3.291 \(\int \frac{x^9}{1-2 x^4+x^8} \, dx\)

Optimal. Leaf size=32 \[ \frac{3 x^2}{4}-\frac{3}{4} \tanh ^{-1}\left (x^2\right )+\frac{x^6}{4 \left (1-x^4\right )} \]

[Out]

(3*x^2)/4 + x^6/(4*(1 - x^4)) - (3*ArcTanh[x^2])/4

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Rubi [A]  time = 0.0377132, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312 \[ \frac{3 x^2}{4}-\frac{3}{4} \tanh ^{-1}\left (x^2\right )+\frac{x^6}{4 \left (1-x^4\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^9/(1 - 2*x^4 + x^8),x]

[Out]

(3*x^2)/4 + x^6/(4*(1 - x^4)) - (3*ArcTanh[x^2])/4

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Rubi in Sympy [A]  time = 8.15939, size = 24, normalized size = 0.75 \[ \frac{x^{6}}{4 \left (- x^{4} + 1\right )} + \frac{3 x^{2}}{4} - \frac{3 \operatorname{atanh}{\left (x^{2} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**9/(x**8-2*x**4+1),x)

[Out]

x**6/(4*(-x**4 + 1)) + 3*x**2/4 - 3*atanh(x**2)/4

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Mathematica [A]  time = 0.0440937, size = 39, normalized size = 1.22 \[ \frac{1}{8} \left (3 \log \left (1-x^2\right )-3 \log \left (x^2+1\right )+2 \left (\frac{1}{1-x^4}+2\right ) x^2\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^9/(1 - 2*x^4 + x^8),x]

[Out]

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

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Maple [A]  time = 0.015, size = 41, normalized size = 1.3 \[{\frac{{x}^{2}}{2}}-{\frac{1}{8\,{x}^{2}-8}}+{\frac{3\,\ln \left ({x}^{2}-1 \right ) }{8}}-{\frac{1}{8\,{x}^{2}+8}}-{\frac{3\,\ln \left ({x}^{2}+1 \right ) }{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^9/(x^8-2*x^4+1),x)

[Out]

1/2*x^2-1/8/(x^2-1)+3/8*ln(x^2-1)-1/8/(x^2+1)-3/8*ln(x^2+1)

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Maxima [A]  time = 0.769749, size = 46, normalized size = 1.44 \[ \frac{1}{2} \, x^{2} - \frac{x^{2}}{4 \,{\left (x^{4} - 1\right )}} - \frac{3}{8} \, \log \left (x^{2} + 1\right ) + \frac{3}{8} \, \log \left (x^{2} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(x^8 - 2*x^4 + 1),x, algorithm="maxima")

[Out]

1/2*x^2 - 1/4*x^2/(x^4 - 1) - 3/8*log(x^2 + 1) + 3/8*log(x^2 - 1)

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Fricas [A]  time = 0.249455, size = 62, normalized size = 1.94 \[ \frac{4 \, x^{6} - 6 \, x^{2} - 3 \,{\left (x^{4} - 1\right )} \log \left (x^{2} + 1\right ) + 3 \,{\left (x^{4} - 1\right )} \log \left (x^{2} - 1\right )}{8 \,{\left (x^{4} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(x^8 - 2*x^4 + 1),x, algorithm="fricas")

[Out]

1/8*(4*x^6 - 6*x^2 - 3*(x^4 - 1)*log(x^2 + 1) + 3*(x^4 - 1)*log(x^2 - 1))/(x^4 -
1)

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Sympy [A]  time = 0.279651, size = 34, normalized size = 1.06 \[ \frac{x^{2}}{2} - \frac{x^{2}}{4 x^{4} - 4} + \frac{3 \log{\left (x^{2} - 1 \right )}}{8} - \frac{3 \log{\left (x^{2} + 1 \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**9/(x**8-2*x**4+1),x)

[Out]

x**2/2 - x**2/(4*x**4 - 4) + 3*log(x**2 - 1)/8 - 3*log(x**2 + 1)/8

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GIAC/XCAS [A]  time = 0.280268, size = 47, normalized size = 1.47 \[ \frac{1}{2} \, x^{2} - \frac{x^{2}}{4 \,{\left (x^{4} - 1\right )}} - \frac{3}{8} \,{\rm ln}\left (x^{2} + 1\right ) + \frac{3}{8} \,{\rm ln}\left ({\left | x^{2} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(x^8 - 2*x^4 + 1),x, algorithm="giac")

[Out]

1/2*x^2 - 1/4*x^2/(x^4 - 1) - 3/8*ln(x^2 + 1) + 3/8*ln(abs(x^2 - 1))