3.355 \(\int \frac{2 a x+x^2}{a^3-x^3} \, dx\)

Optimal. Leaf size=31 \[ -\log (a-x)-\frac{2 \tan ^{-1}\left (\frac{a+2 x}{\sqrt{3} a}\right )}{\sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.0977033, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\log (a-x)-\frac{2 \tan ^{-1}\left (\frac{a+2 x}{\sqrt{3} a}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[(2*a*x + x^2)/(a^3 - x^3),x]

[Out]

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

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Rubi in Sympy [A]  time = 11.6381, size = 31, normalized size = 1. \[ - \log{\left (a - x \right )} - \frac{2 \sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{a}{3} + \frac{2 x}{3}\right )}{a} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*a*x+x**2)/(a**3-x**3),x)

[Out]

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

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Mathematica [A]  time = 0.0263551, size = 58, normalized size = 1.87 \[ \frac{1}{3} \left (-\log \left (x^3-a^3\right )+\log \left (a^2+a x+x^2\right )-2 \log (x-a)-2 \sqrt{3} \tan ^{-1}\left (\frac{a+2 x}{\sqrt{3} a}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(2*a*x + x^2)/(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]
- Log[-a^3 + x^3])/3

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Maple [A]  time = 0.012, size = 29, normalized size = 0.9 \[ -{\frac{2\,\sqrt{3}}{3}\arctan \left ({\frac{ \left ( a+2\,x \right ) \sqrt{3}}{3\,a}} \right ) }-\ln \left ( x-a \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*a*x+x^2)/(a^3-x^3),x)

[Out]

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

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Maxima [A]  time = 1.53961, size = 38, normalized size = 1.23 \[ -\frac{2}{3} \, \sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (a + 2 \, x\right )}}{3 \, a}\right ) - \log \left (-a + x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*a*x + x^2)/(a^3 - x^3),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.258938, size = 43, normalized size = 1.39 \[ -\frac{1}{3} \, \sqrt{3}{\left (\sqrt{3} \log \left (-a + x\right ) + 2 \, \arctan \left (\frac{\sqrt{3}{\left (a + 2 \, x\right )}}{3 \, a}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*a*x + x^2)/(a^3 - x^3),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.599156, size = 54, normalized size = 1.74 \[ - \log{\left (- a + x \right )} + \frac{\sqrt{3} i \log{\left (\frac{a}{2} - \frac{\sqrt{3} i a}{2} + x \right )}}{3} - \frac{\sqrt{3} i \log{\left (\frac{a}{2} + \frac{\sqrt{3} i a}{2} + x \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*a*x+x**2)/(a**3-x**3),x)

[Out]

-log(-a + x) + sqrt(3)*I*log(a/2 - sqrt(3)*I*a/2 + x)/3 - sqrt(3)*I*log(a/2 + sq
rt(3)*I*a/2 + x)/3

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GIAC/XCAS [A]  time = 0.21131, size = 39, normalized size = 1.26 \[ -\frac{2}{3} \, \sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (a + 2 \, x\right )}}{3 \, a}\right ) -{\rm ln}\left ({\left | -a + x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*a*x + x^2)/(a^3 - x^3),x, algorithm="giac")

[Out]

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