3.30 \(\int \frac{1-x^4}{1-6 x^4+x^8} \, dx\)

Optimal. Leaf size=125 \[ \frac{\tan ^{-1}\left (\frac{x}{\sqrt{\sqrt{2}-1}}\right )}{4 \sqrt{2 \left (\sqrt{2}-1\right )}}+\frac{\tan ^{-1}\left (\frac{x}{\sqrt{1+\sqrt{2}}}\right )}{4 \sqrt{2 \left (1+\sqrt{2}\right )}}+\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{\sqrt{2}-1}}\right )}{4 \sqrt{2 \left (\sqrt{2}-1\right )}}+\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{1+\sqrt{2}}}\right )}{4 \sqrt{2 \left (1+\sqrt{2}\right )}} \]

[Out]

ArcTan[x/Sqrt[-1 + Sqrt[2]]]/(4*Sqrt[2*(-1 + Sqrt[2])]) + ArcTan[x/Sqrt[1 + Sqrt
[2]]]/(4*Sqrt[2*(1 + Sqrt[2])]) + ArcTanh[x/Sqrt[-1 + Sqrt[2]]]/(4*Sqrt[2*(-1 +
Sqrt[2])]) + ArcTanh[x/Sqrt[1 + Sqrt[2]]]/(4*Sqrt[2*(1 + Sqrt[2])])

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Rubi [A]  time = 0.146701, antiderivative size = 125, 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 \[ \frac{\tan ^{-1}\left (\frac{x}{\sqrt{\sqrt{2}-1}}\right )}{4 \sqrt{2 \left (\sqrt{2}-1\right )}}+\frac{\tan ^{-1}\left (\frac{x}{\sqrt{1+\sqrt{2}}}\right )}{4 \sqrt{2 \left (1+\sqrt{2}\right )}}+\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{\sqrt{2}-1}}\right )}{4 \sqrt{2 \left (\sqrt{2}-1\right )}}+\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{1+\sqrt{2}}}\right )}{4 \sqrt{2 \left (1+\sqrt{2}\right )}} \]

Antiderivative was successfully verified.

[In]  Int[(1 - x^4)/(1 - 6*x^4 + x^8),x]

[Out]

ArcTan[x/Sqrt[-1 + Sqrt[2]]]/(4*Sqrt[2*(-1 + Sqrt[2])]) + ArcTan[x/Sqrt[1 + Sqrt
[2]]]/(4*Sqrt[2*(1 + Sqrt[2])]) + ArcTanh[x/Sqrt[-1 + Sqrt[2]]]/(4*Sqrt[2*(-1 +
Sqrt[2])]) + ArcTanh[x/Sqrt[1 + Sqrt[2]]]/(4*Sqrt[2*(1 + Sqrt[2])])

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Rubi in Sympy [A]  time = 10.2948, size = 121, normalized size = 0.97 \[ \frac{\sqrt{2} \operatorname{atan}{\left (\frac{x}{\sqrt{-1 + \sqrt{2}}} \right )}}{8 \sqrt{-1 + \sqrt{2}}} + \frac{\sqrt{2} \operatorname{atan}{\left (\frac{x}{\sqrt{1 + \sqrt{2}}} \right )}}{8 \sqrt{1 + \sqrt{2}}} + \frac{\sqrt{2} \operatorname{atanh}{\left (\frac{x}{\sqrt{-1 + \sqrt{2}}} \right )}}{8 \sqrt{-1 + \sqrt{2}}} + \frac{\sqrt{2} \operatorname{atanh}{\left (\frac{x}{\sqrt{1 + \sqrt{2}}} \right )}}{8 \sqrt{1 + \sqrt{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-x**4+1)/(x**8-6*x**4+1),x)

[Out]

sqrt(2)*atan(x/sqrt(-1 + sqrt(2)))/(8*sqrt(-1 + sqrt(2))) + sqrt(2)*atan(x/sqrt(
1 + sqrt(2)))/(8*sqrt(1 + sqrt(2))) + sqrt(2)*atanh(x/sqrt(-1 + sqrt(2)))/(8*sqr
t(-1 + sqrt(2))) + sqrt(2)*atanh(x/sqrt(1 + sqrt(2)))/(8*sqrt(1 + sqrt(2)))

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Mathematica [A]  time = 0.0849075, size = 114, normalized size = 0.91 \[ \frac{\sqrt{1+\sqrt{2}} \tan ^{-1}\left (\frac{x}{\sqrt{\sqrt{2}-1}}\right )+\sqrt{\sqrt{2}-1} \tan ^{-1}\left (\frac{x}{\sqrt{1+\sqrt{2}}}\right )+\sqrt{1+\sqrt{2}} \tanh ^{-1}\left (\frac{x}{\sqrt{\sqrt{2}-1}}\right )+\sqrt{\sqrt{2}-1} \tanh ^{-1}\left (\frac{x}{\sqrt{1+\sqrt{2}}}\right )}{4 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - x^4)/(1 - 6*x^4 + x^8),x]

