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

Optimal. Leaf size=196 \[ -\frac {\log \left (x^2-\sqrt {2 \sqrt {2}-1} x+\sqrt {2}\right )}{4 \sqrt {2 \left (2 \sqrt {2}-1\right )}}+\frac {\log \left (x^2+\sqrt {2 \sqrt {2}-1} x+\sqrt {2}\right )}{4 \sqrt {2 \left (2 \sqrt {2}-1\right )}}-\frac {1}{2} \sqrt {\frac {1}{14} \left (2 \sqrt {2}-1\right )} \tan ^{-1}\left (\frac {\sqrt {2 \sqrt {2}-1}-2 x}{\sqrt {1+2 \sqrt {2}}}\right )+\frac {1}{2} \sqrt {\frac {1}{14} \left (2 \sqrt {2}-1\right )} \tan ^{-1}\left (\frac {2 x+\sqrt {2 \sqrt {2}-1}}{\sqrt {1+2 \sqrt {2}}}\right ) \]

[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)

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Rubi [A]  time = 0.15, antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1094, 634, 618, 204, 628} \[ -\frac {\log \left (x^2-\sqrt {2 \sqrt {2}-1} x+\sqrt {2}\right )}{4 \sqrt {2 \left (2 \sqrt {2}-1\right )}}+\frac {\log \left (x^2+\sqrt {2 \sqrt {2}-1} x+\sqrt {2}\right )}{4 \sqrt {2 \left (2 \sqrt {2}-1\right )}}-\frac {1}{2} \sqrt {\frac {1}{14} \left (2 \sqrt {2}-1\right )} \tan ^{-1}\left (\frac {\sqrt {2 \sqrt {2}-1}-2 x}{\sqrt {1+2 \sqrt {2}}}\right )+\frac {1}{2} \sqrt {\frac {1}{14} \left (2 \sqrt {2}-1\right )} \tan ^{-1}\left (\frac {2 x+\sqrt {2 \sqrt {2}-1}}{\sqrt {1+2 \sqrt {2}}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(2 + x^2 + x^4)^(-1),x]

[Out]

-(Sqrt[(-1 + 2*Sqrt[2])/14]*ArcTan[(Sqrt[-1 + 2*Sqrt[2]] - 2*x)/Sqrt[1 + 2*Sqrt[2]]])/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 204

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

Rule 618

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 628

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 634

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 1094

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+x^2+x^4} \, dx &=\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 {\operatorname {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 {\operatorname {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 {\tan ^{-1}\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 {\tan ^{-1}\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*}

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Mathematica [C]  time = 0.05, size = 91, normalized size = 0.46 \[ \frac {i \tan ^{-1}\left (\frac {x}{\sqrt {\frac {1}{2} \left (1+i \sqrt {7}\right )}}\right )}{\sqrt {\frac {7}{2} \left (1+i \sqrt {7}\right )}}-\frac {i \tan ^{-1}\left (\frac {x}{\sqrt {\frac {1}{2} \left (1-i \sqrt {7}\right )}}\right )}{\sqrt {\frac {7}{2} \left (1-i \sqrt {7}\right )}} \]

Antiderivative was successfully verified.

[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]

