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

Optimal. Leaf size=355 \[ \frac{1}{8} \sqrt{\frac{1}{3} \left (2-\sqrt{3}\right )} \log \left (x^2-\sqrt{2-\sqrt{3}} x+1\right )-\frac{1}{8} \sqrt{\frac{1}{3} \left (2-\sqrt{3}\right )} \log \left (x^2+\sqrt{2-\sqrt{3}} x+1\right )-\frac{1}{8} \sqrt{\frac{1}{3} \left (2+\sqrt{3}\right )} \log \left (x^2-\sqrt{2+\sqrt{3}} x+1\right )+\frac{1}{8} \sqrt{\frac{1}{3} \left (2+\sqrt{3}\right )} \log \left (x^2+\sqrt{2+\sqrt{3}} x+1\right )-\frac{\tan ^{-1}\left (\frac{\sqrt{2-\sqrt{3}}-2 x}{\sqrt{2+\sqrt{3}}}\right )}{4 \sqrt{3 \left (2-\sqrt{3}\right )}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right )}{4 \sqrt{3 \left (2+\sqrt{3}\right )}}+\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2-\sqrt{3}}}{\sqrt{2+\sqrt{3}}}\right )}{4 \sqrt{3 \left (2-\sqrt{3}\right )}}-\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right )}{4 \sqrt{3 \left (2+\sqrt{3}\right )}} \]

[Out]

-ArcTan[(Sqrt[2 - Sqrt[3]] - 2*x)/Sqrt[2 + Sqrt[3]]]/(4*Sqrt[3*(2 - Sqrt[3])]) +
ArcTan[(Sqrt[2 + Sqrt[3]] - 2*x)/Sqrt[2 - Sqrt[3]]]/(4*Sqrt[3*(2 + Sqrt[3])]) +
ArcTan[(Sqrt[2 - Sqrt[3]] + 2*x)/Sqrt[2 + Sqrt[3]]]/(4*Sqrt[3*(2 - Sqrt[3])]) -
ArcTan[(Sqrt[2 + Sqrt[3]] + 2*x)/Sqrt[2 - Sqrt[3]]]/(4*Sqrt[3*(2 + Sqrt[3])]) +
(Sqrt[(2 - Sqrt[3])/3]*Log[1 - Sqrt[2 - Sqrt[3]]*x + x^2])/8 - (Sqrt[(2 - Sqrt[
3])/3]*Log[1 + Sqrt[2 - Sqrt[3]]*x + x^2])/8 - (Sqrt[(2 + Sqrt[3])/3]*Log[1 - Sq
rt[2 + Sqrt[3]]*x + x^2])/8 + (Sqrt[(2 + Sqrt[3])/3]*Log[1 + Sqrt[2 + Sqrt[3]]*x
+ x^2])/8

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Rubi [A]  time = 0.506497, antiderivative size = 355, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ \frac{1}{8} \sqrt{\frac{1}{3} \left (2-\sqrt{3}\right )} \log \left (x^2-\sqrt{2-\sqrt{3}} x+1\right )-\frac{1}{8} \sqrt{\frac{1}{3} \left (2-\sqrt{3}\right )} \log \left (x^2+\sqrt{2-\sqrt{3}} x+1\right )-\frac{1}{8} \sqrt{\frac{1}{3} \left (2+\sqrt{3}\right )} \log \left (x^2-\sqrt{2+\sqrt{3}} x+1\right )+\frac{1}{8} \sqrt{\frac{1}{3} \left (2+\sqrt{3}\right )} \log \left (x^2+\sqrt{2+\sqrt{3}} x+1\right )-\frac{\tan ^{-1}\left (\frac{\sqrt{2-\sqrt{3}}-2 x}{\sqrt{2+\sqrt{3}}}\right )}{4 \sqrt{3 \left (2-\sqrt{3}\right )}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right )}{4 \sqrt{3 \left (2+\sqrt{3}\right )}}+\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2-\sqrt{3}}}{\sqrt{2+\sqrt{3}}}\right )}{4 \sqrt{3 \left (2-\sqrt{3}\right )}}-\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right )}{4 \sqrt{3 \left (2+\sqrt{3}\right )}} \]

Antiderivative was successfully verified.

