\(\int \frac {1-\sqrt {3}+x}{(1+\sqrt {3}+x) \sqrt {-4+4 \sqrt {3} x^2+x^4}} \, dx\) [232]

Optimal result

Integrand size = 40, antiderivative size = 65 \[ \int \frac {1-\sqrt {3}+x}{\left (1+\sqrt {3}+x\right ) \sqrt {-4+4 \sqrt {3} x^2+x^4}} \, dx=\frac {1}{3} \sqrt {-3+2 \sqrt {3}} \text {arctanh}\left (\frac {\left (1-\sqrt {3}+x\right )^2}{\sqrt {3 \left (-3+2 \sqrt {3}\right )} \sqrt {-4+4 \sqrt {3} x^2+x^4}}\right ) \] Output:

1/3*(-3+2*3^(1/2))^(1/2)*arctanh((1+x-3^(1/2))^2/(-9+6*3^(1/2))^(1/2)/(-4+ 
4*3^(1/2)*x^2+x^4)^(1/2))
                                                                                    
                                                                                    
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(192\) vs. \(2(65)=130\).

Time = 34.90 (sec) , antiderivative size = 192, normalized size of antiderivative = 2.95 \[ \int \frac {1-\sqrt {3}+x}{\left (1+\sqrt {3}+x\right ) \sqrt {-4+4 \sqrt {3} x^2+x^4}} \, dx=-\frac {\left (-1+\sqrt {3}+x\right )^2 \sqrt {\frac {6-2 \sqrt {3}-2 \sqrt {3} x+\left (9-5 \sqrt {3}\right ) x^2+\left (3-2 \sqrt {3}\right ) x^3}{\left (-1+\sqrt {3}+x\right )^3}} \arctan \left (\frac {\sqrt {9+6 \sqrt {3}} \left (-1+\sqrt {3}+x\right )^2 \sqrt {\frac {2 \left (1+\sqrt {3}\right )-2 \left (2+\sqrt {3}\right ) x+\left (-1+\sqrt {3}\right ) x^2-x^3}{\left (-1+\sqrt {3}+x\right )^3}}}{2-2 \left (1+\sqrt {3}\right ) x+\left (2+\sqrt {3}\right ) x^2}\right )}{3 \sqrt {-4+4 \sqrt {3} x^2+x^4}} \] Input:

Integrate[(1 - Sqrt[3] + x)/((1 + Sqrt[3] + x)*Sqrt[-4 + 4*Sqrt[3]*x^2 + x 
^4]),x]
 

Output:

-1/3*((-1 + Sqrt[3] + x)^2*Sqrt[(6 - 2*Sqrt[3] - 2*Sqrt[3]*x + (9 - 5*Sqrt 
[3])*x^2 + (3 - 2*Sqrt[3])*x^3)/(-1 + Sqrt[3] + x)^3]*ArcTan[(Sqrt[9 + 6*S 
qrt[3]]*(-1 + Sqrt[3] + x)^2*Sqrt[(2*(1 + Sqrt[3]) - 2*(2 + Sqrt[3])*x + ( 
-1 + Sqrt[3])*x^2 - x^3)/(-1 + Sqrt[3] + x)^3])/(2 - 2*(1 + Sqrt[3])*x + ( 
2 + Sqrt[3])*x^2)])/Sqrt[-4 + 4*Sqrt[3]*x^2 + x^4]
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.12, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {2278, 220}

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.

\(\displaystyle \int \frac {x-\sqrt {3}+1}{\left (x+\sqrt {3}+1\right ) \sqrt {x^4+4 \sqrt {3} x^2-4}} \, dx\)

\(\Big \downarrow \) 2278

\(\displaystyle -4 \left (2-\sqrt {3}\right ) \int \frac {1}{\frac {4 \left (x-\sqrt {3}+1\right )^4}{x^4+4 \sqrt {3} x^2-4}+12 \left (3-2 \sqrt {3}\right )}d\frac {\left (x-\sqrt {3}+1\right )^2}{\sqrt {x^4+4 \sqrt {3} x^2-4}}\)

\(\Big \downarrow \) 220

\(\displaystyle \frac {\left (2-\sqrt {3}\right ) \text {arctanh}\left (\frac {\left (x-\sqrt {3}+1\right )^2}{\sqrt {3 \left (2 \sqrt {3}-3\right )} \sqrt {x^4+4 \sqrt {3} x^2-4}}\right )}{\sqrt {3 \left (2 \sqrt {3}-3\right )}}\)

