\(\int \frac {1}{2+x^2+x^4} \, dx\) [44]

Optimal result

Integrand size = 10, antiderivative size = 196 \[ \int \frac {1}{2+x^2+x^4} \, dx=-\frac {1}{2} \sqrt {\frac {1}{14} \left (-1+2 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {-1+2 \sqrt {2}}-2 x}{\sqrt {1+2 \sqrt {2}}}\right )+\frac {1}{2} \sqrt {\frac {1}{14} \left (-1+2 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {-1+2 \sqrt {2}}+2 x}{\sqrt {1+2 \sqrt {2}}}\right )-\frac {\log \left (\sqrt {2}-\sqrt {-1+2 \sqrt {2}} x+x^2\right )}{4 \sqrt {2 \left (-1+2 \sqrt {2}\right )}}+\frac {\log \left (\sqrt {2}+\sqrt {-1+2 \sqrt {2}} x+x^2\right )}{4 \sqrt {2 \left (-1+2 \sqrt {2}\right )}} \]

[Out]

-1/28*arctan((-2*x+(-1+2*2^(1/2))^(1/2))/(1+2*2^(1/2))^(1/2))*(-14+28*2^(1/2))^(1/2)+1/28*arctan((2*x+(-1+2*2^
(1/2))^(1/2))/(1+2*2^(1/2))^(1/2))*(-14+28*2^(1/2))^(1/2)-1/4*ln(x^2+2^(1/2)-x*(-1+2*2^(1/2))^(1/2))/(-2+4*2^(
1/2))^(1/2)+1/4*ln(x^2+2^(1/2)+x*(-1+2*2^(1/2))^(1/2))/(-2+4*2^(1/2))^(1/2)

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1108, 648, 632, 210, 642} \[ \int \frac {1}{2+x^2+x^4} \, dx=-\frac {1}{2} \sqrt {\frac {1}{14} \left (2 \sqrt {2}-1\right )} \arctan \left (\frac {\sqrt {2 \sqrt {2}-1}-2 x}{\sqrt {1+2 \sqrt {2}}}\right )+\frac {1}{2} \sqrt {\frac {1}{14} \left (2 \sqrt {2}-1\right )} \arctan \left (\frac {2 x+\sqrt {2 \sqrt {2}-1}}{\sqrt {1+2 \sqrt {2}}}\right )-\frac {\log \left (x^2-\sqrt {2 \sqrt {2}-1} x+\sqrt {2}\right )}{4 \sqrt {2 \left (2 \sqrt {2}-1\right )}}+\frac {\log \left (x^2+\sqrt {2 \sqrt {2}-1} x+\sqrt {2}\right )}{4 \sqrt {2 \left (2 \sqrt {2}-1\right )}} \]

[In]

Int[(2 + x^2 + x^4)^(-1),x]

[Out]

-1/2*(Sqrt[(-1 + 2*Sqrt[2])/14]*ArcTan[(Sqrt[-1 + 2*Sqrt[2]] - 2*x)/Sqrt[1 + 2*Sqrt[2]]]) + (Sqrt[(-1 + 2*Sqrt
[2])/14]*ArcTan[(Sqrt[-1 + 2*Sqrt[2]] + 2*x)/Sqrt[1 + 2*Sqrt[2]]])/2 - Log[Sqrt[2] - Sqrt[-1 + 2*Sqrt[2]]*x +
x^2]/(4*Sqrt[2*(-1 + 2*Sqrt[2])]) + Log[Sqrt[2] + Sqrt[-1 + 2*Sqrt[2]]*x + x^2]/(4*Sqrt[2*(-1 + 2*Sqrt[2])])

