Optimal. Leaf size=30 \[ -\frac{3}{4 x^2}-\frac{3}{4} \tan ^{-1}\left (x^2\right )+\frac{1}{4 x^2 \left (x^4+1\right )} \]
[Out]
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Rubi [A] time = 0.0331186, antiderivative size = 30, 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 \[ -\frac{3}{4 x^2}-\frac{3}{4} \tan ^{-1}\left (x^2\right )+\frac{1}{4 x^2 \left (x^4+1\right )} \]
Antiderivative was successfully verified.
[In] Int[1/(x^3*(1 + 2*x^4 + x^8)),x]
[Out]
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Rubi in Sympy [A] time = 7.58688, size = 26, normalized size = 0.87 \[ - \frac{3 \operatorname{atan}{\left (x^{2} \right )}}{4} - \frac{3}{4 x^{2}} + \frac{1}{4 x^{2} \left (x^{4} + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**3/(x**8+2*x**4+1),x)
[Out]
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Mathematica [A] time = 0.0181117, size = 30, normalized size = 1. \[ -\frac{1}{2 x^2}+\frac{3}{4} \tan ^{-1}\left (\frac{1}{x^2}\right )-\frac{x^2}{4 \left (x^4+1\right )} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^3*(1 + 2*x^4 + x^8)),x]
[Out]
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Maple [A] time = 0.015, size = 25, normalized size = 0.8 \[ -{\frac{{x}^{2}}{4\,{x}^{4}+4}}-{\frac{3\,\arctan \left ({x}^{2} \right ) }{4}}-{\frac{1}{2\,{x}^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^3/(x^8+2*x^4+1),x)
[Out]
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Maxima [A] time = 0.851052, size = 34, normalized size = 1.13 \[ -\frac{3 \, x^{4} + 2}{4 \,{\left (x^{6} + x^{2}\right )}} - \frac{3}{4} \, \arctan \left (x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^8 + 2*x^4 + 1)*x^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.251319, size = 42, normalized size = 1.4 \[ -\frac{3 \, x^{4} + 3 \,{\left (x^{6} + x^{2}\right )} \arctan \left (x^{2}\right ) + 2}{4 \,{\left (x^{6} + x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^8 + 2*x^4 + 1)*x^3),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.394401, size = 26, normalized size = 0.87 \[ - \frac{3 x^{4} + 2}{4 x^{6} + 4 x^{2}} - \frac{3 \operatorname{atan}{\left (x^{2} \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**3/(x**8+2*x**4+1),x)
[Out]
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GIAC/XCAS [A] time = 0.299475, size = 34, normalized size = 1.13 \[ -\frac{3 \, x^{4} + 2}{4 \,{\left (x^{6} + x^{2}\right )}} - \frac{3}{4} \, \arctan \left (x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^8 + 2*x^4 + 1)*x^3),x, algorithm="giac")
[Out]