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

Optimal. Leaf size=72 \[ \frac{1}{3} \sqrt{2 \sqrt{3}-3} \tanh ^{-1}\left (\frac{\left (2 x-\sqrt{3}+1\right )^2}{2 \sqrt{3 \left (2 \sqrt{3}-3\right )} \sqrt{4 x^4+4 \sqrt{3} x^2-1}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.196658, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043 \[ \frac{1}{3} \sqrt{2 \sqrt{3}-3} \tanh ^{-1}\left (\frac{\left (2 x-\sqrt{3}+1\right )^2}{2 \sqrt{3 \left (2 \sqrt{3}-3\right )} \sqrt{4 x^4+4 \sqrt{3} x^2-1}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+2*x-3**(1/2))/(1+2*x+3**(1/2))/(-1+4*x**4+4*3**(1/2)*x**2)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 2.40256, size = 623, normalized size = 8.65 \[ \frac{\left (2 x+\sqrt{3}-1\right )^2 \sqrt{\frac{-\frac{4}{2 x+\sqrt{3}-1}+\sqrt{3}+1}{3+\sqrt{3}+i \sqrt{2 \left (2+\sqrt{3}\right )}}} \left (4 \sqrt{3} \sqrt{\frac{2 x^2+\sqrt{3}+2}{\left (2 x+\sqrt{3}-1\right )^2}} \sqrt{\sqrt{2 \left (2+\sqrt{3}\right )}-i \left (\frac{8}{2 x+\sqrt{3}-1}-\sqrt{3}+1\right )} \Pi \left (\frac{2 \sqrt{2 \left (2+\sqrt{3}\right )}}{\sqrt{2 \left (2+\sqrt{3}\right )}+i \left (3+\sqrt{3}\right )};\sin ^{-1}\left (\frac{\sqrt{\sqrt{2 \left (2+\sqrt{3}\right )}-i \left (-\sqrt{3}+1+\frac{8}{2 x+\sqrt{3}-1}\right )}}{2^{3/4} \sqrt [4]{2+\sqrt{3}}}\right )|\frac{2 i \sqrt{2 \left (2+\sqrt{3}\right )}}{3+\sqrt{3}+i \sqrt{2 \left (2+\sqrt{3}\right )}}\right )+\left (\frac{2 \left (2 i \sqrt{3}-\sqrt{2 \left (2+\sqrt{3}\right )}+\sqrt{6 \left (2+\sqrt{3}\right )}\right )}{2 x+\sqrt{3}-1}+i \left (-1+\sqrt{3}+i \sqrt{2 \left (2+\sqrt{3}\right )}\right )\right ) \sqrt{\sqrt{2 \left (2+\sqrt{3}\right )}+i \left (\frac{8}{2 x+\sqrt{3}-1}-\sqrt{3}+1\right )} F\left (\sin ^{-1}\left (\frac{\sqrt{\sqrt{2 \left (2+\sqrt{3}\right )}-i \left (-\sqrt{3}+1+\frac{8}{2 x+\sqrt{3}-1}\right )}}{2^{3/4} \sqrt [4]{2+\sqrt{3}}}\right )|\frac{2 i \sqrt{2 \left (2+\sqrt{3}\right )}}{3+\sqrt{3}+i \sqrt{2 \left (2+\sqrt{3}\right )}}\right )\right )}{\left (\sqrt{2 \left (2+\sqrt{3}\right )}+i \left (3+\sqrt{3}\right )\right ) \sqrt{8 x^4+8 \sqrt{3} x^2-2} \sqrt{\sqrt{2 \left (2+\sqrt{3}\right )}-i \left (\frac{8}{2 x+\sqrt{3}-1}-\sqrt{3}+1\right )}} \]

Warning: Unable to verify antiderivative.

