Integrand size = 46, antiderivative size = 72 \[ \int \frac {1-\sqrt {3}+2 x}{\left (1+\sqrt {3}+2 x\right ) \sqrt {-1+4 \sqrt {3} x^2+4 x^4}} \, dx=\frac {1}{3} \sqrt {-3+2 \sqrt {3}} \text {arctanh}\left (\frac {\left (1-\sqrt {3}+2 x\right )^2}{2 \sqrt {3 \left (-3+2 \sqrt {3}\right )} \sqrt {-1+4 \sqrt {3} x^2+4 x^4}}\right ) \]
1/3*arctanh(1/2*(1+2*x-3^(1/2))^2/(-9+6*3^(1/2))^(1/2)/(-1+4*x^4+4*x^2*3^( 1/2))^(1/2))*(-3+2*3^(1/2))^(1/2)
Time = 8.82 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.12 \[ \int \frac {1-\sqrt {3}+2 x}{\left (1+\sqrt {3}+2 x\right ) \sqrt {-1+4 \sqrt {3} x^2+4 x^4}} \, dx=\frac {1}{3} \sqrt {-3+2 \sqrt {3}} \text {arctanh}\left (\frac {\sqrt {9+6 \sqrt {3}} \sqrt {-1+4 \sqrt {3} x^2+4 x^4}}{1+\left (-2-2 \sqrt {3}\right ) x+\left (4+2 \sqrt {3}\right ) x^2}\right ) \]
(Sqrt[-3 + 2*Sqrt[3]]*ArcTanh[(Sqrt[9 + 6*Sqrt[3]]*Sqrt[-1 + 4*Sqrt[3]*x^2 + 4*x^4])/(1 + (-2 - 2*Sqrt[3])*x + (4 + 2*Sqrt[3])*x^2)])/3
Time = 0.30 (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 |
\(\displaystyle \frac {\left (2-\sqrt {3}\right ) \text {arctanh}\left (\frac {\left (2 x-\sqrt {3}+1\right )^2}{2 \sqrt {3 \left (2 \sqrt {3}-3\right )} \sqrt {4 x^4+4 \sqrt {3} x^2-1}}\right )}{\sqrt {3 \left (2 \sqrt {3}-3\right )}}\) |
((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])]
3.5.10.3.1 Defintions of rubi rules used
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])
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]
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 3.38 (sec) , antiderivative size = 336, normalized size of antiderivative = 4.67
method | result | size |
elliptic | \(\frac {\sqrt {1-\left (2 \sqrt {3}-4\right ) x^{2}}\, \sqrt {1-\left (2 \sqrt {3}+4\right ) x^{2}}\, F\left (x \left (i \sqrt {3}-i\right ), i \sqrt {1+\sqrt {3}\, \left (2 \sqrt {3}+4\right )}\right )}{\left (i \sqrt {3}-i\right ) \sqrt {-1+4 x^{4}+4 x^{2} \sqrt {3}}}-\sqrt {3}\, \left (-\frac {\operatorname {arctanh}\left (\frac {4 \sqrt {3}\, \left (-\frac {1}{2}-\frac {\sqrt {3}}{2}\right )^{2}-2+4 x^{2} \sqrt {3}+8 x^{2} \left (-\frac {1}{2}-\frac {\sqrt {3}}{2}\right )^{2}}{2 \sqrt {4 \left (-\frac {1}{2}-\frac {\sqrt {3}}{2}\right )^{4}+4 \sqrt {3}\, \left (-\frac {1}{2}-\frac {\sqrt {3}}{2}\right )^{2}-1}\, \sqrt {-1+4 x^{4}+4 x^{2} \sqrt {3}}}\right )}{2 \sqrt {4 \left (-\frac {1}{2}-\frac {\sqrt {3}}{2}\right )^{4}+4 \sqrt {3}\, \left (-\frac {1}{2}-\frac {\sqrt {3}}{2}\right )^{2}-1}}-\frac {\sqrt {1-\left (2 \sqrt {3}-4\right ) x^{2}}\, \sqrt {1-\left (2 \sqrt {3}+4\right ) x^{2}}\, \Pi \left (\sqrt {2 \sqrt {3}-4}\, x , \frac {1}{\left (2 \sqrt {3}-4\right ) \left (-\frac {1}{2}-\frac {\sqrt {3}}{2}\right )^{2}}, \frac {\sqrt {2 \sqrt {3}+4}}{\sqrt {2 \sqrt {3}-4}}\right )}{\sqrt {2 \sqrt {3}-4}\, \left (-\frac {1}{2}-\frac {\sqrt {3}}{2}\right ) \sqrt {-1+4 x^{4}+4 x^{2} \sqrt {3}}}\right )\) | \(336\) |
1/(I*3^(1/2)-I)*(1-(2*3^(1/2)-4)*x^2)^(1/2)*(1-(2*3^(1/2)+4)*x^2)^(1/2)/(- 1+4*x^4+4*x^2*3^(1/2))^(1/2)*EllipticF(x*(I*3^(1/2)-I),I*(1+3^(1/2)*(2*3^( 1/2)+4))^(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*x^2*3^ (1/2)+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*x^4+4*x^2*3^(1/2))^(1/2))-1/(2*3^(1/2)-4)^(1 /2)/(-1/2-1/2*3^(1/2))*(1-(2*3^(1/2)-4)*x^2)^(1/2)*(1-(2*3^(1/2)+4)*x^2)^( 1/2)/(-1+4*x^4+4*x^2*3^(1/2))^(1/2)*EllipticPi((2*3^(1/2)-4)^(1/2)*x,1/(2* 3^(1/2)-4)/(-1/2-1/2*3^(1/2))^2,(2*3^(1/2)+4)^(1/2)/(2*3^(1/2)-4)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 328 vs. \(2 (52) = 104\).
