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

Optimal. Leaf size=117 \[ \frac{\tan ^{-1}\left (\frac{x}{\sqrt{\sqrt{2}-1}}\right )}{4 \sqrt{\sqrt{2}-1}}-\frac{\tan ^{-1}\left (\frac{x}{\sqrt{1+\sqrt{2}}}\right )}{4 \sqrt{1+\sqrt{2}}}+\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{\sqrt{2}-1}}\right )}{4 \sqrt{\sqrt{2}-1}}-\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{1+\sqrt{2}}}\right )}{4 \sqrt{1+\sqrt{2}}} \]

[Out]

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

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Rubi [A]  time = 0.113384, antiderivative size = 117, 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 \[ \frac{\tan ^{-1}\left (\frac{x}{\sqrt{\sqrt{2}-1}}\right )}{4 \sqrt{\sqrt{2}-1}}-\frac{\tan ^{-1}\left (\frac{x}{\sqrt{1+\sqrt{2}}}\right )}{4 \sqrt{1+\sqrt{2}}}+\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{\sqrt{2}-1}}\right )}{4 \sqrt{\sqrt{2}-1}}-\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{1+\sqrt{2}}}\right )}{4 \sqrt{1+\sqrt{2}}} \]

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

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Rubi in Sympy [A]  time = 10.3975, size = 100, normalized size = 0.85 \[ \frac{\operatorname{atan}{\left (\frac{x}{\sqrt{-1 + \sqrt{2}}} \right )}}{4 \sqrt{-1 + \sqrt{2}}} - \frac{\operatorname{atan}{\left (\frac{x}{\sqrt{1 + \sqrt{2}}} \right )}}{4 \sqrt{1 + \sqrt{2}}} + \frac{\operatorname{atanh}{\left (\frac{x}{\sqrt{-1 + \sqrt{2}}} \right )}}{4 \sqrt{-1 + \sqrt{2}}} - \frac{\operatorname{atanh}{\left (\frac{x}{\sqrt{1 + \sqrt{2}}} \right )}}{4 \sqrt{1 + \sqrt{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atan(x/sqrt(-1 + sqrt(2)))/(4*sqrt(-1 + sqrt(2))) - atan(x/sqrt(1 + sqrt(2)))/(4
*sqrt(1 + sqrt(2))) + atanh(x/sqrt(-1 + sqrt(2)))/(4*sqrt(-1 + sqrt(2))) - atanh
(x/sqrt(1 + sqrt(2)))/(4*sqrt(1 + sqrt(2)))

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Mathematica [A]  time = 0.0732383, size = 111, normalized size = 0.95 \[ \frac{1}{4} \left (\sqrt{1+\sqrt{2}} \tan ^{-1}\left (\frac{x}{\sqrt{\sqrt{2}-1}}\right )-\sqrt{\sqrt{2}-1} \tan ^{-1}\left (\frac{x}{\sqrt{1+\sqrt{2}}}\right )+\sqrt{1+\sqrt{2}} \tanh ^{-1}\left (\frac{x}{\sqrt{\sqrt{2}-1}}\right )-\sqrt{\sqrt{2}-1} \tanh ^{-1}\left (\frac{x}{\sqrt{1+\sqrt{2}}}\right )\right ) \]

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

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Maple [A]  time = 0.07, size = 78, normalized size = 0.7 \[{\frac{1}{4\,\sqrt{\sqrt{2}-1}}\arctan \left ({\frac{x}{\sqrt{\sqrt{2}-1}}} \right ) }+{\frac{1}{4\,\sqrt{\sqrt{2}-1}}{\it Artanh} \left ({\frac{x}{\sqrt{\sqrt{2}-1}}} \right ) }-{\frac{1}{4\,\sqrt{1+\sqrt{2}}}\arctan \left ({\frac{x}{\sqrt{1+\sqrt{2}}}} \right ) }-{\frac{1}{4\,\sqrt{1+\sqrt{2}}}{\it Artanh} \left ({\frac{x}{\sqrt{1+\sqrt{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^4+1)/(x^8-6*x^4+1),x)

[Out]