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

[Out]

((-1 + Sqrt[3] + 2*x)^2*Sqrt[(1 + Sqrt[3] - 4/(-1 + Sqrt[3] + 2*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] +
2*x))*Sqrt[Sqrt[2*(2 + Sqrt[3])] + I*(1 - Sqrt[3] + 8/(-1 + Sqrt[3] + 2*x))]*Ell
ipticF[ArcSin[Sqrt[Sqrt[2*(2 + Sqrt[3])] - I*(1 - Sqrt[3] + 8/(-1 + Sqrt[3] + 2*
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])])] + 4*Sqrt[3]*Sqrt[(2 + Sqrt[3] + 2*x^2)/(-1 + Sqrt[3]
+ 2*x)^2]*Sqrt[Sqrt[2*(2 + Sqrt[3])] - I*(1 - Sqrt[3] + 8/(-1 + Sqrt[3] + 2*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] + 2*x))]/
(2^(3/4)*(2 + Sqrt[3])^(1/4))], ((2*I)*Sqrt[2*(2 + Sqrt[3])])/(3 + Sqrt[3] + I*S
qrt[2*(2 + Sqrt[3])])]))/((Sqrt[2*(2 + Sqrt[3])] + I*(3 + Sqrt[3]))*Sqrt[-2 + 8*
Sqrt[3]*x^2 + 8*x^4]*Sqrt[Sqrt[2*(2 + Sqrt[3])] - I*(1 - Sqrt[3] + 8/(-1 + Sqrt[
3] + 2*x))])

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Maple [C]  time = 0.225, size = 336, normalized size = 4.7 \[{\frac{{\it EllipticF} \left ( x \left ( i\sqrt{3}-i \right ) ,i\sqrt{1+\sqrt{3} \left ( 2\,\sqrt{3}+4 \right ) } \right ) }{i\sqrt{3}-i}\sqrt{1- \left ( 2\,\sqrt{3}-4 \right ){x}^{2}}\sqrt{1- \left ( 2\,\sqrt{3}+4 \right ){x}^{2}}{\frac{1}{\sqrt{-1+4\,{x}^{4}+4\,{x}^{2}\sqrt{3}}}}}-2\,\sqrt{3} \left ( -1/4\,{\frac{1}{\sqrt{4\, \left ( -1/2-1/2\,\sqrt{3} \right ) ^{4}+4\,\sqrt{3} \left ( -1/2-1/2\,\sqrt{3} \right ) ^{2}-1}}{\it Artanh} \left ( 1/2\,{\frac{4\,\sqrt{3} \left ( -1/2-1/2\,\sqrt{3} \right ) ^{2}-2+4\,{x}^{2}\sqrt{3}+8\,{x}^{2} \left ( -1/2-1/2\,\sqrt{3} \right ) ^{2}}{\sqrt{4\, \left ( -1/2-1/2\,\sqrt{3} \right ) ^{4}+4\,\sqrt{3} \left ( -1/2-1/2\,\sqrt{3} \right ) ^{2}-1}\sqrt{-1+4\,{x}^{4}+4\,{x}^{2}\sqrt{3}}}} \right ) }-1/2\,{\frac{\sqrt{1- \left ( 2\,\sqrt{3}-4 \right ){x}^{2}}\sqrt{1- \left ( 2\,\sqrt{3}+4 \right ){x}^{2}}}{\sqrt{2\,\sqrt{3}-4} \left ( -1/2-1/2\,\sqrt{3} \right ) \sqrt{-1+4\,{x}^{4}+4\,{x}^{2}\sqrt{3}}}{\it EllipticPi} \left ( \sqrt{2\,\sqrt{3}-4}x,{\frac{1}{ \left ( 2\,\sqrt{3}-4 \right ) \left ( -1/2-1/2\,\sqrt{3} \right ) ^{2}}},{\frac{\sqrt{2\,\sqrt{3}+4}}{\sqrt{2\,\sqrt{3}-4}}} \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((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)

[Out]

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))-2*3^(1/2)*(-1/4/(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/(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)*El
lipticPi((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)))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Sympy [A]  time = 11.3049, size = 0, normalized size = 0. \[ \mathrm{NaN} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

nan

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{2 \, x - \sqrt{3} + 1}{\sqrt{4 \, x^{4} + 4 \, \sqrt{3} x^{2} - 1}{\left (2 \, x + \sqrt{3} + 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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