Optimal. Leaf size=22 \[ \frac{1}{2 \left (x^2+1\right )}+\frac{1}{2} \log \left (x^2+1\right ) \]
[Out]
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Rubi [A] time = 0.0294682, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ \frac{1}{2 \left (x^2+1\right )}+\frac{1}{2} \log \left (x^2+1\right ) \]
Antiderivative was successfully verified.
[In] Int[x^3/(1 + 2*x^2 + x^4),x]
[Out]
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Rubi in Sympy [A] time = 4.95612, size = 15, normalized size = 0.68 \[ \frac{\log{\left (x^{2} + 1 \right )}}{2} + \frac{1}{2 \left (x^{2} + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3/(x**4+2*x**2+1),x)
[Out]
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Mathematica [A] time = 0.00729017, size = 18, normalized size = 0.82 \[ \frac{1}{2} \left (\frac{1}{x^2+1}+\log \left (x^2+1\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x^3/(1 + 2*x^2 + x^4),x]
[Out]
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Maple [A] time = 0.008, size = 19, normalized size = 0.9 \[{\frac{1}{2\,{x}^{2}+2}}+{\frac{\ln \left ({x}^{2}+1 \right ) }{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3/(x^4+2*x^2+1),x)
[Out]
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Maxima [A] time = 0.690458, size = 24, normalized size = 1.09 \[ \frac{1}{2 \,{\left (x^{2} + 1\right )}} + \frac{1}{2} \, \log \left (x^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/(x^4 + 2*x^2 + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.251669, size = 31, normalized size = 1.41 \[ \frac{{\left (x^{2} + 1\right )} \log \left (x^{2} + 1\right ) + 1}{2 \,{\left (x^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/(x^4 + 2*x^2 + 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.154311, size = 15, normalized size = 0.68 \[ \frac{\log{\left (x^{2} + 1 \right )}}{2} + \frac{1}{2 x^{2} + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3/(x**4+2*x**2+1),x)
[Out]
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GIAC/XCAS [A] time = 0.270716, size = 24, normalized size = 1.09 \[ \frac{1}{2 \,{\left (x^{2} + 1\right )}} + \frac{1}{2} \,{\rm ln}\left (x^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/(x^4 + 2*x^2 + 1),x, algorithm="giac")
[Out]