\(\int \frac {x}{(1+\sqrt {3}-x) \sqrt {-1+x^3}} \, dx\) [183]

Optimal result

Integrand size = 23, antiderivative size = 164 \[ \int \frac {x}{\left (1+\sqrt {3}-x\right ) \sqrt {-1+x^3}} \, dx=-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {3+2 \sqrt {3}} (1-x)}{\sqrt {-1+x^3}}\right )}{3^{3/4}}+\frac {2 \sqrt {\frac {7}{6}-\frac {2}{\sqrt {3}}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}} \] Output:

-1/3*2^(1/2)*arctanh((3+2*3^(1/2))^(1/2)*(1-x)/(x^3-1)^(1/2))*3^(1/4)+2/3* 
(1/3*6^(1/2)-1/2*2^(1/2))*(1-x)*((x^2+x+1)/(1-3^(1/2)-x)^2)^(1/2)*Elliptic 
F((1+3^(1/2)-x)/(1-3^(1/2)-x),2*I-I*3^(1/2))*3^(3/4)/(-(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 = 10.91 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.40 \[ \int \frac {x}{\left (1+\sqrt {3}-x\right ) \sqrt {-1+x^3}} \, dx=\frac {2 i \sqrt {\frac {1-x}{1+\sqrt [3]{-1}}} \left (\frac {i \sqrt {\frac {\sqrt [3]{-1}+(-1)^{2/3} x}{1+\sqrt [3]{-1}}} \left (3 i+(1+2 i) \sqrt {3}+\left (3+(2+i) \sqrt {3}\right ) x\right ) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right ),\sqrt [3]{-1}\right )}{\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}}+2 \left (1+\sqrt {3}\right ) \sqrt {1+x+x^2} \operatorname {EllipticPi}\left (\frac {2 i \sqrt {3}}{3+(2+i) \sqrt {3}},\arcsin \left (\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right ),\sqrt [3]{-1}\right )\right )}{\left (3+(2+i) \sqrt {3}\right ) \sqrt {-1+x^3}} \] 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 = 0.79 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.14, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2566, 27, 760, 2565, 219}

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.

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

\(\Big \downarrow \) 2566

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 760

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

\(\Big \downarrow \) 2565

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

(-2*ArcTanh[(Sqrt[3 + 2*Sqrt[3]]*(1 - x))/Sqrt[-1 + x^3]])/((3 - Sqrt[3])* 
Sqrt[3 + 2*Sqrt[3]]) + (2*(2 - Sqrt[3])^(3/2)*(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 219

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

rule 760

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 
] && NegQ[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 [A] (verified)

Time = 1.59 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.54

method	result	size
default	\(\frac {2 \left (1+\sqrt {3}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {3}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {x^{3}-1}}-\frac {2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}}\)	\(253\)
elliptic	\(-\frac {2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}}-\frac {2 \left (-1-\sqrt {3}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {3}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {x^{3}-1}}\)	\(255\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.30 \[ \int \frac {x}{\left (1+\sqrt {3}-x\right ) \sqrt {-1+x^3}} \, dx=\frac {1}{3} \, {\left (\sqrt {3} - 3\right )} {\rm weierstrassPInverse}\left (0, 4, x\right ) + \frac {1}{12} \cdot 3^{\frac {1}{4}} \sqrt {2} \log \left (\frac {x^{8} + 16 \, x^{7} + 112 \, x^{6} + 16 \, x^{5} + 112 \, x^{4} - 224 \, x^{3} + 2 \cdot 3^{\frac {1}{4}} \sqrt {2} {\left (x^{6} + 18 \, x^{5} + 12 \, x^{4} + 40 \, x^{3} - 36 \, x^{2} + \sqrt {3} {\left (x^{6} + 6 \, x^{5} + 24 \, x^{4} - 8 \, x^{3} + 12 \, x^{2} - 24 \, x + 16\right )} + 24 \, x - 32\right )} \sqrt {x^{3} - 1} + 64 \, x^{2} + 16 \, \sqrt {3} {\left (x^{7} + 2 \, x^{6} + 6 \, x^{5} - 5 \, x^{4} + 2 \, x^{3} - 6 \, x^{2} + 4 \, x - 4\right )} - 128 \, x + 112}{x^{8} - 8 \, x^{7} + 16 \, x^{6} + 16 \, x^{5} - 56 \, x^{4} - 32 \, x^{3} + 64 \, x^{2} + 64 \, x + 16}\right ) \] Input:

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

Output:

1/3*(sqrt(3) - 3)*weierstrassPInverse(0, 4, x) + 1/12*3^(1/4)*sqrt(2)*log( 
(x^8 + 16*x^7 + 112*x^6 + 16*x^5 + 112*x^4 - 224*x^3 + 2*3^(1/4)*sqrt(2)*( 
x^6 + 18*x^5 + 12*x^4 + 40*x^3 - 36*x^2 + sqrt(3)*(x^6 + 6*x^5 + 24*x^4 - 
8*x^3 + 12*x^2 - 24*x + 16) + 24*x - 32)*sqrt(x^3 - 1) + 64*x^2 + 16*sqrt( 
3)*(x^7 + 2*x^6 + 6*x^5 - 5*x^4 + 2*x^3 - 6*x^2 + 4*x - 4) - 128*x + 112)/ 
(x^8 - 8*x^7 + 16*x^6 + 16*x^5 - 56*x^4 - 32*x^3 + 64*x^2 + 64*x + 16))
 

Sympy [F]

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

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

Output:

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

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro 
unding error%%%{1,[2]%%%} / %%%{%%{[2,4]:[1,0,-3]%%},[2]%%%} Error: Bad Ar 
gument Va
 

Mupad [F(-1)]

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

int(x/((x^3 - 1)^(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)