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

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

[Out]

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

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Rubi [A]  time = 0.0519112, 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 \[ -\sqrt{\frac{1}{26} \left (3+\sqrt{13}\right )} \tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{13}-3}} x\right )-\sqrt{\frac{2}{13 \left (3+\sqrt{13}\right )}} \tanh ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{13}}} x\right ) \]

Antiderivative was successfully verified.

[In]  Int[(-1 - 3*x^2 + x^4)^(-1),x]

[Out]

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

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Rubi in Sympy [A]  time = 1.70357, size = 71, normalized size = 0.97 \[ - \frac{\sqrt{26} \operatorname{atan}{\left (\frac{\sqrt{2} x}{\sqrt{-3 + \sqrt{13}}} \right )}}{13 \sqrt{-3 + \sqrt{13}}} - \frac{\sqrt{26} \operatorname{atanh}{\left (\frac{\sqrt{2} x}{\sqrt{3 + \sqrt{13}}} \right )}}{13 \sqrt{3 + \sqrt{13}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(x**4-3*x**2-1),x)

[Out]

-sqrt(26)*atan(sqrt(2)*x/sqrt(-3 + sqrt(13)))/(13*sqrt(-3 + sqrt(13))) - sqrt(26
)*atanh(sqrt(2)*x/sqrt(3 + sqrt(13)))/(13*sqrt(3 + sqrt(13)))

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Mathematica [A]  time = 0.0404103, size = 68, normalized size = 0.93 \[ -\frac{\sqrt{3+\sqrt{13}} \tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{13}-3}} x\right )+\sqrt{\sqrt{13}-3} \tanh ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{13}}} x\right )}{\sqrt{26}} \]

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]]*Ar
cTanh[Sqrt[2/(3 + Sqrt[13])]*x])/Sqrt[26])

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Maple [A]  time = 0.019, size = 56, normalized size = 0.8 \[ -{\frac{2\,\sqrt{13}}{13\,\sqrt{-6+2\,\sqrt{13}}}\arctan \left ( 2\,{\frac{x}{\sqrt{-6+2\,\sqrt{13}}}} \right ) }-{\frac{2\,\sqrt{13}}{13\,\sqrt{6+2\,\sqrt{13}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{6+2\,\sqrt{13}}}} \right ) } \]

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

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

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 [A]  time = 0.225579, size = 240, normalized size = 3.29 \[ \frac{2}{13} \, \sqrt{\frac{1}{2}} \sqrt{\sqrt{13}{\left (3 \, \sqrt{13} + 13\right )}} \arctan \left (\frac{\sqrt{\frac{1}{2}} \sqrt{\sqrt{13}{\left (3 \, \sqrt{13} + 13\right )}}{\left (\sqrt{13} - 3\right )}}{2 \,{\left (\sqrt{13} \sqrt{\frac{1}{26}} \sqrt{\sqrt{13}{\left (\sqrt{13}{\left (2 \, x^{2} - 3\right )} + 13\right )}} + \sqrt{13} x\right )}}\right ) - \frac{1}{26} \, \sqrt{\frac{1}{2}} \sqrt{-\sqrt{13}{\left (3 \, \sqrt{13} - 13\right )}} \log \left (\sqrt{\frac{1}{2}} \sqrt{-\sqrt{13}{\left (3 \, \sqrt{13} - 13\right )}}{\left (\sqrt{13} + 3\right )} + 2 \, \sqrt{13} x\right ) + \frac{1}{26} \, \sqrt{\frac{1}{2}} \sqrt{-\sqrt{13}{\left (3 \, \sqrt{13} - 13\right )}} \log \left (-\sqrt{\frac{1}{2}} \sqrt{-\sqrt{13}{\left (3 \, \sqrt{13} - 13\right )}}{\left (\sqrt{13} + 3\right )} + 2 \, \sqrt{13} x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/13*sqrt(1/2)*sqrt(sqrt(13)*(3*sqrt(13) + 13))*arctan(1/2*sqrt(1/2)*sqrt(sqrt(1
3)*(3*sqrt(13) + 13))*(sqrt(13) - 3)/(sqrt(13)*sqrt(1/26)*sqrt(sqrt(13)*(sqrt(13
)*(2*x^2 - 3) + 13)) + sqrt(13)*x)) - 1/26*sqrt(1/2)*sqrt(-sqrt(13)*(3*sqrt(13)
- 13))*log(sqrt(1/2)*sqrt(-sqrt(13)*(3*sqrt(13) - 13))*(sqrt(13) + 3) + 2*sqrt(1
3)*x) + 1/26*sqrt(1/2)*sqrt(-sqrt(13)*(3*sqrt(13) - 13))*log(-sqrt(1/2)*sqrt(-sq
rt(13)*(3*sqrt(13) - 13))*(sqrt(13) + 3) + 2*sqrt(13)*x)

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Sympy [A]  time = 0.629646, size = 24, normalized size = 0.33 \[ \operatorname{RootSum}{\left (2704 t^{4} + 156 t^{2} - 1, \left ( t \mapsto t \log{\left (- 312 t^{3} - 22 t + x \right )} \right )\right )} \]

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/XCAS [A]  time = 0.255243, size = 100, normalized size = 1.37 \[ -\frac{1}{26} \, \sqrt{26 \, \sqrt{13} + 78} \arctan \left (\frac{x}{\sqrt{\frac{1}{2} \, \sqrt{13} - \frac{3}{2}}}\right ) - \frac{1}{52} \, \sqrt{26 \, \sqrt{13} - 78}{\rm ln}\left ({\left | x + \sqrt{\frac{1}{2} \, \sqrt{13} + \frac{3}{2}} \right |}\right ) + \frac{1}{52} \, \sqrt{26 \, \sqrt{13} - 78}{\rm ln}\left ({\left | x - \sqrt{\frac{1}{2} \, \sqrt{13} + \frac{3}{2}} \right |}\right ) \]

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*s
qrt(13) - 78)*ln(abs(x + sqrt(1/2*sqrt(13) + 3/2))) + 1/52*sqrt(26*sqrt(13) - 78
)*ln(abs(x - sqrt(1/2*sqrt(13) + 3/2)))