3.93 \(\int \frac{1+3 x^2}{-1-2 x^2-9 x^4} \, dx\)

Optimal. Leaf size=43 \[ \frac{\tan ^{-1}\left (\frac{1-3 x}{\sqrt{2}}\right )}{2 \sqrt{2}}-\frac{\tan ^{-1}\left (\frac{3 x+1}{\sqrt{2}}\right )}{2 \sqrt{2}} \]

[Out]

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

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Rubi [A]  time = 0.0323723, antiderivative size = 43, 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} \[ \frac{\tan ^{-1}\left (\frac{1-3 x}{\sqrt{2}}\right )}{2 \sqrt{2}}-\frac{\tan ^{-1}\left (\frac{3 x+1}{\sqrt{2}}\right )}{2 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[(1 + 3*x^2)/(-1 - 2*x^2 - 9*x^4),x]

[Out]

ArcTan[(1 - 3*x)/Sqrt[2]]/(2*Sqrt[2]) - ArcTan[(1 + 3*x)/Sqrt[2]]/(2*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+3 x^2}{-1-2 x^2-9 x^4} \, dx &=-\left (\frac{1}{6} \int \frac{1}{\frac{1}{3}-\frac{2 x}{3}+x^2} \, dx\right )-\frac{1}{6} \int \frac{1}{\frac{1}{3}+\frac{2 x}{3}+x^2} \, dx\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{-\frac{8}{9}-x^2} \, dx,x,-\frac{2}{3}+2 x\right )+\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{-\frac{8}{9}-x^2} \, dx,x,\frac{2}{3}+2 x\right )\\ &=\frac{\tan ^{-1}\left (\frac{1-3 x}{\sqrt{2}}\right )}{2 \sqrt{2}}-\frac{\tan ^{-1}\left (\frac{1+3 x}{\sqrt{2}}\right )}{2 \sqrt{2}}\\ \end{align*}

Mathematica [C]  time = 0.0329612, size = 99, normalized size = 2.3 \[ -\frac{\left (\sqrt{2}-i\right ) \tan ^{-1}\left (\frac{3 x}{\sqrt{1-2 i \sqrt{2}}}\right )}{2 \sqrt{2 \left (1-2 i \sqrt{2}\right )}}-\frac{\left (\sqrt{2}+i\right ) \tan ^{-1}\left (\frac{3 x}{\sqrt{1+2 i \sqrt{2}}}\right )}{2 \sqrt{2 \left (1+2 i \sqrt{2}\right )}} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + 3*x^2)/(-1 - 2*x^2 - 9*x^4),x]

[Out]

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

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Maple [A]  time = 0.043, size = 34, normalized size = 0.8 \begin{align*} -{\frac{\sqrt{2}}{4}\arctan \left ({\frac{ \left ( 6\,x-2 \right ) \sqrt{2}}{4}} \right ) }-{\frac{\sqrt{2}}{4}\arctan \left ({\frac{ \left ( 6\,x+2 \right ) \sqrt{2}}{4}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^2+1)/(-9*x^4-2*x^2-1),x)

[Out]

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

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Maxima [A]  time = 1.45179, size = 45, normalized size = 1.05 \begin{align*} -\frac{1}{4} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (3 \, x + 1\right )}\right ) - \frac{1}{4} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (3 \, x - 1\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2+1)/(-9*x^4-2*x^2-1),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.45857, size = 113, normalized size = 2.63 \begin{align*} -\frac{1}{4} \, \sqrt{2} \arctan \left (\frac{1}{4} \, \sqrt{2}{\left (9 \, x^{3} + 5 \, x\right )}\right ) - \frac{1}{4} \, \sqrt{2} \arctan \left (\frac{3}{4} \, \sqrt{2} x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2+1)/(-9*x^4-2*x^2-1),x, algorithm="fricas")

[Out]

-1/4*sqrt(2)*arctan(1/4*sqrt(2)*(9*x^3 + 5*x)) - 1/4*sqrt(2)*arctan(3/4*sqrt(2)*x)

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Sympy [A]  time = 0.123907, size = 46, normalized size = 1.07 \begin{align*} - \frac{\sqrt{2} \left (2 \operatorname{atan}{\left (\frac{3 \sqrt{2} x}{4} \right )} + 2 \operatorname{atan}{\left (\frac{9 \sqrt{2} x^{3}}{4} + \frac{5 \sqrt{2} x}{4} \right )}\right )}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x**2+1)/(-9*x**4-2*x**2-1),x)

[Out]

-sqrt(2)*(2*atan(3*sqrt(2)*x/4) + 2*atan(9*sqrt(2)*x**3/4 + 5*sqrt(2)*x/4))/8

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Giac [A]  time = 1.16648, size = 45, normalized size = 1.05 \begin{align*} -\frac{1}{4} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (3 \, x + 1\right )}\right ) - \frac{1}{4} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (3 \, x - 1\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2+1)/(-9*x^4-2*x^2-1),x, algorithm="giac")

[Out]

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