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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 31, 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[(x*(2*a + 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 {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 = 0.01, 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[(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

________________________________________________________________________________________

fricas [A]  time = 0.60, size = 28, 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(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)

________________________________________________________________________________________

giac [A]  time = 0.15, size = 29, normalized size = 0.94 \[ -\frac {2}{3} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (a + 2 \, x\right )}}{3 \, a}\right ) - \log \left ({\left | -a + x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(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))

________________________________________________________________________________________

maple [A]  time = 0.06, size = 29, normalized size = 0.94 \[ -\frac {2 \sqrt {3}\, \arctan \left (\frac {\left (a +2 x \right ) \sqrt {3}}{3 a}\right )}{3}-\ln \left (-a +x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(2*a+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)

________________________________________________________________________________________

maxima [A]  time = 2.84, size = 28, 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(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)

________________________________________________________________________________________

mupad [B]  time = 0.03, size = 27, normalized size = 0.87 \[ \frac {2\,\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}\,a}{a+2\,x}\right )}{3}-\ln \left (x-a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*(2*a + x))/(a^3 - x^3),x)

[Out]

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

________________________________________________________________________________________

sympy [C]  time = 0.17, 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(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

________________________________________________________________________________________