\(\int \frac {1-\sqrt {3}+x}{(1+\sqrt {3}+x) \sqrt {-1-x^3}} \, dx\) [162]

Optimal result

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

-2*arctanh((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 = 1.90 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.16 \[ \int \frac {1-\sqrt {3}+x}{\left (1+\sqrt {3}+x\right ) \sqrt {-1-x^3}} \, dx=2 \sqrt {-1+\frac {2}{\sqrt {3}}} \text {arctanh}\left (\frac {\sqrt {3+2 \sqrt {3}} \sqrt {-1-x^3}}{1-x+x^2}\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.39 (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, 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.

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

\(\Big \downarrow \) 2565

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

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

Defintions of rubi rules used

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 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 = 1.56 (sec) , antiderivative size = 136, normalized size of antiderivative = 3.09

method	result	size
trager	\(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-24 \sqrt {3}+36\right ) \ln \left (-\frac {6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-24 \sqrt {3}+36\right ) x^{2}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-24 \sqrt {3}+36\right ) \sqrt {3}\, x^{2}+4 \sqrt {3}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}-24 \sqrt {3}+36\right ) x -48 \sqrt {-x^{3}-1}\, \sqrt {3}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-24 \sqrt {3}+36\right ) \sqrt {3}+72 \sqrt {-x^{3}-1}+12 \operatorname {RootOf}\left (\textit {\_Z}^{2}-24 \sqrt {3}+36\right )}{\left (\sqrt {3}\, x -x +2\right )^{2}}\right )}{6}\)	\(136\)
default	\(-\frac {2 i \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}-1}}+\frac {4 i \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticPi}\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}+\sqrt {3}}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-x^{3}-1}\, \left (\frac {3}{2}+\frac {i \sqrt {3}}{2}+\sqrt {3}\right )}\)	\(243\)
elliptic	\(-\frac {2 i \sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}-1}}+\frac {4 i \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x +1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticPi}\left (\frac {\sqrt {3}\, \sqrt {i \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}+\sqrt {3}}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-x^{3}-1}\, \left (\frac {3}{2}+\frac {i \sqrt {3}}{2}+\sqrt {3}\right )}\)	\(243\)

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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 204 vs. \(2 (34) = 68\).

Time = 0.13 (sec) , antiderivative size = 204, normalized size of antiderivative = 4.64 \[ \int \frac {1-\sqrt {3}+x}{\left (1+\sqrt {3}+x\right ) \sqrt {-1-x^3}} \, dx=\frac {1}{2} \, \sqrt {\frac {2}{3} \, \sqrt {3} - 1} \log \left (\frac {x^{8} - 16 \, x^{7} + 112 \, x^{6} - 16 \, x^{5} + 112 \, x^{4} + 224 \, x^{3} + 64 \, x^{2} + 4 \, {\left (3 \, x^{6} - 36 \, x^{5} + 54 \, x^{4} - 48 \, x^{3} - 36 \, x^{2} + 2 \, \sqrt {3} {\left (x^{6} - 9 \, x^{5} + 21 \, x^{4} - 4 \, x^{3} + 12 \, x + 4\right )} - 24\right )} \sqrt {-x^{3} - 1} \sqrt {\frac {2}{3} \, \sqrt {3} - 1} - 16 \, \sqrt {3} {\left (x^{7} - 2 \, x^{6} + 6 \, x^{5} + 5 \, x^{4} + 2 \, x^{3} + 6 \, x^{2} + 4 \, x + 4\right )} + 128 \, x + 112}{x^{8} + 8 \, x^{7} + 16 \, x^{6} - 16 \, x^{5} - 56 \, x^{4} + 32 \, x^{3} + 64 \, x^{2} - 64 \, x + 16}\right ) \] Input:

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

Output:

1/2*sqrt(2/3*sqrt(3) - 1)*log((x^8 - 16*x^7 + 112*x^6 - 16*x^5 + 112*x^4 + 
 224*x^3 + 64*x^2 + 4*(3*x^6 - 36*x^5 + 54*x^4 - 48*x^3 - 36*x^2 + 2*sqrt( 
3)*(x^6 - 9*x^5 + 21*x^4 - 4*x^3 + 12*x + 4) - 24)*sqrt(-x^3 - 1)*sqrt(2/3 
*sqrt(3) - 1) - 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]

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

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

Output:

Integral((x - sqrt(3) + 1)/(sqrt(-(x + 1)*(x**2 - x + 1))*(x + 1 + sqrt(3) 
)), 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]%%%} Er
 

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)