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fricas [B]  time = 0.43, size = 297, normalized size = 1.52 \[ -\frac {1}{56} \cdot 8^{\frac {1}{4}} \sqrt {7} \sqrt {2} \sqrt {-4 \, \sqrt {2} + 16} \arctan \left (-\frac {1}{56} \cdot 8^{\frac {3}{4}} \sqrt {7} \sqrt {2} x \sqrt {-4 \, \sqrt {2} + 16} + \frac {1}{112} \cdot 8^{\frac {3}{4}} \sqrt {7} \sqrt {2} \sqrt {4 \, x^{2} + 8^{\frac {1}{4}} x \sqrt {-4 \, \sqrt {2} + 16} + 4 \, \sqrt {2}} \sqrt {-4 \, \sqrt {2} + 16} - \frac {1}{7} \, \sqrt {7} {\left (2 \, \sqrt {2} - 1\right )}\right ) - \frac {1}{56} \cdot 8^{\frac {1}{4}} \sqrt {7} \sqrt {2} \sqrt {-4 \, \sqrt {2} + 16} \arctan \left (-\frac {1}{56} \cdot 8^{\frac {3}{4}} \sqrt {7} \sqrt {2} x \sqrt {-4 \, \sqrt {2} + 16} + \frac {1}{112} \cdot 8^{\frac {3}{4}} \sqrt {7} \sqrt {2} \sqrt {4 \, x^{2} - 8^{\frac {1}{4}} x \sqrt {-4 \, \sqrt {2} + 16} + 4 \, \sqrt {2}} \sqrt {-4 \, \sqrt {2} + 16} + \frac {1}{7} \, \sqrt {7} {\left (2 \, \sqrt {2} - 1\right )}\right ) + \frac {1}{224} \cdot 8^{\frac {1}{4}} {\left (\sqrt {2} + 4\right )} \sqrt {-4 \, \sqrt {2} + 16} \log \left (4 \, x^{2} + 8^{\frac {1}{4}} x \sqrt {-4 \, \sqrt {2} + 16} + 4 \, \sqrt {2}\right ) - \frac {1}{224} \cdot 8^{\frac {1}{4}} {\left (\sqrt {2} + 4\right )} \sqrt {-4 \, \sqrt {2} + 16} \log \left (4 \, x^{2} - 8^{\frac {1}{4}} x \sqrt {-4 \, \sqrt {2} + 16} + 4 \, \sqrt {2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/56*8^(1/4)*sqrt(7)*sqrt(2)*sqrt(-4*sqrt(2) + 16)*arctan(-1/56*8^(3/4)*sqrt(7)*sqrt(2)*x*sqrt(-4*sqrt(2) + 1
6) + 1/112*8^(3/4)*sqrt(7)*sqrt(2)*sqrt(4*x^2 + 8^(1/4)*x*sqrt(-4*sqrt(2) + 16) + 4*sqrt(2))*sqrt(-4*sqrt(2) +
16) - 1/7*sqrt(7)*(2*sqrt(2) - 1)) - 1/56*8^(1/4)*sqrt(7)*sqrt(2)*sqrt(-4*sqrt(2) + 16)*arctan(-1/56*8^(3/4)*
sqrt(7)*sqrt(2)*x*sqrt(-4*sqrt(2) + 16) + 1/112*8^(3/4)*sqrt(7)*sqrt(2)*sqrt(4*x^2 - 8^(1/4)*x*sqrt(-4*sqrt(2)
+ 16) + 4*sqrt(2))*sqrt(-4*sqrt(2) + 16) + 1/7*sqrt(7)*(2*sqrt(2) - 1)) + 1/224*8^(1/4)*(sqrt(2) + 4)*sqrt(-4
*sqrt(2) + 16)*log(4*x^2 + 8^(1/4)*x*sqrt(-4*sqrt(2) + 16) + 4*sqrt(2)) - 1/224*8^(1/4)*(sqrt(2) + 4)*sqrt(-4*
sqrt(2) + 16)*log(4*x^2 - 8^(1/4)*x*sqrt(-4*sqrt(2) + 16) + 4*sqrt(2))

