3.29 \(\int \frac{1-x^4}{1-5 x^4+x^8} \, dx\)

Optimal. Leaf size=169 \[ \frac{\tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{7}-\sqrt{3}}} x\right )}{\sqrt{14 \left (\sqrt{7}-\sqrt{3}\right )}}+\frac{\tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{3}+\sqrt{7}}} x\right )}{\sqrt{14 \left (\sqrt{3}+\sqrt{7}\right )}}+\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{7}-\sqrt{3}}} x\right )}{\sqrt{14 \left (\sqrt{7}-\sqrt{3}\right )}}+\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{3}+\sqrt{7}}} x\right )}{\sqrt{14 \left (\sqrt{3}+\sqrt{7}\right )}} \]

[Out]

ArcTan[Sqrt[2/(-Sqrt[3] + Sqrt[7])]*x]/Sqrt[14*(-Sqrt[3] + Sqrt[7])] + ArcTan[Sq
rt[2/(Sqrt[3] + Sqrt[7])]*x]/Sqrt[14*(Sqrt[3] + Sqrt[7])] + ArcTanh[Sqrt[2/(-Sqr
t[3] + Sqrt[7])]*x]/Sqrt[14*(-Sqrt[3] + Sqrt[7])] + ArcTanh[Sqrt[2/(Sqrt[3] + Sq
rt[7])]*x]/Sqrt[14*(Sqrt[3] + Sqrt[7])]

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Rubi [A]  time = 0.292635, antiderivative size = 169, 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 \[ \frac{\tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{7}-\sqrt{3}}} x\right )}{\sqrt{14 \left (\sqrt{7}-\sqrt{3}\right )}}+\frac{\tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{3}+\sqrt{7}}} x\right )}{\sqrt{14 \left (\sqrt{3}+\sqrt{7}\right )}}+\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{7}-\sqrt{3}}} x\right )}{\sqrt{14 \left (\sqrt{7}-\sqrt{3}\right )}}+\frac{\tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{3}+\sqrt{7}}} x\right )}{\sqrt{14 \left (\sqrt{3}+\sqrt{7}\right )}} \]

Antiderivative was successfully verified.

[In]  Int[(1 - x^4)/(1 - 5*x^4 + x^8),x]

[Out]

ArcTan[Sqrt[2/(-Sqrt[3] + Sqrt[7])]*x]/Sqrt[14*(-Sqrt[3] + Sqrt[7])] + ArcTan[Sq
rt[2/(Sqrt[3] + Sqrt[7])]*x]/Sqrt[14*(Sqrt[3] + Sqrt[7])] + ArcTanh[Sqrt[2/(-Sqr
t[3] + Sqrt[7])]*x]/Sqrt[14*(-Sqrt[3] + Sqrt[7])] + ArcTanh[Sqrt[2/(Sqrt[3] + Sq
rt[7])]*x]/Sqrt[14*(Sqrt[3] + Sqrt[7])]

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Rubi in Sympy [A]  time = 18.8649, size = 168, normalized size = 0.99 \[ \frac{\sqrt{14} \operatorname{atan}{\left (\frac{\sqrt{2} x}{\sqrt{- \sqrt{3} + \sqrt{7}}} \right )}}{14 \sqrt{- \sqrt{3} + \sqrt{7}}} + \frac{\sqrt{14} \operatorname{atan}{\left (\frac{\sqrt{2} x}{\sqrt{\sqrt{3} + \sqrt{7}}} \right )}}{14 \sqrt{\sqrt{3} + \sqrt{7}}} + \frac{\sqrt{14} \operatorname{atanh}{\left (\frac{\sqrt{2} x}{\sqrt{- \sqrt{3} + \sqrt{7}}} \right )}}{14 \sqrt{- \sqrt{3} + \sqrt{7}}} + \frac{\sqrt{14} \operatorname{atanh}{\left (\frac{\sqrt{2} x}{\sqrt{\sqrt{3} + \sqrt{7}}} \right )}}{14 \sqrt{\sqrt{3} + \sqrt{7}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-x**4+1)/(x**8-5*x**4+1),x)

[Out]

sqrt(14)*atan(sqrt(2)*x/sqrt(-sqrt(3) + sqrt(7)))/(14*sqrt(-sqrt(3) + sqrt(7)))
+ sqrt(14)*atan(sqrt(2)*x/sqrt(sqrt(3) + sqrt(7)))/(14*sqrt(sqrt(3) + sqrt(7)))
+ sqrt(14)*atanh(sqrt(2)*x/sqrt(-sqrt(3) + sqrt(7)))/(14*sqrt(-sqrt(3) + sqrt(7)
)) + sqrt(14)*atanh(sqrt(2)*x/sqrt(sqrt(3) + sqrt(7)))/(14*sqrt(sqrt(3) + sqrt(7
)))

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Mathematica [C]  time = 0.0207839, size = 57, normalized size = 0.34 \[ -\frac{1}{4} \text{RootSum}\left [\text{$\#$1}^8-5 \text{$\#$1}^4+1\&,\frac{\text{$\#$1}^4 \log (x-\text{$\#$1})-\log (x-\text{$\#$1})}{2 \text{$\#$1}^7-5 \text{$\#$1}^3}\&\right ] \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - x^4)/(1 - 5*x^4 + x^8),x]

