3.1.15 \(\int \frac {1}{2+2 x^2+x^4} \, dx\) [15]

Optimal. Leaf size=176 \[ -\frac {1}{4} \sqrt {-1+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2 \left (-1+\sqrt {2}\right )}-2 x}{\sqrt {2 \left (1+\sqrt {2}\right )}}\right )+\frac {1}{4} \sqrt {-1+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2 \left (-1+\sqrt {2}\right )}+2 x}{\sqrt {2 \left (1+\sqrt {2}\right )}}\right )-\frac {\log \left (\sqrt {2}-\sqrt {2 \left (-1+\sqrt {2}\right )} x+x^2\right )}{8 \sqrt {-1+\sqrt {2}}}+\frac {\log \left (\sqrt {2}+\sqrt {2 \left (-1+\sqrt {2}\right )} x+x^2\right )}{8 \sqrt {-1+\sqrt {2}}} \]

[Out]

-1/8*ln(x^2+2^(1/2)-x*(-2+2*2^(1/2))^(1/2))/(2^(1/2)-1)^(1/2)+1/8*ln(x^2+2^(1/2)+x*(-2+2*2^(1/2))^(1/2))/(2^(1
/2)-1)^(1/2)-1/4*arctan((-2*x+(-2+2*2^(1/2))^(1/2))/(2+2*2^(1/2))^(1/2))*(2^(1/2)-1)^(1/2)+1/4*arctan((2*x+(-2
+2*2^(1/2))^(1/2))/(2+2*2^(1/2))^(1/2))*(2^(1/2)-1)^(1/2)

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Rubi [A]
time = 0.11, antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {1108, 648, 632, 210, 642} \begin {gather*} -\frac {1}{4} \sqrt {\sqrt {2}-1} \text {ArcTan}\left (\frac {\sqrt {2 \left (\sqrt {2}-1\right )}-2 x}{\sqrt {2 \left (1+\sqrt {2}\right )}}\right )+\frac {1}{4} \sqrt {\sqrt {2}-1} \text {ArcTan}\left (\frac {2 x+\sqrt {2 \left (\sqrt {2}-1\right )}}{\sqrt {2 \left (1+\sqrt {2}\right )}}\right )-\frac {\log \left (x^2-\sqrt {2 \left (\sqrt {2}-1\right )} x+\sqrt {2}\right )}{8 \sqrt {\sqrt {2}-1}}+\frac {\log \left (x^2+\sqrt {2 \left (\sqrt {2}-1\right )} x+\sqrt {2}\right )}{8 \sqrt {\sqrt {2}-1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + 2*x^2 + x^4)^(-1),x]

[Out]