[Out]

(Sqrt[1 + Sqrt[2]]*ArcTan[x/Sqrt[-1 + Sqrt[2]]] + Sqrt[-1 + Sqrt[2]]*ArcTan[x/Sq
rt[1 + Sqrt[2]]] + Sqrt[1 + Sqrt[2]]*ArcTanh[x/Sqrt[-1 + Sqrt[2]]] + Sqrt[-1 + S
qrt[2]]*ArcTanh[x/Sqrt[1 + Sqrt[2]]])/(4*Sqrt[2])

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Maple [A]  time = 0.034, size = 90, normalized size = 0.7 \[{\frac{\sqrt{2}}{8\,\sqrt{\sqrt{2}-1}}\arctan \left ({\frac{x}{\sqrt{\sqrt{2}-1}}} \right ) }+{\frac{\sqrt{2}}{8\,\sqrt{1+\sqrt{2}}}{\it Artanh} \left ({\frac{x}{\sqrt{1+\sqrt{2}}}} \right ) }+{\frac{\sqrt{2}}{8\,\sqrt{1+\sqrt{2}}}\arctan \left ({\frac{x}{\sqrt{1+\sqrt{2}}}} \right ) }+{\frac{\sqrt{2}}{8\,\sqrt{\sqrt{2}-1}}{\it Artanh} \left ({\frac{x}{\sqrt{\sqrt{2}-1}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x^4+1)/(x^8-6*x^4+1),x)

[Out]

1/8*2^(1/2)/(2^(1/2)-1)^(1/2)*arctan(x/(2^(1/2)-1)^(1/2))+1/8*2^(1/2)/(1+2^(1/2)
)^(1/2)*arctanh(x/(1+2^(1/2))^(1/2))+1/8*2^(1/2)/(1+2^(1/2))^(1/2)*arctan(x/(1+2
^(1/2))^(1/2))+1/8*2^(1/2)/(2^(1/2)-1)^(1/2)*arctanh(x/(2^(1/2)-1)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{x^{4} - 1}{x^{8} - 6 \, x^{4} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^4 - 1)/(x^8 - 6*x^4 + 1),x, algorithm="maxima")

[Out]

-integrate((x^4 - 1)/(x^8 - 6*x^4 + 1), x)

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Fricas [A]  time = 0.289652, size = 347, normalized size = 2.78 \[ -\frac{1}{4} \, \sqrt{-\sqrt{2}{\left (\sqrt{2} - 2\right )}} \arctan \left (\frac{\sqrt{-\sqrt{2}{\left (\sqrt{2} - 2\right )}}{\left (\sqrt{2} + 2\right )}}{2 \,{\left (\sqrt{\frac{1}{2}} \sqrt{\sqrt{2}{\left (\sqrt{2}{\left (x^{2} + 1\right )} + 2\right )}} + x\right )}}\right ) + \frac{1}{4} \, \sqrt{\sqrt{2}{\left (\sqrt{2} + 2\right )}} \arctan \left (\frac{\sqrt{\sqrt{2}{\left (\sqrt{2} + 2\right )}}{\left (\sqrt{2} - 2\right )}}{2 \,{\left (\sqrt{\frac{1}{2}} \sqrt{\sqrt{2}{\left (\sqrt{2}{\left (x^{2} - 1\right )} + 2\right )}} + x\right )}}\right ) + \frac{1}{16} \, \sqrt{-\sqrt{2}{\left (\sqrt{2} - 2\right )}} \log \left (\frac{1}{2} \, \sqrt{-\sqrt{2}{\left (\sqrt{2} - 2\right )}}{\left (\sqrt{2} + 2\right )} + x\right ) - \frac{1}{16} \, \sqrt{-\sqrt{2}{\left (\sqrt{2} - 2\right )}} \log \left (-\frac{1}{2} \, \sqrt{-\sqrt{2}{\left (\sqrt{2} - 2\right )}}{\left (\sqrt{2} + 2\right )} + x\right ) - \frac{1}{16} \, \sqrt{\sqrt{2}{\left (\sqrt{2} + 2\right )}} \log \left (\frac{1}{2} \, \sqrt{\sqrt{2}{\left (\sqrt{2} + 2\right )}}{\left (\sqrt{2} - 2\right )} + x\right ) + \frac{1}{16} \, \sqrt{\sqrt{2}{\left (\sqrt{2} + 2\right )}} \log \left (-\frac{1}{2} \, \sqrt{\sqrt{2}{\left (\sqrt{2} + 2\right )}}{\left (\sqrt{2} - 2\right )} + x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^4 - 1)/(x^8 - 6*x^4 + 1),x, algorithm="fricas")

