3.76 \(\int \frac{1-x}{(2+x) \sqrt{-1+x^3}} \, dx\)

Optimal. Leaf size=25 \[ -\frac{2}{3} \tan ^{-1}\left (\frac{(1-x)^2}{3 \sqrt{x^3-1}}\right ) \]

[Out]

(-2*ArcTan[(1 - x)^2/(3*Sqrt[-1 + x^3])])/3

________________________________________________________________________________________

Rubi [A]  time = 0.0588041, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2138, 203} \[ -\frac{2}{3} \tan ^{-1}\left (\frac{(1-x)^2}{3 \sqrt{x^3-1}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(1 - x)/((2 + x)*Sqrt[-1 + x^3]),x]

[Out]

(-2*ArcTan[(1 - x)^2/(3*Sqrt[-1 + x^3])])/3

Rule 2138

Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbol] :> Dist[(-2*e)/d, Subst[Int
[1/(9 - a*x^2), x], x, (1 + (f*x)/e)^2/Sqrt[a + b*x^3]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f,
0] && EqQ[b*c^3 + 8*a*d^3, 0] && EqQ[2*d*e + c*f, 0]

Rule 203

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

Rubi steps

\begin{align*} \int \frac{1-x}{(2+x) \sqrt{-1+x^3}} \, dx &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{9+x^2} \, dx,x,\frac{(1-x)^2}{\sqrt{-1+x^3}}\right )\right )\\ &=-\frac{2}{3} \tan ^{-1}\left (\frac{(1-x)^2}{3 \sqrt{-1+x^3}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0087205, size = 25, normalized size = 1. \[ -\frac{2}{3} \tan ^{-1}\left (\frac{(1-x)^2}{3 \sqrt{x^3-1}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - x)/((2 + x)*Sqrt[-1 + x^3]),x]

[Out]

(-2*ArcTan[(1 - x)^2/(3*Sqrt[-1 + x^3])])/3

________________________________________________________________________________________

Maple [C]  time = 0.023, size = 240, normalized size = 9.6 \begin{align*} -2\,{\frac{-3/2-i/2\sqrt{3}}{\sqrt{{x}^{3}-1}}\sqrt{{\frac{x-1}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2-i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2+i/2\sqrt{3}}{3/2+i/2\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{{\frac{x-1}{-3/2-i/2\sqrt{3}}}},\sqrt{{\frac{3/2+i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}} \right ) }+2\,{\frac{-3/2-i/2\sqrt{3}}{\sqrt{{x}^{3}-1}}\sqrt{{\frac{x-1}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2-i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2+i/2\sqrt{3}}{3/2+i/2\sqrt{3}}}}{\it EllipticPi} \left ( \sqrt{{\frac{x-1}{-3/2-i/2\sqrt{3}}}},1/2+i/6\sqrt{3},\sqrt{{\frac{3/2+i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-x)/(2+x)/(x^3-1)^(1/2),x)

[Out]

-2*(-3/2-1/2*I*3^(1/2))*((x-1)/(-3/2-1/2*I*3^(1/2)))^(1/2)*((x+1/2-1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2)*(
(x+1/2+1/2*I*3^(1/2))/(3/2+1/2*I*3^(1/2)))^(1/2)/(x^3-1)^(1/2)*EllipticF(((x-1)/(-3/2-1/2*I*3^(1/2)))^(1/2),((
3/2+1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2))+2*(-3/2-1/2*I*3^(1/2))*((x-1)/(-3/2-1/2*I*3^(1/2)))^(1/2)*((x+1
/2-1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2)*((x+1/2+1/2*I*3^(1/2))/(3/2+1/2*I*3^(1/2)))^(1/2)/(x^3-1)^(1/2)*E
llipticPi(((x-1)/(-3/2-1/2*I*3^(1/2)))^(1/2),1/2+1/6*I*3^(1/2),((3/2+1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2)
)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{x - 1}{\sqrt{x^{3} - 1}{\left (x + 2\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)/(2+x)/(x^3-1)^(1/2),x, algorithm="maxima")

[Out]

-integrate((x - 1)/(sqrt(x^3 - 1)*(x + 2)), x)

________________________________________________________________________________________

Fricas [B]  time = 2.25901, size = 105, normalized size = 4.2 \begin{align*} -\frac{1}{3} \, \arctan \left (\frac{{\left (x^{3} - 12 \, x^{2} - 6 \, x - 10\right )} \sqrt{x^{3} - 1}}{6 \,{\left (x^{4} - x^{3} - x + 1\right )}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)/(2+x)/(x^3-1)^(1/2),x, algorithm="fricas")

[Out]

-1/3*arctan(1/6*(x^3 - 12*x^2 - 6*x - 10)*sqrt(x^3 - 1)/(x^4 - x^3 - x + 1))

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{x}{x \sqrt{x^{3} - 1} + 2 \sqrt{x^{3} - 1}}\, dx - \int - \frac{1}{x \sqrt{x^{3} - 1} + 2 \sqrt{x^{3} - 1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)/(2+x)/(x**3-1)**(1/2),x)

[Out]

-Integral(x/(x*sqrt(x**3 - 1) + 2*sqrt(x**3 - 1)), x) - Integral(-1/(x*sqrt(x**3 - 1) + 2*sqrt(x**3 - 1)), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{x - 1}{\sqrt{x^{3} - 1}{\left (x + 2\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)/(2+x)/(x^3-1)^(1/2),x, algorithm="giac")

[Out]

integrate(-(x - 1)/(sqrt(x^3 - 1)*(x + 2)), x)