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 1108

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}
, Dist[1/(2*c*q*r), Int[(r - x)/(q - r*x + x^2), x], x] + Dist[1/(2*c*q*r), Int[(r + x)/(q + r*x + x^2), x], x
]]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[b^2 - 4*a*c]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {\sqrt {-1+2 \sqrt {2}}-x}{\sqrt {2}-\sqrt {-1+2 \sqrt {2}} x+x^2} \, dx}{2 \sqrt {2 \left (-1+2 \sqrt {2}\right )}}+\frac {\int \frac {\sqrt {-1+2 \sqrt {2}}+x}{\sqrt {2}+\sqrt {-1+2 \sqrt {2}} x+x^2} \, dx}{2 \sqrt {2 \left (-1+2 \sqrt {2}\right )}} \\ & = \frac {\int \frac {1}{\sqrt {2}-\sqrt {-1+2 \sqrt {2}} x+x^2} \, dx}{4 \sqrt {2}}+\frac {\int \frac {1}{\sqrt {2}+\sqrt {-1+2 \sqrt {2}} x+x^2} \, dx}{4 \sqrt {2}}-\frac {\int \frac {-\sqrt {-1+2 \sqrt {2}}+2 x}{\sqrt {2}-\sqrt {-1+2 \sqrt {2}} x+x^2} \, dx}{4 \sqrt {2 \left (-1+2 \sqrt {2}\right )}}+\frac {\int \frac {\sqrt {-1+2 \sqrt {2}}+2 x}{\sqrt {2}+\sqrt {-1+2 \sqrt {2}} x+x^2} \, dx}{4 \sqrt {2 \left (-1+2 \sqrt {2}\right )}} \\ & = -\frac {\log \left (\sqrt {2}-\sqrt {-1+2 \sqrt {2}} x+x^2\right )}{4 \sqrt {2 \left (-1+2 \sqrt {2}\right )}}+\frac {\log \left (\sqrt {2}+\sqrt {-1+2 \sqrt {2}} x+x^2\right )}{4 \sqrt {2 \left (-1+2 \sqrt {2}\right )}}-\frac {\text {Subst}\left (\int \frac {1}{-1-2 \sqrt {2}-x^2} \, dx,x,-\sqrt {-1+2 \sqrt {2}}+2 x\right )}{2 \sqrt {2}}-\frac {\text {Subst}\left (\int \frac {1}{-1-2 \sqrt {2}-x^2} \, dx,x,\sqrt {-1+2 \sqrt {2}}+2 x\right )}{2 \sqrt {2}} \\ & = -\frac {\arctan \left (\frac {\sqrt {-1+2 \sqrt {2}}-2 x}{\sqrt {1+2 \sqrt {2}}}\right )}{2 \sqrt {2 \left (1+2 \sqrt {2}\right )}}+\frac {\arctan \left (\frac {\sqrt {-1+2 \sqrt {2}}+2 x}{\sqrt {1+2 \sqrt {2}}}\right )}{2 \sqrt {2 \left (1+2 \sqrt {2}\right )}}-\frac {\log \left (\sqrt {2}-\sqrt {-1+2 \sqrt {2}} x+x^2\right )}{4 \sqrt {2 \left (-1+2 \sqrt {2}\right )}}+\frac {\log \left (\sqrt {2}+\sqrt {-1+2 \sqrt {2}} x+x^2\right )}{4 \sqrt {2 \left (-1+2 \sqrt {2}\right )}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.04 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.46 \[ \int \frac {1}{2+x^2+x^4} \, dx=-\frac {i \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \left (1-i \sqrt {7}\right )}}\right )}{\sqrt {\frac {7}{2} \left (1-i \sqrt {7}\right )}}+\frac {i \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \left (1+i \sqrt {7}\right )}}\right )}{\sqrt {\frac {7}{2} \left (1+i \sqrt {7}\right )}} \]

[In]

Integrate[(2 + x^2 + x^4)^(-1),x]

[Out]

((-I)*ArcTan[x/Sqrt[(1 - I*Sqrt[7])/2]])/Sqrt[(7*(1 - I*Sqrt[7]))/2] + (I*ArcTan[x/Sqrt[(1 + I*Sqrt[7])/2]])/S
qrt[(7*(1 + I*Sqrt[7]))/2]

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.14 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.16

method	result	size
risch	\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+\textit {\_Z}^{2}+2\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{3}+\textit {\_R}}\right )}{2}\)	\(31\)
default	\(\frac {\left (-\sqrt {-1+2 \sqrt {2}}\, \sqrt {2}-4 \sqrt {-1+2 \sqrt {2}}\right ) \ln \left (x^{2}+\sqrt {2}-x \sqrt {-1+2 \sqrt {2}}\right )}{56}+\frac {\left (7 \sqrt {2}+\frac {\left (-\sqrt {-1+2 \sqrt {2}}\, \sqrt {2}-4 \sqrt {-1+2 \sqrt {2}}\right ) \sqrt {-1+2 \sqrt {2}}}{2}\right ) \arctan \left (\frac {2 x -\sqrt {-1+2 \sqrt {2}}}{\sqrt {1+2 \sqrt {2}}}\right )}{14 \sqrt {1+2 \sqrt {2}}}+\frac {\left (\sqrt {-1+2 \sqrt {2}}\, \sqrt {2}+4 \sqrt {-1+2 \sqrt {2}}\right ) \ln \left (x^{2}+\sqrt {2}+x \sqrt {-1+2 \sqrt {2}}\right )}{56}+\frac {\left (7 \sqrt {2}-\frac {\left (\sqrt {-1+2 \sqrt {2}}\, \sqrt {2}+4 \sqrt {-1+2 \sqrt {2}}\right ) \sqrt {-1+2 \sqrt {2}}}{2}\right ) \arctan \left (\frac {2 x +\sqrt {-1+2 \sqrt {2}}}{\sqrt {1+2 \sqrt {2}}}\right )}{14 \sqrt {1+2 \sqrt {2}}}\)	\(253\)

