3.387 \(\int \frac{x^9}{1-3 x^4+x^8} \, dx\)

Optimal. Leaf size=90 \[ \frac{x^2}{2}-\frac{1}{2} \sqrt{\frac{1}{5} \left (9+4 \sqrt{5}\right )} \tanh ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x^2\right )+\frac{1}{2} \sqrt{\frac{1}{5} \left (9-4 \sqrt{5}\right )} \tanh ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x^2\right ) \]

[Out]

x^2/2 - (Sqrt[(9 + 4*Sqrt[5])/5]*ArcTanh[Sqrt[2/(3 + Sqrt[5])]*x^2])/2 + (Sqrt[(
9 - 4*Sqrt[5])/5]*ArcTanh[Sqrt[(3 + Sqrt[5])/2]*x^2])/2

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Rubi [A]  time = 0.245514, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{x^2}{2}-\frac{1}{2} \sqrt{\frac{1}{5} \left (9+4 \sqrt{5}\right )} \tanh ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x^2\right )+\frac{1}{2} \sqrt{\frac{1}{5} \left (9-4 \sqrt{5}\right )} \tanh ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x^2\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^9/(1 - 3*x^4 + x^8),x]

[Out]

x^2/2 - (Sqrt[(9 + 4*Sqrt[5])/5]*ArcTanh[Sqrt[2/(3 + Sqrt[5])]*x^2])/2 + (Sqrt[(
9 - 4*Sqrt[5])/5]*ArcTanh[Sqrt[(3 + Sqrt[5])/2]*x^2])/2

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Rubi in Sympy [A]  time = 21.8075, size = 102, normalized size = 1.13 \[ \frac{x^{2}}{2} - \frac{\sqrt{2} \left (- \frac{7 \sqrt{5}}{10} + \frac{3}{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{2} x^{2}}{\sqrt{- \sqrt{5} + 3}} \right )}}{2 \sqrt{- \sqrt{5} + 3}} - \frac{\sqrt{2} \left (\frac{3}{2} + \frac{7 \sqrt{5}}{10}\right ) \operatorname{atanh}{\left (\frac{\sqrt{2} x^{2}}{\sqrt{\sqrt{5} + 3}} \right )}}{2 \sqrt{\sqrt{5} + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**9/(x**8-3*x**4+1),x)

[Out]

x**2/2 - sqrt(2)*(-7*sqrt(5)/10 + 3/2)*atanh(sqrt(2)*x**2/sqrt(-sqrt(5) + 3))/(2
*sqrt(-sqrt(5) + 3)) - sqrt(2)*(3/2 + 7*sqrt(5)/10)*atanh(sqrt(2)*x**2/sqrt(sqrt
(5) + 3))/(2*sqrt(sqrt(5) + 3))

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Mathematica [A]  time = 0.0883431, size = 103, normalized size = 1.14 \[ \frac{1}{20} \left (10 x^2+\left (2 \sqrt{5}-5\right ) \log \left (-2 x^2+\sqrt{5}-1\right )+\left (5+2 \sqrt{5}\right ) \log \left (-2 x^2+\sqrt{5}+1\right )+\left (5-2 \sqrt{5}\right ) \log \left (2 x^2+\sqrt{5}-1\right )-\left (5+2 \sqrt{5}\right ) \log \left (2 x^2+\sqrt{5}+1\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^9/(1 - 3*x^4 + x^8),x]

[Out]

(10*x^2 + (-5 + 2*Sqrt[5])*Log[-1 + Sqrt[5] - 2*x^2] + (5 + 2*Sqrt[5])*Log[1 + S
qrt[5] - 2*x^2] + (5 - 2*Sqrt[5])*Log[-1 + Sqrt[5] + 2*x^2] - (5 + 2*Sqrt[5])*Lo
g[1 + Sqrt[5] + 2*x^2])/20

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Maple [A]  time = 0.008, size = 67, normalized size = 0.7 \[{\frac{{x}^{2}}{2}}+{\frac{\ln \left ({x}^{4}-{x}^{2}-1 \right ) }{4}}-{\frac{\sqrt{5}}{5}{\it Artanh} \left ({\frac{ \left ( 2\,{x}^{2}-1 \right ) \sqrt{5}}{5}} \right ) }-{\frac{\ln \left ({x}^{4}+{x}^{2}-1 \right ) }{4}}-{\frac{\sqrt{5}}{5}{\it Artanh} \left ({\frac{ \left ( 2\,{x}^{2}+1 \right ) \sqrt{5}}{5}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^9/(x^8-3*x^4+1),x)

[Out]

1/2*x^2+1/4*ln(x^4-x^2-1)-1/5*5^(1/2)*arctanh(1/5*(2*x^2-1)*5^(1/2))-1/4*ln(x^4+
x^2-1)-1/5*5^(1/2)*arctanh(1/5*(2*x^2+1)*5^(1/2))

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Maxima [A]  time = 0.820497, size = 124, normalized size = 1.38 \[ \frac{1}{2} \, x^{2} + \frac{1}{10} \, \sqrt{5} \log \left (\frac{2 \, x^{2} - \sqrt{5} + 1}{2 \, x^{2} + \sqrt{5} + 1}\right ) + \frac{1}{10} \, \sqrt{5} \log \left (\frac{2 \, x^{2} - \sqrt{5} - 1}{2 \, x^{2} + \sqrt{5} - 1}\right ) - \frac{1}{4} \, \log \left (x^{4} + x^{2} - 1\right ) + \frac{1}{4} \, \log \left (x^{4} - x^{2} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(x^8 - 3*x^4 + 1),x, algorithm="maxima")

