\(\int \frac {1}{2+2 x^2+x^4} \, dx\) [334]

Optimal result

Integrand size = 12, antiderivative size = 137 \[ \int \frac {1}{2+2 x^2+x^4} \, dx=-\frac {1}{4} \sqrt {-1+\sqrt {2}} \arctan \left (\frac {\sqrt {2 \left (-1+\sqrt {2}\right )}-2 x}{\sqrt {2 \left (1+\sqrt {2}\right )}}\right )+\frac {1}{4} \sqrt {-1+\sqrt {2}} \arctan \left (\frac {\sqrt {2 \left (-1+\sqrt {2}\right )}+2 x}{\sqrt {2 \left (1+\sqrt {2}\right )}}\right )+\frac {\text {arctanh}\left (\frac {\sqrt {2 \left (-1+\sqrt {2}\right )} x}{\sqrt {2}+x^2}\right )}{4 \sqrt {-1+\sqrt {2}}} \] Output:

-1/4*(2^(1/2)-1)^(1/2)*arctan(((-2+2*2^(1/2))^(1/2)-2*x)/(2+2*2^(1/2))^(1/ 
2))+1/4*(2^(1/2)-1)^(1/2)*arctan(((-2+2*2^(1/2))^(1/2)+2*x)/(2+2*2^(1/2))^ 
(1/2))+1/4*arctanh((-2+2*2^(1/2))^(1/2)*x/(x^2+2^(1/2)))/(2^(1/2)-1)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.03 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.30 \[ \int \frac {1}{2+2 x^2+x^4} \, dx=\frac {1}{4} \left ((1-i)^{3/2} \arctan \left (\frac {x}{\sqrt {1-i}}\right )+(1+i)^{3/2} \arctan \left (\frac {x}{\sqrt {1+i}}\right )\right ) \] Input:

Integrate[(2 + 2*x^2 + x^4)^(-1),x]
 

Output:

((1 - I)^(3/2)*ArcTan[x/Sqrt[1 - I]] + (1 + I)^(3/2)*ArcTan[x/Sqrt[1 + I]] 
)/4
 

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.47, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1407, 1142, 25, 1083, 217, 1103}

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.

\(\displaystyle \int \frac {1}{x^4+2 x^2+2} \, dx\)

\(\Big \downarrow \) 1407

\(\displaystyle \frac {\int \frac {\sqrt {2 \left (-1+\sqrt {2}\right )}-x}{x^2-\sqrt {2 \left (-1+\sqrt {2}\right )} x+\sqrt {2}}dx}{4 \sqrt {\sqrt {2}-1}}+\frac {\int \frac {x+\sqrt {2 \left (-1+\sqrt {2}\right )}}{x^2+\sqrt {2 \left (-1+\sqrt {2}\right )} x+\sqrt {2}}dx}{4 \sqrt {\sqrt {2}-1}}\)