[Out]

-1/4*sqrt(-sqrt(2)*(sqrt(2) - 2))*arctan(1/2*sqrt(-sqrt(2)*(sqrt(2) - 2))*(sqrt(
2) + 2)/(sqrt(1/2)*sqrt(sqrt(2)*(sqrt(2)*(x^2 + 1) + 2)) + x)) + 1/4*sqrt(sqrt(2
)*(sqrt(2) + 2))*arctan(1/2*sqrt(sqrt(2)*(sqrt(2) + 2))*(sqrt(2) - 2)/(sqrt(1/2)
*sqrt(sqrt(2)*(sqrt(2)*(x^2 - 1) + 2)) + x)) + 1/16*sqrt(-sqrt(2)*(sqrt(2) - 2))
*log(1/2*sqrt(-sqrt(2)*(sqrt(2) - 2))*(sqrt(2) + 2) + x) - 1/16*sqrt(-sqrt(2)*(s
qrt(2) - 2))*log(-1/2*sqrt(-sqrt(2)*(sqrt(2) - 2))*(sqrt(2) + 2) + x) - 1/16*sqr
t(sqrt(2)*(sqrt(2) + 2))*log(1/2*sqrt(sqrt(2)*(sqrt(2) + 2))*(sqrt(2) - 2) + x)
+ 1/16*sqrt(sqrt(2)*(sqrt(2) + 2))*log(-1/2*sqrt(sqrt(2)*(sqrt(2) + 2))*(sqrt(2)
- 2) + x)

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Sympy [A]  time = 3.15438, size = 51, normalized size = 0.41 \[ - \operatorname{RootSum}{\left (16384 t^{4} - 256 t^{2} - 1, \left ( t \mapsto t \log{\left (65536 t^{5} - 28 t + x \right )} \right )\right )} - \operatorname{RootSum}{\left (16384 t^{4} + 256 t^{2} - 1, \left ( t \mapsto t \log{\left (65536 t^{5} - 28 t + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x**4+1)/(x**8-6*x**4+1),x)

[Out]

-RootSum(16384*_t**4 - 256*_t**2 - 1, Lambda(_t, _t*log(65536*_t**5 - 28*_t + x)
)) - RootSum(16384*_t**4 + 256*_t**2 - 1, Lambda(_t, _t*log(65536*_t**5 - 28*_t
+ x)))

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GIAC/XCAS [A]  time = 0.346548, size = 182, normalized size = 1.46 \[ \frac{1}{8} \, \sqrt{2 \, \sqrt{2} - 2} \arctan \left (\frac{x}{\sqrt{\sqrt{2} + 1}}\right ) + \frac{1}{8} \, \sqrt{2 \, \sqrt{2} + 2} \arctan \left (\frac{x}{\sqrt{\sqrt{2} - 1}}\right ) + \frac{1}{16} \, \sqrt{2 \, \sqrt{2} - 2}{\rm ln}\left ({\left | x + \sqrt{\sqrt{2} + 1} \right |}\right ) - \frac{1}{16} \, \sqrt{2 \, \sqrt{2} - 2}{\rm ln}\left ({\left | x - \sqrt{\sqrt{2} + 1} \right |}\right ) + \frac{1}{16} \, \sqrt{2 \, \sqrt{2} + 2}{\rm ln}\left ({\left | x + \sqrt{\sqrt{2} - 1} \right |}\right ) - \frac{1}{16} \, \sqrt{2 \, \sqrt{2} + 2}{\rm ln}\left ({\left | x - \sqrt{\sqrt{2} - 1} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^4 - 1)/(x^8 - 6*x^4 + 1),x, algorithm="giac")

[Out]

1/8*sqrt(2*sqrt(2) - 2)*arctan(x/sqrt(sqrt(2) + 1)) + 1/8*sqrt(2*sqrt(2) + 2)*ar
ctan(x/sqrt(sqrt(2) - 1)) + 1/16*sqrt(2*sqrt(2) - 2)*ln(abs(x + sqrt(sqrt(2) + 1
))) - 1/16*sqrt(2*sqrt(2) - 2)*ln(abs(x - sqrt(sqrt(2) + 1))) + 1/16*sqrt(2*sqrt
(2) + 2)*ln(abs(x + sqrt(sqrt(2) - 1))) - 1/16*sqrt(2*sqrt(2) + 2)*ln(abs(x - sq
rt(sqrt(2) - 1)))