3.27 \(\int \frac{1-x^4}{1-3 x^4+x^8} \, dx\)

Optimal. Leaf size=129 \[ \frac{\tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )}{\sqrt{10 \left (\sqrt{5}-1\right )}}+\frac{\tan ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{\sqrt{10 \left (1+\sqrt{5}\right )}}+\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )}{\sqrt{10 \left (\sqrt{5}-1\right )}}+\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{\sqrt{10 \left (1+\sqrt{5}\right )}} \]

[Out]

ArcTan[Sqrt[2/(-1 + Sqrt[5])]*x]/Sqrt[10*(-1 + Sqrt[5])] + ArcTan[Sqrt[2/(1 + Sq
rt[5])]*x]/Sqrt[10*(1 + Sqrt[5])] + ArcTanh[Sqrt[2/(-1 + Sqrt[5])]*x]/Sqrt[10*(-
1 + Sqrt[5])] + ArcTanh[Sqrt[2/(1 + Sqrt[5])]*x]/Sqrt[10*(1 + Sqrt[5])]

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Rubi [A]  time = 0.243565, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{\tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )}{\sqrt{10 \left (\sqrt{5}-1\right )}}+\frac{\tan ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{\sqrt{10 \left (1+\sqrt{5}\right )}}+\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )}{\sqrt{10 \left (\sqrt{5}-1\right )}}+\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{\sqrt{10 \left (1+\sqrt{5}\right )}} \]

Antiderivative was successfully verified.

[In]  Int[(1 - x^4)/(1 - 3*x^4 + x^8),x]

[Out]

ArcTan[Sqrt[2/(-1 + Sqrt[5])]*x]/Sqrt[10*(-1 + Sqrt[5])] + ArcTan[Sqrt[2/(1 + Sq
rt[5])]*x]/Sqrt[10*(1 + Sqrt[5])] + ArcTanh[Sqrt[2/(-1 + Sqrt[5])]*x]/Sqrt[10*(-
1 + Sqrt[5])] + ArcTanh[Sqrt[2/(1 + Sqrt[5])]*x]/Sqrt[10*(1 + Sqrt[5])]

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Rubi in Sympy [A]  time = 13.3094, size = 141, normalized size = 1.09 \[ \frac{\sqrt{10} \operatorname{atan}{\left (\frac{\sqrt{2} x}{\sqrt{-1 + \sqrt{5}}} \right )}}{10 \sqrt{-1 + \sqrt{5}}} + \frac{\sqrt{10} \operatorname{atan}{\left (\frac{\sqrt{2} x}{\sqrt{1 + \sqrt{5}}} \right )}}{10 \sqrt{1 + \sqrt{5}}} + \frac{\sqrt{10} \operatorname{atanh}{\left (\frac{\sqrt{2} x}{\sqrt{-1 + \sqrt{5}}} \right )}}{10 \sqrt{-1 + \sqrt{5}}} + \frac{\sqrt{10} \operatorname{atanh}{\left (\frac{\sqrt{2} x}{\sqrt{1 + \sqrt{5}}} \right )}}{10 \sqrt{1 + \sqrt{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-x**4+1)/(x**8-3*x**4+1),x)

[Out]

sqrt(10)*atan(sqrt(2)*x/sqrt(-1 + sqrt(5)))/(10*sqrt(-1 + sqrt(5))) + sqrt(10)*a
tan(sqrt(2)*x/sqrt(1 + sqrt(5)))/(10*sqrt(1 + sqrt(5))) + sqrt(10)*atanh(sqrt(2)
*x/sqrt(-1 + sqrt(5)))/(10*sqrt(-1 + sqrt(5))) + sqrt(10)*atanh(sqrt(2)*x/sqrt(1
+ sqrt(5)))/(10*sqrt(1 + sqrt(5)))

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Mathematica [A]  time = 0.122379, size = 129, normalized size = 1. \[ \frac{\tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )}{\sqrt{10 \left (\sqrt{5}-1\right )}}+\frac{\tan ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{\sqrt{10 \left (1+\sqrt{5}\right )}}+\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )}{\sqrt{10 \left (\sqrt{5}-1\right )}}+\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{\sqrt{10 \left (1+\sqrt{5}\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - x^4)/(1 - 3*x^4 + x^8),x]

[Out]

ArcTan[Sqrt[2/(-1 + Sqrt[5])]*x]/Sqrt[10*(-1 + Sqrt[5])] + ArcTan[Sqrt[2/(1 + Sq
rt[5])]*x]/Sqrt[10*(1 + Sqrt[5])] + ArcTanh[Sqrt[2/(-1 + Sqrt[5])]*x]/Sqrt[10*(-
1 + Sqrt[5])] + ArcTanh[Sqrt[2/(1 + Sqrt[5])]*x]/Sqrt[10*(1 + Sqrt[5])]

