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

Optimal. Leaf size=131 \[ \frac{\tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )}{\sqrt{2 \left (\sqrt{5}-1\right )}}-\frac{\tan ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{\sqrt{2 \left (1+\sqrt{5}\right )}}+\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )}{\sqrt{2 \left (\sqrt{5}-1\right )}}-\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{\sqrt{2 \left (1+\sqrt{5}\right )}} \]

[Out]

ArcTan[Sqrt[2/(-1 + Sqrt[5])]*x]/Sqrt[2*(-1 + Sqrt[5])] - ArcTan[Sqrt[2/(1 + Sqr
t[5])]*x]/Sqrt[2*(1 + Sqrt[5])] + ArcTanh[Sqrt[2/(-1 + Sqrt[5])]*x]/Sqrt[2*(-1 +
Sqrt[5])] - ArcTanh[Sqrt[2/(1 + Sqrt[5])]*x]/Sqrt[2*(1 + Sqrt[5])]

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Rubi [A]  time = 0.181362, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{\tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )}{\sqrt{2 \left (\sqrt{5}-1\right )}}-\frac{\tan ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{\sqrt{2 \left (1+\sqrt{5}\right )}}+\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )}{\sqrt{2 \left (\sqrt{5}-1\right )}}-\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{\sqrt{2 \left (1+\sqrt{5}\right )}} \]

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[2*(-1 + Sqrt[5])] - ArcTan[Sqrt[2/(1 + Sqr
t[5])]*x]/Sqrt[2*(1 + Sqrt[5])] + ArcTanh[Sqrt[2/(-1 + Sqrt[5])]*x]/Sqrt[2*(-1 +
Sqrt[5])] - ArcTanh[Sqrt[2/(1 + Sqrt[5])]*x]/Sqrt[2*(1 + Sqrt[5])]

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Rubi in Sympy [A]  time = 14.183, size = 141, normalized size = 1.08 \[ \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{\sqrt{-1 + \sqrt{5}}} \right )}}{2 \sqrt{-1 + \sqrt{5}}} - \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{\sqrt{1 + \sqrt{5}}} \right )}}{2 \sqrt{1 + \sqrt{5}}} + \frac{\sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} x}{\sqrt{-1 + \sqrt{5}}} \right )}}{2 \sqrt{-1 + \sqrt{5}}} - \frac{\sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} x}{\sqrt{1 + \sqrt{5}}} \right )}}{2 \sqrt{1 + \sqrt{5}}} \]

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(2)*atan(sqrt(2)*x/sqrt(-1 + sqrt(5)))/(2*sqrt(-1 + sqrt(5))) - sqrt(2)*atan
(sqrt(2)*x/sqrt(1 + sqrt(5)))/(2*sqrt(1 + sqrt(5))) + sqrt(2)*atanh(sqrt(2)*x/sq
rt(-1 + sqrt(5)))/(2*sqrt(-1 + sqrt(5))) - sqrt(2)*atanh(sqrt(2)*x/sqrt(1 + sqrt
(5)))/(2*sqrt(1 + sqrt(5)))

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Mathematica [A]  time = 0.124403, size = 131, normalized size = 1. \[ \frac{\tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )}{\sqrt{2 \left (\sqrt{5}-1\right )}}-\frac{\tan ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{\sqrt{2 \left (1+\sqrt{5}\right )}}+\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )}{\sqrt{2 \left (\sqrt{5}-1\right )}}-\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{\sqrt{2 \left (1+\sqrt{5}\right )}} \]

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[2*(-1 + Sqrt[5])] - ArcTan[Sqrt[2/(1 + Sqr
t[5])]*x]/Sqrt[2*(1 + Sqrt[5])] + ArcTanh[Sqrt[2/(-1 + Sqrt[5])]*x]/Sqrt[2*(-1 +
Sqrt[5])] - ArcTanh[Sqrt[2/(1 + Sqrt[5])]*x]/Sqrt[2*(1 + Sqrt[5])]

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Maple [A]  time = 0.045, size = 96, normalized size = 0.7 \[ -{\frac{1}{\sqrt{2\,\sqrt{5}+2}}\arctan \left ( 2\,{\frac{x}{\sqrt{2\,\sqrt{5}+2}}} \right ) }+{\frac{1}{\sqrt{-2+2\,\sqrt{5}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }+{\frac{1}{\sqrt{-2+2\,\sqrt{5}}}\arctan \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }-{\frac{1}{\sqrt{2\,\sqrt{5}+2}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{2\,\sqrt{5}+2}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^4+1)/(x^8-3*x^4+1),x)

[Out]

