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

Optimal. Leaf size=27 \[ -\frac{2}{3} \tanh ^{-1}\left (\frac{(1-x)^2}{3 \sqrt{1-x^3}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0650662, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {2138, 206} \[ -\frac{2}{3} \tanh ^{-1}\left (\frac{(1-x)^2}{3 \sqrt{1-x^3}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*ArcTanh[(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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[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} \tanh ^{-1}\left (\frac{(1-x)^2}{3 \sqrt{1-x^3}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0098922, size = 54, normalized size = 2. \[ \frac{1}{3} \log \left (3-\frac{(1-x)^2}{\sqrt{1-x^3}}\right )-\frac{1}{3} \log \left (\frac{(1-x)^2}{\sqrt{1-x^3}}+3\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.024, size = 240, normalized size = 8.9 \begin{align*}{{\frac{2\,i}{3}}\sqrt{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{x-1}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x+{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticF} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}+1}}}}-{\frac{2\,i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{x-1}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x+{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}+1}}}} \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*I*3^(1/2)*(I*(x+1/2-1/2*I*3^(1/2))*3^(1/2))^(1/2)*((x-1)/(-3/2+1/2*I*3^(1/2)))^(1/2)*(-I*(x+1/2+1/2*I*3^(1
/2))*3^(1/2))^(1/2)/(-x^3+1)^(1/2)*EllipticF(1/3*3^(1/2)*(I*(x+1/2-1/2*I*3^(1/2))*3^(1/2))^(1/2),(I*3^(1/2)/(-
3/2+1/2*I*3^(1/2)))^(1/2))-2*I*3^(1/2)*(I*(x+1/2-1/2*I*3^(1/2))*3^(1/2))^(1/2)*((x-1)/(-3/2+1/2*I*3^(1/2)))^(1
/2)*(-I*(x+1/2+1/2*I*3^(1/2))*3^(1/2))^(1/2)/(-x^3+1)^(1/2)/(3/2+1/2*I*3^(1/2))*EllipticPi(1/3*3^(1/2)*(I*(x+1
/2-1/2*I*3^(1/2))*3^(1/2))^(1/2),I*3^(1/2)/(3/2+1/2*I*3^(1/2)),(I*3^(1/2)/(-3/2+1/2*I*3^(1/2)))^(1/2))

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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)

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Fricas [B]  time = 1.90357, size = 120, normalized size = 4.44 \begin{align*} \frac{1}{3} \, \log \left (-\frac{x^{3} - 12 \, x^{2} - 6 \, \sqrt{-x^{3} + 1}{\left (x - 1\right )} - 6 \, x - 10}{x^{3} + 6 \, x^{2} + 12 \, x + 8}\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*log(-(x^3 - 12*x^2 - 6*sqrt(-x^3 + 1)*(x - 1) - 6*x - 10)/(x^3 + 6*x^2 + 12*x + 8))

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

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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)