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]

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

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Rubi [A]  time = 0.03, antiderivative size = 29, normalized size of antiderivative = 1.00, 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 31

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

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])

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 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]

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*}

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Mathematica [A]  time = 0.01, size = 57, normalized size = 1.97 \[ \frac {1}{3} \left (-\log \left (a^3+x^3\right )+\log \left (a^2-a x+x^2\right )-2 \log (a+x)+2 \sqrt {3} \tan ^{-1}\left (\frac {2 x-a}{\sqrt {3} a}\right )\right ) \]

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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fricas [A]  time = 0.75, size = 26, normalized size = 0.90 \[ \frac {2}{3} \, \sqrt {3} \arctan \left (-\frac {\sqrt {3} {\left (a - 2 \, x\right )}}{3 \, a}\right ) - \log \left (a + x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*a-x)*x/(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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giac [A]  time = 0.15, size = 27, normalized size = 0.93 \[ \frac {2}{3} \, \sqrt {3} \arctan \left (-\frac {\sqrt {3} {\left (a - 2 \, x\right )}}{3 \, a}\right ) - \log \left ({\left | a + x \right |}\right ) \]

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))

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maple [A]  time = 0.05, size = 29, normalized size = 1.00 \[ \frac {2 \sqrt {3}\, \arctan \left (\frac {\left (-a +2 x \right ) \sqrt {3}}{3 a}\right )}{3}-\ln \left (a +x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 2.97, size = 26, normalized size = 0.90 \[ \frac {2}{3} \, \sqrt {3} \arctan \left (-\frac {\sqrt {3} {\left (a - 2 \, x\right )}}{3 \, a}\right ) - \log \left (a + x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*a-x)*x/(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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mupad [B]  time = 0.03, size = 26, normalized size = 0.90 \[ -\ln \left (a+x\right )-\frac {2\,\sqrt {3}\,\mathrm {atan}\left (-\frac {\sqrt {3}\,a}{a-2\,x}\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*(2*a - x))/(a^3 + x^3),x)

[Out]

- log(a + x) - (2*3^(1/2)*atan(-(3^(1/2)*a)/(a - 2*x)))/3

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sympy [C]  time = 0.17, size = 54, normalized size = 1.86 \[ - \log {\left (a + x \right )} - \frac {\sqrt {3} i \log {\left (- \frac {a}{2} - \frac {\sqrt {3} i a}{2} + x \right )}}{3} + \frac {\sqrt {3} i \log {\left (- \frac {a}{2} + \frac {\sqrt {3} i a}{2} + x \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*a-x)*x/(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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