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

Optimal. Leaf size=429 \[ -\frac{\sqrt [4]{3+\sqrt{5}} \log \left (\sqrt{2} x^2-2^{3/4} \sqrt [4]{3-\sqrt{5}} x+\sqrt{3-\sqrt{5}}\right )}{4\ 2^{3/4}}+\frac{\sqrt [4]{3+\sqrt{5}} \log \left (\sqrt{2} x^2+2^{3/4} \sqrt [4]{3-\sqrt{5}} x+\sqrt{3-\sqrt{5}}\right )}{4\ 2^{3/4}}+\frac{\sqrt [4]{3-\sqrt{5}} \log \left (\sqrt{2} x^2-2^{3/4} \sqrt [4]{3+\sqrt{5}} x+\sqrt{3+\sqrt{5}}\right )}{4\ 2^{3/4}}-\frac{\sqrt [4]{3-\sqrt{5}} \log \left (\sqrt{2} x^2+2^{3/4} \sqrt [4]{3+\sqrt{5}} x+\sqrt{3+\sqrt{5}}\right )}{4\ 2^{3/4}}-\frac{\sqrt [4]{3+\sqrt{5}} \tan ^{-1}\left (1-\frac{2^{3/4} x}{\sqrt [4]{3-\sqrt{5}}}\right )}{2\ 2^{3/4}}+\frac{\sqrt [4]{3+\sqrt{5}} \tan ^{-1}\left (\frac{2^{3/4} x}{\sqrt [4]{3-\sqrt{5}}}+1\right )}{2\ 2^{3/4}}+\frac{\sqrt [4]{3-\sqrt{5}} \tan ^{-1}\left (1-\frac{2^{3/4} x}{\sqrt [4]{3+\sqrt{5}}}\right )}{2\ 2^{3/4}}-\frac{\sqrt [4]{3-\sqrt{5}} \tan ^{-1}\left (\frac{2^{3/4} x}{\sqrt [4]{3+\sqrt{5}}}+1\right )}{2\ 2^{3/4}} \]

[Out]

-((3 + Sqrt[5])^(1/4)*ArcTan[1 - (2^(3/4)*x)/(3 - Sqrt[5])^(1/4)])/(2*2^(3/4)) +
((3 + Sqrt[5])^(1/4)*ArcTan[1 + (2^(3/4)*x)/(3 - Sqrt[5])^(1/4)])/(2*2^(3/4)) +
((3 - Sqrt[5])^(1/4)*ArcTan[1 - (2^(3/4)*x)/(3 + Sqrt[5])^(1/4)])/(2*2^(3/4)) -
((3 - Sqrt[5])^(1/4)*ArcTan[1 + (2^(3/4)*x)/(3 + Sqrt[5])^(1/4)])/(2*2^(3/4)) -
((3 + Sqrt[5])^(1/4)*Log[Sqrt[3 - Sqrt[5]] - 2^(3/4)*(3 - Sqrt[5])^(1/4)*x + Sq
rt[2]*x^2])/(4*2^(3/4)) + ((3 + Sqrt[5])^(1/4)*Log[Sqrt[3 - Sqrt[5]] + 2^(3/4)*(
3 - Sqrt[5])^(1/4)*x + Sqrt[2]*x^2])/(4*2^(3/4)) + ((3 - Sqrt[5])^(1/4)*Log[Sqrt
[3 + Sqrt[5]] - 2^(3/4)*(3 + Sqrt[5])^(1/4)*x + Sqrt[2]*x^2])/(4*2^(3/4)) - ((3
- Sqrt[5])^(1/4)*Log[Sqrt[3 + Sqrt[5]] + 2^(3/4)*(3 + Sqrt[5])^(1/4)*x + Sqrt[2]
*x^2])/(4*2^(3/4))