Input:

Int[(1 - Sqrt[3] + x)/((1 + Sqrt[3] + x)*Sqrt[-4 + 4*Sqrt[3]*x^2 + x^4]),x 
]
 

Output:

((2 - Sqrt[3])*ArcTanh[(1 - Sqrt[3] + x)^2/(Sqrt[3*(-3 + 2*Sqrt[3])]*Sqrt[ 
-4 + 4*Sqrt[3]*x^2 + x^4])])/Sqrt[3*(-3 + 2*Sqrt[3])]
 

Defintions of rubi rules used

rule 220

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(- 
1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && 
 (LtQ[a, 0] || GtQ[b, 0])
 

rule 2278

Int[((A_) + (B_.)*(x_))/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2 + (c_ 
.)*(x_)^4]), x_Symbol] :> Simp[(-A^2)*((B*d + A*e)/e)   Subst[Int[1/(6*A^3* 
B*d + 3*A^4*e - a*e*x^2), x], x, (A + B*x)^2/Sqrt[a + b*x^2 + c*x^4]], x] / 
; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[B*d - A*e, 0] && EqQ[c^2*d^6 + a*e 
^4*(13*c*d^2 + b*e^2), 0] && EqQ[b^2*e^4 - 12*c*d^2*(c*d^2 - b*e^2), 0] && 
EqQ[4*A*c*d*e + B*(2*c*d^2 - b*e^2), 0]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3.

Time = 2.04 (sec) , antiderivative size = 327, normalized size of antiderivative = 5.03

method	result	size
elliptic	\(\frac {\sqrt {1-\left (-1+\frac {\sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (1+\frac {\sqrt {3}}{2}\right ) x^{2}}\, \operatorname {EllipticF}\left (x \left (\frac {i \sqrt {3}}{2}-\frac {i}{2}\right ), i \sqrt {1+4 \sqrt {3}\, \left (1+\frac {\sqrt {3}}{2}\right )}\right )}{\left (\frac {i \sqrt {3}}{2}-\frac {i}{2}\right ) \sqrt {-4+4 \sqrt {3}\, x^{2}+x^{4}}}-2 \sqrt {3}\, \left (-\frac {\operatorname {arctanh}\left (\frac {4 \sqrt {3}\, \left (-1-\sqrt {3}\right )^{2}-8+4 \sqrt {3}\, x^{2}+2 x^{2} \left (-1-\sqrt {3}\right )^{2}}{2 \sqrt {\left (-1-\sqrt {3}\right )^{4}+4 \sqrt {3}\, \left (-1-\sqrt {3}\right )^{2}-4}\, \sqrt {-4+4 \sqrt {3}\, x^{2}+x^{4}}}\right )}{2 \sqrt {\left (-1-\sqrt {3}\right )^{4}+4 \sqrt {3}\, \left (-1-\sqrt {3}\right )^{2}-4}}-\frac {\sqrt {1-\left (-1+\frac {\sqrt {3}}{2}\right ) x^{2}}\, \sqrt {1-\left (1+\frac {\sqrt {3}}{2}\right ) x^{2}}\, \operatorname {EllipticPi}\left (\sqrt {-1+\frac {\sqrt {3}}{2}}\, x , \frac {1}{\left (-1+\frac {\sqrt {3}}{2}\right ) \left (-1-\sqrt {3}\right )^{2}}, \frac {\sqrt {1+\frac {\sqrt {3}}{2}}}{\sqrt {-1+\frac {\sqrt {3}}{2}}}\right )}{\sqrt {-1+\frac {\sqrt {3}}{2}}\, \left (-1-\sqrt {3}\right ) \sqrt {-4+4 \sqrt {3}\, x^{2}+x^{4}}}\right )\)	\(327\)

Input:

int((1-3^(1/2)+x)/(1+3^(1/2)+x)/(-4+4*3^(1/2)*x^2+x^4)^(1/2),x,method=_RET 
URNVERBOSE)
 

Output:

1/(1/2*I*3^(1/2)-1/2*I)*(1-(-1+1/2*3^(1/2))*x^2)^(1/2)*(1-(1+1/2*3^(1/2))* 
x^2)^(1/2)/(-4+4*3^(1/2)*x^2+x^4)^(1/2)*EllipticF(x*(1/2*I*3^(1/2)-1/2*I), 
I*(1+4*3^(1/2)*(1+1/2*3^(1/2)))^(1/2))-2*3^(1/2)*(-1/2/((-1-3^(1/2))^4+4*3 
^(1/2)*(-1-3^(1/2))^2-4)^(1/2)*arctanh(1/2*(4*3^(1/2)*(-1-3^(1/2))^2-8+4*3 
^(1/2)*x^2+2*x^2*(-1-3^(1/2))^2)/((-1-3^(1/2))^4+4*3^(1/2)*(-1-3^(1/2))^2- 
4)^(1/2)/(-4+4*3^(1/2)*x^2+x^4)^(1/2))-1/(-1+1/2*3^(1/2))^(1/2)/(-1-3^(1/2 
))*(1-(-1+1/2*3^(1/2))*x^2)^(1/2)*(1-(1+1/2*3^(1/2))*x^2)^(1/2)/(-4+4*3^(1 
/2)*x^2+x^4)^(1/2)*EllipticPi((-1+1/2*3^(1/2))^(1/2)*x,1/(-1+1/2*3^(1/2))/ 
(-1-3^(1/2))^2,(1+1/2*3^(1/2))^(1/2)/(-1+1/2*3^(1/2))^(1/2)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 323 vs. \(2 (47) = 94\).

Time = 0.31 (sec) , antiderivative size = 323, normalized size of antiderivative = 4.97 \[ \int \frac {1-\sqrt {3}+x}{\left (1+\sqrt {3}+x\right ) \sqrt {-4+4 \sqrt {3} x^2+x^4}} \, dx=\frac {1}{12} \, \sqrt {2 \, \sqrt {3} - 3} \log \left (-\frac {37 \, x^{12} - 204 \, x^{11} + 804 \, x^{10} - 2408 \, x^{9} + 3708 \, x^{8} - 5472 \, x^{7} + 6432 \, x^{6} + 10944 \, x^{5} + 14832 \, x^{4} + 19264 \, x^{3} + 12864 \, x^{2} + {\left (54 \, x^{10} - 300 \, x^{9} + 1026 \, x^{8} - 2232 \, x^{7} + 3024 \, x^{6} - 3024 \, x^{5} - 1008 \, x^{4} - 2016 \, x^{3} - 2592 \, x^{2} + \sqrt {3} {\left (31 \, x^{10} - 176 \, x^{9} + 576 \, x^{8} - 1320 \, x^{7} + 1848 \, x^{6} - 1008 \, x^{5} + 1344 \, x^{4} + 1632 \, x^{3} + 1008 \, x^{2} + 832 \, x + 256\right )} - 1152 \, x - 480\right )} \sqrt {x^{4} + 4 \, \sqrt {3} x^{2} - 4} \sqrt {2 \, \sqrt {3} - 3} + 3 \, \sqrt {3} {\left (7 \, x^{12} - 40 \, x^{11} + 160 \, x^{10} - 400 \, x^{9} + 924 \, x^{8} - 960 \, x^{7} - 1920 \, x^{5} - 3696 \, x^{4} - 3200 \, x^{3} - 2560 \, x^{2} - 1280 \, x - 448\right )} + 6528 \, x + 2368}{x^{12} + 12 \, x^{11} + 48 \, x^{10} + 40 \, x^{9} - 180 \, x^{8} - 288 \, x^{7} + 384 \, x^{6} + 576 \, x^{5} - 720 \, x^{4} - 320 \, x^{3} + 768 \, x^{2} - 384 \, x + 64}\right ) \] Input:

integrate((1-3^(1/2)+x)/(1+3^(1/2)+x)/(-4+4*x^2*3^(1/2)+x^4)^(1/2),x, algo 
rithm="fricas")
 

Output:

