\(\int \frac {1+\sqrt {3}+x}{(1-\sqrt {3}+x) \sqrt {-1-x^3}} \, dx\) [150]

Optimal result

Integrand size = 32, antiderivative size = 44 \[ \int \frac {1+\sqrt {3}+x}{\left (1-\sqrt {3}+x\right ) \sqrt {-1-x^3}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt {-3+2 \sqrt {3}} (1+x)}{\sqrt {-1-x^3}}\right )}{\sqrt {-3+2 \sqrt {3}}} \] Output:

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

Mathematica [A] (verified)

Time = 1.85 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.16 \[ \int \frac {1+\sqrt {3}+x}{\left (1-\sqrt {3}+x\right ) \sqrt {-1-x^3}} \, dx=2 \sqrt {1+\frac {2}{\sqrt {3}}} \arctan \left (\frac {\sqrt {-3+2 \sqrt {3}} \sqrt {-1-x^3}}{1-x+x^2}\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {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.

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

\(\Big \downarrow \) 2565

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

\(\Big \downarrow \) 216

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

Input:

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

Output:

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

Defintions of rubi rules used

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 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]
 

Maple [C] (verified)

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

Time = 1.70 (sec) , antiderivative size = 134, normalized size of antiderivative = 3.05

method	result	size
trager	\(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+24 \sqrt {3}+36\right ) \ln \left (-\frac {6 \operatorname {RootOf}\left (\textit {\_Z}^{2}+24 \sqrt {3}+36\right ) x^{2}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+24 \sqrt {3}+36\right ) \sqrt {3}\, x^{2}-4 \sqrt {3}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+24 \sqrt {3}+36\right ) x +48 \sqrt {-x^{3}-1}\, \sqrt {3}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+24 \sqrt {3}+36\right ) \sqrt {3}+72 \sqrt {-x^{3}-1}+12 \operatorname {RootOf}\left (\textit {\_Z}^{2}+24 \sqrt {3}+36\right )}{\left (\sqrt {3}\, x +x -2\right )^{2}}\right )}{6}\)	\(134\)
default	\(-\frac {2 i \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}-1}}-\frac {4 i \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticPi}\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}-\sqrt {3}}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-x^{3}-1}\, \left (\frac {3}{2}+\frac {i \sqrt {3}}{2}-\sqrt {3}\right )}\)	\(247\)
elliptic	\(-\frac {2 i \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}-1}}-\frac {4 i \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticPi}\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}-\sqrt {3}}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-x^{3}-1}\, \left (\frac {3}{2}+\frac {i \sqrt {3}}{2}-\sqrt {3}\right )}\)	\(247\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.23 \[ \int \frac {1+\sqrt {3}+x}{\left (1-\sqrt {3}+x\right ) \sqrt {-1-x^3}} \, dx=\sqrt {\frac {2}{3} \, \sqrt {3} + 1} \arctan \left (\frac {\sqrt {-x^{3} - 1} {\left (x^{2} + 2 \, \sqrt {3} {\left (x + 1\right )} - 4 \, x - 2\right )} \sqrt {\frac {2}{3} \, \sqrt {3} + 1}}{2 \, {\left (x^{3} + 1\right )}}\right ) \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

integrate((1+3^(1/2)+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,-1]:[1,0,-3]%%},[2]%%%} / %%%{%%{[-2,4]:[1,0,-3]%%} 
,[2]%%%}
 

Mupad [F(-1)]

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

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

Output:

\text{Hanged}
 

Reduce [F]

\[ \int \frac {1+\sqrt {3}+x}{\left (1-\sqrt {3}+x\right ) \sqrt {-1-x^3}} \, dx=\frac {i \left (-4 \sqrt {3}\, \left (\int \frac {\sqrt {x^{3}+1}}{x^{5}+2 x^{4}-2 x^{3}+x^{2}+2 x -2}d x \right )-\sqrt {3}\, \left (\int \frac {\sqrt {x^{3}+1}\, x^{2}}{x^{5}+2 x^{4}-2 x^{3}+x^{2}+2 x -2}d x \right )-2 \sqrt {3}\, \left (\int \frac {\sqrt {x^{3}+1}\, x}{x^{5}+2 x^{4}-2 x^{3}+x^{2}+2 x -2}d x \right )-6 \left (\int \frac {\sqrt {x^{3}+1}}{x^{5}+2 x^{4}-2 x^{3}+x^{2}+2 x -2}d x \right )-6 \left (\int \frac {\sqrt {x^{3}+1}\, x}{x^{5}+2 x^{4}-2 x^{3}+x^{2}+2 x -2}d x \right )\right )}{\sqrt {3}} \] Input:

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

Output:

(i*( - 4*sqrt(3)*int(sqrt(x**3 + 1)/(x**5 + 2*x**4 - 2*x**3 + x**2 + 2*x - 
 2),x) - sqrt(3)*int((sqrt(x**3 + 1)*x**2)/(x**5 + 2*x**4 - 2*x**3 + x**2 
+ 2*x - 2),x) - 2*sqrt(3)*int((sqrt(x**3 + 1)*x)/(x**5 + 2*x**4 - 2*x**3 + 
 x**2 + 2*x - 2),x) - 6*int(sqrt(x**3 + 1)/(x**5 + 2*x**4 - 2*x**3 + x**2 
+ 2*x - 2),x) - 6*int((sqrt(x**3 + 1)*x)/(x**5 + 2*x**4 - 2*x**3 + x**2 + 
2*x - 2),x)))/sqrt(3)