\(\int \frac {1-\sqrt {3}+2 x}{(1+\sqrt {3}+2 x) \sqrt {-1+4 \sqrt {3} x^2+4 x^4}} \, dx\) [234]

Optimal result

Integrand size = 46, antiderivative size = 72 \[ \int \frac {1-\sqrt {3}+2 x}{\left (1+\sqrt {3}+2 x\right ) \sqrt {-1+4 \sqrt {3} x^2+4 x^4}} \, dx=\frac {1}{3} \sqrt {-3+2 \sqrt {3}} \text {arctanh}\left (\frac {\left (1-\sqrt {3}+2 x\right )^2}{2 \sqrt {3 \left (-3+2 \sqrt {3}\right )} \sqrt {-1+4 \sqrt {3} x^2+4 x^4}}\right ) \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(207\) vs. \(2(72)=144\).

Time = 21.99 (sec) , antiderivative size = 207, normalized size of antiderivative = 2.88 \[ \int \frac {1-\sqrt {3}+2 x}{\left (1+\sqrt {3}+2 x\right ) \sqrt {-1+4 \sqrt {3} x^2+4 x^4}} \, dx=-\frac {\sqrt {-\frac {3}{2}+\sqrt {3}} \left (-1+\sqrt {3}+2 x\right )^2 \sqrt {\frac {1+\sqrt {3}-2 \left (2+\sqrt {3}\right ) x+2 \left (-1+\sqrt {3}\right ) x^2-4 x^3}{\left (-1+\sqrt {3}+2 x\right )^3}} \arctan \left (\frac {\sqrt {\frac {9}{2}+3 \sqrt {3}} \left (-1+\sqrt {3}+2 x\right )^2 \sqrt {\frac {1+\sqrt {3}-2 \left (2+\sqrt {3}\right ) x+2 \left (-1+\sqrt {3}\right ) x^2-4 x^3}{\left (-1+\sqrt {3}+2 x\right )^3}}}{1-2 \left (1+\sqrt {3}\right ) x+2 \left (2+\sqrt {3}\right ) x^2}\right )}{3 \sqrt {-1+4 \sqrt {3} x^2+4 x^4}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.11, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, 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.

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

\(\Big \downarrow \) 2278

\(\displaystyle -4 \left (2-\sqrt {3}\right ) \int \frac {1}{\frac {2 \left (2 x-\sqrt {3}+1\right )^4}{4 x^4+4 \sqrt {3} x^2-1}+24 \left (3-2 \sqrt {3}\right )}d\frac {\left (2 x-\sqrt {3}+1\right )^2}{\sqrt {4 x^4+4 \sqrt {3} x^2-1}}\)

\(\Big \downarrow \) 220

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

Input:

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

Output:

((2 - Sqrt[3])*ArcTanh[(1 - Sqrt[3] + 2*x)^2/(2*Sqrt[3*(-3 + 2*Sqrt[3])]*S 
qrt[-1 + 4*Sqrt[3]*x^2 + 4*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 = 1.97 (sec) , antiderivative size = 336, normalized size of antiderivative = 4.67

method	result	size
elliptic	\(\frac {\sqrt {1-\left (-4+2 \sqrt {3}\right ) x^{2}}\, \sqrt {1-\left (4+2 \sqrt {3}\right ) x^{2}}\, \operatorname {EllipticF}\left (x \left (i \sqrt {3}-i\right ), i \sqrt {1+\sqrt {3}\, \left (4+2 \sqrt {3}\right )}\right )}{\left (i \sqrt {3}-i\right ) \sqrt {-1+4 \sqrt {3}\, x^{2}+4 x^{4}}}-\sqrt {3}\, \left (-\frac {\operatorname {arctanh}\left (\frac {4 \sqrt {3}\, \left (-\frac {1}{2}-\frac {\sqrt {3}}{2}\right )^{2}-2+4 \sqrt {3}\, x^{2}+8 x^{2} \left (-\frac {1}{2}-\frac {\sqrt {3}}{2}\right )^{2}}{2 \sqrt {4 \left (-\frac {1}{2}-\frac {\sqrt {3}}{2}\right )^{4}+4 \sqrt {3}\, \left (-\frac {1}{2}-\frac {\sqrt {3}}{2}\right )^{2}-1}\, \sqrt {-1+4 \sqrt {3}\, x^{2}+4 x^{4}}}\right )}{2 \sqrt {4 \left (-\frac {1}{2}-\frac {\sqrt {3}}{2}\right )^{4}+4 \sqrt {3}\, \left (-\frac {1}{2}-\frac {\sqrt {3}}{2}\right )^{2}-1}}-\frac {\sqrt {1-\left (-4+2 \sqrt {3}\right ) x^{2}}\, \sqrt {1-\left (4+2 \sqrt {3}\right ) x^{2}}\, \operatorname {EllipticPi}\left (\sqrt {-4+2 \sqrt {3}}\, x , \frac {1}{\left (-4+2 \sqrt {3}\right ) \left (-\frac {1}{2}-\frac {\sqrt {3}}{2}\right )^{2}}, \frac {\sqrt {4+2 \sqrt {3}}}{\sqrt {-4+2 \sqrt {3}}}\right )}{\sqrt {-4+2 \sqrt {3}}\, \left (-\frac {1}{2}-\frac {\sqrt {3}}{2}\right ) \sqrt {-1+4 \sqrt {3}\, x^{2}+4 x^{4}}}\right )\)	\(336\)