1/12*sqrt(2*sqrt(3) - 3)*log(-(37*x^12 - 204*x^11 + 804*x^10 - 2408*x^9 + 
3708*x^8 - 5472*x^7 + 6432*x^6 + 10944*x^5 + 14832*x^4 + 19264*x^3 + 12864 
*x^2 + (54*x^10 - 300*x^9 + 1026*x^8 - 2232*x^7 + 3024*x^6 - 3024*x^5 - 10 
08*x^4 - 2016*x^3 - 2592*x^2 + sqrt(3)*(31*x^10 - 176*x^9 + 576*x^8 - 1320 
*x^7 + 1848*x^6 - 1008*x^5 + 1344*x^4 + 1632*x^3 + 1008*x^2 + 832*x + 256) 
 - 1152*x - 480)*sqrt(x^4 + 4*sqrt(3)*x^2 - 4)*sqrt(2*sqrt(3) - 3) + 3*sqr 
t(3)*(7*x^12 - 40*x^11 + 160*x^10 - 400*x^9 + 924*x^8 - 960*x^7 - 1920*x^5 
 - 3696*x^4 - 3200*x^3 - 2560*x^2 - 1280*x - 448) + 6528*x + 2368)/(x^12 + 
 12*x^11 + 48*x^10 + 40*x^9 - 180*x^8 - 288*x^7 + 384*x^6 + 576*x^5 - 720* 
x^4 - 320*x^3 + 768*x^2 - 384*x + 64))
 

Sympy [F]

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

integrate((1-3**(1/2)+x)/(1+3**(1/2)+x)/(-4+4*x**2*3**(1/2)+x**4)**(1/2),x 
)
                                                                                    
                                                                                    
 

Output:

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

Maxima [F]

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

integrate((1-3^(1/2)+x)/(1+3^(1/2)+x)/(-4+4*x^2*3^(1/2)+x^4)^(1/2),x, algo 
rithm="maxima")
 

Output:

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

Giac [F]

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

integrate((1-3^(1/2)+x)/(1+3^(1/2)+x)/(-4+4*x^2*3^(1/2)+x^4)^(1/2),x, algo 
rithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {1-\sqrt {3}+x}{\left (1+\sqrt {3}+x\right ) \sqrt {-4+4 \sqrt {3} x^2+x^4}} \, dx=-4 \sqrt {3}\, \left (\int \frac {\sqrt {4 \sqrt {3}\, x^{2}+x^{4}-4}}{x^{8}-56 x^{4}+16}d x \right )-2 \sqrt {3}\, \left (\int \frac {\sqrt {4 \sqrt {3}\, x^{2}+x^{4}-4}\, x^{3}}{x^{8}-56 x^{4}+16}d x \right )-2 \sqrt {3}\, \left (\int \frac {\sqrt {4 \sqrt {3}\, x^{2}+x^{4}-4}\, x^{2}}{x^{8}-56 x^{4}+16}d x \right )-8 \sqrt {3}\, \left (\int \frac {\sqrt {4 \sqrt {3}\, x^{2}+x^{4}-4}\, x}{x^{8}-56 x^{4}+16}d x \right )+8 \left (\int \frac {\sqrt {4 \sqrt {3}\, x^{2}+x^{4}-4}}{x^{8}-56 x^{4}+16}d x \right )+\int \frac {\sqrt {4 \sqrt {3}\, x^{2}+x^{4}-4}\, x^{4}}{x^{8}-56 x^{4}+16}d x +6 \left (\int \frac {\sqrt {4 \sqrt {3}\, x^{2}+x^{4}-4}\, x^{2}}{x^{8}-56 x^{4}+16}d x \right )+12 \left (\int \frac {\sqrt {4 \sqrt {3}\, x^{2}+x^{4}-4}\, x}{x^{8}-56 x^{4}+16}d x \right ) \] Input:

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

Output:

 - 4*sqrt(3)*int(sqrt(4*sqrt(3)*x**2 + x**4 - 4)/(x**8 - 56*x**4 + 16),x) 
- 2*sqrt(3)*int((sqrt(4*sqrt(3)*x**2 + x**4 - 4)*x**3)/(x**8 - 56*x**4 + 1 
6),x) - 2*sqrt(3)*int((sqrt(4*sqrt(3)*x**2 + x**4 - 4)*x**2)/(x**8 - 56*x* 
*4 + 16),x) - 8*sqrt(3)*int((sqrt(4*sqrt(3)*x**2 + x**4 - 4)*x)/(x**8 - 56 
*x**4 + 16),x) + 8*int(sqrt(4*sqrt(3)*x**2 + x**4 - 4)/(x**8 - 56*x**4 + 1 
6),x) + int((sqrt(4*sqrt(3)*x**2 + x**4 - 4)*x**4)/(x**8 - 56*x**4 + 16),x 
) + 6*int((sqrt(4*sqrt(3)*x**2 + x**4 - 4)*x**2)/(x**8 - 56*x**4 + 16),x) 
+ 12*int((sqrt(4*sqrt(3)*x**2 + x**4 - 4)*x)/(x**8 - 56*x**4 + 16),x)