt[3*(1 + Sqrt[2])]*ArcTanh[x/Sqrt[-1 + Sqrt[2]]])/(8*S qrt[6])
Time = 0.67 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.20, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, 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.
| \(\displaystyle \int \frac {1}{4-\left (x^2+1\right )^4} \, dx\) |
| \(\Big \downarrow \) 2460 |
| \(\displaystyle \int \left (\frac {1}{4 \left (x^4+2 x^2+3\right )}-\frac {1}{4 \left (x^4+2 x^2-1\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {1}{16} \sqrt {\frac {1}{3} \left (\sqrt {3}-1\right )} \arctan \left (\frac {\sqrt {2 \left (\sqrt {3}-1\right )}-2 x}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )+\frac {\arctan \left (\frac {x}{\sqrt {1+\sqrt {2}}}\right )}{8 \sqrt {2 \left (1+\sqrt {2}\right )}}+\frac {1}{16} \sqrt {\frac {1}{3} \left (\sqrt {3}-1\right )} \arctan \left (\frac {2 x+\sqrt {2 \left (\sqrt {3}-1\right )}}{\sqrt {2 \left (1+\sqrt {3}\right )}}\right )+\frac {\text {arctanh}\left (\frac {x}{\sqrt {\sqrt {2}-1}}\right )}{8 \sqrt {2 \left (\sqrt {2}-1\right )}}-\frac {\log \left (x^2-\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )}{16 \sqrt {6 \left (\sqrt {3}-1\right )}}+\frac {\log \left (x^2+\sqrt {2 \left (\sqrt {3}-1\right )} x+\sqrt {3}\right )}{16 \sqrt {6 \left (\sqrt {3}-1\right )}}\) |
Input:
Int[(4 - (1 + x^2)^4)^(-1),x]
Output:
-1/16*(Sqrt[(-1 + Sqrt[3])/3]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] - 2*x)/Sqrt[2 *(1 + Sqrt[3])]]) + ArcTan[x/Sqrt[1 + Sqrt[2]]]/(8*Sqrt[2*(1 + Sqrt[2])]) + (Sqrt[(-1 + Sqrt[3])/3]*ArcTan[(Sqrt[2*(-1 + Sqrt[3])] + 2*x)/Sqrt[2*(1 + Sqrt[3])]])/16 + ArcTanh[x/Sqrt[-1 + Sqrt[2]]]/(8*Sqrt[2*(-1 + Sqrt[2])] ) - Log[Sqrt[3] - Sqrt[2*(-1 + Sqrt[3])]*x + x^2]/(16*Sqrt[6*(-1 + Sqrt[3] )]) + Log[Sqrt[3] + Sqrt[2*(-1 + Sqrt[3])]*x + x^2]/(16*Sqrt[6*(-1 + Sqrt[ 3])])
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 = 0.09 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.30
| method | result | size |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (4 \textit {\_Z}^{4}-4 \textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (-2 \textit {\_R}^{3}+3 \textit {\_R} +x \right )\right )}{16}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (12 \textit {\_Z}^{4}-4 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (6 \textit {\_R}^{3}+\textit {\_R} +x \right )\right )}{16}\) | \(62\) |
| default | \(-\frac {\left (\sqrt {-2+2 \sqrt {3}}\, \sqrt {3}+3 \sqrt {-2+2 \sqrt {3}}\right ) \ln \left (x^{2}-x \sqrt {-2+2 \sqrt {3}}+\sqrt {3}\right )}{192}-\frac {\left (-4 \sqrt {3}+\frac {\left (\sqrt {-2+2 \sqrt {3}}\, \sqrt {3}+3 \sqrt {-2+2 \sqrt {3}}\right ) \sqrt {-2+2 \sqrt {3}}}{2}\right ) \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {3}}}{\sqrt {2+2 \sqrt {3}}}\right )}{48 \sqrt {2+2 \sqrt {3}}}+\frac {\left (\sqrt {-2+2 \sqrt {3}}\, \sqrt {3}+3 \sqrt {-2+2 \sqrt {3}}\right ) \ln \left (x^{2}+x \sqrt {-2+2 \sqrt {3}}+\sqrt {3}\right )}{192}+\frac {\left (4 \sqrt {3}-\frac {\left (\sqrt {-2+2 \sqrt {3}}\, \sqrt {3}+3 \sqrt {-2+2 \sqrt {3}}\right ) \sqrt {-2+2 \sqrt {3}}}{2}\right ) \arctan \left (\frac {\sqrt {-2+2 \sqrt {3}}+2 x}{\sqrt {2+2 \sqrt {3}}}\right )}{48 \sqrt {2+2 \sqrt {3}}}+\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {x}{\sqrt {\sqrt {2}-1}}\right )}{16 \sqrt {\sqrt {2}-1}}+\frac {\sqrt {2}\, \arctan \left (\frac {x}{\sqrt {1+\sqrt {2}}}\right )}{16 \sqrt {1+\sqrt {2}}}\) | \(295\) |
Input:
int(1/(4-(x^2+1)^4),x,method=_RETURNVERBOSE)
Output:
1/16*sum(_R*ln(-2*_R^3+3*_R+x),_R=RootOf(4*_Z^4-4*_Z^2-1))+1/16*sum(_R*ln( 6*_R^3+_R+x),_R=RootOf(12*_Z^4-4*_Z^2+1))
Time = 0.08 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.28 \[ \int \frac {1}{4-\left (1+x^2\right )^4} \, dx=\frac {1}{16} \, \sqrt {\frac {1}{3} \, \sqrt {3} - \frac {1}{3}} \arctan \left (\sqrt {3} x \sqrt {\frac {1}{3} \, \sqrt {3} - \frac {1}{3}} + \frac {3}{2} \, {\left (\sqrt {3} - 1\right )} \sqrt {\frac {1}{3} \, \sqrt {3} + \frac {1}{3}} \sqrt {\frac {1}{3} \, \sqrt {3} - \frac {1}{3}}\right ) - \frac {1}{16} \, \sqrt {\frac {1}{3} \, \sqrt {3} - \frac {1}{3}} \arctan \left (-\sqrt {3} x \sqrt {\frac {1}{3} \, \sqrt {3} - \frac {1}{3}} + \frac {3}{2} \, {\left (\sqrt {3} - 1\right )} \sqrt {\frac {1}{3} \, \sqrt {3} + \frac {1}{3}} \sqrt {\frac {1}{3} \, \sqrt {3} - \frac {1}{3}}\right ) + \frac {1}{8} \, \sqrt {\frac {1}{2} \, \sqrt {2} - \frac {1}{2}} \arctan \left (\sqrt {2} x \sqrt {\frac {1}{2} \, \sqrt {2} - \frac {1}{2}}\right ) - \frac {1}{32} \, \sqrt {\frac {1}{3} \, \sqrt {3} + \frac {1}{3}} \log \left (x^{2} + {\left (\sqrt {3} x - 3 \, x\right )} \sqrt {\frac {1}{3} \, \sqrt {3} + \frac {1}{3}} + \sqrt {3}\right ) + \frac {1}{32} \, \sqrt {\frac {1}{3} \, \sqrt {3} + \frac {1}{3}} \log \left (x^{2} - {\left (\sqrt {3} x - 3 \, x\right )} \sqrt {\frac {1}{3} \, \sqrt {3} + \frac {1}{3}} + \sqrt {3}\right ) - \frac {1}{16} \, \sqrt {\frac {1}{2} \, \sqrt {2} + \frac {1}{2}} \log \left ({\left (\sqrt {2} - 2\right )} \sqrt {\frac {1}{2} \, \sqrt {2} + \frac {1}{2}} + x\right ) + \frac {1}{16} \, \sqrt {\frac {1}{2} \, \sqrt {2} + \frac {1}{2}} \log \left (-{\left (\sqrt {2} - 2\right )} \sqrt {\frac {1}{2} \, \sqrt {2} + \frac {1}{2}} + x\right ) \] Input:
integrate(1/(4-(x^2+1)^4),x, algorithm="fricas")
Output:
