3.119 \(\int \frac{x}{a^3+x^3} \, dx\)

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]

-(ArcTan[(a - 2*x)/(Sqrt[3]*a)]/(Sqrt[3]*a)) - Log[a + x]/(3*a) + Log[a^2 - a*x
+ x^2]/(6*a)

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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]

-(ArcTan[(a - 2*x)/(Sqrt[3]*a)]/(Sqrt[3]*a)) - Log[a + x]/(3*a) + Log[a^2 - a*x
+ x^2]/(6*a)

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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]

-log(a + x)/(3*a) + log(a**2 - a*x + x**2)/(6*a) - sqrt(3)*atan(sqrt(3)*(a/3 - 2
*x/3)/a)/(3*a)

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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]

(2*Sqrt[3]*ArcTan[(-a + 2*x)/(Sqrt[3]*a)] - 2*Log[a + x] + Log[a^2 - a*x + x^2])
/(6*a)

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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]

-1/3*ln(a+x)/a+1/6*ln(a^2-a*x+x^2)/a+1/3*3^(1/2)/a*arctan(1/3*(2*x-a)*3^(1/2)/a)

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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]

1/3*sqrt(3)*arctan(-1/3*sqrt(3)*(a - 2*x)/a)/a + 1/6*log(a^2 - a*x + x^2)/a - 1/
3*log(a + x)/a

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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]

1/18*sqrt(3)*(sqrt(3)*log(a^2 - a*x + x^2) - 2*sqrt(3)*log(a + x) + 6*arctan(-1/
3*sqrt(3)*(a - 2*x)/a))/a

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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]

(-log(a + x)/3 + (1/6 - sqrt(3)*I/6)*log(9*a*(1/6 - sqrt(3)*I/6)**2 + x) + (1/6
+ sqrt(3)*I/6)*log(9*a*(1/6 + sqrt(3)*I/6)**2 + x))/a

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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]

1/3*sqrt(3)*arctan(-1/3*sqrt(3)*(a - 2*x)/a)/a + 1/6*ln(a^2 - a*x + x^2)/a - 1/3
*ln(abs(a + x))/a