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

Optimal. Leaf size=65 \[ \frac{1}{3} \sqrt{2 \sqrt{3}-3} \tanh ^{-1}\left (\frac{\left (x-\sqrt{3}+1\right )^2}{\sqrt{3 \left (2 \sqrt{3}-3\right )} \sqrt{x^4+4 \sqrt{3} x^2-4}}\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.128968, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {1740, 207} \[ \frac{1}{3} \sqrt{2 \sqrt{3}-3} \tanh ^{-1}\left (\frac{\left (x-\sqrt{3}+1\right )^2}{\sqrt{3 \left (2 \sqrt{3}-3\right )} \sqrt{x^4+4 \sqrt{3} x^2-4}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1740

Int[((A_) + (B_.)*(x_))/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> -Dist[(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]

Rule 207

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

Rubi steps

\begin{align*} \int \frac{1-\sqrt{3}+x}{\left (1+\sqrt{3}+x\right ) \sqrt{-4+4 \sqrt{3} x^2+x^4}} \, dx &=-\left (\left (4 \left (2-\sqrt{3}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{3 \left (1-\sqrt{3}\right )^4+6 \left (1-\sqrt{3}\right )^3 \left (1+\sqrt{3}\right )+4 x^2} \, dx,x,\frac{\left (1-\sqrt{3}+x\right )^2}{\sqrt{-4+4 \sqrt{3} x^2+x^4}}\right )\right )\\ &=\frac{1}{3} \sqrt{-3+2 \sqrt{3}} \tanh ^{-1}\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 )\\ \end{align*}

Mathematica [C]  time = 3.20489, size = 685, normalized size = 10.54 \[ \frac{\left (x+\sqrt{3}-1\right )^2 \sqrt{-x^3+\left (\sqrt{3}-1\right ) x^2-2 \left (2+\sqrt{3}\right ) x+2 \left (1+\sqrt{3}\right )} \sqrt{\frac{-\frac{4}{x+\sqrt{3}-1}+\sqrt{3}+1}{3+\sqrt{3}+i \sqrt{2 \left (2+\sqrt{3}\right )}}} \left (\left (\frac{2 \left (2 i \sqrt{3}-\sqrt{2 \left (2+\sqrt{3}\right )}+\sqrt{6 \left (2+\sqrt{3}\right )}\right )}{x+\sqrt{3}-1}+i \left (-1+\sqrt{3}+i \sqrt{2 \left (2+\sqrt{3}\right )}\right )\right ) \sqrt{\sqrt{2 \left (2+\sqrt{3}\right )}+i \left (\frac{8}{x+\sqrt{3}-1}-\sqrt{3}+1\right )} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\sqrt{2 \left (2+\sqrt{3}\right )}-i \left (\frac{8}{x+\sqrt{3}-1}-\sqrt{3}+1\right )}}{2^{3/4} \sqrt [4]{2+\sqrt{3}}}\right ),\frac{2 i \sqrt{2 \left (2+\sqrt{3}\right )}}{3+\sqrt{3}+i \sqrt{2 \left (2+\sqrt{3}\right )}}\right )+2 \sqrt{6} \sqrt{\frac{x^2+2 \sqrt{3}+4}{\left (x+\sqrt{3}-1\right )^2}} \sqrt{\sqrt{2 \left (2+\sqrt{3}\right )}-i \left (\frac{8}{x+\sqrt{3}-1}-\sqrt{3}+1\right )} \Pi \left (\frac{2 \sqrt{2 \left (2+\sqrt{3}\right )}}{\sqrt{2 \left (2+\sqrt{3}\right )}+i \left (3+\sqrt{3}\right )};\sin ^{-1}\left (\frac{\sqrt{\sqrt{2 \left (2+\sqrt{3}\right )}-i \left (-\sqrt{3}+1+\frac{8}{x+\sqrt{3}-1}\right )}}{2^{3/4} \sqrt [4]{2+\sqrt{3}}}\right )|\frac{2 i \sqrt{2 \left (2+\sqrt{3}\right )}}{3+\sqrt{3}+i \sqrt{2 \left (2+\sqrt{3}\right )}}\right )\right )}{\left (\sqrt{2 \left (2+\sqrt{3}\right )}+i \left (3+\sqrt{3}\right )\right ) \sqrt{-\frac{x^3}{2}+\frac{1}{2} \left (\sqrt{3}-1\right ) x^2-\left (2+\sqrt{3}\right ) x+\sqrt{3}+1} \sqrt{x^4+4 \sqrt{3} x^2-4} \sqrt{\sqrt{2 \left (2+\sqrt{3}\right )}-i \left (\frac{8}{x+\sqrt{3}-1}-\sqrt{3}+1\right )}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-1 + Sqrt[3] + x)^2*Sqrt[2*(1 + Sqrt[3]) - 2*(2 + Sqrt[3])*x + (-1 + Sqrt[3])*x^2 - x^3]*Sqrt[(1 + Sqrt[3] -
4/(-1 + Sqrt[3] + x))/(3 + Sqrt[3] + I*Sqrt[2*(2 + Sqrt[3])])]*((I*(-1 + Sqrt[3] + I*Sqrt[2*(2 + Sqrt[3])]) +
(2*((2*I)*Sqrt[3] - Sqrt[2*(2 + Sqrt[3])] + Sqrt[6*(2 + Sqrt[3])]))/(-1 + Sqrt[3] + x))*Sqrt[Sqrt[2*(2 + Sqrt
[3])] + I*(1 - Sqrt[3] + 8/(-1 + Sqrt[3] + x))]*EllipticF[ArcSin[Sqrt[Sqrt[2*(2 + Sqrt[3])] - I*(1 - Sqrt[3] +
8/(-1 + Sqrt[3] + x))]/(2^(3/4)*(2 + Sqrt[3])^(1/4))], ((2*I)*Sqrt[2*(2 + Sqrt[3])])/(3 + Sqrt[3] + I*Sqrt[2*
(2 + Sqrt[3])])] + 2*Sqrt[6]*Sqrt[(4 + 2*Sqrt[3] + x^2)/(-1 + Sqrt[3] + x)^2]*Sqrt[Sqrt[2*(2 + Sqrt[3])] - I*(
1 - Sqrt[3] + 8/(-1 + Sqrt[3] + x))]*EllipticPi[(2*Sqrt[2*(2 + Sqrt[3])])/(Sqrt[2*(2 + Sqrt[3])] + I*(3 + Sqrt
[3])), ArcSin[Sqrt[Sqrt[2*(2 + Sqrt[3])] - I*(1 - Sqrt[3] + 8/(-1 + Sqrt[3] + x))]/(2^(3/4)*(2 + Sqrt[3])^(1/4
))], ((2*I)*Sqrt[2*(2 + Sqrt[3])])/(3 + Sqrt[3] + I*Sqrt[2*(2 + Sqrt[3])])]))/((Sqrt[2*(2 + Sqrt[3])] + I*(3 +
Sqrt[3]))*Sqrt[1 + Sqrt[3] - (2 + Sqrt[3])*x + ((-1 + Sqrt[3])*x^2)/2 - x^3/2]*Sqrt[-4 + 4*Sqrt[3]*x^2 + x^4]
*Sqrt[Sqrt[2*(2 + Sqrt[3])] - I*(1 - Sqrt[3] + 8/(-1 + Sqrt[3] + x))])

