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

Optimal. Leaf size=29 \[ -\log (a+x)-\frac{2 \tan ^{-1}\left (\frac{a-2 x}{\sqrt{3} a}\right )}{\sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.0347537, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {1868, 31, 617, 204} \[ -\log (a+x)-\frac{2 \tan ^{-1}\left (\frac{a-2 x}{\sqrt{3} a}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[((2*a - x)*x)/(a^3 + x^3),x]

[Out]

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

Rule 1868

Int[(P2_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x,
2]}, With[{q = Rt[a/b, 3]}, Dist[C/b, Int[1/(q + x), x], x] + Dist[(B + C*q)/b, Int[1/(q^2 - q*x + x^2), x],
x]] /; EqQ[A - Rt[a/b, 3]*B - 2*Rt[a/b, 3]^2*C, 0]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{(2 a-x) x}{a^3+x^3} \, dx &=a \int \frac{1}{a^2-a x+x^2} \, dx-\int \frac{1}{a+x} \, dx\\ &=-\log (a+x)+2 \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 x}{a}\right )\\ &=-\frac{2 \tan ^{-1}\left (\frac{a-2 x}{\sqrt{3} a}\right )}{\sqrt{3}}-\log (a+x)\\ \end{align*}

Mathematica [A]  time = 0.0052049, size = 57, normalized size = 1.97 \[ \frac{1}{3} \left (\log \left (a^2-a x+x^2\right )-\log \left (a^3+x^3\right )-2 \log (a+x)+2 \sqrt{3} \tan ^{-1}\left (\frac{2 x-a}{\sqrt{3} a}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

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Maple [A]  time = 0.006, size = 29, normalized size = 1. \begin{align*} -\ln \left ( a+x \right ) +{\frac{2\,\sqrt{3}}{3}\arctan \left ({\frac{ \left ( 2\,x-a \right ) \sqrt{3}}{3\,a}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*a-x)*x/(a^3+x^3),x)

[Out]

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

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Maxima [A]  time = 1.42322, size = 35, normalized size = 1.21 \begin{align*} \frac{2}{3} \, \sqrt{3} \arctan \left (-\frac{\sqrt{3}{\left (a - 2 \, x\right )}}{3 \, a}\right ) - \log \left (a + x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*a-x)*x/(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 = 1.00505, size = 80, normalized size = 2.76 \begin{align*} \frac{2}{3} \, \sqrt{3} \arctan \left (-\frac{\sqrt{3}{\left (a - 2 \, x\right )}}{3 \, a}\right ) - \log \left (a + x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C]  time = 0.286357, size = 54, normalized size = 1.86 \begin{align*} - \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*a-x)*x/(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 + sqrt(3)*I*a/2 + x)/3

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Giac [A]  time = 1.07708, size = 36, normalized size = 1.24 \begin{align*} \frac{2}{3} \, \sqrt{3} \arctan \left (-\frac{\sqrt{3}{\left (a - 2 \, x\right )}}{3 \, a}\right ) - \log \left ({\left | a + x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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