1/16*sqrt(1/3*sqrt(3) - 1/3)*arctan(sqrt(3)*x*sqrt(1/3*sqrt(3) - 1/3) + 3/ 2*(sqrt(3) - 1)*sqrt(1/3*sqrt(3) + 1/3)*sqrt(1/3*sqrt(3) - 1/3)) - 1/16*sq rt(1/3*sqrt(3) - 1/3)*arctan(-sqrt(3)*x*sqrt(1/3*sqrt(3) - 1/3) + 3/2*(sqr t(3) - 1)*sqrt(1/3*sqrt(3) + 1/3)*sqrt(1/3*sqrt(3) - 1/3)) + 1/8*sqrt(1/2* sqrt(2) - 1/2)*arctan(sqrt(2)*x*sqrt(1/2*sqrt(2) - 1/2)) - 1/32*sqrt(1/3*s qrt(3) + 1/3)*log(x^2 + (sqrt(3)*x - 3*x)*sqrt(1/3*sqrt(3) + 1/3) + sqrt(3 )) + 1/32*sqrt(1/3*sqrt(3) + 1/3)*log(x^2 - (sqrt(3)*x - 3*x)*sqrt(1/3*sqr t(3) + 1/3) + sqrt(3)) - 1/16*sqrt(1/2*sqrt(2) + 1/2)*log((sqrt(2) - 2)*sq rt(1/2*sqrt(2) + 1/2) + x) + 1/16*sqrt(1/2*sqrt(2) + 1/2)*log(-(sqrt(2) - 2)*sqrt(1/2*sqrt(2) + 1/2) + x)
Time = 1.02 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.34 \[ \int \frac {1}{4-\left (1+x^2\right )^4} \, dx=- \operatorname {RootSum} {\left (262144 t^{4} - 1024 t^{2} - 1, \left ( t \mapsto t \log {\left (4831838208 t^{7} - 22020096 t^{5} + 2048 t^{3} - 36 t + x \right )} \right )\right )} - \operatorname {RootSum} {\left (786432 t^{4} - 1024 t^{2} + 1, \left ( t \mapsto t \log {\left (4831838208 t^{7} - 22020096 t^{5} + 2048 t^{3} - 36 t + x \right )} \right )\right )} \] Input:
integrate(1/(4-(x**2+1)**4),x)
Output:
-RootSum(262144*_t**4 - 1024*_t**2 - 1, Lambda(_t, _t*log(4831838208*_t**7 - 22020096*_t**5 + 2048*_t**3 - 36*_t + x))) - RootSum(786432*_t**4 - 102 4*_t**2 + 1, Lambda(_t, _t*log(4831838208*_t**7 - 22020096*_t**5 + 2048*_t **3 - 36*_t + x)))
\[ \int \frac {1}{4-\left (1+x^2\right )^4} \, dx=\int { -\frac {1}{{\left (x^{2} + 1\right )}^{4} - 4} \,d x } \] Input:
integrate(1/(4-(x^2+1)^4),x, algorithm="maxima")
Output:
-integrate(1/((x^2 + 1)^4 - 4), x)
Leaf count of result is larger than twice the leaf count of optimal. 317 vs. \(2 (148) = 296\).
Time = 0.44 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.52 \[ \int \frac {1}{4-\left (1+x^2\right )^4} \, dx=\frac {1}{288} \, \sqrt {2} {\left (3^{\frac {1}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} + 3^{\frac {1}{4}} \sqrt {-6 \, \sqrt {3} + 18}\right )} \arctan \left (\frac {3^{\frac {3}{4}} {\left (x + 3^{\frac {1}{4}} \sqrt {-\frac {1}{6} \, \sqrt {3} + \frac {1}{2}}\right )}}{3 \, \sqrt {\frac {1}{6} \, \sqrt {3} + \frac {1}{2}}}\right ) + \frac {1}{288} \, \sqrt {2} {\left (3^{\frac {1}{4}} \sqrt {2} \sqrt {6 \, \sqrt {3} + 18} + 3^{\frac {1}{4}} \sqrt {-6 \, \sqrt {3} + 18}\right )} \arctan \left (\frac {3^{\frac {3}{4}} {\left (x - 3^{\frac {1}{4}} \sqrt {-\frac {1}{6} \, \sqrt {3} + \frac {1}{2}}\right )}}{3 \, \sqrt {\frac {1}{6} \, \sqrt {3} + \frac {1}{2}}}\right ) + \frac {1}{576} \, \sqrt {2} {\left (3^{\frac {1}{4}} \sqrt {2} \sqrt {-6 \, \sqrt {3} + 18} - 3^{\frac {1}{4}} \sqrt {6 \, \sqrt {3} + 18}\right )} \log \left (x^{2} + 2 \cdot 3^{\frac {1}{4}} x \sqrt {-\frac {1}{6} \, \sqrt {3} + \frac {1}{2}} + \sqrt {3}\right ) - \frac {1}{576} \, \sqrt {2} {\left (3^{\frac {1}{4}} \sqrt {2} \sqrt {-6 \, \sqrt {3} + 18} - 3^{\frac {1}{4}} \sqrt {6 \, \sqrt {3} + 18}\right )} \log \left (x^{2} - 2 \cdot 3^{\frac {1}{4}} x \sqrt {-\frac {1}{6} \, \sqrt {3} + \frac {1}{2}} + \sqrt {3}\right ) + \frac {1}{16} \, \sqrt {2 \, \sqrt {2} - 2} \arctan \left (\frac {x}{\sqrt {\sqrt {2} + 1}}\right ) + \frac {1}{32} \, \sqrt {2 \, \sqrt {2} + 2} \log \left ({\left | x + \sqrt {\sqrt {2} - 1} \right |}\right ) - \frac {1}{32} \, \sqrt {2 \, \sqrt {2} + 2} \log \left ({\left | x - \sqrt {\sqrt {2} - 1} \right |}\right ) \] Input:
integrate(1/(4-(x^2+1)^4),x, algorithm="giac")
Output:
1/288*sqrt(2)*(3^(1/4)*sqrt(2)*sqrt(6*sqrt(3) + 18) + 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)) + 1/288*sqrt(2)*(3^(1/4)*sqrt(2)*sqrt(6*sqrt(3) + 18) + 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)) + 1/576*sqrt(2)*(3^(1/4)*sqrt(2) *sqrt(-6*sqrt(3) + 18) - 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)) - 1/576*sqrt(2)*(3^(1/4)*sqrt(2)*sq rt(-6*sqrt(3) + 18) - 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(2*sqrt(2) - 2)*arctan(x/sq rt(sqrt(2) + 1)) + 1/32*sqrt(2*sqrt(2) + 2)*log(abs(x + sqrt(sqrt(2) - 1)) ) - 1/32*sqrt(2*sqrt(2) + 2)*log(abs(x - sqrt(sqrt(2) - 1)))
Time = 9.90 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.25 \[ \int \frac {1}{4-\left (1+x^2\right )^4} \, dx=\frac {\mathrm {atan}\left (\frac {x\,\sqrt {6-\sqrt {2}\,6{}\mathrm {i}}\,10{}\mathrm {i}}{81\,\left (-\frac {4}{27}+\frac {\sqrt {2}\,10{}\mathrm {i}}{27}\right )}-\frac {2\,\sqrt {2}\,x\,\sqrt {6-\sqrt {2}\,6{}\mathrm {i}}}{81\,\left (-\frac {4}{27}+\frac {\sqrt {2}\,10{}\mathrm {i}}{27}\right )}\right )\,\sqrt {6-\sqrt {2}\,6{}\mathrm {i}}\,1{}\mathrm {i}}{48}-\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {x\,\sqrt {1-\sqrt {2}}\,8{}\mathrm {i}}{6\,\sqrt {2}-8}-\frac {\sqrt {2}\,x\,\sqrt {1-\sqrt {2}}\,6{}\mathrm {i}}{6\,\sqrt {2}-8}\right )\,\sqrt {1-\sqrt {2}}\,1{}\mathrm {i}}{16}-\frac {\sqrt {6}\,\mathrm {atan}\left (\frac {\sqrt {6}\,x\,\sqrt {1+\sqrt {2}\,1{}\mathrm {i}}\,10{}\mathrm {i}}{81\,\left (\frac {4}{27}+\frac {\sqrt {2}\,10{}\mathrm {i}}{27}\right )}+\frac {2\,\sqrt {2}\,\sqrt {6}\,x\,\sqrt {1+\sqrt {2}\,1{}\mathrm {i}}}{81\,\left (\frac {4}{27}+\frac {\sqrt {2}\,10{}\mathrm {i}}{27}\right )}\right )\,\sqrt {1+\sqrt {2}\,1{}\mathrm {i}}\,1{}\mathrm {i}}{48}-\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {x\,\sqrt {\sqrt {2}+1}\,8{}\mathrm {i}}{6\,\sqrt {2}+8}+\frac {\sqrt {2}\,x\,\sqrt {\sqrt {2}+1}\,6{}\mathrm {i}}{6\,\sqrt {2}+8}\right )\,\sqrt {\sqrt {2}+1}\,1{}\mathrm {i}}{16} \] Input:
int(-1/((x^2 + 1)^4 - 4),x)
Output:
(atan((x*(6 - 2^(1/2)*6i)^(1/2)*10i)/(81*((2^(1/2)*10i)/27 - 4/27)) - (2*2 ^(1/2)*x*(6 - 2^(1/2)*6i)^(1/2))/(81*((2^(1/2)*10i)/27 - 4/27)))*(6 - 2^(1 /2)*6i)^(1/2)*1i)/48 - (2^(1/2)*atan((x*(1 - 2^(1/2))^(1/2)*8i)/(6*2^(1/2) - 8) - (2^(1/2)*x*(1 - 2^(1/2))^(1/2)*6i)/(6*2^(1/2) - 8))*(1 - 2^(1/2))^ (1/2)*1i)/16 - (6^(1/2)*atan((6^(1/2)*x*(2^(1/2)*1i + 1)^(1/2)*10i)/(81*(( 2^(1/2)*10i)/27 + 4/27)) + (2*2^(1/2)*6^(1/2)*x*(2^(1/2)*1i + 1)^(1/2))/(8 1*((2^(1/2)*10i)/27 + 4/27)))*(2^(1/2)*1i + 1)^(1/2)*1i)/48 - (2^(1/2)*ata n((x*(2^(1/2) + 1)^(1/2)*8i)/(6*2^(1/2) + 8) + (2^(1/2)*x*(2^(1/2) + 1)^(1 /2)*6i)/(6*2^(1/2) + 8))*(2^(1/2) + 1)^(1/2)*1i)/16