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

[Out]

-ArcTan[(Sqrt[2 - Sqrt[3]] - 2*x)/Sqrt[2 + Sqrt[3]]]/(4*Sqrt[3*(2 - Sqrt[3])]) +
ArcTan[(Sqrt[2 + Sqrt[3]] - 2*x)/Sqrt[2 - Sqrt[3]]]/(4*Sqrt[3*(2 + Sqrt[3])]) +
ArcTan[(Sqrt[2 - Sqrt[3]] + 2*x)/Sqrt[2 + Sqrt[3]]]/(4*Sqrt[3*(2 - Sqrt[3])]) -
ArcTan[(Sqrt[2 + Sqrt[3]] + 2*x)/Sqrt[2 - Sqrt[3]]]/(4*Sqrt[3*(2 + Sqrt[3])]) +
(Sqrt[(2 - Sqrt[3])/3]*Log[1 - Sqrt[2 - Sqrt[3]]*x + x^2])/8 - (Sqrt[(2 - Sqrt[
3])/3]*Log[1 + Sqrt[2 - Sqrt[3]]*x + x^2])/8 - (Sqrt[(2 + Sqrt[3])/3]*Log[1 - Sq
rt[2 + Sqrt[3]]*x + x^2])/8 + (Sqrt[(2 + Sqrt[3])/3]*Log[1 + Sqrt[2 + Sqrt[3]]*x
+ x^2])/8