\(\Big \downarrow \) 1142

\(\displaystyle \frac {\sqrt {\frac {1}{2} \left (\sqrt {2}-1\right )} \int \frac {1}{x^2-\sqrt {2 \left (-1+\sqrt {2}\right )} x+\sqrt {2}}dx-\frac {1}{2} \int -\frac {\sqrt {2 \left (-1+\sqrt {2}\right )}-2 x}{x^2-\sqrt {2 \left (-1+\sqrt {2}\right )} x+\sqrt {2}}dx}{4 \sqrt {\sqrt {2}-1}}+\frac {\sqrt {\frac {1}{2} \left (\sqrt {2}-1\right )} \int \frac {1}{x^2+\sqrt {2 \left (-1+\sqrt {2}\right )} x+\sqrt {2}}dx+\frac {1}{2} \int \frac {2 x+\sqrt {2 \left (-1+\sqrt {2}\right )}}{x^2+\sqrt {2 \left (-1+\sqrt {2}\right )} x+\sqrt {2}}dx}{4 \sqrt {\sqrt {2}-1}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {\frac {1}{2} \left (\sqrt {2}-1\right )} \int \frac {1}{x^2-\sqrt {2 \left (-1+\sqrt {2}\right )} x+\sqrt {2}}dx+\frac {1}{2} \int \frac {\sqrt {2 \left (-1+\sqrt {2}\right )}-2 x}{x^2-\sqrt {2 \left (-1+\sqrt {2}\right )} x+\sqrt {2}}dx}{4 \sqrt {\sqrt {2}-1}}+\frac {\sqrt {\frac {1}{2} \left (\sqrt {2}-1\right )} \int \frac {1}{x^2+\sqrt {2 \left (-1+\sqrt {2}\right )} x+\sqrt {2}}dx+\frac {1}{2} \int \frac {2 x+\sqrt {2 \left (-1+\sqrt {2}\right )}}{x^2+\sqrt {2 \left (-1+\sqrt {2}\right )} x+\sqrt {2}}dx}{4 \sqrt {\sqrt {2}-1}}\)

\(\Big \downarrow \) 1083

\(\displaystyle \frac {\frac {1}{2} \int \frac {\sqrt {2 \left (-1+\sqrt {2}\right )}-2 x}{x^2-\sqrt {2 \left (-1+\sqrt {2}\right )} x+\sqrt {2}}dx-\sqrt {2 \left (\sqrt {2}-1\right )} \int \frac {1}{-\left (2 x-\sqrt {2 \left (-1+\sqrt {2}\right )}\right )^2-2 \left (1+\sqrt {2}\right )}d\left (2 x-\sqrt {2 \left (-1+\sqrt {2}\right )}\right )}{4 \sqrt {\sqrt {2}-1}}+\frac {\frac {1}{2} \int \frac {2 x+\sqrt {2 \left (-1+\sqrt {2}\right )}}{x^2+\sqrt {2 \left (-1+\sqrt {2}\right )} x+\sqrt {2}}dx-\sqrt {2 \left (\sqrt {2}-1\right )} \int \frac {1}{-\left (2 x+\sqrt {2 \left (-1+\sqrt {2}\right )}\right )^2-2 \left (1+\sqrt {2}\right )}d\left (2 x+\sqrt {2 \left (-1+\sqrt {2}\right )}\right )}{4 \sqrt {\sqrt {2}-1}}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {\frac {1}{2} \int \frac {\sqrt {2 \left (-1+\sqrt {2}\right )}-2 x}{x^2-\sqrt {2 \left (-1+\sqrt {2}\right )} x+\sqrt {2}}dx+\sqrt {\frac {\sqrt {2}-1}{1+\sqrt {2}}} \arctan \left (\frac {2 x-\sqrt {2 \left (\sqrt {2}-1\right )}}{\sqrt {2 \left (1+\sqrt {2}\right )}}\right )}{4 \sqrt {\sqrt {2}-1}}+\frac {\frac {1}{2} \int \frac {2 x+\sqrt {2 \left (-1+\sqrt {2}\right )}}{x^2+\sqrt {2 \left (-1+\sqrt {2}\right )} x+\sqrt {2}}dx+\sqrt {\frac {\sqrt {2}-1}{1+\sqrt {2}}} \arctan \left (\frac {2 x+\sqrt {2 \left (\sqrt {2}-1\right )}}{\sqrt {2 \left (1+\sqrt {2}\right )}}\right )}{4 \sqrt {\sqrt {2}-1}}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {\sqrt {\frac {\sqrt {2}-1}{1+\sqrt {2}}} \arctan \left (\frac {2 x-\sqrt {2 \left (\sqrt {2}-1\right )}}{\sqrt {2 \left (1+\sqrt {2}\right )}}\right )-\frac {1}{2} \log \left (x^2-\sqrt {2 \left (\sqrt {2}-1\right )} x+\sqrt {2}\right )}{4 \sqrt {\sqrt {2}-1}}+\frac {\sqrt {\frac {\sqrt {2}-1}{1+\sqrt {2}}} \arctan \left (\frac {2 x+\sqrt {2 \left (\sqrt {2}-1\right )}}{\sqrt {2 \left (1+\sqrt {2}\right )}}\right )+\frac {1}{2} \log \left (x^2+\sqrt {2 \left (\sqrt {2}-1\right )} x+\sqrt {2}\right )}{4 \sqrt {\sqrt {2}-1}}\)