[Out]

1/2*x^2 + 1/10*sqrt(5)*log((2*x^2 - sqrt(5) + 1)/(2*x^2 + sqrt(5) + 1)) + 1/10*s
qrt(5)*log((2*x^2 - sqrt(5) - 1)/(2*x^2 + sqrt(5) - 1)) - 1/4*log(x^4 + x^2 - 1)
+ 1/4*log(x^4 - x^2 - 1)

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Fricas [A]  time = 0.270642, size = 166, normalized size = 1.84 \[ \frac{1}{20} \, \sqrt{5}{\left (2 \, \sqrt{5} x^{2} - \sqrt{5} \log \left (x^{4} + x^{2} - 1\right ) + \sqrt{5} \log \left (x^{4} - x^{2} - 1\right ) + 2 \, \log \left (-\frac{10 \, x^{2} - \sqrt{5}{\left (2 \, x^{4} + 2 \, x^{2} + 3\right )} + 5}{x^{4} + x^{2} - 1}\right ) + 2 \, \log \left (-\frac{10 \, x^{2} - \sqrt{5}{\left (2 \, x^{4} - 2 \, x^{2} + 3\right )} - 5}{x^{4} - x^{2} - 1}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(x^8 - 3*x^4 + 1),x, algorithm="fricas")

[Out]

1/20*sqrt(5)*(2*sqrt(5)*x^2 - sqrt(5)*log(x^4 + x^2 - 1) + sqrt(5)*log(x^4 - x^2
- 1) + 2*log(-(10*x^2 - sqrt(5)*(2*x^4 + 2*x^2 + 3) + 5)/(x^4 + x^2 - 1)) + 2*l
og(-(10*x^2 - sqrt(5)*(2*x^4 - 2*x^2 + 3) - 5)/(x^4 - x^2 - 1)))

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Sympy [A]  time = 1.75372, size = 170, normalized size = 1.89 \[ \frac{x^{2}}{2} + \left (- \frac{1}{4} - \frac{\sqrt{5}}{10}\right ) \log{\left (x^{2} - \frac{47}{8} - \frac{47 \sqrt{5}}{20} - 120 \left (- \frac{1}{4} - \frac{\sqrt{5}}{10}\right )^{3} \right )} + \left (- \frac{1}{4} + \frac{\sqrt{5}}{10}\right ) \log{\left (x^{2} - \frac{47}{8} - 120 \left (- \frac{1}{4} + \frac{\sqrt{5}}{10}\right )^{3} + \frac{47 \sqrt{5}}{20} \right )} + \left (- \frac{\sqrt{5}}{10} + \frac{1}{4}\right ) \log{\left (x^{2} - \frac{47 \sqrt{5}}{20} - 120 \left (- \frac{\sqrt{5}}{10} + \frac{1}{4}\right )^{3} + \frac{47}{8} \right )} + \left (\frac{\sqrt{5}}{10} + \frac{1}{4}\right ) \log{\left (x^{2} - 120 \left (\frac{\sqrt{5}}{10} + \frac{1}{4}\right )^{3} + \frac{47 \sqrt{5}}{20} + \frac{47}{8} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**9/(x**8-3*x**4+1),x)

[Out]

x**2/2 + (-1/4 - sqrt(5)/10)*log(x**2 - 47/8 - 47*sqrt(5)/20 - 120*(-1/4 - sqrt(
5)/10)**3) + (-1/4 + sqrt(5)/10)*log(x**2 - 47/8 - 120*(-1/4 + sqrt(5)/10)**3 +
47*sqrt(5)/20) + (-sqrt(5)/10 + 1/4)*log(x**2 - 47*sqrt(5)/20 - 120*(-sqrt(5)/10
+ 1/4)**3 + 47/8) + (sqrt(5)/10 + 1/4)*log(x**2 - 120*(sqrt(5)/10 + 1/4)**3 + 4
7*sqrt(5)/20 + 47/8)

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GIAC/XCAS [A]  time = 0.31495, size = 131, normalized size = 1.46 \[ \frac{1}{2} \, x^{2} + \frac{1}{10} \, \sqrt{5}{\rm ln}\left (\frac{{\left | 2 \, x^{2} - \sqrt{5} + 1 \right |}}{2 \, x^{2} + \sqrt{5} + 1}\right ) + \frac{1}{10} \, \sqrt{5}{\rm ln}\left (\frac{{\left | 2 \, x^{2} - \sqrt{5} - 1 \right |}}{{\left | 2 \, x^{2} + \sqrt{5} - 1 \right |}}\right ) - \frac{1}{4} \,{\rm ln}\left ({\left | x^{4} + x^{2} - 1 \right |}\right ) + \frac{1}{4} \,{\rm ln}\left ({\left | x^{4} - x^{2} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(x^8 - 3*x^4 + 1),x, algorithm="giac")

[Out]

1/2*x^2 + 1/10*sqrt(5)*ln(abs(2*x^2 - sqrt(5) + 1)/(2*x^2 + sqrt(5) + 1)) + 1/10
*sqrt(5)*ln(abs(2*x^2 - sqrt(5) - 1)/abs(2*x^2 + sqrt(5) - 1)) - 1/4*ln(abs(x^4
+ x^2 - 1)) + 1/4*ln(abs(x^4 - x^2 - 1))