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Rubi in Sympy [A]  time = 78.6325, size = 495, normalized size = 1.39 \[ \frac{\sqrt{3} \left (- \frac{\sqrt{3}}{2} + 1\right ) \log{\left (x^{2} - x \sqrt{- \sqrt{3} + 2} + 1 \right )}}{12 \sqrt{- \sqrt{3} + 2}} - \frac{\sqrt{3} \left (- \frac{\sqrt{3}}{2} + 1\right ) \log{\left (x^{2} + x \sqrt{- \sqrt{3} + 2} + 1 \right )}}{12 \sqrt{- \sqrt{3} + 2}} - \frac{\sqrt{3} \left (\frac{\sqrt{3}}{2} + 1\right ) \log{\left (x^{2} - x \sqrt{\sqrt{3} + 2} + 1 \right )}}{12 \sqrt{\sqrt{3} + 2}} + \frac{\sqrt{3} \left (\frac{\sqrt{3}}{2} + 1\right ) \log{\left (x^{2} + x \sqrt{\sqrt{3} + 2} + 1 \right )}}{12 \sqrt{\sqrt{3} + 2}} + \frac{\sqrt{3} \left (- \frac{\left (\sqrt{3} + 2\right )^{\frac{3}{2}}}{2} + \sqrt{3} \sqrt{\sqrt{3} + 2}\right ) \operatorname{atan}{\left (\frac{2 x - \sqrt{\sqrt{3} + 2}}{\sqrt{- \sqrt{3} + 2}} \right )}}{6 \sqrt{- \sqrt{3} + 2} \sqrt{\sqrt{3} + 2}} + \frac{\sqrt{3} \left (- \frac{\left (\sqrt{3} + 2\right )^{\frac{3}{2}}}{2} + \sqrt{3} \sqrt{\sqrt{3} + 2}\right ) \operatorname{atan}{\left (\frac{2 x + \sqrt{\sqrt{3} + 2}}{\sqrt{- \sqrt{3} + 2}} \right )}}{6 \sqrt{- \sqrt{3} + 2} \sqrt{\sqrt{3} + 2}} + \frac{\sqrt{3} \left (\frac{\left (- \sqrt{3} + 2\right )^{\frac{3}{2}}}{2} + \sqrt{3} \sqrt{- \sqrt{3} + 2}\right ) \operatorname{atan}{\left (\frac{2 x - \sqrt{- \sqrt{3} + 2}}{\sqrt{\sqrt{3} + 2}} \right )}}{6 \sqrt{- \sqrt{3} + 2} \sqrt{\sqrt{3} + 2}} + \frac{\sqrt{3} \left (\frac{\left (- \sqrt{3} + 2\right )^{\frac{3}{2}}}{2} + \sqrt{3} \sqrt{- \sqrt{3} + 2}\right ) \operatorname{atan}{\left (\frac{2 x + \sqrt{- \sqrt{3} + 2}}{\sqrt{\sqrt{3} + 2}} \right )}}{6 \sqrt{- \sqrt{3} + 2} \sqrt{\sqrt{3} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(3)*(-sqrt(3)/2 + 1)*log(x**2 - x*sqrt(-sqrt(3) + 2) + 1)/(12*sqrt(-sqrt(3)
+ 2)) - sqrt(3)*(-sqrt(3)/2 + 1)*log(x**2 + x*sqrt(-sqrt(3) + 2) + 1)/(12*sqrt(-
sqrt(3) + 2)) - sqrt(3)*(sqrt(3)/2 + 1)*log(x**2 - x*sqrt(sqrt(3) + 2) + 1)/(12*
sqrt(sqrt(3) + 2)) + sqrt(3)*(sqrt(3)/2 + 1)*log(x**2 + x*sqrt(sqrt(3) + 2) + 1)
/(12*sqrt(sqrt(3) + 2)) + sqrt(3)*(-(sqrt(3) + 2)**(3/2)/2 + sqrt(3)*sqrt(sqrt(3
) + 2))*atan((2*x - sqrt(sqrt(3) + 2))/sqrt(-sqrt(3) + 2))/(6*sqrt(-sqrt(3) + 2)
*sqrt(sqrt(3) + 2)) + sqrt(3)*(-(sqrt(3) + 2)**(3/2)/2 + sqrt(3)*sqrt(sqrt(3) +
2))*atan((2*x + sqrt(sqrt(3) + 2))/sqrt(-sqrt(3) + 2))/(6*sqrt(-sqrt(3) + 2)*sqr
t(sqrt(3) + 2)) + sqrt(3)*((-sqrt(3) + 2)**(3/2)/2 + sqrt(3)*sqrt(-sqrt(3) + 2))
*atan((2*x - sqrt(-sqrt(3) + 2))/sqrt(sqrt(3) + 2))/(6*sqrt(-sqrt(3) + 2)*sqrt(s
qrt(3) + 2)) + sqrt(3)*((-sqrt(3) + 2)**(3/2)/2 + sqrt(3)*sqrt(-sqrt(3) + 2))*at
an((2*x + sqrt(-sqrt(3) + 2))/sqrt(sqrt(3) + 2))/(6*sqrt(-sqrt(3) + 2)*sqrt(sqrt
(3) + 2))

