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

Optimal. Leaf size=46 \[ \frac {\tan ^{-1}\left (\frac {4 x+\sqrt {5}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\tan ^{-1}\left (\frac {\sqrt {5}-4 x}{\sqrt {3}}\right )}{\sqrt {3}} \]

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Rubi [A]  time = 0.04, antiderivative size = 46, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {\tan ^{-1}\left (\frac {4 x+\sqrt {5}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\tan ^{-1}\left (\frac {\sqrt {5}-4 x}{\sqrt {3}}\right )}{\sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + 2*x^2)/(1 - x^2 + 4*x^4),x]

[Out]

-(ArcTan[(Sqrt[5] - 4*x)/Sqrt[3]]/Sqrt[3]) + ArcTan[(Sqrt[5] + 4*x)/Sqrt[3]]/Sqrt[3]

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 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]))

Rubi steps

\begin {align*} \int \frac {1+2 x^2}{1-x^2+4 x^4} \, dx &=\frac {1}{4} \int \frac {1}{\frac {1}{2}-\frac {\sqrt {5} x}{2}+x^2} \, dx+\frac {1}{4} \int \frac {1}{\frac {1}{2}+\frac {\sqrt {5} x}{2}+x^2} \, dx\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{-\frac {3}{4}-x^2} \, dx,x,-\frac {\sqrt {5}}{2}+2 x\right )\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{-\frac {3}{4}-x^2} \, dx,x,\frac {\sqrt {5}}{2}+2 x\right )\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {5}-4 x}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\tan ^{-1}\left (\frac {\sqrt {5}+4 x}{\sqrt {3}}\right )}{\sqrt {3}}\\ \end {align*}

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Mathematica [C]  time = 0.27, size = 101, normalized size = 2.20 \begin {gather*} \frac {\left (\sqrt {15}-5 i\right ) \tan ^{-1}\left (\frac {2 x}{\sqrt {\frac {1}{2} \left (-1-i \sqrt {15}\right )}}\right )}{\sqrt {30 \left (-1-i \sqrt {15}\right )}}+\frac {\left (\sqrt {15}+5 i\right ) \tan ^{-1}\left (\frac {2 x}{\sqrt {\frac {1}{2} \left (-1+i \sqrt {15}\right )}}\right )}{\sqrt {30 \left (-1+i \sqrt {15}\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-5*I + Sqrt[15])*ArcTan[(2*x)/Sqrt[(-1 - I*Sqrt[15])/2]])/Sqrt[30*(-1 - I*Sqrt[15])] + ((5*I + Sqrt[15])*Arc
Tan[(2*x)/Sqrt[(-1 + I*Sqrt[15])/2]])/Sqrt[30*(-1 + I*Sqrt[15])]

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+2 x^2}{1-x^2+4 x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.63, size = 31, normalized size = 0.67 \begin {gather*} \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (4 \, x^{3} + x\right )}\right ) + \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.24, size = 52, normalized size = 1.13 \begin {gather*} \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {6} \left (\frac {1}{4}\right )^{\frac {3}{4}} {\left (4 \, x + \sqrt {10} \left (\frac {1}{4}\right )^{\frac {1}{4}}\right )}\right ) + \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {6} \left (\frac {1}{4}\right )^{\frac {3}{4}} {\left (4 \, x - \sqrt {10} \left (\frac {1}{4}\right )^{\frac {1}{4}}\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*sqrt(3)*arctan(2/3*sqrt(6)*(1/4)^(3/4)*(4*x + sqrt(10)*(1/4)^(1/4))) + 1/3*sqrt(3)*arctan(2/3*sqrt(6)*(1/4
)^(3/4)*(4*x - sqrt(10)*(1/4)^(1/4)))

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maple [A]  time = 0.03, size = 40, normalized size = 0.87 \begin {gather*} \frac {\sqrt {3}\, \arctan \left (\frac {\left (4 x -\sqrt {5}\right ) \sqrt {3}}{3}\right )}{3}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (4 x +\sqrt {5}\right ) \sqrt {3}}{3}\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*arctan(1/3*(4*x+5^(1/2))*3^(1/2))*3^(1/2)+1/3*3^(1/2)*arctan(1/3*(4*x-5^(1/2))*3^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 4.37, size = 29, normalized size = 0.63 \begin {gather*} \frac {\sqrt {3}\,\left (\mathrm {atan}\left (\frac {4\,\sqrt {3}\,x^3}{3}+\frac {\sqrt {3}\,x}{3}\right )+\mathrm {atan}\left (\frac {2\,\sqrt {3}\,x}{3}\right )\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^2 + 1)/(4*x^4 - x^2 + 1),x)

[Out]

(3^(1/2)*(atan((3^(1/2)*x)/3 + (4*3^(1/2)*x^3)/3) + atan((2*3^(1/2)*x)/3)))/3

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sympy [A]  time = 0.14, size = 42, normalized size = 0.91 \begin {gather*} \frac {\sqrt {3} \left (2 \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{3} \right )} + 2 \operatorname {atan}{\left (\frac {4 \sqrt {3} x^{3}}{3} + \frac {\sqrt {3} x}{3} \right )}\right )}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(3)*(2*atan(2*sqrt(3)*x/3) + 2*atan(4*sqrt(3)*x**3/3 + sqrt(3)*x/3))/6

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