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

Optimal. Leaf size=73 \[ -\sqrt{\frac{2}{13 \left (3+\sqrt{13}\right )}} \tan ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{13}}} x\right )-\sqrt{\frac{1}{26} \left (3+\sqrt{13}\right )} \tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{13}-3}} x\right ) \]

[Out]

-(Sqrt[2/(13*(3 + Sqrt[13]))]*ArcTan[Sqrt[2/(3 + Sqrt[13])]*x]) - Sqrt[(3 + Sqrt[13])/26]*ArcTanh[Sqrt[2/(-3 +
Sqrt[13])]*x]

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Rubi [A]  time = 0.0737312, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {1093, 207, 203} \[ -\sqrt{\frac{2}{13 \left (3+\sqrt{13}\right )}} \tan ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{13}}} x\right )-\sqrt{\frac{1}{26} \left (3+\sqrt{13}\right )} \tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{13}-3}} x\right ) \]

Antiderivative was successfully verified.

[In]

Int[(-1 + 3*x^2 + x^4)^(-1),x]

[Out]

-(Sqrt[2/(13*(3 + Sqrt[13]))]*ArcTan[Sqrt[2/(3 + Sqrt[13])]*x]) - Sqrt[(3 + Sqrt[13])/26]*ArcTanh[Sqrt[2/(-3 +
Sqrt[13])]*x]

Rule 1093

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[1/(b/
2 - q/2 + c*x^2), x], x] - Dist[c/q, Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*
a*c, 0] && PosQ[b^2 - 4*a*c]

Rule 207

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

Rule 203

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

Rubi steps

\begin{align*} \int \frac{1}{-1+3 x^2+x^4} \, dx &=\frac{\int \frac{1}{\frac{3}{2}-\frac{\sqrt{13}}{2}+x^2} \, dx}{\sqrt{13}}-\frac{\int \frac{1}{\frac{3}{2}+\frac{\sqrt{13}}{2}+x^2} \, dx}{\sqrt{13}}\\ &=-\sqrt{\frac{2}{13 \left (3+\sqrt{13}\right )}} \tan ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{13}}} x\right )-\sqrt{\frac{1}{26} \left (3+\sqrt{13}\right )} \tanh ^{-1}\left (\sqrt{\frac{2}{-3+\sqrt{13}}} x\right )\\ \end{align*}

Mathematica [A]  time = 0.0531403, size = 68, normalized size = 0.93 \[ -\frac{\sqrt{\sqrt{13}-3} \tan ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{13}}} x\right )+\sqrt{3+\sqrt{13}} \tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{13}-3}} x\right )}{\sqrt{26}} \]

Antiderivative was successfully verified.

[In]

Integrate[(-1 + 3*x^2 + x^4)^(-1),x]

[Out]

-((Sqrt[-3 + Sqrt[13]]*ArcTan[Sqrt[2/(3 + Sqrt[13])]*x] + Sqrt[3 + Sqrt[13]]*ArcTanh[Sqrt[2/(-3 + Sqrt[13])]*x
])/Sqrt[26])

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Maple [A]  time = 0.033, size = 56, normalized size = 0.8 \begin{align*} -{\frac{2\,\sqrt{13}}{13\,\sqrt{-6+2\,\sqrt{13}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{-6+2\,\sqrt{13}}}} \right ) }-{\frac{2\,\sqrt{13}}{13\,\sqrt{6+2\,\sqrt{13}}}\arctan \left ( 2\,{\frac{x}{\sqrt{6+2\,\sqrt{13}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/13*13^(1/2)/(-6+2*13^(1/2))^(1/2)*arctanh(2*x/(-6+2*13^(1/2))^(1/2))-2/13*13^(1/2)/(6+2*13^(1/2))^(1/2)*arc
tan(2*x/(6+2*13^(1/2))^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{4} + 3 \, x^{2} - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(x^4 + 3*x^2 - 1), x)

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Fricas [B]  time = 1.98269, size = 475, normalized size = 6.51 \begin{align*} \frac{1}{13} \, \sqrt{26} \sqrt{\sqrt{13} - 3} \arctan \left (\frac{1}{52} \, \sqrt{26} \sqrt{13} \sqrt{2} \sqrt{2 \, x^{2} + \sqrt{13} + 3} \sqrt{\sqrt{13} - 3} - \frac{1}{26} \, \sqrt{26} \sqrt{13} x \sqrt{\sqrt{13} - 3}\right ) + \frac{1}{52} \, \sqrt{26} \sqrt{\sqrt{13} + 3} \log \left (\sqrt{26}{\left (3 \, \sqrt{13} - 13\right )} \sqrt{\sqrt{13} + 3} + 52 \, x\right ) - \frac{1}{52} \, \sqrt{26} \sqrt{\sqrt{13} + 3} \log \left (-\sqrt{26}{\left (3 \, \sqrt{13} - 13\right )} \sqrt{\sqrt{13} + 3} + 52 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*sqrt(26)*sqrt(sqrt(13) - 3)*arctan(1/52*sqrt(26)*sqrt(13)*sqrt(2)*sqrt(2*x^2 + sqrt(13) + 3)*sqrt(sqrt(13
) - 3) - 1/26*sqrt(26)*sqrt(13)*x*sqrt(sqrt(13) - 3)) + 1/52*sqrt(26)*sqrt(sqrt(13) + 3)*log(sqrt(26)*(3*sqrt(
13) - 13)*sqrt(sqrt(13) + 3) + 52*x) - 1/52*sqrt(26)*sqrt(sqrt(13) + 3)*log(-sqrt(26)*(3*sqrt(13) - 13)*sqrt(s
qrt(13) + 3) + 52*x)

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Sympy [A]  time = 0.350315, size = 24, normalized size = 0.33 \begin{align*} \operatorname{RootSum}{\left (2704 t^{4} - 156 t^{2} - 1, \left ( t \mapsto t \log{\left (312 t^{3} - 22 t + x \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(2704*_t**4 - 156*_t**2 - 1, Lambda(_t, _t*log(312*_t**3 - 22*_t + x)))

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Giac [A]  time = 1.14783, size = 100, normalized size = 1.37 \begin{align*} -\frac{1}{26} \, \sqrt{26 \, \sqrt{13} - 78} \arctan \left (\frac{x}{\sqrt{\frac{1}{2} \, \sqrt{13} + \frac{3}{2}}}\right ) - \frac{1}{52} \, \sqrt{26 \, \sqrt{13} + 78} \log \left ({\left | x + \sqrt{\frac{1}{2} \, \sqrt{13} - \frac{3}{2}} \right |}\right ) + \frac{1}{52} \, \sqrt{26 \, \sqrt{13} + 78} \log \left ({\left | x - \sqrt{\frac{1}{2} \, \sqrt{13} - \frac{3}{2}} \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/26*sqrt(26*sqrt(13) - 78)*arctan(x/sqrt(1/2*sqrt(13) + 3/2)) - 1/52*sqrt(26*sqrt(13) + 78)*log(abs(x + sqrt
(1/2*sqrt(13) - 3/2))) + 1/52*sqrt(26*sqrt(13) + 78)*log(abs(x - sqrt(1/2*sqrt(13) - 3/2)))