\(\int \frac {1-\sqrt {3}+x}{(1+\sqrt {3}+x) \sqrt {1+x^3}} \, dx\) [159]

Optimal result

Integrand size = 30, antiderivative size = 42 \[ \int \frac {1-\sqrt {3}+x}{\left (1+\sqrt {3}+x\right ) \sqrt {1+x^3}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt {3+2 \sqrt {3}} (1+x)}{\sqrt {1+x^3}}\right )}{\sqrt {3+2 \sqrt {3}}} \] 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 = 1.91 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.17 \[ \int \frac {1-\sqrt {3}+x}{\left (1+\sqrt {3}+x\right ) \sqrt {1+x^3}} \, dx=-2 \sqrt {-1+\frac {2}{\sqrt {3}}} \arctan \left (\frac {\sqrt {3+2 \sqrt {3}} \sqrt {1+x^3}}{1-x+x^2}\right ) \] 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 = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, 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

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

\(\Big \downarrow \) 216

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

Input:

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

Output:

(-2*ArcTan[(Sqrt[3 + 2*Sqrt[3]]*(1 + x))/Sqrt[1 + x^3]])/Sqrt[3 + 2*Sqrt[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 = 1.66 (sec) , antiderivative size = 132, normalized size of antiderivative = 3.14

method	result	size
trager	\(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-36+24 \sqrt {3}\right ) \ln \left (-\frac {6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-36+24 \sqrt {3}\right ) x^{2}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-36+24 \sqrt {3}\right ) \sqrt {3}\, x^{2}+4 \sqrt {3}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}-36+24 \sqrt {3}\right ) x -48 \sqrt {x^{3}+1}\, \sqrt {3}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-36+24 \sqrt {3}\right ) \sqrt {3}+72 \sqrt {x^{3}+1}+12 \operatorname {RootOf}\left (\textit {\_Z}^{2}-36+24 \sqrt {3}\right )}{\left (\sqrt {3}\, x -x +2\right )^{2}}\right )}{6}\)	\(132\)
default	\(\frac {2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}-\frac {4 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}\)	\(245\)
elliptic	\(\frac {2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}-\frac {4 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}\)	\(245\)

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

Fricas [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.10 \[ \int \frac {1-\sqrt {3}+x}{\left (1+\sqrt {3}+x\right ) \sqrt {1+x^3}} \, dx=-\sqrt {\frac {2}{3} \, \sqrt {3} - 1} \arctan \left (-\frac {{\left (x^{2} - 2 \, \sqrt {3} {\left (x + 1\right )} - 4 \, x - 2\right )} \sqrt {\frac {2}{3} \, \sqrt {3} - 1}}{2 \, \sqrt {x^{3} + 1}}\right ) \] 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*(x^2 - 2*sqrt(3)*(x + 1) - 4*x - 2)*sqr 
t(2/3*sqrt(3) - 1)/sqrt(x^3 + 1))
 

Sympy [F]

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

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

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

Output:

(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)