Time = 0.24 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.64 \[ \int \frac {1}{4-\left (1+x^2\right )^4} \, dx=\frac {\sqrt {\sqrt {3}+1}\, \sqrt {6}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {3}-1}\, \sqrt {2}-2 x}{\sqrt {\sqrt {3}+1}\, \sqrt {2}}\right )}{96}-\frac {\sqrt {\sqrt {3}+1}\, \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {3}-1}\, \sqrt {2}-2 x}{\sqrt {\sqrt {3}+1}\, \sqrt {2}}\right )}{32}-\frac {\sqrt {\sqrt {3}+1}\, \sqrt {6}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {3}-1}\, \sqrt {2}+2 x}{\sqrt {\sqrt {3}+1}\, \sqrt {2}}\right )}{96}+\frac {\sqrt {\sqrt {3}+1}\, \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {3}-1}\, \sqrt {2}+2 x}{\sqrt {\sqrt {3}+1}\, \sqrt {2}}\right )}{32}-\frac {\sqrt {\sqrt {2}+1}\, \sqrt {2}\, \mathit {atan} \left (\frac {x}{\sqrt {\sqrt {2}+1}}\right )}{16}+\frac {\sqrt {\sqrt {2}+1}\, \mathit {atan} \left (\frac {x}{\sqrt {\sqrt {2}+1}}\right )}{8}-\frac {\sqrt {\sqrt {3}-1}\, \sqrt {6}\, \mathrm {log}\left (-\sqrt {\sqrt {3}-1}\, \sqrt {2}\, x +\sqrt {3}+x^{2}\right )}{192}+\frac {\sqrt {\sqrt {3}-1}\, \sqrt {6}\, \mathrm {log}\left (\sqrt {\sqrt {3}-1}\, \sqrt {2}\, x +\sqrt {3}+x^{2}\right )}{192}-\frac {\sqrt {\sqrt {3}-1}\, \sqrt {2}\, \mathrm {log}\left (-\sqrt {\sqrt {3}-1}\, \sqrt {2}\, x +\sqrt {3}+x^{2}\right )}{64}+\frac {\sqrt {\sqrt {3}-1}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {\sqrt {3}-1}\, \sqrt {2}\, x +\sqrt {3}+x^{2}\right )}{64}-\frac {\sqrt {\sqrt {2}-1}\, \sqrt {2}\, \mathrm {log}\left (-\sqrt {\sqrt {2}-1}+x \right )}{32}+\frac {\sqrt {\sqrt {2}-1}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {\sqrt {2}-1}+x \right )}{32}-\frac {\sqrt {\sqrt {2}-1}\, \mathrm {log}\left (-\sqrt {\sqrt {2}-1}+x \right )}{16}+\frac {\sqrt {\sqrt {2}-1}\, \mathrm {log}\left (\sqrt {\sqrt {2}-1}+x \right )}{16} \] Input:
int(1/(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))) - 6*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))) + 6 *sqrt(sqrt(3) + 1)*sqrt(2)*atan((sqrt(sqrt(3) - 1)*sqrt(2) + 2*x)/(sqrt(sq rt(3) + 1)*sqrt(2))) - 12*sqrt(sqrt(2) + 1)*sqrt(2)*atan(x/sqrt(sqrt(2) + 1)) + 24*sqrt(sqrt(2) + 1)*atan(x/sqrt(sqrt(2) + 1)) - sqrt(sqrt(3) - 1)*s qrt(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) - 3*sqrt(s qrt(3) - 1)*sqrt(2)*log( - sqrt(sqrt(3) - 1)*sqrt(2)*x + sqrt(3) + x**2) + 3*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*sqrt( sqrt(2) - 1)*sqrt(2)*log(sqrt(sqrt(2) - 1) + x) - 12*sqrt(sqrt(2) - 1)*log ( - sqrt(sqrt(2) - 1) + x) + 12*sqrt(sqrt(2) - 1)*log(sqrt(sqrt(2) - 1) + x))/192