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

Optimal. Leaf size=46 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{3+2 \sqrt{3}} (1-x)}{\sqrt{1-x^3}}\right )}{\sqrt{3+2 \sqrt{3}}} \]

[Out]

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

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Rubi [A]  time = 0.101986, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {2140, 203} \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{3+2 \sqrt{3}} (1-x)}{\sqrt{1-x^3}}\right )}{\sqrt{3+2 \sqrt{3}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2140

Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbol] :> With[{k = Simplify[(d*e
+ 2*c*f)/(c*f)]}, Dist[((1 + k)*e)/d, Subst[Int[1/(1 + (3 + 2*k)*a*x^2), x], x, (1 + ((1 + k)*d*x)/c)/Sqrt[a +
b*x^3]], x]] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && EqQ[b^2*c^6 - 20*a*b*c^3*d^3 - 8*a^2*d^6
, 0] && EqQ[6*a*d^4*e - c*f*(b*c^3 - 22*a*d^3), 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-\sqrt{3}-x}{\left (1+\sqrt{3}-x\right ) \sqrt{1-x^3}} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{1+\left (3+2 \sqrt{3}\right ) x^2} \, dx,x,\frac{1-x}{\sqrt{1-x^3}}\right )\\ &=\frac{2 \tan ^{-1}\left (\frac{\sqrt{3+2 \sqrt{3}} (1-x)}{\sqrt{1-x^3}}\right )}{\sqrt{3+2 \sqrt{3}}}\\ \end{align*}

Mathematica [C]  time = 0.434934, size = 267, normalized size = 5.8 \[ \frac{2 \sqrt{6} \sqrt{\frac{i (x-1)}{\sqrt{3}-3 i}} \left (\sqrt{2 i x+\sqrt{3}+i} \left (\left (\sqrt{3}+(2+i)\right ) x+i \sqrt{3}+(1+2 i)\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{-2 i x+\sqrt{3}-i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{-3 i+\sqrt{3}}\right )-4 i \sqrt{-2 i x+\sqrt{3}-i} \sqrt{x^2+x+1} \Pi \left (\frac{2 i \sqrt{3}}{3+(2+i) \sqrt{3}};\sin ^{-1}\left (\frac{\sqrt{-2 i x+\sqrt{3}-i}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{-3 i+\sqrt{3}}\right )\right )}{\left (3+(2+i) \sqrt{3}\right ) \sqrt{-2 i x+\sqrt{3}-i} \sqrt{1-x^3}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*Sqrt[6]*Sqrt[(I*(-1 + x))/(-3*I + Sqrt[3])]*(Sqrt[I + Sqrt[3] + (2*I)*x]*((1 + 2*I) + I*Sqrt[3] + ((2 + I)
+ Sqrt[3])*x)*EllipticF[ArcSin[Sqrt[-I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(-3*I + Sqrt[3])]
- (4*I)*Sqrt[-I + Sqrt[3] - (2*I)*x]*Sqrt[1 + x + x^2]*EllipticPi[((2*I)*Sqrt[3])/(3 + (2 + I)*Sqrt[3]), ArcSi
n[Sqrt[-I + Sqrt[3] - (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(-3*I + Sqrt[3])]))/((3 + (2 + I)*Sqrt[3])*Sqrt
[-I + Sqrt[3] - (2*I)*x]*Sqrt[1 - x^3])

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Maple [C]  time = 0.025, size = 247, normalized size = 5.4 \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{4\,i}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{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 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{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-3^(1/2))/(1-x+3^(1/2))/(-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))-4*I*(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)-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)-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 + \sqrt{3} - 1}{\sqrt{-x^{3} + 1}{\left (x - \sqrt{3} - 1\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.67513, size = 170, normalized size = 3.7 \begin{align*} \frac{1}{3} \, \sqrt{3} \sqrt{2 \, \sqrt{3} - 3} \arctan \left (\frac{\sqrt{-x^{3} + 1}{\left (\sqrt{3}{\left (x^{2} + 4 \, x - 2\right )} + 6 \, x - 6\right )} \sqrt{2 \, \sqrt{3} - 3}}{6 \,{\left (x^{3} - 1\right )}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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