\(\int \frac {1}{(2^{2/3}-x) \sqrt {-1+x^3}} \, dx\) [120]

Optimal result

Integrand size = 21, antiderivative size = 163 \[ \int \frac {1}{\left (2^{2/3}-x\right ) \sqrt {-1+x^3}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} x\right )}{\sqrt {-1+x^3}}\right )}{3 \sqrt {3}}-\frac {2 \sqrt [3]{2} \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right ),-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}} \] Output:

-2/9*arctanh(3^(1/2)*(1-2^(1/3)*x)/(x^3-1)^(1/2))*3^(1/2)-2/9*2^(1/3)*(1/2 
*6^(1/2)-1/2*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),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 = 20.31 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.90 \[ \int \frac {1}{\left (2^{2/3}-x\right ) \sqrt {-1+x^3}} \, dx=-\frac {4 i \sqrt {2} \sqrt {-\frac {i (-1+x)}{3 i+\sqrt {3}}} \sqrt {1+x+x^2} \operatorname {EllipticPi}\left (\frac {2 \sqrt {3}}{i+2 i 2^{2/3}+\sqrt {3}},\arcsin \left (\frac {\sqrt {i+\sqrt {3}+2 i x}}{\sqrt {2} \sqrt [4]{3}}\right ),\frac {2 \sqrt {3}}{3 i+\sqrt {3}}\right )}{\left (1+2\ 2^{2/3}-i \sqrt {3}\right ) \sqrt {-1+x^3}} \] Input:

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

Output:

((-4*I)*Sqrt[2]*Sqrt[((-I)*(-1 + x))/(3*I + Sqrt[3])]*Sqrt[1 + x + x^2]*El 
lipticPi[(2*Sqrt[3])/(I + (2*I)*2^(2/3) + Sqrt[3]), ArcSin[Sqrt[I + Sqrt[3 
] + (2*I)*x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])])/((1 + 2*2^( 
2/3) - I*Sqrt[3])*Sqrt[-1 + x^3])
 

Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2559, 27, 760, 2562, 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.

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

\(\Big \downarrow \) 2559

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 760

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

\(\Big \downarrow \) 2562

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

(-2*ArcTanh[(Sqrt[3]*(1 - 2^(1/3)*x))/Sqrt[-1 + x^3]])/(3*Sqrt[3]) - (2*2^ 
(1/3)*Sqrt[2 - Sqrt[3]]*(1 - x)*Sqrt[(1 + x + x^2)/(1 - Sqrt[3] - x)^2]*El 
lipticF[ArcSin[(1 + Sqrt[3] - x)/(1 - Sqrt[3] - x)], -7 + 4*Sqrt[3]])/(3*3 
^(1/4)*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 2559

Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbol] :> Simp[2/ 
(3*c)   Int[1/Sqrt[a + b*x^3], x], x] + Simp[1/(3*c)   Int[(c - 2*d*x)/((c 
+ d*x)*Sqrt[a + b*x^3]), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^3 - 4* 
a*d^3, 0]
 

rule 2562

Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_ 
Symbol] :> Simp[2*(e/d)   Subst[Int[1/(1 + 3*a*x^2), x], x, (1 + 2*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*c^3 - 4*a*d^3, 0] && EqQ[2*d*e + c*f, 0]
 

Maple [A] (verified)

Time = 1.64 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.88

method	result	size
default	\(-\frac {2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-2^{\frac {2}{3}}+1}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}\, \left (-2^{\frac {2}{3}}+1\right )}\)	\(143\)
elliptic	\(-\frac {2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-2^{\frac {2}{3}}+1}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}\, \left (-2^{\frac {2}{3}}+1\right )}\)	\(143\)

Input:

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

Output:

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{\left (2^{2/3}-x\right ) \sqrt {-1+x^3}} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >>  Error detected within library code:   catd 
ef: division by zero
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-integrate(1/(sqrt(x^3 - 1)*(x - 2^(2/3))), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {1}{\left (2^{2/3}-x\right ) \sqrt {-1+x^3}} \, dx=\text {Exception raised: RuntimeError} \] Input:

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

Output:

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:Unable to divide, perhaps due to rounding error%%%{1,[2 
]%%%} / %%%{%%{[2,0]:[1,0,0,-2]%%},[2]%%%} Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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