Integrand size = 27, antiderivative size = 153 \[ \int \frac {\left (1+x^2\right )^3}{1+4 x^2+6 x^4+4 x^6} \, dx=\frac {x}{4}-\frac {1}{8} \sqrt {1+\sqrt {2}} \arctan \left (\frac {\sqrt {-1+\sqrt {2}}-2 x}{\sqrt {1+\sqrt {2}}}\right )+\frac {\arctan \left (\sqrt {2} x\right )}{4 \sqrt {2}}+\frac {1}{8} \sqrt {1+\sqrt {2}} \arctan \left (\frac {\sqrt {-1+\sqrt {2}}+2 x}{\sqrt {1+\sqrt {2}}}\right )+\frac {1}{8} \sqrt {-1+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2 \left (-1+\sqrt {2}\right )} x}{1+\sqrt {2} x^2}\right ) \] Output:
1/4*x-1/8*(1+2^(1/2))^(1/2)*arctan(((2^(1/2)-1)^(1/2)-2*x)/(1+2^(1/2))^(1/ 2))+1/8*arctan(x*2^(1/2))*2^(1/2)+1/8*(1+2^(1/2))^(1/2)*arctan(((2^(1/2)-1 )^(1/2)+2*x)/(1+2^(1/2))^(1/2))+1/8*(2^(1/2)-1)^(1/2)*arctanh((-2+2*2^(1/2 ))^(1/2)*x/(1+x^2*2^(1/2)))
Result contains complex when optimal does not.
Time = 0.07 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.45 \[ \int \frac {\left (1+x^2\right )^3}{1+4 x^2+6 x^4+4 x^6} \, dx=\frac {\sqrt {2} x+\sqrt {1-i} \arctan \left (\frac {x}{\sqrt {\frac {1}{2}-\frac {i}{2}}}\right )+\sqrt {1+i} \arctan \left (\frac {x}{\sqrt {\frac {1}{2}+\frac {i}{2}}}\right )+\arctan \left (\sqrt {2} x\right )}{4 \sqrt {2}} \] Input:
Integrate[(1 + x^2)^3/(1 + 4*x^2 + 6*x^4 + 4*x^6),x]
Output:
(Sqrt[2]*x + Sqrt[1 - I]*ArcTan[x/Sqrt[1/2 - I/2]] + Sqrt[1 + I]*ArcTan[x/ Sqrt[1/2 + I/2]] + ArcTan[Sqrt[2]*x])/(4*Sqrt[2])
Time = 0.39 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.25, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, 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.
\(\displaystyle \int \frac {\left (x^2+1\right )^3}{4 x^6+6 x^4+4 x^2+1} \, dx\) |
\(\Big \downarrow \) 2460 |
\(\displaystyle \int \left (\frac {1}{4 \left (2 x^2+1\right )}+\frac {x^2+1}{2 \left (2 x^4+2 x^2+1\right )}+\frac {1}{4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\arctan \left (\frac {\sqrt {\sqrt {2}-1}-2 x}{\sqrt {1+\sqrt {2}}}\right )}{8 \sqrt {\sqrt {2}-1}}+\frac {\arctan \left (\sqrt {2} x\right )}{4 \sqrt {2}}+\frac {\arctan \left (\frac {2 x+\sqrt {\sqrt {2}-1}}{\sqrt {1+\sqrt {2}}}\right )}{8 \sqrt {\sqrt {2}-1}}-\frac {1}{16} \sqrt {\sqrt {2}-1} \log \left (2 x^2-2 \sqrt {\sqrt {2}-1} x+\sqrt {2}\right )+\frac {1}{16} \sqrt {\sqrt {2}-1} \log \left (\sqrt {2} x^2+\sqrt {2 \left (\sqrt {2}-1\right )} x+1\right )+\frac {x}{4}\) |
Input:
Int[(1 + x^2)^3/(1 + 4*x^2 + 6*x^4 + 4*x^6),x]
Output:
x/4 - ArcTan[(Sqrt[-1 + Sqrt[2]] - 2*x)/Sqrt[1 + Sqrt[2]]]/(8*Sqrt[-1 + Sq rt[2]]) + ArcTan[Sqrt[2]*x]/(4*Sqrt[2]) + ArcTan[(Sqrt[-1 + Sqrt[2]] + 2*x )/Sqrt[1 + Sqrt[2]]]/(8*Sqrt[-1 + Sqrt[2]]) - (Sqrt[-1 + Sqrt[2]]*Log[Sqrt [2] - 2*Sqrt[-1 + Sqrt[2]]*x + 2*x^2])/16 + (Sqrt[-1 + Sqrt[2]]*Log[1 + Sq rt[2*(-1 + Sqrt[2])]*x + Sqrt[2]*x^2])/16
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 = 0.22 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.26
method | result | size |
risch | \(\frac {x}{4}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (x +\textit {\_R} \right )\right )}{8}+\frac {\arctan \left (\sqrt {2}\, x \right ) \sqrt {2}}{8}\) | \(40\) |
default | \(\frac {x}{4}+\frac {\sqrt {2}\, \left (\frac {\sqrt {-2+2 \sqrt {2}}\, \ln \left (\sqrt {2}+\sqrt {-2+2 \sqrt {2}}\, \sqrt {2}\, x +2 x^{2}\right )}{4}+\frac {\left (-\frac {\left (-2+2 \sqrt {2}\right ) \sqrt {2}}{4}+2\right ) \arctan \left (\frac {\sqrt {2}\, \sqrt {-2+2 \sqrt {2}}+4 x}{2 \sqrt {1+\sqrt {2}}}\right )}{\sqrt {1+\sqrt {2}}}\right )}{8}+\frac {\sqrt {2}\, \left (-\frac {\sqrt {-2+2 \sqrt {2}}\, \ln \left (-\sqrt {-2+2 \sqrt {2}}\, \sqrt {2}\, x +2 x^{2}+\sqrt {2}\right )}{4}+\frac {\left (-\frac {\left (-2+2 \sqrt {2}\right ) \sqrt {2}}{4}+2\right ) \arctan \left (\frac {-\sqrt {2}\, \sqrt {-2+2 \sqrt {2}}+4 x}{2 \sqrt {1+\sqrt {2}}}\right )}{\sqrt {1+\sqrt {2}}}\right )}{8}+\frac {\arctan \left (\sqrt {2}\, x \right ) \sqrt {2}}{8}\) | \(198\) |
Input:
int((x^2+1)^3/(4*x^6+6*x^4+4*x^2+1),x,method=_RETURNVERBOSE)
Output:
1/4*x+1/8*sum(_R*ln(x+_R),_R=RootOf(2*_Z^4+2*_Z^2+1))+1/8*arctan(2^(1/2)*x )*2^(1/2)
Time = 0.08 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.95 \[ \int \frac {\left (1+x^2\right )^3}{1+4 x^2+6 x^4+4 x^6} \, dx=\frac {1}{8} \, \sqrt {2} \arctan \left (\sqrt {2} x\right ) + \frac {1}{8} \, \sqrt {\sqrt {2} + 1} \arctan \left ({\left (2 \, \sqrt {2} x + {\left (\sqrt {2} - 1\right )}^{\frac {3}{2}} - 2 \, x\right )} \sqrt {\sqrt {2} + 1}\right ) - \frac {1}{8} \, \sqrt {\sqrt {2} + 1} \arctan \left (-{\left (2 \, \sqrt {2} x - {\left (\sqrt {2} - 1\right )}^{\frac {3}{2}} - 2 \, x\right )} \sqrt {\sqrt {2} + 1}\right ) + \frac {1}{16} \, \sqrt {\sqrt {2} - 1} \log \left (2 \, x^{2} + 2 \, x \sqrt {\sqrt {2} - 1} + \sqrt {2}\right ) - \frac {1}{16} \, \sqrt {\sqrt {2} - 1} \log \left (2 \, x^{2} - 2 \, x \sqrt {\sqrt {2} - 1} + \sqrt {2}\right ) + \frac {1}{4} \, x \] Input:
integrate((x^2+1)^3/(4*x^6+6*x^4+4*x^2+1),x, algorithm="fricas")
Output:
1/8*sqrt(2)*arctan(sqrt(2)*x) + 1/8*sqrt(sqrt(2) + 1)*arctan((2*sqrt(2)*x + (sqrt(2) - 1)^(3/2) - 2*x)*sqrt(sqrt(2) + 1)) - 1/8*sqrt(sqrt(2) + 1)*ar ctan(-(2*sqrt(2)*x - (sqrt(2) - 1)^(3/2) - 2*x)*sqrt(sqrt(2) + 1)) + 1/16* sqrt(sqrt(2) - 1)*log(2*x^2 + 2*x*sqrt(sqrt(2) - 1) + sqrt(2)) - 1/16*sqrt (sqrt(2) - 1)*log(2*x^2 - 2*x*sqrt(sqrt(2) - 1) + sqrt(2)) + 1/4*x
Time = 0.79 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.11 \[ \int \frac {\left (1+x^2\right )^3}{1+4 x^2+6 x^4+4 x^6} \, dx=\frac {x}{4} + \frac {\sqrt {2} \operatorname {atan}{\left (\sqrt {2} x \right )}}{8} \] Input:
integrate((x**2+1)**3/(4*x**6+6*x**4+4*x**2+1),x)
Output:
x/4 + sqrt(2)*atan(sqrt(2)*x)/8
\[ \int \frac {\left (1+x^2\right )^3}{1+4 x^2+6 x^4+4 x^6} \, dx=\int { \frac {{\left (x^{2} + 1\right )}^{3}}{4 \, x^{6} + 6 \, x^{4} + 4 \, x^{2} + 1} \,d x } \] Input:
integrate((x^2+1)^3/(4*x^6+6*x^4+4*x^2+1),x, algorithm="maxima")
Output:
1/8*sqrt(2)*arctan(sqrt(2)*x) + 1/4*x + 1/2*integrate((x^2 + 1)/(2*x^4 + 2 *x^2 + 1), x)
Time = 0.62 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.03 \[ \int \frac {\left (1+x^2\right )^3}{1+4 x^2+6 x^4+4 x^6} \, dx=\frac {1}{8} \, \sqrt {2} \arctan \left (\sqrt {2} x\right ) + \frac {1}{8} \, \sqrt {\sqrt {2} + 1} \arctan \left (\frac {2 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (2 \, x + \left (\frac {1}{2}\right )^{\frac {1}{4}} \sqrt {-\sqrt {2} + 2}\right )}}{\sqrt {\sqrt {2} + 2}}\right ) + \frac {1}{8} \, \sqrt {\sqrt {2} + 1} \arctan \left (\frac {2 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (2 \, x - \left (\frac {1}{2}\right )^{\frac {1}{4}} \sqrt {-\sqrt {2} + 2}\right )}}{\sqrt {\sqrt {2} + 2}}\right ) + \frac {1}{16} \, \sqrt {\sqrt {2} - 1} \log \left (x^{2} + \left (\frac {1}{2}\right )^{\frac {1}{4}} x \sqrt {-\sqrt {2} + 2} + \sqrt {\frac {1}{2}}\right ) - \frac {1}{16} \, \sqrt {\sqrt {2} - 1} \log \left (x^{2} - \left (\frac {1}{2}\right )^{\frac {1}{4}} x \sqrt {-\sqrt {2} + 2} + \sqrt {\frac {1}{2}}\right ) + \frac {1}{4} \, x \] Input:
integrate((x^2+1)^3/(4*x^6+6*x^4+4*x^2+1),x, algorithm="giac")
Output:
1/8*sqrt(2)*arctan(sqrt(2)*x) + 1/8*sqrt(sqrt(2) + 1)*arctan(2*(1/2)^(3/4) *(2*x + (1/2)^(1/4)*sqrt(-sqrt(2) + 2))/sqrt(sqrt(2) + 2)) + 1/8*sqrt(sqrt (2) + 1)*arctan(2*(1/2)^(3/4)*(2*x - (1/2)^(1/4)*sqrt(-sqrt(2) + 2))/sqrt( sqrt(2) + 2)) + 1/16*sqrt(sqrt(2) - 1)*log(x^2 + (1/2)^(1/4)*x*sqrt(-sqrt( 2) + 2) + sqrt(1/2)) - 1/16*sqrt(sqrt(2) - 1)*log(x^2 - (1/2)^(1/4)*x*sqrt (-sqrt(2) + 2) + sqrt(1/2)) + 1/4*x