1/4*arctan(x/(2^(1/2)-1)^(1/2))/(2^(1/2)-1)^(1/2)+1/4*arctanh(x/(2^(1/2)-1)^(1/2
))/(2^(1/2)-1)^(1/2)-1/4*arctan(x/(1+2^(1/2))^(1/2))/(1+2^(1/2))^(1/2)-1/4*arcta
nh(x/(1+2^(1/2))^(1/2))/(1+2^(1/2))^(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 = 0.285506, size = 390, normalized size = 3.33 \[ \frac{1}{2} \, \sqrt{\frac{1}{2}} \sqrt{-\sqrt{2}{\left (\sqrt{2} - 2\right )}} \arctan \left (\frac{\sqrt{\frac{1}{2}} \sqrt{-\sqrt{2}{\left (\sqrt{2} - 2\right )}}{\left (\sqrt{2} + 1\right )}}{\sqrt{\frac{1}{2}} \sqrt{\sqrt{2}{\left (\sqrt{2}{\left (x^{2} + 1\right )} + 2\right )}} + x}\right ) - \frac{1}{2} \, \sqrt{\frac{1}{2}} \sqrt{\sqrt{2}{\left (\sqrt{2} + 2\right )}} \arctan \left (\frac{\sqrt{\frac{1}{2}} \sqrt{\sqrt{2}{\left (\sqrt{2} + 2\right )}}{\left (\sqrt{2} - 1\right )}}{\sqrt{\frac{1}{2}} \sqrt{\sqrt{2}{\left (\sqrt{2}{\left (x^{2} - 1\right )} + 2\right )}} + x}\right ) - \frac{1}{8} \, \sqrt{\frac{1}{2}} \sqrt{-\sqrt{2}{\left (\sqrt{2} - 2\right )}} \log \left (\sqrt{\frac{1}{2}} \sqrt{-\sqrt{2}{\left (\sqrt{2} - 2\right )}}{\left (\sqrt{2} + 1\right )} + x\right ) + \frac{1}{8} \, \sqrt{\frac{1}{2}} \sqrt{-\sqrt{2}{\left (\sqrt{2} - 2\right )}} \log \left (-\sqrt{\frac{1}{2}} \sqrt{-\sqrt{2}{\left (\sqrt{2} - 2\right )}}{\left (\sqrt{2} + 1\right )} + x\right ) + \frac{1}{8} \, \sqrt{\frac{1}{2}} \sqrt{\sqrt{2}{\left (\sqrt{2} + 2\right )}} \log \left (\sqrt{\frac{1}{2}} \sqrt{\sqrt{2}{\left (\sqrt{2} + 2\right )}}{\left (\sqrt{2} - 1\right )} + x\right ) - \frac{1}{8} \, \sqrt{\frac{1}{2}} \sqrt{\sqrt{2}{\left (\sqrt{2} + 2\right )}} \log \left (-\sqrt{\frac{1}{2}} \sqrt{\sqrt{2}{\left (\sqrt{2} + 2\right )}}{\left (\sqrt{2} - 1\right )} + x\right ) \]

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]

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

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Sympy [A]  time = 3.17126, size = 49, normalized size = 0.42 \[ \operatorname{RootSum}{\left (4096 t^{4} - 128 t^{2} - 1, \left ( t \mapsto t \log{\left (16384 t^{5} - 20 t + x \right )} \right )\right )} + \operatorname{RootSum}{\left (4096 t^{4} + 128 t^{2} - 1, \left ( t \mapsto t \log{\left (16384 t^{5} - 20 t + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(4096*_t**4 - 128*_t**2 - 1, Lambda(_t, _t*log(16384*_t**5 - 20*_t + x)))
+ RootSum(4096*_t**4 + 128*_t**2 - 1, Lambda(_t, _t*log(16384*_t**5 - 20*_t + x
)))

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GIAC/XCAS [A]  time = 0.346108, size = 166, normalized size = 1.42 \[ -\frac{1}{4} \, \sqrt{\sqrt{2} - 1} \arctan \left (\frac{x}{\sqrt{\sqrt{2} + 1}}\right ) + \frac{1}{4} \, \sqrt{\sqrt{2} + 1} \arctan \left (\frac{x}{\sqrt{\sqrt{2} - 1}}\right ) - \frac{1}{8} \, \sqrt{\sqrt{2} - 1}{\rm ln}\left ({\left | x + \sqrt{\sqrt{2} + 1} \right |}\right ) + \frac{1}{8} \, \sqrt{\sqrt{2} - 1}{\rm ln}\left ({\left | x - \sqrt{\sqrt{2} + 1} \right |}\right ) + \frac{1}{8} \, \sqrt{\sqrt{2} + 1}{\rm ln}\left ({\left | x + \sqrt{\sqrt{2} - 1} \right |}\right ) - \frac{1}{8} \, \sqrt{\sqrt{2} + 1}{\rm ln}\left ({\left | x - \sqrt{\sqrt{2} - 1} \right |}\right ) \]

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]

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