Input:

Int[(2 + 2*x^2 + x^4)^(-1),x]
 

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 217

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 1083

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

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[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 1142

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[(2*c*d - b*e)/(2*c)   Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) 
Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
 

rule 1407

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]}, Simp[1/(2*c*q*r)   Int[(r - x)/(q - r* 
x + x^2), x], x] + Simp[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]
 

Maple [C] (verified)

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

Time = 0.17 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.23

method	result	size
risch	\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+2 \textit {\_Z}^{2}+2\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}+\textit {\_R}}\right )}{4}\)	\(31\)
default	\(\frac {\left (-\sqrt {-2+2 \sqrt {2}}\, \sqrt {2}-2 \sqrt {-2+2 \sqrt {2}}\right ) \ln \left (x^{2}-x \sqrt {-2+2 \sqrt {2}}+\sqrt {2}\right )}{16}+\frac {\left (2 \sqrt {2}+\frac {\left (-\sqrt {-2+2 \sqrt {2}}\, \sqrt {2}-2 \sqrt {-2+2 \sqrt {2}}\right ) \sqrt {-2+2 \sqrt {2}}}{2}\right ) \arctan \left (\frac {2 x -\sqrt {-2+2 \sqrt {2}}}{\sqrt {2+2 \sqrt {2}}}\right )}{4 \sqrt {2+2 \sqrt {2}}}+\frac {\left (\sqrt {-2+2 \sqrt {2}}\, \sqrt {2}+2 \sqrt {-2+2 \sqrt {2}}\right ) \ln \left (x^{2}+x \sqrt {-2+2 \sqrt {2}}+\sqrt {2}\right )}{16}+\frac {\left (2 \sqrt {2}-\frac {\left (\sqrt {-2+2 \sqrt {2}}\, \sqrt {2}+2 \sqrt {-2+2 \sqrt {2}}\right ) \sqrt {-2+2 \sqrt {2}}}{2}\right ) \arctan \left (\frac {\sqrt {-2+2 \sqrt {2}}+2 x}{\sqrt {2+2 \sqrt {2}}}\right )}{4 \sqrt {2+2 \sqrt {2}}}\)	\(253\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.07 \[ \int \frac {1}{2+2 x^2+x^4} \, dx=\frac {1}{4} \, \sqrt {\sqrt {2} - 1} \arctan \left (\sqrt {2} x \sqrt {\sqrt {2} - 1} + \sqrt {\sqrt {2} + 1} {\left (\sqrt {2} - 1\right )}^{\frac {3}{2}}\right ) - \frac {1}{4} \, \sqrt {\sqrt {2} - 1} \arctan \left (-\sqrt {2} x \sqrt {\sqrt {2} - 1} + \sqrt {\sqrt {2} + 1} {\left (\sqrt {2} - 1\right )}^{\frac {3}{2}}\right ) - \frac {1}{8} \, \sqrt {\sqrt {2} + 1} \log \left (x^{2} + {\left (\sqrt {2} x - 2 \, x\right )} \sqrt {\sqrt {2} + 1} + \sqrt {2}\right ) + \frac {1}{8} \, \sqrt {\sqrt {2} + 1} \log \left (x^{2} - {\left (\sqrt {2} x - 2 \, x\right )} \sqrt {\sqrt {2} + 1} + \sqrt {2}\right ) \] Input:

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

Output:

