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

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

[Out]

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

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Rubi [A]  time = 0.0674888, antiderivative size = 25, 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, 203} \[ \frac{2}{3} \tan ^{-1}\left (\frac{(x+1)^2}{3 \sqrt{-x^3-1}}\right ) \]

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

Mathematica [A]  time = 0.0088987, size = 25, normalized size = 1. \[ \frac{2}{3} \tan ^{-1}\left (\frac{(x+1)^2}{3 \sqrt{-x^3-1}}\right ) \]

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

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Maple [C]  time = 0.023, size = 240, normalized size = 9.6 \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{1+x}{{\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{1+x}{{\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)*((1+x)/(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)*((1+x)/(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 [A]  time = 2.09533, size = 107, normalized size = 4.28 \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))

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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)

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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)