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Rubi [A]  time = 0.704994, antiderivative size = 411, normalized size of antiderivative = 0.96, number of steps used = 19, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ -\frac{\sqrt [4]{3+\sqrt{5}} \log \left (2 x^2-2 \sqrt [4]{2 \left (3-\sqrt{5}\right )} x+\sqrt{2 \left (3-\sqrt{5}\right )}\right )}{4\ 2^{3/4}}+\frac{\sqrt [4]{3+\sqrt{5}} \log \left (2 x^2+2 \sqrt [4]{2 \left (3-\sqrt{5}\right )} x+\sqrt{2 \left (3-\sqrt{5}\right )}\right )}{4\ 2^{3/4}}+\frac{\sqrt [4]{3-\sqrt{5}} \log \left (2 x^2-2 \sqrt [4]{2 \left (3+\sqrt{5}\right )} x+\sqrt{2 \left (3+\sqrt{5}\right )}\right )}{4\ 2^{3/4}}-\frac{\sqrt [4]{3-\sqrt{5}} \log \left (2 x^2+2 \sqrt [4]{2 \left (3+\sqrt{5}\right )} x+\sqrt{2 \left (3+\sqrt{5}\right )}\right )}{4\ 2^{3/4}}-\frac{\sqrt [4]{3+\sqrt{5}} \tan ^{-1}\left (1-\frac{2^{3/4} x}{\sqrt [4]{3-\sqrt{5}}}\right )}{2\ 2^{3/4}}+\frac{\sqrt [4]{3+\sqrt{5}} \tan ^{-1}\left (\frac{2^{3/4} x}{\sqrt [4]{3-\sqrt{5}}}+1\right )}{2\ 2^{3/4}}+\frac{\sqrt [4]{3-\sqrt{5}} \tan ^{-1}\left (1-\frac{2^{3/4} x}{\sqrt [4]{3+\sqrt{5}}}\right )}{2\ 2^{3/4}}-\frac{\sqrt [4]{3-\sqrt{5}} \tan ^{-1}\left (\frac{2^{3/4} x}{\sqrt [4]{3+\sqrt{5}}}+1\right )}{2\ 2^{3/4}} \]

Warning: Unable to verify antiderivative.

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

[Out]

-((3 + Sqrt[5])^(1/4)*ArcTan[1 - (2^(3/4)*x)/(3 - Sqrt[5])^(1/4)])/(2*2^(3/4)) +
((3 + Sqrt[5])^(1/4)*ArcTan[1 + (2^(3/4)*x)/(3 - Sqrt[5])^(1/4)])/(2*2^(3/4)) +
((3 - Sqrt[5])^(1/4)*ArcTan[1 - (2^(3/4)*x)/(3 + Sqrt[5])^(1/4)])/(2*2^(3/4)) -
((3 - Sqrt[5])^(1/4)*ArcTan[1 + (2^(3/4)*x)/(3 + Sqrt[5])^(1/4)])/(2*2^(3/4)) -
((3 + Sqrt[5])^(1/4)*Log[Sqrt[2*(3 - Sqrt[5])] - 2*(2*(3 - Sqrt[5]))^(1/4)*x +
2*x^2])/(4*2^(3/4)) + ((3 + Sqrt[5])^(1/4)*Log[Sqrt[2*(3 - Sqrt[5])] + 2*(2*(3 -
Sqrt[5]))^(1/4)*x + 2*x^2])/(4*2^(3/4)) + ((3 - Sqrt[5])^(1/4)*Log[Sqrt[2*(3 +
Sqrt[5])] - 2*(2*(3 + Sqrt[5]))^(1/4)*x + 2*x^2])/(4*2^(3/4)) - ((3 - Sqrt[5])^(
1/4)*Log[Sqrt[2*(3 + Sqrt[5])] + 2*(2*(3 + Sqrt[5]))^(1/4)*x + 2*x^2])/(4*2^(3/4
))