[In]

int(1/(x^4+x^2+2),x,method=_RETURNVERBOSE)

[Out]

1/2*sum(1/(2*_R^3+_R)*ln(x-_R),_R=RootOf(_Z^4+_Z^2+2))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.26 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.71 \[ \int \frac {1}{2+x^2+x^4} \, dx=\frac {1}{28} \, \sqrt {7} \sqrt {i \, \sqrt {7} + 1} \log \left ({\left (\sqrt {7} + i\right )} \sqrt {i \, \sqrt {7} + 1} + 4 \, x\right ) - \frac {1}{28} \, \sqrt {7} \sqrt {i \, \sqrt {7} + 1} \log \left (-{\left (\sqrt {7} + i\right )} \sqrt {i \, \sqrt {7} + 1} + 4 \, x\right ) + \frac {1}{28} \, \sqrt {7} \sqrt {-i \, \sqrt {7} + 1} \log \left ({\left (\sqrt {7} - i\right )} \sqrt {-i \, \sqrt {7} + 1} + 4 \, x\right ) - \frac {1}{28} \, \sqrt {7} \sqrt {-i \, \sqrt {7} + 1} \log \left (-{\left (\sqrt {7} - i\right )} \sqrt {-i \, \sqrt {7} + 1} + 4 \, x\right ) \]

[In]

integrate(1/(x^4+x^2+2),x, algorithm="fricas")

[Out]

1/28*sqrt(7)*sqrt(I*sqrt(7) + 1)*log((sqrt(7) + I)*sqrt(I*sqrt(7) + 1) + 4*x) - 1/28*sqrt(7)*sqrt(I*sqrt(7) +
1)*log(-(sqrt(7) + I)*sqrt(I*sqrt(7) + 1) + 4*x) + 1/28*sqrt(7)*sqrt(-I*sqrt(7) + 1)*log((sqrt(7) - I)*sqrt(-I
*sqrt(7) + 1) + 4*x) - 1/28*sqrt(7)*sqrt(-I*sqrt(7) + 1)*log(-(sqrt(7) - I)*sqrt(-I*sqrt(7) + 1) + 4*x)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 994 vs. \(2 (158) = 316\).