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giac [A]  time = 1.81, size = 248, normalized size = 1.27 \[ \frac {1}{224} \, \sqrt {7} {\left (2 \, \sqrt {7} 2^{\frac {3}{4}} \sqrt {\sqrt {2} + 4} - 2^{\frac {1}{4}} \sqrt {-8 \, \sqrt {2} + 32}\right )} \arctan \left (\frac {2 \cdot 2^{\frac {3}{4}} \sqrt {\frac {1}{2}} {\left (x + 2^{\frac {1}{4}} \sqrt {-\frac {1}{8} \, \sqrt {2} + \frac {1}{2}}\right )}}{\sqrt {\sqrt {2} + 4}}\right ) + \frac {1}{224} \, \sqrt {7} {\left (2 \, \sqrt {7} 2^{\frac {3}{4}} \sqrt {\sqrt {2} + 4} - 2^{\frac {1}{4}} \sqrt {-8 \, \sqrt {2} + 32}\right )} \arctan \left (\frac {2 \cdot 2^{\frac {3}{4}} \sqrt {\frac {1}{2}} {\left (x - 2^{\frac {1}{4}} \sqrt {-\frac {1}{8} \, \sqrt {2} + \frac {1}{2}}\right )}}{\sqrt {\sqrt {2} + 4}}\right ) + \frac {1}{448} \, \sqrt {7} {\left (2 \cdot 2^{\frac {3}{4}} \sqrt {\sqrt {2} + 4} + \sqrt {7} 2^{\frac {1}{4}} \sqrt {-8 \, \sqrt {2} + 32}\right )} \log \left (x^{2} + 2 \cdot 2^{\frac {1}{4}} x \sqrt {-\frac {1}{8} \, \sqrt {2} + \frac {1}{2}} + \sqrt {2}\right ) - \frac {1}{448} \, \sqrt {7} {\left (2 \cdot 2^{\frac {3}{4}} \sqrt {\sqrt {2} + 4} + \sqrt {7} 2^{\frac {1}{4}} \sqrt {-8 \, \sqrt {2} + 32}\right )} \log \left (x^{2} - 2 \cdot 2^{\frac {1}{4}} x \sqrt {-\frac {1}{8} \, \sqrt {2} + \frac {1}{2}} + \sqrt {2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/224*sqrt(7)*(2*sqrt(7)*2^(3/4)*sqrt(sqrt(2) + 4) - 2^(1/4)*sqrt(-8*sqrt(2) + 32))*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)*(2*sqrt(7)*2^(3/4)*sqrt(sqrt(2) + 4
) - 2^(1/4)*sqrt(-8*sqrt(2) + 32))*arctan(2*2^(3/4)*sqrt(1/2)*(x - 2^(1/4)*sqrt(-1/8*sqrt(2) + 1/2))/sqrt(sqrt
(2) + 4)) + 1/448*sqrt(7)*(2*2^(3/4)*sqrt(sqrt(2) + 4) + sqrt(7)*2^(1/4)*sqrt(-8*sqrt(2) + 32))*log(x^2 + 2*2^
(1/4)*x*sqrt(-1/8*sqrt(2) + 1/2) + sqrt(2)) - 1/448*sqrt(7)*(2*2^(3/4)*sqrt(sqrt(2) + 4) + sqrt(7)*2^(1/4)*sqr
t(-8*sqrt(2) + 32))*log(x^2 - 2*2^(1/4)*x*sqrt(-1/8*sqrt(2) + 1/2) + sqrt(2))