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Mathematica [C]  time = 0.0213857, size = 57, normalized size = 0.16 \[ -\frac{1}{4} \text{RootSum}\left [\text{$\#$1}^8-\text{$\#$1}^4+1\&,\frac{\text{$\#$1}^4 \log (x-\text{$\#$1})-\log (x-\text{$\#$1})}{2 \text{$\#$1}^7-\text{$\#$1}^3}\&\right ] \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.001, size = 44, normalized size = 0.1 \[{\frac{1}{4}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}-{{\it \_Z}}^{4}+1 \right ) }{\frac{ \left ( -{{\it \_R}}^{4}+1 \right ) \ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{7}-{{\it \_R}}^{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x^4+1)/(x^8-x^4+1),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.279543, size = 1223, normalized size = 3.45 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*(4*(7*sqrt(3) + 12)*sqrt((sqrt(3) + 2)/(4*sqrt(3) + 7))*arctan(1/(2*(sqrt(3
) - 2)*sqrt(-(97*x^2 - 56*sqrt(3)*(x^2 + 1) + (209*sqrt(3)*x - 362*x)*sqrt((sqrt
(3) - 2)/(4*sqrt(3) - 7)) + 97)/(56*sqrt(3) - 97))*sqrt((sqrt(3) - 2)/(4*sqrt(3)
- 7)) + 2*(sqrt(3)*x - 2*x)*sqrt((sqrt(3) - 2)/(4*sqrt(3) - 7)) - sqrt(3) + 2))
+ 4*(7*sqrt(3) + 12)*sqrt((sqrt(3) + 2)/(4*sqrt(3) + 7))*arctan(1/(2*(sqrt(3) -
2)*sqrt(-(97*x^2 - 56*sqrt(3)*(x^2 + 1) - (209*sqrt(3)*x - 362*x)*sqrt((sqrt(3)
- 2)/(4*sqrt(3) - 7)) + 97)/(56*sqrt(3) - 97))*sqrt((sqrt(3) - 2)/(4*sqrt(3) -
7)) + 2*(sqrt(3)*x - 2*x)*sqrt((sqrt(3) - 2)/(4*sqrt(3) - 7)) + sqrt(3) - 2)) +
(2*sqrt(3) + 3)*sqrt((sqrt(3) - 2)/(4*sqrt(3) - 7))*log(194*x^2 + 112*sqrt(3)*(x
^2 + 1) + 2*(209*sqrt(3)*x + 362*x)*sqrt((sqrt(3) + 2)/(4*sqrt(3) + 7)) + 194) -
(2*sqrt(3) + 3)*sqrt((sqrt(3) - 2)/(4*sqrt(3) - 7))*log(194*x^2 + 112*sqrt(3)*(
x^2 + 1) - 2*(209*sqrt(3)*x + 362*x)*sqrt((sqrt(3) + 2)/(4*sqrt(3) + 7)) + 194)
- (2*sqrt(3) + 3)*sqrt((sqrt(3) + 2)/(4*sqrt(3) + 7))*log(-194*x^2 + 112*sqrt(3)
*(x^2 + 1) + 2*(209*sqrt(3)*x - 362*x)*sqrt((sqrt(3) - 2)/(4*sqrt(3) - 7)) - 194
) + (2*sqrt(3) + 3)*sqrt((sqrt(3) + 2)/(4*sqrt(3) + 7))*log(-194*x^2 + 112*sqrt(
3)*(x^2 + 1) - 2*(209*sqrt(3)*x - 362*x)*sqrt((sqrt(3) - 2)/(4*sqrt(3) - 7)) - 1
94) + 4*sqrt(3)*sqrt((sqrt(3) - 2)/(4*sqrt(3) - 7))*arctan(1/(2*(sqrt(3) + 2)*sq
rt((97*x^2 + 56*sqrt(3)*(x^2 + 1) + (209*sqrt(3)*x + 362*x)*sqrt((sqrt(3) + 2)/(
4*sqrt(3) + 7)) + 97)/(56*sqrt(3) + 97))*sqrt((sqrt(3) + 2)/(4*sqrt(3) + 7)) + 2
*(sqrt(3)*x + 2*x)*sqrt((sqrt(3) + 2)/(4*sqrt(3) + 7)) + sqrt(3) + 2)) + 4*sqrt(
3)*sqrt((sqrt(3) - 2)/(4*sqrt(3) - 7))*arctan(1/(2*(sqrt(3) + 2)*sqrt((97*x^2 +
56*sqrt(3)*(x^2 + 1) - (209*sqrt(3)*x + 362*x)*sqrt((sqrt(3) + 2)/(4*sqrt(3) + 7
)) + 97)/(56*sqrt(3) + 97))*sqrt((sqrt(3) + 2)/(4*sqrt(3) + 7)) + 2*(sqrt(3)*x +
2*x)*sqrt((sqrt(3) + 2)/(4*sqrt(3) + 7)) - sqrt(3) - 2)))/((sqrt(3) + 2)*sqrt((
sqrt(3) + 2)/(4*sqrt(3) + 7))*sqrt((sqrt(3) - 2)/(4*sqrt(3) - 7)))