Time = 0.68 (sec) , antiderivative size = 994, normalized size of antiderivative = 5.07 \[ \int \frac {1}{2+x^2+x^4} \, dx=\sqrt {\frac {1}{224} + \frac {\sqrt {2}}{112}} \log {\left (x^{2} + x \left (- \frac {4 \sqrt {7} \sqrt {1 + 2 \sqrt {2}}}{7} + \frac {5 \sqrt {14} \sqrt {1 + 2 \sqrt {2}}}{28} + \frac {3 \sqrt {14} \sqrt {1 + 2 \sqrt {2}} \sqrt {4 \sqrt {2} + 9}}{28}\right ) - \frac {33 \sqrt {4 \sqrt {2} + 9}}{28} - \frac {11}{28} + \frac {11 \sqrt {2} \sqrt {4 \sqrt {2} + 9}}{28} + \frac {83 \sqrt {2}}{28} \right )} - \sqrt {\frac {1}{224} + \frac {\sqrt {2}}{112}} \log {\left (x^{2} + x \left (- \frac {3 \sqrt {14} \sqrt {1 + 2 \sqrt {2}} \sqrt {4 \sqrt {2} + 9}}{28} - \frac {5 \sqrt {14} \sqrt {1 + 2 \sqrt {2}}}{28} + \frac {4 \sqrt {7} \sqrt {1 + 2 \sqrt {2}}}{7}\right ) - \frac {33 \sqrt {4 \sqrt {2} + 9}}{28} - \frac {11}{28} + \frac {11 \sqrt {2} \sqrt {4 \sqrt {2} + 9}}{28} + \frac {83 \sqrt {2}}{28} \right )} + 2 \sqrt {- \frac {\sqrt {4 \sqrt {2} + 9}}{112} + \frac {1}{224} + \frac {3 \sqrt {2}}{112}} \operatorname {atan}{\left (\frac {4 \sqrt {14} x}{\sqrt {4 \sqrt {2} + 9} \sqrt {- 2 \sqrt {4 \sqrt {2} + 9} + 1 + 6 \sqrt {2}} + 7 \sqrt {- 2 \sqrt {4 \sqrt {2} + 9} + 1 + 6 \sqrt {2}}} - \frac {8 \sqrt {2} \sqrt {1 + 2 \sqrt {2}}}{\sqrt {4 \sqrt {2} + 9} \sqrt {- 2 \sqrt {4 \sqrt {2} + 9} + 1 + 6 \sqrt {2}} + 7 \sqrt {- 2 \sqrt {4 \sqrt {2} + 9} + 1 + 6 \sqrt {2}}} + \frac {5 \sqrt {1 + 2 \sqrt {2}}}{\sqrt {4 \sqrt {2} + 9} \sqrt {- 2 \sqrt {4 \sqrt {2} + 9} + 1 + 6 \sqrt {2}} + 7 \sqrt {- 2 \sqrt {4 \sqrt {2} + 9} + 1 + 6 \sqrt {2}}} + \frac {3 \sqrt {1 + 2 \sqrt {2}} \sqrt {4 \sqrt {2} + 9}}{\sqrt {4 \sqrt {2} + 9} \sqrt {- 2 \sqrt {4 \sqrt {2} + 9} + 1 + 6 \sqrt {2}} + 7 \sqrt {- 2 \sqrt {4 \sqrt {2} + 9} + 1 + 6 \sqrt {2}}} \right )} + 2 \sqrt {- \frac {\sqrt {4 \sqrt {2} + 9}}{112} + \frac {1}{224} + \frac {3 \sqrt {2}}{112}} \operatorname {atan}{\left (\frac {4 \sqrt {14} x}{\sqrt {4 \sqrt {2} + 9} \sqrt {- 2 \sqrt {4 \sqrt {2} + 9} + 1 + 6 \sqrt {2}} + 7 \sqrt {- 2 \sqrt {4 \sqrt {2} + 9} + 1 + 6 \sqrt {2}}} - \frac {3 \sqrt {1 + 2 \sqrt {2}} \sqrt {4 \sqrt {2} + 9}}{\sqrt {4 \sqrt {2} + 9} \sqrt {- 2 \sqrt {4 \sqrt {2} + 9} + 1 + 6 \sqrt {2}} + 7 \sqrt {- 2 \sqrt {4 \sqrt {2} + 9} + 1 + 6 \sqrt {2}}} - \frac {5 \sqrt {1 + 2 \sqrt {2}}}{\sqrt {4 \sqrt {2} + 9} \sqrt {- 2 \sqrt {4 \sqrt {2} + 9} + 1 + 6 \sqrt {2}} + 7 \sqrt {- 2 \sqrt {4 \sqrt {2} + 9} + 1 + 6 \sqrt {2}}} + \frac {8 \sqrt {2} \sqrt {1 + 2 \sqrt {2}}}{\sqrt {4 \sqrt {2} + 9} \sqrt {- 2 \sqrt {4 \sqrt {2} + 9} + 1 + 6 \sqrt {2}} + 7 \sqrt {- 2 \sqrt {4 \sqrt {2} + 9} + 1 + 6 \sqrt {2}}} \right )} \]

[In]

integrate(1/(x**4+x**2+2),x)