Input:

int((1-3^(1/2)+2*x)/(1+3^(1/2)+2*x)/(-1+4*3^(1/2)*x^2+4*x^4)^(1/2),x,metho 
d=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 328 vs. \(2 (52) = 104\).

Time = 0.24 (sec) , antiderivative size = 328, normalized size of antiderivative = 4.56 \[ \int \frac {1-\sqrt {3}+2 x}{\left (1+\sqrt {3}+2 x\right ) \sqrt {-1+4 \sqrt {3} x^2+4 x^4}} \, dx=\frac {1}{12} \, \sqrt {2 \, \sqrt {3} - 3} \log \left (-\frac {2368 \, x^{12} - 6528 \, x^{11} + 12864 \, x^{10} - 19264 \, x^{9} + 14832 \, x^{8} - 10944 \, x^{7} + 6432 \, x^{6} + 5472 \, x^{5} + 3708 \, x^{4} + 2408 \, x^{3} + 804 \, x^{2} + {\left (1728 \, x^{10} - 4800 \, x^{9} + 8208 \, x^{8} - 8928 \, x^{7} + 6048 \, x^{6} - 3024 \, x^{5} - 504 \, x^{4} - 504 \, x^{3} - 324 \, x^{2} + 2 \, \sqrt {3} {\left (496 \, x^{10} - 1408 \, x^{9} + 2304 \, x^{8} - 2640 \, x^{7} + 1848 \, x^{6} - 504 \, x^{5} + 336 \, x^{4} + 204 \, x^{3} + 63 \, x^{2} + 26 \, x + 4\right )} - 72 \, x - 15\right )} \sqrt {4 \, x^{4} + 4 \, \sqrt {3} x^{2} - 1} \sqrt {2 \, \sqrt {3} - 3} + 3 \, \sqrt {3} {\left (448 \, x^{12} - 1280 \, x^{11} + 2560 \, x^{10} - 3200 \, x^{9} + 3696 \, x^{8} - 1920 \, x^{7} - 960 \, x^{5} - 924 \, x^{4} - 400 \, x^{3} - 160 \, x^{2} - 40 \, x - 7\right )} + 204 \, x + 37}{64 \, x^{12} + 384 \, x^{11} + 768 \, x^{10} + 320 \, x^{9} - 720 \, x^{8} - 576 \, x^{7} + 384 \, x^{6} + 288 \, x^{5} - 180 \, x^{4} - 40 \, x^{3} + 48 \, x^{2} - 12 \, x + 1}\right ) \] Input:

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

Output:

1/12*sqrt(2*sqrt(3) - 3)*log(-(2368*x^12 - 6528*x^11 + 12864*x^10 - 19264* 
x^9 + 14832*x^8 - 10944*x^7 + 6432*x^6 + 5472*x^5 + 3708*x^4 + 2408*x^3 + 
804*x^2 + (1728*x^10 - 4800*x^9 + 8208*x^8 - 8928*x^7 + 6048*x^6 - 3024*x^ 
5 - 504*x^4 - 504*x^3 - 324*x^2 + 2*sqrt(3)*(496*x^10 - 1408*x^9 + 2304*x^ 
8 - 2640*x^7 + 1848*x^6 - 504*x^5 + 336*x^4 + 204*x^3 + 63*x^2 + 26*x + 4) 
 - 72*x - 15)*sqrt(4*x^4 + 4*sqrt(3)*x^2 - 1)*sqrt(2*sqrt(3) - 3) + 3*sqrt 
(3)*(448*x^12 - 1280*x^11 + 2560*x^10 - 3200*x^9 + 3696*x^8 - 1920*x^7 - 9 
60*x^5 - 924*x^4 - 400*x^3 - 160*x^2 - 40*x - 7) + 204*x + 37)/(64*x^12 + 
384*x^11 + 768*x^10 + 320*x^9 - 720*x^8 - 576*x^7 + 384*x^6 + 288*x^5 - 18 
0*x^4 - 40*x^3 + 48*x^2 - 12*x + 1))
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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