-1/(2*5^(1/2)+2)^(1/2)*arctan(2*x/(2*5^(1/2)+2)^(1/2))+1/(-2+2*5^(1/2))^(1/2)*ar
ctanh(2*x/(-2+2*5^(1/2))^(1/2))+1/(-2+2*5^(1/2))^(1/2)*arctan(2*x/(-2+2*5^(1/2))
^(1/2))-1/(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 = 0.312941, size = 289, normalized size = 2.21 \[ \frac{1}{8} \, \sqrt{2}{\left (4 \, \sqrt{\sqrt{5} - 1} \arctan \left (\frac{{\left (\sqrt{5} + 1\right )} \sqrt{\sqrt{5} - 1}}{2 \,{\left (\sqrt{2} x + \sqrt{2 \, x^{2} + \sqrt{5} + 1}\right )}}\right ) - 4 \, \sqrt{\sqrt{5} + 1} \arctan \left (\frac{\sqrt{\sqrt{5} + 1}{\left (\sqrt{5} - 1\right )}}{2 \,{\left (\sqrt{2} x + \sqrt{2 \, x^{2} + \sqrt{5} - 1}\right )}}\right ) - \sqrt{\sqrt{5} - 1} \log \left (2 \, \sqrt{2} x +{\left (\sqrt{5} + 1\right )} \sqrt{\sqrt{5} - 1}\right ) + \sqrt{\sqrt{5} - 1} \log \left (2 \, \sqrt{2} x -{\left (\sqrt{5} + 1\right )} \sqrt{\sqrt{5} - 1}\right ) + \sqrt{\sqrt{5} + 1} \log \left (2 \, \sqrt{2} x + \sqrt{\sqrt{5} + 1}{\left (\sqrt{5} - 1\right )}\right ) - \sqrt{\sqrt{5} + 1} \log \left (2 \, \sqrt{2} x - \sqrt{\sqrt{5} + 1}{\left (\sqrt{5} - 1\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*sqrt(2)*(4*sqrt(sqrt(5) - 1)*arctan(1/2*(sqrt(5) + 1)*sqrt(sqrt(5) - 1)/(sqr
t(2)*x + sqrt(2*x^2 + sqrt(5) + 1))) - 4*sqrt(sqrt(5) + 1)*arctan(1/2*sqrt(sqrt(
5) + 1)*(sqrt(5) - 1)/(sqrt(2)*x + sqrt(2*x^2 + sqrt(5) - 1))) - sqrt(sqrt(5) -
1)*log(2*sqrt(2)*x + (sqrt(5) + 1)*sqrt(sqrt(5) - 1)) + sqrt(sqrt(5) - 1)*log(2*
sqrt(2)*x - (sqrt(5) + 1)*sqrt(sqrt(5) - 1)) + sqrt(sqrt(5) + 1)*log(2*sqrt(2)*x
+ sqrt(sqrt(5) + 1)*(sqrt(5) - 1)) - sqrt(sqrt(5) + 1)*log(2*sqrt(2)*x - sqrt(s
qrt(5) + 1)*(sqrt(5) - 1)))

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Sympy [A]  time = 3.19619, size = 49, normalized size = 0.37 \[ \operatorname{RootSum}{\left (256 t^{4} - 16 t^{2} - 1, \left ( t \mapsto t \log{\left (1024 t^{5} - 8 t + x \right )} \right )\right )} + \operatorname{RootSum}{\left (256 t^{4} + 16 t^{2} - 1, \left ( t \mapsto t \log{\left (1024 t^{5} - 8 t + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(256*_t**4 - 16*_t**2 - 1, Lambda(_t, _t*log(1024*_t**5 - 8*_t + x))) + R
ootSum(256*_t**4 + 16*_t**2 - 1, Lambda(_t, _t*log(1024*_t**5 - 8*_t + x)))

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GIAC/XCAS [A]  time = 0.341627, size = 198, normalized size = 1.51 \[ -\frac{1}{4} \, \sqrt{2 \, \sqrt{5} - 2} \arctan \left (\frac{x}{\sqrt{\frac{1}{2} \, \sqrt{5} + \frac{1}{2}}}\right ) + \frac{1}{4} \, \sqrt{2 \, \sqrt{5} + 2} \arctan \left (\frac{x}{\sqrt{\frac{1}{2} \, \sqrt{5} - \frac{1}{2}}}\right ) - \frac{1}{8} \, \sqrt{2 \, \sqrt{5} - 2}{\rm ln}\left ({\left | x + \sqrt{\frac{1}{2} \, \sqrt{5} + \frac{1}{2}} \right |}\right ) + \frac{1}{8} \, \sqrt{2 \, \sqrt{5} - 2}{\rm ln}\left ({\left | x - \sqrt{\frac{1}{2} \, \sqrt{5} + \frac{1}{2}} \right |}\right ) + \frac{1}{8} \, \sqrt{2 \, \sqrt{5} + 2}{\rm ln}\left ({\left | x + \sqrt{\frac{1}{2} \, \sqrt{5} - \frac{1}{2}} \right |}\right ) - \frac{1}{8} \, \sqrt{2 \, \sqrt{5} + 2}{\rm ln}\left ({\left | x - \sqrt{\frac{1}{2} \, \sqrt{5} - \frac{1}{2}} \right |}\right ) \]

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]

-1/4*sqrt(2*sqrt(5) - 2)*arctan(x/sqrt(1/2*sqrt(5) + 1/2)) + 1/4*sqrt(2*sqrt(5)
+ 2)*arctan(x/sqrt(1/2*sqrt(5) - 1/2)) - 1/8*sqrt(2*sqrt(5) - 2)*ln(abs(x + sqrt
(1/2*sqrt(5) + 1/2))) + 1/8*sqrt(2*sqrt(5) - 2)*ln(abs(x - sqrt(1/2*sqrt(5) + 1/
2))) + 1/8*sqrt(2*sqrt(5) + 2)*ln(abs(x + sqrt(1/2*sqrt(5) - 1/2))) - 1/8*sqrt(2
*sqrt(5) + 2)*ln(abs(x - sqrt(1/2*sqrt(5) - 1/2)))