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maple [B]  time = 0.12, size = 386, normalized size = 1.97 \[ -\frac {\left (-1+2 \sqrt {2}\right ) \sqrt {2}\, \arctan \left (\frac {2 x -\sqrt {-1+2 \sqrt {2}}}{\sqrt {1+2 \sqrt {2}}}\right )}{28 \sqrt {1+2 \sqrt {2}}}-\frac {\left (-1+2 \sqrt {2}\right ) \arctan \left (\frac {2 x -\sqrt {-1+2 \sqrt {2}}}{\sqrt {1+2 \sqrt {2}}}\right )}{7 \sqrt {1+2 \sqrt {2}}}+\frac {\sqrt {2}\, \arctan \left (\frac {2 x -\sqrt {-1+2 \sqrt {2}}}{\sqrt {1+2 \sqrt {2}}}\right )}{2 \sqrt {1+2 \sqrt {2}}}-\frac {\left (-1+2 \sqrt {2}\right ) \sqrt {2}\, \arctan \left (\frac {2 x +\sqrt {-1+2 \sqrt {2}}}{\sqrt {1+2 \sqrt {2}}}\right )}{28 \sqrt {1+2 \sqrt {2}}}-\frac {\left (-1+2 \sqrt {2}\right ) \arctan \left (\frac {2 x +\sqrt {-1+2 \sqrt {2}}}{\sqrt {1+2 \sqrt {2}}}\right )}{7 \sqrt {1+2 \sqrt {2}}}+\frac {\sqrt {2}\, \arctan \left (\frac {2 x +\sqrt {-1+2 \sqrt {2}}}{\sqrt {1+2 \sqrt {2}}}\right )}{2 \sqrt {1+2 \sqrt {2}}}-\frac {\sqrt {-1+2 \sqrt {2}}\, \sqrt {2}\, \ln \left (x^{2}-\sqrt {-1+2 \sqrt {2}}\, x +\sqrt {2}\right )}{56}-\frac {\sqrt {-1+2 \sqrt {2}}\, \ln \left (x^{2}-\sqrt {-1+2 \sqrt {2}}\, x +\sqrt {2}\right )}{14}+\frac {\sqrt {-1+2 \sqrt {2}}\, \sqrt {2}\, \ln \left (x^{2}+\sqrt {-1+2 \sqrt {2}}\, x +\sqrt {2}\right )}{56}+\frac {\sqrt {-1+2 \sqrt {2}}\, \ln \left (x^{2}+\sqrt {-1+2 \sqrt {2}}\, x +\sqrt {2}\right )}{14} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/56*ln(x^2+2^(1/2)+x*(-1+2*2^(1/2))^(1/2))*(-1+2*2^(1/2))^(1/2)*2^(1/2)+1/14*ln(x^2+2^(1/2)+x*(-1+2*2^(1/2))^
(1/2))*(-1+2*2^(1/2))^(1/2)-1/28/(1+2*2^(1/2))^(1/2)*arctan((2*x+(-1+2*2^(1/2))^(1/2))/(1+2*2^(1/2))^(1/2))*(-
1+2*2^(1/2))*2^(1/2)-1/7/(1+2*2^(1/2))^(1/2)*arctan((2*x+(-1+2*2^(1/2))^(1/2))/(1+2*2^(1/2))^(1/2))*(-1+2*2^(1
/2))+1/2/(1+2*2^(1/2))^(1/2)*arctan((2*x+(-1+2*2^(1/2))^(1/2))/(1+2*2^(1/2))^(1/2))*2^(1/2)-1/56*ln(x^2+2^(1/2
)-x*(-1+2*2^(1/2))^(1/2))*(-1+2*2^(1/2))^(1/2)*2^(1/2)-1/14*ln(x^2+2^(1/2)-x*(-1+2*2^(1/2))^(1/2))*(-1+2*2^(1/
2))^(1/2)-1/28/(1+2*2^(1/2))^(1/2)*arctan((2*x-(-1+2*2^(1/2))^(1/2))/(1+2*2^(1/2))^(1/2))*(-1+2*2^(1/2))*2^(1/
2)-1/7/(1+2*2^(1/2))^(1/2)*arctan((2*x-(-1+2*2^(1/2))^(1/2))/(1+2*2^(1/2))^(1/2))*(-1+2*2^(1/2))+1/2/(1+2*2^(1
/2))^(1/2)*arctan((2*x-(-1+2*2^(1/2))^(1/2))/(1+2*2^(1/2))^(1/2))*2^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{4} + x^{2} + 2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(x^4 + x^2 + 2), x)

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mupad [B]  time = 0.22, size = 61, normalized size = 0.31 \[ \frac {\mathrm {atan}\left (\frac {\sqrt {7}\,x\,\sqrt {7-\sqrt {7}\,7{}\mathrm {i}}}{14}\right )\,\sqrt {7-\sqrt {7}\,7{}\mathrm {i}}\,1{}\mathrm {i}}{14}-\frac {\sqrt {7}\,\mathrm {atan}\left (\frac {x\,\sqrt {1+\sqrt {7}\,1{}\mathrm {i}}}{2}\right )\,\sqrt {1+\sqrt {7}\,1{}\mathrm {i}}\,1{}\mathrm {i}}{14} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [B]  time = 1.14, size = 994, normalized size = 5.07 \[ \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 )} \]

Verification of antiderivative is not currently implemented for this CAS.

[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))))

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