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Maple [A]  time = 0.034, size = 110, normalized size = 0.9 \[{\frac{\sqrt{5}}{5\,\sqrt{2\,\sqrt{5}+2}}\arctan \left ( 2\,{\frac{x}{\sqrt{2\,\sqrt{5}+2}}} \right ) }+{\frac{\sqrt{5}}{5\,\sqrt{-2+2\,\sqrt{5}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }+{\frac{\sqrt{5}}{5\,\sqrt{-2+2\,\sqrt{5}}}\arctan \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }+{\frac{\sqrt{5}}{5\,\sqrt{2\,\sqrt{5}+2}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{2\,\sqrt{5}+2}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x^4+1)/(x^8-3*x^4+1),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{x^{4} - 1}{x^{8} - 3 \, x^{4} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^4 - 1)/(x^8 - 3*x^4 + 1),x, algorithm="maxima")

[Out]

-integrate((x^4 - 1)/(x^8 - 3*x^4 + 1), x)

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Fricas [A]  time = 0.280235, size = 401, normalized size = 3.11 \[ -\frac{1}{5} \, \sqrt{\frac{1}{2}} \sqrt{-\sqrt{5}{\left (\sqrt{5} - 5\right )}} \arctan \left (\frac{\sqrt{\frac{1}{2}} \sqrt{-\sqrt{5}{\left (\sqrt{5} - 5\right )}}{\left (\sqrt{5} + 5\right )}}{10 \,{\left (\sqrt{\frac{1}{10}} \sqrt{\sqrt{5}{\left (\sqrt{5}{\left (2 \, x^{2} + 1\right )} + 5\right )}} + x\right )}}\right ) + \frac{1}{5} \, \sqrt{\frac{1}{2}} \sqrt{\sqrt{5}{\left (\sqrt{5} + 5\right )}} \arctan \left (\frac{\sqrt{\frac{1}{2}} \sqrt{\sqrt{5}{\left (\sqrt{5} + 5\right )}}{\left (\sqrt{5} - 5\right )}}{10 \,{\left (\sqrt{\frac{1}{10}} \sqrt{\sqrt{5}{\left (\sqrt{5}{\left (2 \, x^{2} - 1\right )} + 5\right )}} + x\right )}}\right ) + \frac{1}{20} \, \sqrt{\frac{1}{2}} \sqrt{-\sqrt{5}{\left (\sqrt{5} - 5\right )}} \log \left (\frac{1}{10} \, \sqrt{\frac{1}{2}} \sqrt{-\sqrt{5}{\left (\sqrt{5} - 5\right )}}{\left (\sqrt{5} + 5\right )} + x\right ) - \frac{1}{20} \, \sqrt{\frac{1}{2}} \sqrt{-\sqrt{5}{\left (\sqrt{5} - 5\right )}} \log \left (-\frac{1}{10} \, \sqrt{\frac{1}{2}} \sqrt{-\sqrt{5}{\left (\sqrt{5} - 5\right )}}{\left (\sqrt{5} + 5\right )} + x\right ) - \frac{1}{20} \, \sqrt{\frac{1}{2}} \sqrt{\sqrt{5}{\left (\sqrt{5} + 5\right )}} \log \left (\frac{1}{10} \, \sqrt{\frac{1}{2}} \sqrt{\sqrt{5}{\left (\sqrt{5} + 5\right )}}{\left (\sqrt{5} - 5\right )} + x\right ) + \frac{1}{20} \, \sqrt{\frac{1}{2}} \sqrt{\sqrt{5}{\left (\sqrt{5} + 5\right )}} \log \left (-\frac{1}{10} \, \sqrt{\frac{1}{2}} \sqrt{\sqrt{5}{\left (\sqrt{5} + 5\right )}}{\left (\sqrt{5} - 5\right )} + x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^4 - 1)/(x^8 - 3*x^4 + 1),x, algorithm="fricas")

[Out]