[Out]

sqrt(1/224 + sqrt(2)/112)*log(x**2 + x*(-4*sqrt(7)*sqrt(1 + 2*sqrt(2))/7 + 5*sqrt(14)*sqrt(1 + 2*sqrt(2))/28 +
3*sqrt(14)*sqrt(1 + 2*sqrt(2))*sqrt(4*sqrt(2) + 9)/28) - 33*sqrt(4*sqrt(2) + 9)/28 - 11/28 + 11*sqrt(2)*sqrt(
4*sqrt(2) + 9)/28 + 83*sqrt(2)/28) - sqrt(1/224 + sqrt(2)/112)*log(x**2 + x*(-3*sqrt(14)*sqrt(1 + 2*sqrt(2))*s
qrt(4*sqrt(2) + 9)/28 - 5*sqrt(14)*sqrt(1 + 2*sqrt(2))/28 + 4*sqrt(7)*sqrt(1 + 2*sqrt(2))/7) - 33*sqrt(4*sqrt(
2) + 9)/28 - 11/28 + 11*sqrt(2)*sqrt(4*sqrt(2) + 9)/28 + 83*sqrt(2)/28) + 2*sqrt(-sqrt(4*sqrt(2) + 9)/112 + 1/
224 + 3*sqrt(2)/112)*atan(4*sqrt(14)*x/(sqrt(4*sqrt(2) + 9)*sqrt(-2*sqrt(4*sqrt(2) + 9) + 1 + 6*sqrt(2)) + 7*s
qrt(-2*sqrt(4*sqrt(2) + 9) + 1 + 6*sqrt(2))) - 8*sqrt(2)*sqrt(1 + 2*sqrt(2))/(sqrt(4*sqrt(2) + 9)*sqrt(-2*sqrt
(4*sqrt(2) + 9) + 1 + 6*sqrt(2)) + 7*sqrt(-2*sqrt(4*sqrt(2) + 9) + 1 + 6*sqrt(2))) + 5*sqrt(1 + 2*sqrt(2))/(sq
rt(4*sqrt(2) + 9)*sqrt(-2*sqrt(4*sqrt(2) + 9) + 1 + 6*sqrt(2)) + 7*sqrt(-2*sqrt(4*sqrt(2) + 9) + 1 + 6*sqrt(2)
)) + 3*sqrt(1 + 2*sqrt(2))*sqrt(4*sqrt(2) + 9)/(sqrt(4*sqrt(2) + 9)*sqrt(-2*sqrt(4*sqrt(2) + 9) + 1 + 6*sqrt(2
)) + 7*sqrt(-2*sqrt(4*sqrt(2) + 9) + 1 + 6*sqrt(2)))) + 2*sqrt(-sqrt(4*sqrt(2) + 9)/112 + 1/224 + 3*sqrt(2)/11
2)*atan(4*sqrt(14)*x/(sqrt(4*sqrt(2) + 9)*sqrt(-2*sqrt(4*sqrt(2) + 9) + 1 + 6*sqrt(2)) + 7*sqrt(-2*sqrt(4*sqrt
(2) + 9) + 1 + 6*sqrt(2))) - 3*sqrt(1 + 2*sqrt(2))*sqrt(4*sqrt(2) + 9)/(sqrt(4*sqrt(2) + 9)*sqrt(-2*sqrt(4*sqr
t(2) + 9) + 1 + 6*sqrt(2)) + 7*sqrt(-2*sqrt(4*sqrt(2) + 9) + 1 + 6*sqrt(2))) - 5*sqrt(1 + 2*sqrt(2))/(sqrt(4*s
qrt(2) + 9)*sqrt(-2*sqrt(4*sqrt(2) + 9) + 1 + 6*sqrt(2)) + 7*sqrt(-2*sqrt(4*sqrt(2) + 9) + 1 + 6*sqrt(2))) + 8
*sqrt(2)*sqrt(1 + 2*sqrt(2))/(sqrt(4*sqrt(2) + 9)*sqrt(-2*sqrt(4*sqrt(2) + 9) + 1 + 6*sqrt(2)) + 7*sqrt(-2*sqr
t(4*sqrt(2) + 9) + 1 + 6*sqrt(2))))

Maxima [F]

\[ \int \frac {1}{2+x^2+x^4} \, dx=\int { \frac {1}{x^{4} + x^{2} + 2} \,d x } \]

[In]

integrate(1/(x^4+x^2+2),x, algorithm="maxima")

[Out]

integrate(1/(x^4 + x^2 + 2), x)