-1/4*(Sqrt[-1 + Sqrt[2]]*ArcTan[(Sqrt[2*(-1 + Sqrt[2])] - 2*x)/Sqrt[2*(1 + Sqrt[2])]]) + (Sqrt[-1 + Sqrt[2]]*A
rcTan[(Sqrt[2*(-1 + Sqrt[2])] + 2*x)/Sqrt[2*(1 + Sqrt[2])]])/4 - Log[Sqrt[2] - Sqrt[2*(-1 + Sqrt[2])]*x + x^2]
/(8*Sqrt[-1 + Sqrt[2]]) + Log[Sqrt[2] + Sqrt[2*(-1 + Sqrt[2])]*x + x^2]/(8*Sqrt[-1 + 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*} \int \frac {1}{2+2 x^2+x^4} \, dx &=\frac {\int \frac {\sqrt {2 \left (-1+\sqrt {2}\right )}-x}{\sqrt {2}-\sqrt {2 \left (-1+\sqrt {2}\right )} x+x^2} \, dx}{4 \sqrt {-1+\sqrt {2}}}+\frac {\int \frac {\sqrt {2 \left (-1+\sqrt {2}\right )}+x}{\sqrt {2}+\sqrt {2 \left (-1+\sqrt {2}\right )} x+x^2} \, dx}{4 \sqrt {-1+\sqrt {2}}}\\ &=\frac {\int \frac {1}{\sqrt {2}-\sqrt {2 \left (-1+\sqrt {2}\right )} x+x^2} \, dx}{4 \sqrt {2}}+\frac {\int \frac {1}{\sqrt {2}+\sqrt {2 \left (-1+\sqrt {2}\right )} x+x^2} \, dx}{4 \sqrt {2}}-\frac {\int \frac {-\sqrt {2 \left (-1+\sqrt {2}\right )}+2 x}{\sqrt {2}-\sqrt {2 \left (-1+\sqrt {2}\right )} x+x^2} \, dx}{8 \sqrt {-1+\sqrt {2}}}+\frac {\int \frac {\sqrt {2 \left (-1+\sqrt {2}\right )}+2 x}{\sqrt {2}+\sqrt {2 \left (-1+\sqrt {2}\right )} x+x^2} \, dx}{8 \sqrt {-1+\sqrt {2}}}\\ &=-\frac {\log \left (\sqrt {2}-\sqrt {2 \left (-1+\sqrt {2}\right )} x+x^2\right )}{8 \sqrt {-1+\sqrt {2}}}+\frac {\log \left (\sqrt {2}+\sqrt {2 \left (-1+\sqrt {2}\right )} x+x^2\right )}{8 \sqrt {-1+\sqrt {2}}}-\frac {\text {Subst}\left (\int \frac {1}{-2 \left (1+\sqrt {2}\right )-x^2} \, dx,x,-\sqrt {2 \left (-1+\sqrt {2}\right )}+2 x\right )}{2 \sqrt {2}}-\frac {\text {Subst}\left (\int \frac {1}{-2 \left (1+\sqrt {2}\right )-x^2} \, dx,x,\sqrt {2 \left (-1+\sqrt {2}\right )}+2 x\right )}{2 \sqrt {2}}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {2 \left (-1+\sqrt {2}\right )}-2 x}{\sqrt {2 \left (1+\sqrt {2}\right )}}\right )}{4 \sqrt {1+\sqrt {2}}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2 \left (-1+\sqrt {2}\right )}+2 x}{\sqrt {2 \left (1+\sqrt {2}\right )}}\right )}{4 \sqrt {1+\sqrt {2}}}-\frac {\log \left (\sqrt {2}-\sqrt {2 \left (-1+\sqrt {2}\right )} x+x^2\right )}{8 \sqrt {-1+\sqrt {2}}}+\frac {\log \left (\sqrt {2}+\sqrt {2 \left (-1+\sqrt {2}\right )} x+x^2\right )}{8 \sqrt {-1+\sqrt {2}}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.03, size = 41, normalized size = 0.23 \begin {gather*} \frac {1}{4} \left ((1-i)^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {1-i}}\right )+(1+i)^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {1+i}}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + 2*x^2 + x^4)^(-1),x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(252\) vs. \(2(124)=248\).
time = 0.06, size = 253, normalized size = 1.44

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*((-2+2*2^(1/2))^(1/2)*2^(1/2)+2*(-2+2*2^(1/2))^(1/2))*ln(x^2+2^(1/2)+x*(-2+2*2^(1/2))^(1/2))+1/4*(2*2^(1/
2)-1/2*((-2+2*2^(1/2))^(1/2)*2^(1/2)+2*(-2+2*2^(1/2))^(1/2))*(-2+2*2^(1/2))^(1/2))/(2+2*2^(1/2))^(1/2)*arctan(
(2*x+(-2+2*2^(1/2))^(1/2))/(2+2*2^(1/2))^(1/2))+1/16*(-(-2+2*2^(1/2))^(1/2)*2^(1/2)-2*(-2+2*2^(1/2))^(1/2))*ln
(x^2+2^(1/2)-x*(-2+2*2^(1/2))^(1/2))+1/4*(2*2^(1/2)+1/2*(-(-2+2*2^(1/2))^(1/2)*2^(1/2)-2*(-2+2*2^(1/2))^(1/2))
*(-2+2*2^(1/2))^(1/2))/(2+2*2^(1/2))^(1/2)*arctan((2*x-(-2+2*2^(1/2))^(1/2))/(2+2*2^(1/2))^(1/2))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(x^4 + 2*x^2 + 2), x)