Time = 0.19 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.88 \[ \int \frac {\left (1+x^2\right )^3}{1+4 x^2+6 x^4+4 x^6} \, dx=\frac {x}{4}+\frac {\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,x\right )}{8}+\mathrm {atan}\left (\sqrt {2}\,x\,\sqrt {-\frac {\sqrt {2}}{256}-\frac {1}{256}}\,8{}\mathrm {i}-\sqrt {2}\,x\,\sqrt {\frac {\sqrt {2}}{256}-\frac {1}{256}}\,8{}\mathrm {i}\right )\,\left (\sqrt {-\frac {\sqrt {2}}{256}-\frac {1}{256}}\,2{}\mathrm {i}+\sqrt {\frac {\sqrt {2}}{256}-\frac {1}{256}}\,2{}\mathrm {i}\right )+\mathrm {atan}\left (\sqrt {2}\,x\,\sqrt {-\frac {\sqrt {2}}{256}-\frac {1}{256}}\,8{}\mathrm {i}+\sqrt {2}\,x\,\sqrt {\frac {\sqrt {2}}{256}-\frac {1}{256}}\,8{}\mathrm {i}\right )\,\left (\sqrt {-\frac {\sqrt {2}}{256}-\frac {1}{256}}\,2{}\mathrm {i}-\sqrt {\frac {\sqrt {2}}{256}-\frac {1}{256}}\,2{}\mathrm {i}\right ) \] Input:
int((x^2 + 1)^3/(4*x^2 + 6*x^4 + 4*x^6 + 1),x)
Output:
x/4 + (2^(1/2)*atan(2^(1/2)*x))/8 + atan(2^(1/2)*x*(- 2^(1/2)/256 - 1/256) ^(1/2)*8i - 2^(1/2)*x*(2^(1/2)/256 - 1/256)^(1/2)*8i)*((- 2^(1/2)/256 - 1/ 256)^(1/2)*2i + (2^(1/2)/256 - 1/256)^(1/2)*2i) + atan(2^(1/2)*x*(- 2^(1/2 )/256 - 1/256)^(1/2)*8i + 2^(1/2)*x*(2^(1/2)/256 - 1/256)^(1/2)*8i)*((- 2^ (1/2)/256 - 1/256)^(1/2)*2i - (2^(1/2)/256 - 1/256)^(1/2)*2i)
Time = 0.19 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.76 \[ \int \frac {\left (1+x^2\right )^3}{1+4 x^2+6 x^4+4 x^6} \, dx=-\frac {\sqrt {\sqrt {2}+1}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {2}-1}-2 x}{\sqrt {\sqrt {2}+1}}\right )}{8}+\frac {\sqrt {\sqrt {2}+1}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {2}-1}+2 x}{\sqrt {\sqrt {2}+1}}\right )}{8}+\frac {\sqrt {2}\, \mathit {atan} \left (\frac {2 x}{\sqrt {2}}\right )}{8}-\frac {\sqrt {\sqrt {2}-1}\, \mathrm {log}\left (-\sqrt {\sqrt {2}-1}\, \sqrt {2}\, x +\sqrt {2}\, x^{2}+1\right )}{16}+\frac {\sqrt {\sqrt {2}-1}\, \mathrm {log}\left (\sqrt {\sqrt {2}-1}\, \sqrt {2}\, x +\sqrt {2}\, x^{2}+1\right )}{16}+\frac {x}{4} \] Input:
int((x^2+1)^3/(4*x^6+6*x^4+4*x^2+1),x)
Output:
( - 2*sqrt(sqrt(2) + 1)*atan((sqrt(sqrt(2) - 1) - 2*x)/sqrt(sqrt(2) + 1)) + 2*sqrt(sqrt(2) + 1)*atan((sqrt(sqrt(2) - 1) + 2*x)/sqrt(sqrt(2) + 1)) + 2*sqrt(2)*atan((2*x)/sqrt(2)) - sqrt(sqrt(2) - 1)*log( - sqrt(sqrt(2) - 1) *sqrt(2)*x + sqrt(2)*x**2 + 1) + sqrt(sqrt(2) - 1)*log(sqrt(sqrt(2) - 1)*s qrt(2)*x + sqrt(2)*x**2 + 1) + 4*x)/16