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

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

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Rubi [A]  time = 0.211471, 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 \[ \frac{1}{3} \sqrt{2 \sqrt{3}-3} \tanh ^{-1}\left (\frac{\left (x-\sqrt{3}+1\right )^2}{\sqrt{3 \left (2 \sqrt{3}-3\right )} \sqrt{x^4+4 \sqrt{3} x^2-4}}\right ) \]

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

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

[Out]

Timed out

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Mathematica [C]  time = 4.03084, size = 685, normalized size = 10.54 \[ \frac{\left (x+\sqrt{3}-1\right )^2 \sqrt{-x^3+\left (\sqrt{3}-1\right ) x^2-2 \left (2+\sqrt{3}\right ) x+2 \left (1+\sqrt{3}\right )} \sqrt{\frac{-\frac{4}{x+\sqrt{3}-1}+\sqrt{3}+1}{3+\sqrt{3}+i \sqrt{2 \left (2+\sqrt{3}\right )}}} \left (2 \sqrt{6} \sqrt{\frac{x^2+2 \sqrt{3}+4}{\left (x+\sqrt{3}-1\right )^2}} \sqrt{\sqrt{2 \left (2+\sqrt{3}\right )}-i \left (\frac{8}{x+\sqrt{3}-1}-\sqrt{3}+1\right )} \Pi \left (\frac{2 \sqrt{2 \left (2+\sqrt{3}\right )}}{\sqrt{2 \left (2+\sqrt{3}\right )}+i \left (3+\sqrt{3}\right )};\sin ^{-1}\left (\frac{\sqrt{\sqrt{2 \left (2+\sqrt{3}\right )}-i \left (-\sqrt{3}+1+\frac{8}{x+\sqrt{3}-1}\right )}}{2^{3/4} \sqrt [4]{2+\sqrt{3}}}\right )|\frac{2 i \sqrt{2 \left (2+\sqrt{3}\right )}}{3+\sqrt{3}+i \sqrt{2 \left (2+\sqrt{3}\right )}}\right )+\left (\frac{2 \left (2 i \sqrt{3}-\sqrt{2 \left (2+\sqrt{3}\right )}+\sqrt{6 \left (2+\sqrt{3}\right )}\right )}{x+\sqrt{3}-1}+i \left (-1+\sqrt{3}+i \sqrt{2 \left (2+\sqrt{3}\right )}\right )\right ) \sqrt{\sqrt{2 \left (2+\sqrt{3}\right )}+i \left (\frac{8}{x+\sqrt{3}-1}-\sqrt{3}+1\right )} F\left (\sin ^{-1}\left (\frac{\sqrt{\sqrt{2 \left (2+\sqrt{3}\right )}-i \left (-\sqrt{3}+1+\frac{8}{x+\sqrt{3}-1}\right )}}{2^{3/4} \sqrt [4]{2+\sqrt{3}}}\right )|\frac{2 i \sqrt{2 \left (2+\sqrt{3}\right )}}{3+\sqrt{3}+i \sqrt{2 \left (2+\sqrt{3}\right )}}\right )\right )}{\left (\sqrt{2 \left (2+\sqrt{3}\right )}+i \left (3+\sqrt{3}\right )\right ) \sqrt{-\frac{x^3}{2}+\frac{1}{2} \left (\sqrt{3}-1\right ) x^2-\left (2+\sqrt{3}\right ) x+\sqrt{3}+1} \sqrt{x^4+4 \sqrt{3} x^2-4} \sqrt{\sqrt{2 \left (2+\sqrt{3}\right )}-i \left (\frac{8}{x+\sqrt{3}-1}-\sqrt{3}+1\right )}} \]

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 + Sq
rt[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 + Sq
rt[3])] - I*(1 - Sqrt[3] + 8/(-1 + Sqrt[3] + x))])

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Maple [C]  time = 0.24, size = 327, normalized size = 5. \[{\frac{{\it EllipticF} \left ( x \left ({\frac{i}{2}}\sqrt{3}-{\frac{i}{2}} \right ) ,i\sqrt{1+4\,\sqrt{3} \left ( 1+1/2\,\sqrt{3} \right ) } \right ) }{{\frac{i}{2}}\sqrt{3}-{\frac{i}{2}}}\sqrt{1- \left ( -1+{\frac{\sqrt{3}}{2}} \right ){x}^{2}}\sqrt{1- \left ( 1+{\frac{\sqrt{3}}{2}} \right ){x}^{2}}{\frac{1}{\sqrt{-4+{x}^{4}+4\,\sqrt{3}{x}^{2}}}}}-2\,\sqrt{3} \left ( -1/2\,{\frac{1}{\sqrt{ \left ( -1-\sqrt{3} \right ) ^{4}+4\,\sqrt{3} \left ( -1-\sqrt{3} \right ) ^{2}-4}}{\it Artanh} \left ( 1/2\,{\frac{4\,\sqrt{3} \left ( -1-\sqrt{3} \right ) ^{2}-8+4\,\sqrt{3}{x}^{2}+2\,{x}^{2} \left ( -1-\sqrt{3} \right ) ^{2}}{\sqrt{ \left ( -1-\sqrt{3} \right ) ^{4}+4\,\sqrt{3} \left ( -1-\sqrt{3} \right ) ^{2}-4}\sqrt{-4+{x}^{4}+4\,\sqrt{3}{x}^{2}}}} \right ) }-{\frac{\sqrt{1- \left ( -1+1/2\,\sqrt{3} \right ){x}^{2}}\sqrt{1- \left ( 1+1/2\,\sqrt{3} \right ){x}^{2}}}{\sqrt{-1+1/2\,\sqrt{3}} \left ( -1-\sqrt{3} \right ) \sqrt{-4+{x}^{4}+4\,\sqrt{3}{x}^{2}}}{\it EllipticPi} \left ( \sqrt{-1+1/2\,\sqrt{3}}x,{\frac{1}{ \left ( -1+1/2\,\sqrt{3} \right ) \left ( -1-\sqrt{3} \right ) ^{2}}},{\frac{\sqrt{1+1/2\,\sqrt{3}}}{\sqrt{-1+1/2\,\sqrt{3}}}} \right ) } \right ) \]

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)))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

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)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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