\(\int \frac {1-\sqrt {3}-x}{(1+\sqrt {3}-x) \sqrt {1-x^3}} \, dx\) [114]

Optimal result

Integrand size = 36, antiderivative size = 46 \[ \int \frac {1-\sqrt {3}-x}{\left (1+\sqrt {3}-x\right ) \sqrt {1-x^3}} \, dx=\frac {2 \arctan \left (\frac {\sqrt {3+2 \sqrt {3}} (1-x)}{\sqrt {1-x^3}}\right )}{\sqrt {3+2 \sqrt {3}}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2165, 209} \[ \int \frac {1-\sqrt {3}-x}{\left (1+\sqrt {3}-x\right ) \sqrt {1-x^3}} \, dx=\frac {2 \arctan \left (\frac {\sqrt {3+2 \sqrt {3}} (1-x)}{\sqrt {1-x^3}}\right )}{\sqrt {3+2 \sqrt {3}}} \]

[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 209

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

Rule 2165

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]

Rubi steps \begin{align*} \text {integral}& = 2 \text {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 [A] (verified)

Time = 1.79 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.07 \[ \int \frac {1-\sqrt {3}-x}{\left (1+\sqrt {3}-x\right ) \sqrt {1-x^3}} \, dx=2 \sqrt {-1+\frac {2}{\sqrt {3}}} \arctan \left (\frac {\sqrt {3+2 \sqrt {3}} \sqrt {1-x^3}}{1+x+x^2}\right ) \]

[In]

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

[Out]

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

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 2.47 (sec) , antiderivative size = 135, normalized size of antiderivative = 2.93

method	result	size
trager	\(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-36+24 \sqrt {3}\right ) \ln \left (\frac {6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-36+24 \sqrt {3}\right ) x^{2}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-36+24 \sqrt {3}\right ) \sqrt {3}\, x^{2}-4 \sqrt {3}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}-36+24 \sqrt {3}\right ) x +48 \sqrt {-x^{3}+1}\, \sqrt {3}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-36+24 \sqrt {3}\right ) \sqrt {3}-72 \sqrt {-x^{3}+1}+12 \operatorname {RootOf}\left (\textit {\_Z}^{2}-36+24 \sqrt {3}\right )}{\left (x \sqrt {3}-x -2\right )^{2}}\right )}{6}\)	\(135\)
default	\(-\frac {2 i \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, F\left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}+1}}-\frac {4 i \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \Pi \left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{-\frac {3}{2}-\sqrt {3}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-x^{3}+1}\, \left (-\frac {3}{2}-\sqrt {3}+\frac {i \sqrt {3}}{2}\right )}\)	\(247\)
elliptic	\(-\frac {2 i \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, F\left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}+1}}-\frac {4 i \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \Pi \left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{-\frac {3}{2}-\sqrt {3}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-x^{3}+1}\, \left (-\frac {3}{2}-\sqrt {3}+\frac {i \sqrt {3}}{2}\right )}\)	\(247\)

[In]

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

[Out]

-1/6*RootOf(_Z^2-36+24*3^(1/2))*ln((6*RootOf(_Z^2-36+24*3^(1/2))*x^2-4*RootOf(_Z^2-36+24*3^(1/2))*3^(1/2)*x^2-
4*3^(1/2)*RootOf(_Z^2-36+24*3^(1/2))*x+48*(-x^3+1)^(1/2)*3^(1/2)-4*RootOf(_Z^2-36+24*3^(1/2))*3^(1/2)-72*(-x^3
+1)^(1/2)+12*RootOf(_Z^2-36+24*3^(1/2)))/(x*3^(1/2)-x-2)^2)

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.28 \[ \int \frac {1-\sqrt {3}-x}{\left (1+\sqrt {3}-x\right ) \sqrt {1-x^3}} \, dx=\frac {1}{3} \, \sqrt {3} \sqrt {2 \, \sqrt {3} - 3} \arctan \left (\frac {\sqrt {-x^{3} + 1} {\left (\sqrt {3} {\left (x^{2} + 4 \, x - 2\right )} + 6 \, x - 6\right )} \sqrt {2 \, \sqrt {3} - 3}}{6 \, {\left (x^{3} - 1\right )}}\right ) \]

[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))

Sympy [F]

\[ \int \frac {1-\sqrt {3}-x}{\left (1+\sqrt {3}-x\right ) \sqrt {1-x^3}} \, dx=\int \frac {x - 1 + \sqrt {3}}{\sqrt {- \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x - \sqrt {3} - 1\right )}\, dx \]

[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)

Maxima [F]

\[ \int \frac {1-\sqrt {3}-x}{\left (1+\sqrt {3}-x\right ) \sqrt {1-x^3}} \, dx=\int { \frac {x + \sqrt {3} - 1}{\sqrt {-x^{3} + 1} {\left (x - \sqrt {3} - 1\right )}} \,d x } \]

[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)

Giac [F(-2)]

Exception generated. \[ \int \frac {1-\sqrt {3}-x}{\left (1+\sqrt {3}-x\right ) \sqrt {1-x^3}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Unable to divide, perhaps due to rounding error%%%{%%{[1,-1]:[1,0,-3]%%},[2]%%%} / %%%{%%{[2,4]:[1,0,-3]%%}
,[2]%%%} Er

Mupad [F(-1)]

Timed out. \[ \int \frac {1-\sqrt {3}-x}{\left (1+\sqrt {3}-x\right ) \sqrt {1-x^3}} \, dx=\text {Hanged} \]

[In]

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

[Out]

\text{Hanged}