Time = 0.39 (sec) , antiderivative size = 328, normalized size of antiderivative = 4.56 \[ \int \frac {1-\sqrt {3}+2 x}{\left (1+\sqrt {3}+2 x\right ) \sqrt {-1+4 \sqrt {3} x^2+4 x^4}} \, dx=\frac {1}{12} \, \sqrt {2 \, \sqrt {3} - 3} \log \left (-\frac {2368 \, x^{12} - 6528 \, x^{11} + 12864 \, x^{10} - 19264 \, x^{9} + 14832 \, x^{8} - 10944 \, x^{7} + 6432 \, x^{6} + 5472 \, x^{5} + 3708 \, x^{4} + 2408 \, x^{3} + 804 \, x^{2} + {\left (1728 \, x^{10} - 4800 \, x^{9} + 8208 \, x^{8} - 8928 \, x^{7} + 6048 \, x^{6} - 3024 \, x^{5} - 504 \, x^{4} - 504 \, x^{3} - 324 \, x^{2} + 2 \, \sqrt {3} {\left (496 \, x^{10} - 1408 \, x^{9} + 2304 \, x^{8} - 2640 \, x^{7} + 1848 \, x^{6} - 504 \, x^{5} + 336 \, x^{4} + 204 \, x^{3} + 63 \, x^{2} + 26 \, x + 4\right )} - 72 \, x - 15\right )} \sqrt {4 \, x^{4} + 4 \, \sqrt {3} x^{2} - 1} \sqrt {2 \, \sqrt {3} - 3} + 3 \, \sqrt {3} {\left (448 \, x^{12} - 1280 \, x^{11} + 2560 \, x^{10} - 3200 \, x^{9} + 3696 \, x^{8} - 1920 \, x^{7} - 960 \, x^{5} - 924 \, x^{4} - 400 \, x^{3} - 160 \, x^{2} - 40 \, x - 7\right )} + 204 \, x + 37}{64 \, x^{12} + 384 \, x^{11} + 768 \, x^{10} + 320 \, x^{9} - 720 \, x^{8} - 576 \, x^{7} + 384 \, x^{6} + 288 \, x^{5} - 180 \, x^{4} - 40 \, x^{3} + 48 \, x^{2} - 12 \, x + 1}\right ) \]
integrate((1+2*x-3^(1/2))/(1+2*x+3^(1/2))/(-1+4*x^4+4*x^2*3^(1/2))^(1/2),x , algorithm="fricas")
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))
\[ \int \frac {1-\sqrt {3}+2 x}{\left (1+\sqrt {3}+2 x\right ) \sqrt {-1+4 \sqrt {3} x^2+4 x^4}} \, dx=\int \frac {2 x - \sqrt {3} + 1}{\left (2 x + 1 + \sqrt {3}\right ) \sqrt {4 x^{4} + 4 \sqrt {3} x^{2} - 1}}\, dx \]
\[ \int \frac {1-\sqrt {3}+2 x}{\left (1+\sqrt {3}+2 x\right ) \sqrt {-1+4 \sqrt {3} x^2+4 x^4}} \, dx=\int { \frac {2 \, x - \sqrt {3} + 1}{\sqrt {4 \, x^{4} + 4 \, \sqrt {3} x^{2} - 1} {\left (2 \, x + \sqrt {3} + 1\right )}} \,d x } \]
integrate((1+2*x-3^(1/2))/(1+2*x+3^(1/2))/(-1+4*x^4+4*x^2*3^(1/2))^(1/2),x , algorithm="maxima")
\[ \int \frac {1-\sqrt {3}+2 x}{\left (1+\sqrt {3}+2 x\right ) \sqrt {-1+4 \sqrt {3} x^2+4 x^4}} \, dx=\int { \frac {2 \, x - \sqrt {3} + 1}{\sqrt {4 \, x^{4} + 4 \, \sqrt {3} x^{2} - 1} {\left (2 \, x + \sqrt {3} + 1\right )}} \,d x } \]
integrate((1+2*x-3^(1/2))/(1+2*x+3^(1/2))/(-1+4*x^4+4*x^2*3^(1/2))^(1/2),x , algorithm="giac")
Timed out. \[ \int \frac {1-\sqrt {3}+2 x}{\left (1+\sqrt {3}+2 x\right ) \sqrt {-1+4 \sqrt {3} x^2+4 x^4}} \, dx=\int \frac {2\,x-\sqrt {3}+1}{\sqrt {4\,x^4+4\,\sqrt {3}\,x^2-1}\,\left (2\,x+\sqrt {3}+1\right )} \,d x \]