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

Optimal. Leaf size=44 \[ \frac{\tan ^{-1}\left (2 \sqrt{2} x+\sqrt{3}\right )}{\sqrt{2}}-\frac{\tan ^{-1}\left (\sqrt{3}-2 \sqrt{2} x\right )}{\sqrt{2}} \]

[Out]

-(ArcTan[Sqrt[3] - 2*Sqrt[2]*x]/Sqrt[2]) + ArcTan[Sqrt[3] + 2*Sqrt[2]*x]/Sqrt[2]

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Rubi [A]  time = 0.0336888, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {1161, 618, 204} \[ \frac{\tan ^{-1}\left (2 \sqrt{2} x+\sqrt{3}\right )}{\sqrt{2}}-\frac{\tan ^{-1}\left (\sqrt{3}-2 \sqrt{2} x\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(ArcTan[Sqrt[3] - 2*Sqrt[2]*x]/Sqrt[2]) + ArcTan[Sqrt[3] + 2*Sqrt[2]*x]/Sqrt[2]

Rule 1161

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e - b/c, 2]},
Dist[e/(2*c), Int[1/Simp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /
; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] && (GtQ[(2*d)/e - b/c, 0] || ( !Lt
Q[(2*d)/e - b/c, 0] && EqQ[d - e*Rt[a/c, 2], 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 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])

Rubi steps

\begin{align*} \int \frac{1+2 x^2}{1-2 x^2+4 x^4} \, dx &=\frac{1}{4} \int \frac{1}{\frac{1}{2}-\sqrt{\frac{3}{2}} x+x^2} \, dx+\frac{1}{4} \int \frac{1}{\frac{1}{2}+\sqrt{\frac{3}{2}} x+x^2} \, dx\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{2}-x^2} \, dx,x,-\sqrt{\frac{3}{2}}+2 x\right )\right )-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{2}-x^2} \, dx,x,\sqrt{\frac{3}{2}}+2 x\right )\\ &=-\frac{\tan ^{-1}\left (\sqrt{3}-2 \sqrt{2} x\right )}{\sqrt{2}}+\frac{\tan ^{-1}\left (\sqrt{3}+2 \sqrt{2} x\right )}{\sqrt{2}}\\ \end{align*}

Mathematica [C]  time = 0.103253, size = 99, normalized size = 2.25 \[ \frac{\left (\sqrt{3}-3 i\right ) \tan ^{-1}\left (\frac{2 x}{\sqrt{-1-i \sqrt{3}}}\right )}{2 \sqrt{3 \left (-1-i \sqrt{3}\right )}}+\frac{\left (\sqrt{3}+3 i\right ) \tan ^{-1}\left (\frac{2 x}{\sqrt{-1+i \sqrt{3}}}\right )}{2 \sqrt{3 \left (-1+i \sqrt{3}\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.063, size = 40, normalized size = 0.9 \begin{align*}{\frac{\sqrt{2}}{2}\arctan \left ({\frac{ \left ( 4\,x-\sqrt{6} \right ) \sqrt{2}}{2}} \right ) }+{\frac{\sqrt{2}}{2}\arctan \left ({\frac{ \left ( 4\,x+\sqrt{6} \right ) \sqrt{2}}{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{2 \, x^{2} + 1}{4 \, x^{4} - 2 \, x^{2} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.32637, size = 90, normalized size = 2.05 \begin{align*} \frac{1}{2} \, \sqrt{2} \arctan \left (2 \, \sqrt{2} x^{3}\right ) + \frac{1}{2} \, \sqrt{2} \arctan \left (\sqrt{2} x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(2)*arctan(2*sqrt(2)*x^3) + 1/2*sqrt(2)*arctan(sqrt(2)*x)

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Sympy [A]  time = 0.107194, size = 29, normalized size = 0.66 \begin{align*} \frac{\sqrt{2} \left (2 \operatorname{atan}{\left (\sqrt{2} x \right )} + 2 \operatorname{atan}{\left (2 \sqrt{2} x^{3} \right )}\right )}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(2)*(2*atan(sqrt(2)*x) + 2*atan(2*sqrt(2)*x**3))/4

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{2 \, x^{2} + 1}{4 \, x^{4} - 2 \, x^{2} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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