Optimal. Leaf size=56 \[ \frac{\log \left (a^2-a x+x^2\right )}{6 a}-\frac{\log (a+x)}{3 a}-\frac{\tan ^{-1}\left (\frac{a-2 x}{\sqrt{3} a}\right )}{\sqrt{3} a} \]
[Out]
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Rubi [A] time = 0.06278, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.546 \[ \frac{\log \left (a^2-a x+x^2\right )}{6 a}-\frac{\log (a+x)}{3 a}-\frac{\tan ^{-1}\left (\frac{a-2 x}{\sqrt{3} a}\right )}{\sqrt{3} a} \]
Antiderivative was successfully verified.
[In] Int[x/(a^3 + x^3),x]
[Out]
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Rubi in Sympy [A] time = 5.40662, size = 48, normalized size = 0.86 \[ - \frac{\log{\left (a + x \right )}}{3 a} + \frac{\log{\left (a^{2} - a x + x^{2} \right )}}{6 a} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{a}{3} - \frac{2 x}{3}\right )}{a} \right )}}{3 a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(a**3+x**3),x)
[Out]
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Mathematica [A] time = 0.00929327, size = 50, normalized size = 0.89 \[ \frac{\log \left (a^2-a x+x^2\right )-2 \log (a+x)+2 \sqrt{3} \tan ^{-1}\left (\frac{2 x-a}{\sqrt{3} a}\right )}{6 a} \]
Antiderivative was successfully verified.
[In] Integrate[x/(a^3 + x^3),x]
[Out]
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Maple [A] time = 0.009, size = 52, normalized size = 0.9 \[ -{\frac{\ln \left ( a+x \right ) }{3\,a}}+{\frac{\ln \left ({a}^{2}-ax+{x}^{2} \right ) }{6\,a}}+{\frac{\sqrt{3}}{3\,a}\arctan \left ({\frac{ \left ( 2\,x-a \right ) \sqrt{3}}{3\,a}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(a^3+x^3),x)
[Out]
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Maxima [A] time = 1.49684, size = 66, normalized size = 1.18 \[ \frac{\sqrt{3} \arctan \left (-\frac{\sqrt{3}{\left (a - 2 \, x\right )}}{3 \, a}\right )}{3 \, a} + \frac{\log \left (a^{2} - a x + x^{2}\right )}{6 \, a} - \frac{\log \left (a + x\right )}{3 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(a^3 + x^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.208996, size = 68, normalized size = 1.21 \[ \frac{\sqrt{3}{\left (\sqrt{3} \log \left (a^{2} - a x + x^{2}\right ) - 2 \, \sqrt{3} \log \left (a + x\right ) + 6 \, \arctan \left (-\frac{\sqrt{3}{\left (a - 2 \, x\right )}}{3 \, a}\right )\right )}}{18 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(a^3 + x^3),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.12903, size = 71, normalized size = 1.27 \[ \frac{- \frac{\log{\left (a + x \right )}}{3} + \left (\frac{1}{6} - \frac{\sqrt{3} i}{6}\right ) \log{\left (9 a \left (\frac{1}{6} - \frac{\sqrt{3} i}{6}\right )^{2} + x \right )} + \left (\frac{1}{6} + \frac{\sqrt{3} i}{6}\right ) \log{\left (9 a \left (\frac{1}{6} + \frac{\sqrt{3} i}{6}\right )^{2} + x \right )}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(a**3+x**3),x)
[Out]
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GIAC/XCAS [A] time = 0.208759, size = 68, normalized size = 1.21 \[ \frac{\sqrt{3} \arctan \left (-\frac{\sqrt{3}{\left (a - 2 \, x\right )}}{3 \, a}\right )}{3 \, a} + \frac{{\rm ln}\left (a^{2} - a x + x^{2}\right )}{6 \, a} - \frac{{\rm ln}\left ({\left | a + x \right |}\right )}{3 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(a^3 + x^3),x, algorithm="giac")
[Out]