[Out]

-RootSum[1 - 5*#1^4 + #1^8 & , (-Log[x - #1] + Log[x - #1]*#1^4)/(-5*#1^3 + 2*#1
^7) & ]/4

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Maple [C]  time = 0.01, size = 44, normalized size = 0.3 \[{\frac{1}{4}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}-5\,{{\it \_Z}}^{4}+1 \right ) }{\frac{ \left ( -{{\it \_R}}^{4}+1 \right ) \ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{7}-5\,{{\it \_R}}^{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x^4+1)/(x^8-5*x^4+1),x)

[Out]

1/4*sum((-_R^4+1)/(2*_R^7-5*_R^3)*ln(x-_R),_R=RootOf(_Z^8-5*_Z^4+1))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{x^{4} - 1}{x^{8} - 5 \, x^{4} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^4 - 1)/(x^8 - 5*x^4 + 1),x, algorithm="maxima")

[Out]

-integrate((x^4 - 1)/(x^8 - 5*x^4 + 1), x)

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Fricas [A]  time = 0.310394, size = 693, normalized size = 4.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^4 - 1)/(x^8 - 5*x^4 + 1),x, algorithm="fricas")

[Out]

-sqrt(1/7)*sqrt(sqrt(1/14)*sqrt(sqrt(7)*(5*sqrt(7) - 7*sqrt(3))))*arctan(7/2*sqr
t(1/7)*sqrt(sqrt(1/14)*sqrt(sqrt(7)*(5*sqrt(7) - 7*sqrt(3))))*(sqrt(7) + sqrt(3)
)/(sqrt(7)*x + sqrt(7)*sqrt(1/2*sqrt(1/14)*sqrt(sqrt(7)*(5*sqrt(7) - 7*sqrt(3)))
*(sqrt(7)*sqrt(3) + 5) + x^2))) - sqrt(1/7)*sqrt(sqrt(1/14)*sqrt(sqrt(7)*(5*sqrt
(7) + 7*sqrt(3))))*arctan(7/2*sqrt(1/7)*sqrt(sqrt(1/14)*sqrt(sqrt(7)*(5*sqrt(7)
+ 7*sqrt(3))))*(sqrt(7) - sqrt(3))/(sqrt(7)*x + sqrt(7)*sqrt(-1/2*sqrt(1/14)*sqr
t(sqrt(7)*(5*sqrt(7) + 7*sqrt(3)))*(sqrt(7)*sqrt(3) - 5) + x^2))) + 1/4*sqrt(1/7
)*sqrt(sqrt(1/14)*sqrt(sqrt(7)*(5*sqrt(7) - 7*sqrt(3))))*log(7/2*sqrt(1/7)*sqrt(
sqrt(1/14)*sqrt(sqrt(7)*(5*sqrt(7) - 7*sqrt(3))))*(sqrt(7) + sqrt(3)) + sqrt(7)*
x) - 1/4*sqrt(1/7)*sqrt(sqrt(1/14)*sqrt(sqrt(7)*(5*sqrt(7) - 7*sqrt(3))))*log(-7
/2*sqrt(1/7)*sqrt(sqrt(1/14)*sqrt(sqrt(7)*(5*sqrt(7) - 7*sqrt(3))))*(sqrt(7) + s
qrt(3)) + sqrt(7)*x) + 1/4*sqrt(1/7)*sqrt(sqrt(1/14)*sqrt(sqrt(7)*(5*sqrt(7) + 7
*sqrt(3))))*log(7/2*sqrt(1/7)*sqrt(sqrt(1/14)*sqrt(sqrt(7)*(5*sqrt(7) + 7*sqrt(3
))))*(sqrt(7) - sqrt(3)) + sqrt(7)*x) - 1/4*sqrt(1/7)*sqrt(sqrt(1/14)*sqrt(sqrt(
7)*(5*sqrt(7) + 7*sqrt(3))))*log(-7/2*sqrt(1/7)*sqrt(sqrt(1/14)*sqrt(sqrt(7)*(5*
sqrt(7) + 7*sqrt(3))))*(sqrt(7) - sqrt(3)) + sqrt(7)*x)

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Sympy [A]  time = 0.57599, size = 26, normalized size = 0.15 \[ - \operatorname{RootSum}{\left (157351936 t^{8} - 62720 t^{4} + 1, \left ( t \mapsto t \log{\left (50176 t^{5} - 24 t + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x**4+1)/(x**8-5*x**4+1),x)

[Out]

-RootSum(157351936*_t**8 - 62720*_t**4 + 1, Lambda(_t, _t*log(50176*_t**5 - 24*_
t + x)))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{x^{4} - 1}{x^{8} - 5 \, x^{4} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^4 - 1)/(x^8 - 5*x^4 + 1),x, algorithm="giac")

[Out]

integrate(-(x^4 - 1)/(x^8 - 5*x^4 + 1), x)