________________________________________________________________________________________

Maple [C]  time = 0.132, size = 327, normalized size = 5. \begin{align*}{\frac{{\it EllipticF} \left ( x \left ({\frac{i}{2}}\sqrt{3}-{\frac{i}{2}} \right ) ,i\sqrt{1+4\,\sqrt{3} \left ( 1+1/2\,\sqrt{3} \right ) } \right ) }{{\frac{i}{2}}\sqrt{3}-{\frac{i}{2}}}\sqrt{1- \left ( -1+{\frac{\sqrt{3}}{2}} \right ){x}^{2}}\sqrt{1- \left ( 1+{\frac{\sqrt{3}}{2}} \right ){x}^{2}}{\frac{1}{\sqrt{-4+{x}^{4}+4\,\sqrt{3}{x}^{2}}}}}-2\,\sqrt{3} \left ( -1/2\,{\frac{1}{\sqrt{ \left ( -1-\sqrt{3} \right ) ^{4}+4\,\sqrt{3} \left ( -1-\sqrt{3} \right ) ^{2}-4}}{\it Artanh} \left ( 1/2\,{\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}}{\sqrt{ \left ( -1-\sqrt{3} \right ) ^{4}+4\,\sqrt{3} \left ( -1-\sqrt{3} \right ) ^{2}-4}\sqrt{-4+{x}^{4}+4\,\sqrt{3}{x}^{2}}}} \right ) }-{\frac{\sqrt{1- \left ( -1+1/2\,\sqrt{3} \right ){x}^{2}}\sqrt{1- \left ( 1+1/2\,\sqrt{3} \right ){x}^{2}}}{\sqrt{-1+1/2\,\sqrt{3}} \left ( -1-\sqrt{3} \right ) \sqrt{-4+{x}^{4}+4\,\sqrt{3}{x}^{2}}}{\it EllipticPi} \left ( \sqrt{-1+1/2\,\sqrt{3}}x,{\frac{1}{ \left ( -1+1/2\,\sqrt{3} \right ) \left ( -1-\sqrt{3} \right ) ^{2}}},{\frac{\sqrt{1+1/2\,\sqrt{3}}}{\sqrt{-1+1/2\,\sqrt{3}}}} \right ) } \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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+x^4+4*3^(1/2)*x^2)^(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+x^4+4*3^(1/2)*x^2)^(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+x^4+4*3^(1/2)*x^2)^(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)))

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x - \sqrt{3} + 1}{\sqrt{x^{4} + 4 \, \sqrt{3} x^{2} - 4}{\left (x + \sqrt{3} + 1\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [B]  time = 3.4669, size = 953, normalized size = 14.66 \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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 + 109
44*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 -
1008*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
*sqrt(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]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x - \sqrt{3} + 1}{\left (x + 1 + \sqrt{3}\right ) \sqrt{x^{4} + 4 \sqrt{3} x^{2} - 4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x - \sqrt{3} + 1}{\sqrt{x^{4} + 4 \, \sqrt{3} x^{2} - 4}{\left (x + \sqrt{3} + 1\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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