1/4*sqrt(sqrt(2) - 1)*arctan(sqrt(2)*x*sqrt(sqrt(2) - 1) + sqrt(sqrt(2) + 
1)*(sqrt(2) - 1)^(3/2)) - 1/4*sqrt(sqrt(2) - 1)*arctan(-sqrt(2)*x*sqrt(sqr 
t(2) - 1) + sqrt(sqrt(2) + 1)*(sqrt(2) - 1)^(3/2)) - 1/8*sqrt(sqrt(2) + 1) 
*log(x^2 + (sqrt(2)*x - 2*x)*sqrt(sqrt(2) + 1) + sqrt(2)) + 1/8*sqrt(sqrt( 
2) + 1)*log(x^2 - (sqrt(2)*x - 2*x)*sqrt(sqrt(2) + 1) + sqrt(2))
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 899 vs. \(2 (116) = 232\).

Time = 0.53 (sec) , antiderivative size = 899, normalized size of antiderivative = 6.56 \[ \int \frac {1}{2+2 x^2+x^4} \, dx =\text {Too large to display} \] Input:

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

Output:

sqrt(1/64 + sqrt(2)/64)*log(x**2 + x*(-4*sqrt(2)*sqrt(1 + sqrt(2)) - sqrt( 
1 + sqrt(2)) + 3*sqrt(1 + sqrt(2))*sqrt(2*sqrt(2) + 3)) - 15*sqrt(2*sqrt(2 
) + 3) - 7*sqrt(2)*sqrt(2*sqrt(2) + 3) + 29 + 23*sqrt(2)) - sqrt(1/64 + sq 
rt(2)/64)*log(x**2 + x*(-3*sqrt(1 + sqrt(2))*sqrt(2*sqrt(2) + 3) + sqrt(1 
+ sqrt(2)) + 4*sqrt(2)*sqrt(1 + sqrt(2))) - 15*sqrt(2*sqrt(2) + 3) - 7*sqr 
t(2)*sqrt(2*sqrt(2) + 3) + 29 + 23*sqrt(2)) + 2*sqrt(-sqrt(2*sqrt(2) + 3)/ 
32 + 1/64 + 3*sqrt(2)/64)*atan(2*x/(sqrt(-2*sqrt(2*sqrt(2) + 3) + 1 + 3*sq 
rt(2)) + sqrt(2*sqrt(2) + 3)*sqrt(-2*sqrt(2*sqrt(2) + 3) + 1 + 3*sqrt(2))) 
 - 4*sqrt(2)*sqrt(1 + sqrt(2))/(sqrt(-2*sqrt(2*sqrt(2) + 3) + 1 + 3*sqrt(2 
)) + sqrt(2*sqrt(2) + 3)*sqrt(-2*sqrt(2*sqrt(2) + 3) + 1 + 3*sqrt(2))) - s 
qrt(1 + sqrt(2))/(sqrt(-2*sqrt(2*sqrt(2) + 3) + 1 + 3*sqrt(2)) + sqrt(2*sq 
rt(2) + 3)*sqrt(-2*sqrt(2*sqrt(2) + 3) + 1 + 3*sqrt(2))) + 3*sqrt(1 + sqrt 
(2))*sqrt(2*sqrt(2) + 3)/(sqrt(-2*sqrt(2*sqrt(2) + 3) + 1 + 3*sqrt(2)) + s 
qrt(2*sqrt(2) + 3)*sqrt(-2*sqrt(2*sqrt(2) + 3) + 1 + 3*sqrt(2)))) + 2*sqrt 
(-sqrt(2*sqrt(2) + 3)/32 + 1/64 + 3*sqrt(2)/64)*atan(2*x/(sqrt(-2*sqrt(2*s 
qrt(2) + 3) + 1 + 3*sqrt(2)) + sqrt(2*sqrt(2) + 3)*sqrt(-2*sqrt(2*sqrt(2) 
+ 3) + 1 + 3*sqrt(2))) - 3*sqrt(1 + sqrt(2))*sqrt(2*sqrt(2) + 3)/(sqrt(-2* 
sqrt(2*sqrt(2) + 3) + 1 + 3*sqrt(2)) + sqrt(2*sqrt(2) + 3)*sqrt(-2*sqrt(2* 
sqrt(2) + 3) + 1 + 3*sqrt(2))) + sqrt(1 + sqrt(2))/(sqrt(-2*sqrt(2*sqrt(2) 
 + 3) + 1 + 3*sqrt(2)) + sqrt(2*sqrt(2) + 3)*sqrt(-2*sqrt(2*sqrt(2) + 3...
 

