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

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 = 0.0367465, antiderivative size = 31, 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[(x*(2*a + 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{x (2 a+x)}{a^3-x^3} \, dx &=-\left (a \int \frac{1}{a^2+a x+x^2} \, dx\right )-\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 = 0.0057692, size = 58, normalized size = 1.87 \[ \frac{1}{3} \left (\log \left (a^2+a x+x^2\right )-\log \left (x^3-a^3\right )-2 \log (x-a)-2 \sqrt{3} \tan ^{-1}\left (\frac{a+2 x}{\sqrt{3} a}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(x*(2*a + 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 = 0.005, size = 29, normalized size = 0.9 \begin{align*} -{\frac{2\,\sqrt{3}}{3}\arctan \left ({\frac{ \left ( a+2\,x \right ) \sqrt{3}}{3\,a}} \right ) }-\ln \left ( -a+x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.41686, size = 38, normalized size = 1.23 \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(x*(2*a+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 = 0.941276, size = 81, normalized size = 2.61 \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(x*(2*a+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 = 0.288601, size = 54, normalized size = 1.74 \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(x*(2*a+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 = 1.05676, size = 39, normalized size = 1.26 \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(x*(2*a+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))