Optimal. Leaf size=23 \[ -\frac{\tanh ^{-1}\left (\frac{2 x^4+3}{\sqrt{5}}\right )}{2 \sqrt{5}} \]
[Out]
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Rubi [A] time = 0.0499458, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ -\frac{\tanh ^{-1}\left (\frac{2 x^4+3}{\sqrt{5}}\right )}{2 \sqrt{5}} \]
Antiderivative was successfully verified.
[In] Int[x^3/(1 + 3*x^4 + x^8),x]
[Out]
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Rubi in Sympy [A] time = 5.65705, size = 24, normalized size = 1.04 \[ - \frac{\sqrt{5} \operatorname{atanh}{\left (\sqrt{5} \left (\frac{2 x^{4}}{5} + \frac{3}{5}\right ) \right )}}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3/(x**8+3*x**4+1),x)
[Out]
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Mathematica [A] time = 0.0155355, size = 38, normalized size = 1.65 \[ \frac{\log \left (-2 x^4+\sqrt{5}-3\right )-\log \left (2 x^4+\sqrt{5}+3\right )}{4 \sqrt{5}} \]
Antiderivative was successfully verified.
[In] Integrate[x^3/(1 + 3*x^4 + x^8),x]
[Out]
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Maple [A] time = 0.003, size = 19, normalized size = 0.8 \[ -{\frac{\sqrt{5}}{10}{\it Artanh} \left ({\frac{ \left ( 2\,{x}^{4}+3 \right ) \sqrt{5}}{5}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3/(x^8+3*x^4+1),x)
[Out]
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Maxima [A] time = 0.824023, size = 42, normalized size = 1.83 \[ \frac{1}{20} \, \sqrt{5} \log \left (\frac{2 \, x^{4} - \sqrt{5} + 3}{2 \, x^{4} + \sqrt{5} + 3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/(x^8 + 3*x^4 + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.257924, size = 59, normalized size = 2.57 \[ \frac{1}{20} \, \sqrt{5} \log \left (-\frac{10 \, x^{4} - \sqrt{5}{\left (2 \, x^{8} + 6 \, x^{4} + 7\right )} + 15}{x^{8} + 3 \, x^{4} + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/(x^8 + 3*x^4 + 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.260736, size = 42, normalized size = 1.83 \[ \frac{\sqrt{5} \log{\left (x^{4} - \frac{\sqrt{5}}{2} + \frac{3}{2} \right )}}{20} - \frac{\sqrt{5} \log{\left (x^{4} + \frac{\sqrt{5}}{2} + \frac{3}{2} \right )}}{20} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3/(x**8+3*x**4+1),x)
[Out]
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GIAC/XCAS [A] time = 0.279537, size = 42, normalized size = 1.83 \[ \frac{1}{20} \, \sqrt{5}{\rm ln}\left (\frac{2 \, x^{4} - \sqrt{5} + 3}{2 \, x^{4} + \sqrt{5} + 3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/(x^8 + 3*x^4 + 1),x, algorithm="giac")
[Out]