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.04 \[ \int \frac {1}{2+2 x^2+x^4} \, dx=\frac {1}{4} \, \sqrt {\sqrt {2} - 1} \arctan \left (\frac {2^{\frac {3}{4}} {\left (2 \, x + 2^{\frac {1}{4}} \sqrt {-\sqrt {2} + 2}\right )}}{2 \, \sqrt {\sqrt {2} + 2}}\right ) + \frac {1}{4} \, \sqrt {\sqrt {2} - 1} \arctan \left (\frac {2^{\frac {3}{4}} {\left (2 \, x - 2^{\frac {1}{4}} \sqrt {-\sqrt {2} + 2}\right )}}{2 \, \sqrt {\sqrt {2} + 2}}\right ) + \frac {1}{8} \, \sqrt {\sqrt {2} + 1} \log \left (x^{2} + 2^{\frac {1}{4}} x \sqrt {-\sqrt {2} + 2} + \sqrt {2}\right ) - \frac {1}{8} \, \sqrt {\sqrt {2} + 1} \log \left (x^{2} - 2^{\frac {1}{4}} x \sqrt {-\sqrt {2} + 2} + \sqrt {2}\right ) \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 17.25 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.53 \[ \int \frac {1}{2+2 x^2+x^4} \, dx=\mathrm {atanh}\left (\frac {4\,\sqrt {2}\,x\,\sqrt {\frac {1}{64}-\frac {\sqrt {2}}{64}}}{64\,\sqrt {\frac {1}{64}-\frac {\sqrt {2}}{64}}\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{64}}-1}+\frac {4\,\sqrt {2}\,x\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{64}}}{64\,\sqrt {\frac {1}{64}-\frac {\sqrt {2}}{64}}\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{64}}-1}\right )\,\left (2\,\sqrt {\frac {1}{64}-\frac {\sqrt {2}}{64}}-2\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{64}}\right )-\mathrm {atanh}\left (\frac {4\,\sqrt {2}\,x\,\sqrt {\frac {1}{64}-\frac {\sqrt {2}}{64}}}{64\,\sqrt {\frac {1}{64}-\frac {\sqrt {2}}{64}}\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{64}}+1}-\frac {4\,\sqrt {2}\,x\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{64}}}{64\,\sqrt {\frac {1}{64}-\frac {\sqrt {2}}{64}}\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{64}}+1}\right )\,\left (2\,\sqrt {\frac {1}{64}-\frac {\sqrt {2}}{64}}+2\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{64}}\right ) \] Input:

int(1/(2*x^2 + x^4 + 2),x)
 

Output:

atanh((4*2^(1/2)*x*(1/64 - 2^(1/2)/64)^(1/2))/(64*(1/64 - 2^(1/2)/64)^(1/2 
)*(2^(1/2)/64 + 1/64)^(1/2) - 1) + (4*2^(1/2)*x*(2^(1/2)/64 + 1/64)^(1/2)) 
/(64*(1/64 - 2^(1/2)/64)^(1/2)*(2^(1/2)/64 + 1/64)^(1/2) - 1))*(2*(1/64 - 
2^(1/2)/64)^(1/2) - 2*(2^(1/2)/64 + 1/64)^(1/2)) - atanh((4*2^(1/2)*x*(1/6 
4 - 2^(1/2)/64)^(1/2))/(64*(1/64 - 2^(1/2)/64)^(1/2)*(2^(1/2)/64 + 1/64)^( 
1/2) + 1) - (4*2^(1/2)*x*(2^(1/2)/64 + 1/64)^(1/2))/(64*(1/64 - 2^(1/2)/64 
)^(1/2)*(2^(1/2)/64 + 1/64)^(1/2) + 1))*(2*(1/64 - 2^(1/2)/64)^(1/2) + 2*( 
2^(1/2)/64 + 1/64)^(1/2))
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.69 \[ \int \frac {1}{2+2 x^2+x^4} \, dx=-\frac {\sqrt {\sqrt {2}+1}\, \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {2}-1}\, \sqrt {2}-2 x}{\sqrt {\sqrt {2}+1}\, \sqrt {2}}\right )}{4}+\frac {\sqrt {\sqrt {2}+1}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {2}-1}\, \sqrt {2}-2 x}{\sqrt {\sqrt {2}+1}\, \sqrt {2}}\right )}{4}+\frac {\sqrt {\sqrt {2}+1}\, \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {2}-1}\, \sqrt {2}+2 x}{\sqrt {\sqrt {2}+1}\, \sqrt {2}}\right )}{4}-\frac {\sqrt {\sqrt {2}+1}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {2}-1}\, \sqrt {2}+2 x}{\sqrt {\sqrt {2}+1}\, \sqrt {2}}\right )}{4}-\frac {\sqrt {\sqrt {2}-1}\, \sqrt {2}\, \mathrm {log}\left (-\sqrt {\sqrt {2}-1}\, \sqrt {2}\, x +\sqrt {2}+x^{2}\right )}{8}+\frac {\sqrt {\sqrt {2}-1}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {\sqrt {2}-1}\, \sqrt {2}\, x +\sqrt {2}+x^{2}\right )}{8}-\frac {\sqrt {\sqrt {2}-1}\, \mathrm {log}\left (-\sqrt {\sqrt {2}-1}\, \sqrt {2}\, x +\sqrt {2}+x^{2}\right )}{8}+\frac {\sqrt {\sqrt {2}-1}\, \mathrm {log}\left (\sqrt {\sqrt {2}-1}\, \sqrt {2}\, x +\sqrt {2}+x^{2}\right )}{8} \] Input:

int(1/(x^4+2*x^2+2),x)
 

Output:

( - 2*sqrt(sqrt(2) + 1)*sqrt(2)*atan((sqrt(sqrt(2) - 1)*sqrt(2) - 2*x)/(sq 
rt(sqrt(2) + 1)*sqrt(2))) + 2*sqrt(sqrt(2) + 1)*atan((sqrt(sqrt(2) - 1)*sq 
rt(2) - 2*x)/(sqrt(sqrt(2) + 1)*sqrt(2))) + 2*sqrt(sqrt(2) + 1)*sqrt(2)*at 
an((sqrt(sqrt(2) - 1)*sqrt(2) + 2*x)/(sqrt(sqrt(2) + 1)*sqrt(2))) - 2*sqrt 
(sqrt(2) + 1)*atan((sqrt(sqrt(2) - 1)*sqrt(2) + 2*x)/(sqrt(sqrt(2) + 1)*sq 
rt(2))) - sqrt(sqrt(2) - 1)*sqrt(2)*log( - sqrt(sqrt(2) - 1)*sqrt(2)*x + s 
qrt(2) + x**2) + sqrt(sqrt(2) - 1)*sqrt(2)*log(sqrt(sqrt(2) - 1)*sqrt(2)*x 
 + sqrt(2) + x**2) - sqrt(sqrt(2) - 1)*log( - sqrt(sqrt(2) - 1)*sqrt(2)*x 
+ sqrt(2) + x**2) + sqrt(sqrt(2) - 1)*log(sqrt(sqrt(2) - 1)*sqrt(2)*x + sq 
rt(2) + x**2))/8