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Sympy [A]  time = 4.64108, size = 26, normalized size = 0.07 \[ - \operatorname{RootSum}{\left (5308416 t^{8} - 2304 t^{4} + 1, \left ( t \mapsto t \log{\left (9216 t^{5} - 8 t + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-RootSum(5308416*_t**8 - 2304*_t**4 + 1, Lambda(_t, _t*log(9216*_t**5 - 8*_t + x
)))

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GIAC/XCAS [A]  time = 0.305732, size = 342, normalized size = 0.96 \[ \frac{1}{24} \,{\left (\sqrt{6} + 3 \, \sqrt{2}\right )} \arctan \left (\frac{4 \, x + \sqrt{6} - \sqrt{2}}{\sqrt{6} + \sqrt{2}}\right ) + \frac{1}{24} \,{\left (\sqrt{6} + 3 \, \sqrt{2}\right )} \arctan \left (\frac{4 \, x - \sqrt{6} + \sqrt{2}}{\sqrt{6} + \sqrt{2}}\right ) + \frac{1}{24} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )} \arctan \left (\frac{4 \, x + \sqrt{6} + \sqrt{2}}{\sqrt{6} - \sqrt{2}}\right ) + \frac{1}{24} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )} \arctan \left (\frac{4 \, x - \sqrt{6} - \sqrt{2}}{\sqrt{6} - \sqrt{2}}\right ) + \frac{1}{48} \,{\left (\sqrt{6} + 3 \, \sqrt{2}\right )}{\rm ln}\left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{6} + \sqrt{2}\right )} + 1\right ) - \frac{1}{48} \,{\left (\sqrt{6} + 3 \, \sqrt{2}\right )}{\rm ln}\left (x^{2} - \frac{1}{2} \, x{\left (\sqrt{6} + \sqrt{2}\right )} + 1\right ) + \frac{1}{48} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )}{\rm ln}\left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{6} - \sqrt{2}\right )} + 1\right ) - \frac{1}{48} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )}{\rm ln}\left (x^{2} - \frac{1}{2} \, x{\left (\sqrt{6} - \sqrt{2}\right )} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*(sqrt(6) + 3*sqrt(2))*arctan((4*x + sqrt(6) - sqrt(2))/(sqrt(6) + sqrt(2)))
+ 1/24*(sqrt(6) + 3*sqrt(2))*arctan((4*x - sqrt(6) + sqrt(2))/(sqrt(6) + sqrt(2
))) + 1/24*(sqrt(6) - 3*sqrt(2))*arctan((4*x + sqrt(6) + sqrt(2))/(sqrt(6) - sqr
t(2))) + 1/24*(sqrt(6) - 3*sqrt(2))*arctan((4*x - sqrt(6) - sqrt(2))/(sqrt(6) -
sqrt(2))) + 1/48*(sqrt(6) + 3*sqrt(2))*ln(x^2 + 1/2*x*(sqrt(6) + sqrt(2)) + 1) -
1/48*(sqrt(6) + 3*sqrt(2))*ln(x^2 - 1/2*x*(sqrt(6) + sqrt(2)) + 1) + 1/48*(sqrt
(6) - 3*sqrt(2))*ln(x^2 + 1/2*x*(sqrt(6) - sqrt(2)) + 1) - 1/48*(sqrt(6) - 3*sqr
t(2))*ln(x^2 - 1/2*x*(sqrt(6) - sqrt(2)) + 1)