\(\int \frac {x}{(1+\sqrt {3}-x) \sqrt {1-x^3}} \, dx\) [182]

Optimal result

Integrand size = 25, antiderivative size = 152 \[ \int \frac {x}{\left (1+\sqrt {3}-x\right ) \sqrt {1-x^3}} \, dx=-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {3+2 \sqrt {3}} (1-x)}{\sqrt {1-x^3}}\right )}{3^{3/4}}+\frac {\sqrt {2} (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}-x}{1+\sqrt {3}-x}\right ),-7-4 \sqrt {3}\right )}{3^{3/4} \sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}} \sqrt {1-x^3}} \] Output:

-1/3*2^(1/2)*arctan((3+2*3^(1/2))^(1/2)*(1-x)/(-x^3+1)^(1/2))*3^(1/4)+1/3* 
2^(1/2)*(1-x)*((x^2+x+1)/(1+3^(1/2)-x)^2)^(1/2)*EllipticF((1-3^(1/2)-x)/(1 
+3^(1/2)-x),I*3^(1/2)+2*I)*3^(1/4)/((1-x)/(1+3^(1/2)-x)^2)^(1/2)/(-x^3+1)^ 
(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 10.59 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.53 \[ \int \frac {x}{\left (1+\sqrt {3}-x\right ) \sqrt {1-x^3}} \, dx=\frac {2 i \sqrt {\frac {1-x}{1+\sqrt [3]{-1}}} \left (\frac {i \sqrt {\frac {\sqrt [3]{-1}+(-1)^{2/3} x}{1+\sqrt [3]{-1}}} \left (3 i+(1+2 i) \sqrt {3}+\left (3+(2+i) \sqrt {3}\right ) x\right ) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right ),\sqrt [3]{-1}\right )}{\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}}+2 \left (1+\sqrt {3}\right ) \sqrt {1+x+x^2} \operatorname {EllipticPi}\left (\frac {2 i \sqrt {3}}{3+(2+i) \sqrt {3}},\arcsin \left (\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right ),\sqrt [3]{-1}\right )\right )}{\left (3+(2+i) \sqrt {3}\right ) \sqrt {1-x^3}} \] Input:

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

Output:

((2*I)*Sqrt[(1 - x)/(1 + (-1)^(1/3))]*((I*Sqrt[((-1)^(1/3) + (-1)^(2/3)*x) 
/(1 + (-1)^(1/3))]*(3*I + (1 + 2*I)*Sqrt[3] + (3 + (2 + I)*Sqrt[3])*x)*Ell 
ipticF[ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/Sqr 
t[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))] + 2*(1 + Sqrt[3])*Sqrt[1 + x + x^2] 
*EllipticPi[((2*I)*Sqrt[3])/(3 + (2 + I)*Sqrt[3]), ArcSin[Sqrt[(1 - (-1)^( 
2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)]))/((3 + (2 + I)*Sqrt[3])*Sqrt[1 - 
x^3])
 

Rubi [A] (warning: unable to verify)

Time = 0.80 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.27, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2566, 27, 759, 2565, 216}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 2566

\(\displaystyle \frac {\int -\frac {6 \left (-x-\sqrt {3}+1\right )}{\left (-x+\sqrt {3}+1\right ) \sqrt {1-x^3}}dx}{6 \left (3-\sqrt {3}\right )}-\frac {\left (2-\sqrt {3}\right ) \int \frac {1}{\sqrt {1-x^3}}dx}{3-\sqrt {3}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\left (2-\sqrt {3}\right ) \int \frac {1}{\sqrt {1-x^3}}dx}{3-\sqrt {3}}-\frac {\int \frac {-x-\sqrt {3}+1}{\left (-x+\sqrt {3}+1\right ) \sqrt {1-x^3}}dx}{3-\sqrt {3}}\)

\(\Big \downarrow \) 759

\(\displaystyle \frac {2 \left (2-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \left (3-\sqrt {3}\right ) \sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {1-x^3}}-\frac {\int \frac {-x-\sqrt {3}+1}{\left (-x+\sqrt {3}+1\right ) \sqrt {1-x^3}}dx}{3-\sqrt {3}}\)

\(\Big \downarrow \) 2565

\(\displaystyle \frac {2 \left (2-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \left (3-\sqrt {3}\right ) \sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {1-x^3}}-\frac {2 \int \frac {1}{\frac {\left (3+2 \sqrt {3}\right ) (1-x)^2}{1-x^3}+1}d\frac {1-x}{\sqrt {1-x^3}}}{3-\sqrt {3}}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {2 \left (2-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \left (3-\sqrt {3}\right ) \sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {1-x^3}}-\frac {2 \arctan \left (\frac {\sqrt {3+2 \sqrt {3}} (1-x)}{\sqrt {1-x^3}}\right )}{\left (3-\sqrt {3}\right ) \sqrt {3+2 \sqrt {3}}}\)

Input:

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

Output:

(-2*ArcTan[(Sqrt[3 + 2*Sqrt[3]]*(1 - x))/Sqrt[1 - x^3]])/((3 - Sqrt[3])*Sq 
rt[3 + 2*Sqrt[3]]) + (2*(2 - Sqrt[3])*Sqrt[2 + Sqrt[3]]*(1 - x)*Sqrt[(1 + 
x + x^2)/(1 + Sqrt[3] - x)^2]*EllipticF[ArcSin[(1 - Sqrt[3] - x)/(1 + Sqrt 
[3] - x)], -7 - 4*Sqrt[3]])/(3^(1/4)*(3 - Sqrt[3])*Sqrt[(1 - x)/(1 + Sqrt[ 
3] - x)^2]*Sqrt[1 - x^3])
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 216

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

rule 759

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], 
s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 + Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s 
*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[s* 
((s + r*x)/((1 + Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 - Sqrt[3])*s 
+ r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x] & 
& PosQ[a]
 

rule 2565

Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_ 
Symbol] :> With[{k = Simplify[(d*e + 2*c*f)/(c*f)]}, Simp[(1 + k)*(e/d)   S 
ubst[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 2566

Int[((e_.) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x 
_Symbol] :> Simp[-(6*a*d^4*e - c*f*(b*c^3 - 22*a*d^3))/(c*d*(b*c^3 - 28*a*d 
^3))   Int[1/Sqrt[a + b*x^3], x], x] + Simp[(d*e - c*f)/(c*d*(b*c^3 - 28*a* 
d^3))   Int[(c*(b*c^3 - 22*a*d^3) + 6*a*d^4*x)/((c + d*x)*Sqrt[a + b*x^3]), 
 x], 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] && NeQ[6*a*d^4*e - c*f*(b*c^3 - 22*a*d^3) 
, 0]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 254 vs. \(2 (124 ) = 248\).

Time = 1.56 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.68

method	result	size
default	\(\frac {2 i \left (1+\sqrt {3}\right ) \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticPi}\left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}-\sqrt {3}}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}+1}\, \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}-\sqrt {3}\right )}+\frac {2 i \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}+1}}\)	\(255\)
elliptic	\(\frac {2 i \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}+1}}-\frac {2 i \left (-1-\sqrt {3}\right ) \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticPi}\left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}-\sqrt {3}}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}+1}\, \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}-\sqrt {3}\right )}\)	\(257\)

Input:

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

Output:

2/3*I*(1+3^(1/2))*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)-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))+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))
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.45 \[ \int \frac {x}{\left (1+\sqrt {3}-x\right ) \sqrt {1-x^3}} \, dx=-\frac {1}{3} \, {\left (i \, \sqrt {3} - 3 i\right )} {\rm weierstrassPInverse}\left (0, 4, x\right ) + \frac {1}{6} \cdot 3^{\frac {1}{4}} \sqrt {2} \arctan \left (-\frac {3^{\frac {1}{4}} \sqrt {2} \sqrt {-x^{3} + 1} {\left (3 \, x^{2} - \sqrt {3} {\left (x^{2} - 2 \, x + 4\right )} + 6 \, x\right )}}{12 \, {\left (x^{3} - 1\right )}}\right ) \] Input:

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

Output:

-1/3*(I*sqrt(3) - 3*I)*weierstrassPInverse(0, 4, x) + 1/6*3^(1/4)*sqrt(2)* 
arctan(-1/12*3^(1/4)*sqrt(2)*sqrt(-x^3 + 1)*(3*x^2 - sqrt(3)*(x^2 - 2*x + 
4) + 6*x)/(x^3 - 1))
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

\text{Hanged}
 

Reduce [F]

\[ \int \frac {x}{\left (1+\sqrt {3}-x\right ) \sqrt {1-x^3}} \, dx=\frac {\sqrt {3}\, \left (\int \frac {\sqrt {-x^{3}+1}\, x^{2}}{x^{5}-2 x^{4}-2 x^{3}-x^{2}+2 x +2}d x \right )-\sqrt {3}\, \left (\int \frac {\sqrt {-x^{3}+1}\, x}{x^{5}-2 x^{4}-2 x^{3}-x^{2}+2 x +2}d x \right )+3 \left (\int \frac {\sqrt {-x^{3}+1}\, x}{x^{5}-2 x^{4}-2 x^{3}-x^{2}+2 x +2}d x \right )}{\sqrt {3}} \] Input:

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

Output:

(sqrt(3)*int((sqrt( - x**3 + 1)*x**2)/(x**5 - 2*x**4 - 2*x**3 - x**2 + 2*x 
 + 2),x) - sqrt(3)*int((sqrt( - x**3 + 1)*x)/(x**5 - 2*x**4 - 2*x**3 - x** 
2 + 2*x + 2),x) + 3*int((sqrt( - x**3 + 1)*x)/(x**5 - 2*x**4 - 2*x**3 - x* 
*2 + 2*x + 2),x))/sqrt(3)