\(\int \frac {x}{(2-x) \sqrt {-1-x^3}} \, dx\) [141]

Optimal result

Integrand size = 20, antiderivative size = 140 \[ \int \frac {x}{(2-x) \sqrt {-1-x^3}} \, dx=\frac {4}{9} \arctan \left (\frac {(1+x)^2}{3 \sqrt {-1-x^3}}\right )-\frac {2 \sqrt {2-\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1-\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}+x}{1-\sqrt {3}+x}\right ),-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {-\frac {1+x}{\left (1-\sqrt {3}+x\right )^2}} \sqrt {-1-x^3}} \] Output:

4/9*arctan(1/3*(1+x)^2/(-x^3-1)^(1/2))-2/9*(1/2*6^(1/2)-1/2*2^(1/2))*(1+x) 
*((x^2-x+1)/(1+x-3^(1/2))^2)^(1/2)*EllipticF((1+x+3^(1/2))/(1+x-3^(1/2)),2 
*I-I*3^(1/2))*3^(3/4)/(-(1+x)/(1+x-3^(1/2))^2)^(1/2)/(-x^3-1)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 10.26 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.39 \[ \int \frac {x}{(2-x) \sqrt {-1-x^3}} \, dx=\frac {2 \sqrt {\frac {1+x}{1+\sqrt [3]{-1}}} \left (\frac {\left (\sqrt [3]{-1}-x\right ) \sqrt {\frac {\sqrt [3]{-1}-(-1)^{2/3} x}{1+\sqrt [3]{-1}}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {1+(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right ),\sqrt [3]{-1}\right )}{\sqrt {\frac {1+(-1)^{2/3} x}{1+\sqrt [3]{-1}}}}+\frac {2 i \sqrt {1-x+x^2} \operatorname {EllipticPi}\left (\frac {2 \sqrt {3}}{3 i+\sqrt {3}},\arcsin \left (\sqrt {\frac {1+(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right ),\sqrt [3]{-1}\right )}{-2+\sqrt [3]{-1}}\right )}{\sqrt {-1-x^3}} \] Input:

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

Output:

(2*Sqrt[(1 + x)/(1 + (-1)^(1/3))]*((((-1)^(1/3) - x)*Sqrt[((-1)^(1/3) - (- 
1)^(2/3)*x)/(1 + (-1)^(1/3))]*EllipticF[ArcSin[Sqrt[(1 + (-1)^(2/3)*x)/(1 
+ (-1)^(1/3))]], (-1)^(1/3)])/Sqrt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))] + 
((2*I)*Sqrt[1 - x + x^2]*EllipticPi[(2*Sqrt[3])/(3*I + Sqrt[3]), ArcSin[Sq 
rt[(1 + (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(-2 + (-1)^(1/3)))) 
/Sqrt[-1 - x^3]
 

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2564, 27, 760, 2563, 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.

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

\(\Big \downarrow \) 2564

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 760

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

\(\Big \downarrow \) 2563

\(\displaystyle \frac {4}{3} \int \frac {1}{\frac {(x+1)^4}{-x^3-1}+9}d\frac {(x+1)^2}{\sqrt {-x^3-1}}-\frac {2 \sqrt {2-\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x+\sqrt {3}+1}{x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {-\frac {x+1}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {-x^3-1}}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {4}{9} \arctan \left (\frac {(x+1)^2}{3 \sqrt {-x^3-1}}\right )-\frac {2 \sqrt {2-\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x+\sqrt {3}+1}{x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {-\frac {x+1}{\left (x-\sqrt {3}+1\right )^2}} \sqrt {-x^3-1}}\)

Input:

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

Output:

(4*ArcTan[(1 + x)^2/(3*Sqrt[-1 - x^3])])/9 - (2*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*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 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 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 2563

Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_ 
Symbol] :> Simp[-2*(e/d)   Subst[Int[1/(9 - a*x^2), x], x, (1 + f*(x/e))^2/ 
Sqrt[a + b*x^3]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] & 
& EqQ[b*c^3 + 8*a*d^3, 0] && EqQ[2*d*e + c*f, 0]
 

rule 2564

Int[((e_.) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x 
_Symbol] :> Simp[(2*d*e + c*f)/(3*c*d)   Int[1/Sqrt[a + b*x^3], x], x] + Si 
mp[(d*e - c*f)/(3*c*d)   Int[(c - 2*d*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*c^3 - 4*a* 
d^3, 0] || EqQ[b*c^3 + 8*a*d^3, 0]) && NeQ[2*d*e + c*f, 0]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 239 vs. \(2 (113 ) = 226\).

Time = 0.49 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.71

method	result	size
default	\(\frac {2 i \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}-1}}+\frac {4 i \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticPi}\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}-1}\, \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right )}\)	\(240\)
elliptic	\(\frac {2 i \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}-1}}+\frac {4 i \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticPi}\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}-1}\, \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right )}\)	\(240\)

Input:

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

Output:

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)*Elli 
pticF(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))+4/3*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)/(-3/2+1/2*I*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)),(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 = 45, normalized size of antiderivative = 0.32 \[ \int \frac {x}{(2-x) \sqrt {-1-x^3}} \, dx=-\frac {2}{9} \, \arctan \left (\frac {{\left (x^{3} + 12 \, x^{2} - 6 \, x + 10\right )} \sqrt {-x^{3} - 1}}{6 \, {\left (x^{4} + x^{3} + x + 1\right )}}\right ) + \frac {2}{3} i \, {\rm weierstrassPInverse}\left (0, -4, x\right ) \] Input:

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

Output:

-2/9*arctan(1/6*(x^3 + 12*x^2 - 6*x + 10)*sqrt(-x^3 - 1)/(x^4 + x^3 + x + 
1)) + 2/3*I*weierstrassPInverse(0, -4, x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.59 \[ \int \frac {x}{(2-x) \sqrt {-1-x^3}} \, dx=-\frac {\left (3+\sqrt {3}\,1{}\mathrm {i}\right )\,\sqrt {x^3+1}\,\sqrt {\frac {x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {\frac {1}{2}-x+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\left (3\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )-2\,\Pi \left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6};\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )\right )}{3\,\sqrt {-x^3-1}\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \] Input:

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

Output:

-((3^(1/2)*1i + 3)*(x^3 + 1)^(1/2)*((x + (3^(1/2)*1i)/2 - 1/2)/((3^(1/2)*1 
i)/2 - 3/2))^(1/2)*((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*(((3^(1/2)*1i)/2 
 - x + 1/2)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*(3*ellipticF(asin(((x + 1)/((3^( 
1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((3^(1/2)*1i)/2 - 3/2)) 
 - 2*ellipticPi((3^(1/2)*1i)/6 + 1/2, asin(((x + 1)/((3^(1/2)*1i)/2 + 3/2) 
)^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((3^(1/2)*1i)/2 - 3/2))))/(3*(- x^3 - 1) 
^(1/2)*(x^3 - x*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1) - ((3^ 
(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2))^(1/2))
 

Reduce [F]

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

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

Output:

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