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Fricas [A]
time = 0.37, size = 249, normalized size = 1.41 \begin {gather*} \frac {1}{16} \cdot 2^{\frac {1}{4}} {\left (\sqrt {2} + 1\right )} \sqrt {-2 \, \sqrt {2} + 4} \log \left (2 \cdot 2^{\frac {3}{4}} x \sqrt {-2 \, \sqrt {2} + 4} + 4 \, x^{2} + 4 \, \sqrt {2}\right ) - \frac {1}{16} \cdot 2^{\frac {1}{4}} {\left (\sqrt {2} + 1\right )} \sqrt {-2 \, \sqrt {2} + 4} \log \left (-2 \cdot 2^{\frac {3}{4}} x \sqrt {-2 \, \sqrt {2} + 4} + 4 \, x^{2} + 4 \, \sqrt {2}\right ) - \frac {1}{4} \cdot 2^{\frac {1}{4}} \sqrt {-2 \, \sqrt {2} + 4} \arctan \left (-\frac {1}{2} \cdot 2^{\frac {3}{4}} x \sqrt {-2 \, \sqrt {2} + 4} + \frac {1}{4} \cdot 2^{\frac {3}{4}} \sqrt {2 \cdot 2^{\frac {3}{4}} x \sqrt {-2 \, \sqrt {2} + 4} + 4 \, x^{2} + 4 \, \sqrt {2}} \sqrt {-2 \, \sqrt {2} + 4} - \sqrt {2} + 1\right ) - \frac {1}{4} \cdot 2^{\frac {1}{4}} \sqrt {-2 \, \sqrt {2} + 4} \arctan \left (-\frac {1}{2} \cdot 2^{\frac {3}{4}} x \sqrt {-2 \, \sqrt {2} + 4} + \frac {1}{4} \cdot 2^{\frac {3}{4}} \sqrt {-2 \cdot 2^{\frac {3}{4}} x \sqrt {-2 \, \sqrt {2} + 4} + 4 \, x^{2} + 4 \, \sqrt {2}} \sqrt {-2 \, \sqrt {2} + 4} + \sqrt {2} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*2^(1/4)*(sqrt(2) + 1)*sqrt(-2*sqrt(2) + 4)*log(2*2^(3/4)*x*sqrt(-2*sqrt(2) + 4) + 4*x^2 + 4*sqrt(2)) - 1/
16*2^(1/4)*(sqrt(2) + 1)*sqrt(-2*sqrt(2) + 4)*log(-2*2^(3/4)*x*sqrt(-2*sqrt(2) + 4) + 4*x^2 + 4*sqrt(2)) - 1/4
*2^(1/4)*sqrt(-2*sqrt(2) + 4)*arctan(-1/2*2^(3/4)*x*sqrt(-2*sqrt(2) + 4) + 1/4*2^(3/4)*sqrt(2*2^(3/4)*x*sqrt(-
2*sqrt(2) + 4) + 4*x^2 + 4*sqrt(2))*sqrt(-2*sqrt(2) + 4) - sqrt(2) + 1) - 1/4*2^(1/4)*sqrt(-2*sqrt(2) + 4)*arc
tan(-1/2*2^(3/4)*x*sqrt(-2*sqrt(2) + 4) + 1/4*2^(3/4)*sqrt(-2*2^(3/4)*x*sqrt(-2*sqrt(2) + 4) + 4*x^2 + 4*sqrt(
2))*sqrt(-2*sqrt(2) + 4) + sqrt(2) - 1)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 899 vs. \(2 (151) = 302\).
time = 0.62, size = 899, normalized size = 5.11 \begin {gather*} \sqrt {\frac {1}{64} + \frac {\sqrt {2}}{64}} \log {\left (x^{2} + x \left (- 4 \sqrt {2} \sqrt {1 + \sqrt {2}} - \sqrt {1 + \sqrt {2}} + 3 \sqrt {1 + \sqrt {2}} \sqrt {2 \sqrt {2} + 3}\right ) - 15 \sqrt {2 \sqrt {2} + 3} - 7 \sqrt {2} \sqrt {2 \sqrt {2} + 3} + 29 + 23 \sqrt {2} \right )} - \sqrt {\frac {1}{64} + \frac {\sqrt {2}}{64}} \log {\left (x^{2} + x \left (- 3 \sqrt {1 + \sqrt {2}} \sqrt {2 \sqrt {2} + 3} + \sqrt {1 + \sqrt {2}} + 4 \sqrt {2} \sqrt {1 + \sqrt {2}}\right ) - 15 \sqrt {2 \sqrt {2} + 3} - 7 \sqrt {2} \sqrt {2 \sqrt {2} + 3} + 29 + 23 \sqrt {2} \right )} + 2 \sqrt {- \frac {\sqrt {2 \sqrt {2} + 3}}{32} + \frac {1}{64} + \frac {3 \sqrt {2}}{64}} \operatorname {atan}{\left (\frac {2 x}{\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}}} - \frac {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}}} - \frac {\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}}} + \frac {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}}} \right )} + 2 \sqrt {- \frac {\sqrt {2 \sqrt {2} + 3}}{32} + \frac {1}{64} + \frac {3 \sqrt {2}}{64}} \operatorname {atan}{\left (\frac {2 x}{\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}}} - \frac {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}}} + \frac {\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}}} + \frac {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}}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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))*s
qrt(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 +
sqrt(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 + sq
rt(2))) - 15*sqrt(2*sqrt(2) + 3) - 7*sqrt(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*sqrt(2)) + sqrt(2*sqrt(2) + 3)*sqr
t(-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))) - 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))) + 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*s
qrt(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*sqrt(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)*sqr
t(-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) + 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))))

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Giac [A]
time = 3.16, size = 143, normalized size = 0.81 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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))

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Mupad [B]
time = 4.21, size = 210, normalized size = 1.19 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(2*x^2 + x^4 + 2),x)

[Out]

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*(1/64 - 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))

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