-1/5*sqrt(1/2)*sqrt(-sqrt(5)*(sqrt(5) - 5))*arctan(1/10*sqrt(1/2)*sqrt(-sqrt(5)*
(sqrt(5) - 5))*(sqrt(5) + 5)/(sqrt(1/10)*sqrt(sqrt(5)*(sqrt(5)*(2*x^2 + 1) + 5))
+ x)) + 1/5*sqrt(1/2)*sqrt(sqrt(5)*(sqrt(5) + 5))*arctan(1/10*sqrt(1/2)*sqrt(sq
rt(5)*(sqrt(5) + 5))*(sqrt(5) - 5)/(sqrt(1/10)*sqrt(sqrt(5)*(sqrt(5)*(2*x^2 - 1)
+ 5)) + x)) + 1/20*sqrt(1/2)*sqrt(-sqrt(5)*(sqrt(5) - 5))*log(1/10*sqrt(1/2)*sq
rt(-sqrt(5)*(sqrt(5) - 5))*(sqrt(5) + 5) + x) - 1/20*sqrt(1/2)*sqrt(-sqrt(5)*(sq
rt(5) - 5))*log(-1/10*sqrt(1/2)*sqrt(-sqrt(5)*(sqrt(5) - 5))*(sqrt(5) + 5) + x)
- 1/20*sqrt(1/2)*sqrt(sqrt(5)*(sqrt(5) + 5))*log(1/10*sqrt(1/2)*sqrt(sqrt(5)*(sq
rt(5) + 5))*(sqrt(5) - 5) + x) + 1/20*sqrt(1/2)*sqrt(sqrt(5)*(sqrt(5) + 5))*log(
-1/10*sqrt(1/2)*sqrt(sqrt(5)*(sqrt(5) + 5))*(sqrt(5) - 5) + x)

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Sympy [A]  time = 3.18749, size = 51, normalized size = 0.4 \[ - \operatorname{RootSum}{\left (6400 t^{4} - 80 t^{2} - 1, \left ( t \mapsto t \log{\left (25600 t^{5} - 16 t + x \right )} \right )\right )} - \operatorname{RootSum}{\left (6400 t^{4} + 80 t^{2} - 1, \left ( t \mapsto t \log{\left (25600 t^{5} - 16 t + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x**4+1)/(x**8-3*x**4+1),x)

[Out]

-RootSum(6400*_t**4 - 80*_t**2 - 1, Lambda(_t, _t*log(25600*_t**5 - 16*_t + x)))
- RootSum(6400*_t**4 + 80*_t**2 - 1, Lambda(_t, _t*log(25600*_t**5 - 16*_t + x)
))

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GIAC/XCAS [A]  time = 0.343793, size = 198, normalized size = 1.53 \[ \frac{1}{20} \, \sqrt{10 \, \sqrt{5} - 10} \arctan \left (\frac{x}{\sqrt{\frac{1}{2} \, \sqrt{5} + \frac{1}{2}}}\right ) + \frac{1}{20} \, \sqrt{10 \, \sqrt{5} + 10} \arctan \left (\frac{x}{\sqrt{\frac{1}{2} \, \sqrt{5} - \frac{1}{2}}}\right ) + \frac{1}{40} \, \sqrt{10 \, \sqrt{5} - 10}{\rm ln}\left ({\left | x + \sqrt{\frac{1}{2} \, \sqrt{5} + \frac{1}{2}} \right |}\right ) - \frac{1}{40} \, \sqrt{10 \, \sqrt{5} - 10}{\rm ln}\left ({\left | x - \sqrt{\frac{1}{2} \, \sqrt{5} + \frac{1}{2}} \right |}\right ) + \frac{1}{40} \, \sqrt{10 \, \sqrt{5} + 10}{\rm ln}\left ({\left | x + \sqrt{\frac{1}{2} \, \sqrt{5} - \frac{1}{2}} \right |}\right ) - \frac{1}{40} \, \sqrt{10 \, \sqrt{5} + 10}{\rm ln}\left ({\left | x - \sqrt{\frac{1}{2} \, \sqrt{5} - \frac{1}{2}} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^4 - 1)/(x^8 - 3*x^4 + 1),x, algorithm="giac")

[Out]

1/20*sqrt(10*sqrt(5) - 10)*arctan(x/sqrt(1/2*sqrt(5) + 1/2)) + 1/20*sqrt(10*sqrt
(5) + 10)*arctan(x/sqrt(1/2*sqrt(5) - 1/2)) + 1/40*sqrt(10*sqrt(5) - 10)*ln(abs(
x + sqrt(1/2*sqrt(5) + 1/2))) - 1/40*sqrt(10*sqrt(5) - 10)*ln(abs(x - sqrt(1/2*s
qrt(5) + 1/2))) + 1/40*sqrt(10*sqrt(5) + 10)*ln(abs(x + sqrt(1/2*sqrt(5) - 1/2))
) - 1/40*sqrt(10*sqrt(5) + 10)*ln(abs(x - sqrt(1/2*sqrt(5) - 1/2)))