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Rubi in Sympy [A]  time = 85.7917, size = 590, normalized size = 1.38 \[ \frac{2^{\frac{3}{4}} \sqrt{- 2 \sqrt{5} + 6} \left (- \frac{\sqrt{5}}{2} + \frac{1}{2}\right ) \log{\left (2 x^{2} - 2 \sqrt [4]{2} x \sqrt [4]{- \sqrt{5} + 3} + \sqrt{- 2 \sqrt{5} + 6} \right )}}{8 \left (- \sqrt{5} + 3\right )^{\frac{5}{4}}} - \frac{2^{\frac{3}{4}} \sqrt{- 2 \sqrt{5} + 6} \left (- \frac{\sqrt{5}}{2} + \frac{1}{2}\right ) \log{\left (2 x^{2} + 2 \sqrt [4]{2} x \sqrt [4]{- \sqrt{5} + 3} + \sqrt{- 2 \sqrt{5} + 6} \right )}}{8 \left (- \sqrt{5} + 3\right )^{\frac{5}{4}}} + \frac{2^{\frac{3}{4}} \left (\frac{1}{2} + \frac{\sqrt{5}}{2}\right ) \sqrt{2 \sqrt{5} + 6} \log{\left (2 x^{2} - 2 \sqrt [4]{2} x \sqrt [4]{\sqrt{5} + 3} + \sqrt{2 \sqrt{5} + 6} \right )}}{8 \left (\sqrt{5} + 3\right )^{\frac{5}{4}}} - \frac{2^{\frac{3}{4}} \left (\frac{1}{2} + \frac{\sqrt{5}}{2}\right ) \sqrt{2 \sqrt{5} + 6} \log{\left (2 x^{2} + 2 \sqrt [4]{2} x \sqrt [4]{\sqrt{5} + 3} + \sqrt{2 \sqrt{5} + 6} \right )}}{8 \left (\sqrt{5} + 3\right )^{\frac{5}{4}}} - \frac{2^{\frac{3}{4}} \left (- \frac{\sqrt{5}}{2} + \frac{1}{2}\right ) \operatorname{atan}{\left (\frac{2^{\frac{3}{4}} \left (x - \frac{\sqrt [4]{- 2 \sqrt{5} + 6}}{2}\right )}{\sqrt [4]{- \sqrt{5} + 3}} \right )}}{2 \sqrt{- 2 \sqrt{5} + 6} \sqrt [4]{- \sqrt{5} + 3}} - \frac{2^{\frac{3}{4}} \left (- \frac{\sqrt{5}}{2} + \frac{1}{2}\right ) \operatorname{atan}{\left (\frac{2^{\frac{3}{4}} \left (x + \frac{\sqrt [4]{- 2 \sqrt{5} + 6}}{2}\right )}{\sqrt [4]{- \sqrt{5} + 3}} \right )}}{2 \sqrt{- 2 \sqrt{5} + 6} \sqrt [4]{- \sqrt{5} + 3}} - \frac{2^{\frac{3}{4}} \left (\frac{1}{2} + \frac{\sqrt{5}}{2}\right ) \operatorname{atan}{\left (\frac{2^{\frac{3}{4}} \left (x - \frac{\sqrt [4]{2 \sqrt{5} + 6}}{2}\right )}{\sqrt [4]{\sqrt{5} + 3}} \right )}}{2 \sqrt [4]{\sqrt{5} + 3} \sqrt{2 \sqrt{5} + 6}} - \frac{2^{\frac{3}{4}} \left (\frac{1}{2} + \frac{\sqrt{5}}{2}\right ) \operatorname{atan}{\left (\frac{2^{\frac{3}{4}} \left (x + \frac{\sqrt [4]{2 \sqrt{5} + 6}}{2}\right )}{\sqrt [4]{\sqrt{5} + 3}} \right )}}{2 \sqrt [4]{\sqrt{5} + 3} \sqrt{2 \sqrt{5} + 6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2**(3/4)*sqrt(-2*sqrt(5) + 6)*(-sqrt(5)/2 + 1/2)*log(2*x**2 - 2*2**(1/4)*x*(-sqr
t(5) + 3)**(1/4) + sqrt(-2*sqrt(5) + 6))/(8*(-sqrt(5) + 3)**(5/4)) - 2**(3/4)*sq
rt(-2*sqrt(5) + 6)*(-sqrt(5)/2 + 1/2)*log(2*x**2 + 2*2**(1/4)*x*(-sqrt(5) + 3)**
(1/4) + sqrt(-2*sqrt(5) + 6))/(8*(-sqrt(5) + 3)**(5/4)) + 2**(3/4)*(1/2 + sqrt(5
)/2)*sqrt(2*sqrt(5) + 6)*log(2*x**2 - 2*2**(1/4)*x*(sqrt(5) + 3)**(1/4) + sqrt(2
*sqrt(5) + 6))/(8*(sqrt(5) + 3)**(5/4)) - 2**(3/4)*(1/2 + sqrt(5)/2)*sqrt(2*sqrt
(5) + 6)*log(2*x**2 + 2*2**(1/4)*x*(sqrt(5) + 3)**(1/4) + sqrt(2*sqrt(5) + 6))/(
8*(sqrt(5) + 3)**(5/4)) - 2**(3/4)*(-sqrt(5)/2 + 1/2)*atan(2**(3/4)*(x - (-2*sqr
t(5) + 6)**(1/4)/2)/(-sqrt(5) + 3)**(1/4))/(2*sqrt(-2*sqrt(5) + 6)*(-sqrt(5) + 3
)**(1/4)) - 2**(3/4)*(-sqrt(5)/2 + 1/2)*atan(2**(3/4)*(x + (-2*sqrt(5) + 6)**(1/
4)/2)/(-sqrt(5) + 3)**(1/4))/(2*sqrt(-2*sqrt(5) + 6)*(-sqrt(5) + 3)**(1/4)) - 2*
*(3/4)*(1/2 + sqrt(5)/2)*atan(2**(3/4)*(x - (2*sqrt(5) + 6)**(1/4)/2)/(sqrt(5) +
3)**(1/4))/(2*(sqrt(5) + 3)**(1/4)*sqrt(2*sqrt(5) + 6)) - 2**(3/4)*(1/2 + sqrt(
5)/2)*atan(2**(3/4)*(x + (2*sqrt(5) + 6)**(1/4)/2)/(sqrt(5) + 3)**(1/4))/(2*(sqr
t(5) + 3)**(1/4)*sqrt(2*sqrt(5) + 6))