Giac [A] (verification not implemented)

none

Time = 0.41 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.29 \[ \int \frac {1}{2+x^2+x^4} \, dx=\frac {1}{112} \, \sqrt {7} {\left (\sqrt {7} 2^{\frac {1}{4}} \sqrt {2 \, \sqrt {2} + 8} - 2^{\frac {1}{4}} \sqrt {-2 \, \sqrt {2} + 8}\right )} \arctan \left (\frac {2 \cdot 2^{\frac {3}{4}} \sqrt {\frac {1}{2}} {\left (x + 2^{\frac {1}{4}} \sqrt {-\frac {1}{8} \, \sqrt {2} + \frac {1}{2}}\right )}}{\sqrt {\sqrt {2} + 4}}\right ) + \frac {1}{112} \, \sqrt {7} {\left (\sqrt {7} 2^{\frac {1}{4}} \sqrt {2 \, \sqrt {2} + 8} - 2^{\frac {1}{4}} \sqrt {-2 \, \sqrt {2} + 8}\right )} \arctan \left (\frac {2 \cdot 2^{\frac {3}{4}} \sqrt {\frac {1}{2}} {\left (x - 2^{\frac {1}{4}} \sqrt {-\frac {1}{8} \, \sqrt {2} + \frac {1}{2}}\right )}}{\sqrt {\sqrt {2} + 4}}\right ) + \frac {1}{224} \, \sqrt {7} {\left (\sqrt {7} 2^{\frac {1}{4}} \sqrt {-2 \, \sqrt {2} + 8} + 2^{\frac {1}{4}} \sqrt {2 \, \sqrt {2} + 8}\right )} \log \left (x^{2} + 2 \cdot 2^{\frac {1}{4}} x \sqrt {-\frac {1}{8} \, \sqrt {2} + \frac {1}{2}} + \sqrt {2}\right ) - \frac {1}{224} \, \sqrt {7} {\left (\sqrt {7} 2^{\frac {1}{4}} \sqrt {-2 \, \sqrt {2} + 8} + 2^{\frac {1}{4}} \sqrt {2 \, \sqrt {2} + 8}\right )} \log \left (x^{2} - 2 \cdot 2^{\frac {1}{4}} x \sqrt {-\frac {1}{8} \, \sqrt {2} + \frac {1}{2}} + \sqrt {2}\right ) \]

[In]

integrate(1/(x^4+x^2+2),x, algorithm="giac")

[Out]

1/112*sqrt(7)*(sqrt(7)*2^(1/4)*sqrt(2*sqrt(2) + 8) - 2^(1/4)*sqrt(-2*sqrt(2) + 8))*arctan(2*2^(3/4)*sqrt(1/2)*
(x + 2^(1/4)*sqrt(-1/8*sqrt(2) + 1/2))/sqrt(sqrt(2) + 4)) + 1/112*sqrt(7)*(sqrt(7)*2^(1/4)*sqrt(2*sqrt(2) + 8)
- 2^(1/4)*sqrt(-2*sqrt(2) + 8))*arctan(2*2^(3/4)*sqrt(1/2)*(x - 2^(1/4)*sqrt(-1/8*sqrt(2) + 1/2))/sqrt(sqrt(2
) + 4)) + 1/224*sqrt(7)*(sqrt(7)*2^(1/4)*sqrt(-2*sqrt(2) + 8) + 2^(1/4)*sqrt(2*sqrt(2) + 8))*log(x^2 + 2*2^(1/
4)*x*sqrt(-1/8*sqrt(2) + 1/2) + sqrt(2)) - 1/224*sqrt(7)*(sqrt(7)*2^(1/4)*sqrt(-2*sqrt(2) + 8) + 2^(1/4)*sqrt(
2*sqrt(2) + 8))*log(x^2 - 2*2^(1/4)*x*sqrt(-1/8*sqrt(2) + 1/2) + sqrt(2))

Mupad [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.31 \[ \int \frac {1}{2+x^2+x^4} \, dx=\frac {\mathrm {atan}\left (\frac {\sqrt {7}\,x\,\sqrt {7-\sqrt {7}\,7{}\mathrm {i}}}{14}\right )\,\sqrt {7-\sqrt {7}\,7{}\mathrm {i}}\,1{}\mathrm {i}}{14}-\frac {\sqrt {7}\,\mathrm {atan}\left (\frac {x\,\sqrt {1+\sqrt {7}\,1{}\mathrm {i}}}{2}\right )\,\sqrt {1+\sqrt {7}\,1{}\mathrm {i}}\,1{}\mathrm {i}}{14} \]

[In]

int(1/(x^2 + x^4 + 2),x)

[Out]

(atan((7^(1/2)*x*(7 - 7^(1/2)*7i)^(1/2))/14)*(7 - 7^(1/2)*7i)^(1/2)*1i)/14 - (7^(1/2)*atan((x*(7^(1/2)*1i + 1)
^(1/2))/2)*(7^(1/2)*1i + 1)^(1/2)*1i)/14