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Mathematica [C]  time = 0.0214814, size = 57, normalized size = 0.13 \[ -\frac{1}{4} \text{RootSum}\left [\text{$\#$1}^8+3 \text{$\#$1}^4+1\&,\frac{\text{$\#$1}^4 \log (x-\text{$\#$1})-\log (x-\text{$\#$1})}{2 \text{$\#$1}^7+3 \text{$\#$1}^3}\&\right ] \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.009, size = 44, normalized size = 0.1 \[{\frac{1}{4}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}+3\,{{\it \_Z}}^{4}+1 \right ) }{\frac{ \left ( -{{\it \_R}}^{4}+1 \right ) \ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{7}+3\,{{\it \_R}}^{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x^4+1)/(x^8+3*x^4+1),x)

[Out]

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

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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 = 0.295184, size = 1007, normalized size = 2.35 \[ \text{result too large to display} \]

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]

-1/16*2^(3/4)*(4*sqrt(2)*(sqrt(5) + 3)^(1/4)*arctan((sqrt(5)*sqrt(2) - sqrt(2))*
(sqrt(5) + 3)^(1/4)/(4*2^(1/4)*x + (sqrt(5)*sqrt(2) - sqrt(2))*(sqrt(5) + 3)^(1/
4) + 2*2^(1/4)*sqrt(sqrt(2)*(2*sqrt(2)*x^2 - sqrt(sqrt(5) + 3)*(sqrt(5) - 3) + (
sqrt(5)*2^(3/4)*x - 2^(3/4)*x)*(sqrt(5) + 3)^(1/4))))) + 4*sqrt(2)*(sqrt(5) + 3)
^(1/4)*arctan((sqrt(5)*sqrt(2) - sqrt(2))*(sqrt(5) + 3)^(1/4)/(4*2^(1/4)*x - (sq
rt(5)*sqrt(2) - sqrt(2))*(sqrt(5) + 3)^(1/4) + 2*2^(1/4)*sqrt(sqrt(2)*(2*sqrt(2)
*x^2 - sqrt(sqrt(5) + 3)*(sqrt(5) - 3) - (sqrt(5)*2^(3/4)*x - 2^(3/4)*x)*(sqrt(5
) + 3)^(1/4))))) - 4*sqrt(2)*(-sqrt(5) + 3)^(1/4)*arctan((sqrt(5)*sqrt(2) + sqrt
(2))*(-sqrt(5) + 3)^(1/4)/(4*2^(1/4)*x + (sqrt(5)*sqrt(2) + sqrt(2))*(-sqrt(5) +
3)^(1/4) + 2*2^(1/4)*sqrt(sqrt(2)*(2*sqrt(2)*x^2 + (sqrt(5) + 3)*sqrt(-sqrt(5)
+ 3) + (sqrt(5)*2^(3/4)*x + 2^(3/4)*x)*(-sqrt(5) + 3)^(1/4))))) - 4*sqrt(2)*(-sq
rt(5) + 3)^(1/4)*arctan((sqrt(5)*sqrt(2) + sqrt(2))*(-sqrt(5) + 3)^(1/4)/(4*2^(1
/4)*x - (sqrt(5)*sqrt(2) + sqrt(2))*(-sqrt(5) + 3)^(1/4) + 2*2^(1/4)*sqrt(sqrt(2
)*(2*sqrt(2)*x^2 + (sqrt(5) + 3)*sqrt(-sqrt(5) + 3) - (sqrt(5)*2^(3/4)*x + 2^(3/
4)*x)*(-sqrt(5) + 3)^(1/4))))) - sqrt(2)*(sqrt(5) + 3)^(1/4)*log(2*sqrt(2)*x^2 -
sqrt(sqrt(5) + 3)*(sqrt(5) - 3) + (sqrt(5)*2^(3/4)*x - 2^(3/4)*x)*(sqrt(5) + 3)
^(1/4)) + sqrt(2)*(sqrt(5) + 3)^(1/4)*log(2*sqrt(2)*x^2 - sqrt(sqrt(5) + 3)*(sqr
t(5) - 3) - (sqrt(5)*2^(3/4)*x - 2^(3/4)*x)*(sqrt(5) + 3)^(1/4)) + sqrt(2)*(-sqr
t(5) + 3)^(1/4)*log(2*sqrt(2)*x^2 + (sqrt(5) + 3)*sqrt(-sqrt(5) + 3) + (sqrt(5)*
2^(3/4)*x + 2^(3/4)*x)*(-sqrt(5) + 3)^(1/4)) - sqrt(2)*(-sqrt(5) + 3)^(1/4)*log(
2*sqrt(2)*x^2 + (sqrt(5) + 3)*sqrt(-sqrt(5) + 3) - (sqrt(5)*2^(3/4)*x + 2^(3/4)*
x)*(-sqrt(5) + 3)^(1/4)))

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Sympy [A]  time = 3.80062, size = 26, normalized size = 0.06 \[ - \operatorname{RootSum}{\left (65536 t^{8} + 768 t^{4} + 1, \left ( t \mapsto t \log{\left (1024 t^{5} + 8 t + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-RootSum(65536*_t**8 + 768*_t**4 + 1, Lambda(_t, _t*log(1024*_t**5 + 8*_t + x)))

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GIAC/XCAS [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="giac")

[Out]

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