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

Optimal. Leaf size=63 \[ -\frac{1}{3} \sqrt{3+2 \sqrt{3}} \tan ^{-1}\left (\frac{\left (x+\sqrt{3}+1\right )^2}{\sqrt{3 \left (3+2 \sqrt{3}\right )} \sqrt{x^4-4 \sqrt{3} x^2-4}}\right ) \]

[Out]

-(Sqrt[3 + 2*Sqrt[3]]*ArcTan[(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 = 0.12776, antiderivative size = 63, 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, 203} \[ -\frac{1}{3} \sqrt{3+2 \sqrt{3}} \tan ^{-1}\left (\frac{\left (x+\sqrt{3}+1\right )^2}{\sqrt{3 \left (3+2 \sqrt{3}\right )} \sqrt{x^4-4 \sqrt{3} x^2-4}}\right ) \]

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]]*ArcTan[(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 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[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}{6 \left (1-\sqrt{3}\right ) \left (1+\sqrt{3}\right )^3+3 \left (1+\sqrt{3}\right )^4+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}} \tan ^{-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 = 7.54576, size = 876, normalized size = 13.9 \[ -\frac{\sqrt{2} \sqrt{\frac{\sqrt{3}-1-\frac{4}{-x+\sqrt{3}+1}}{-3+\sqrt{3}-i \sqrt{4-2 \sqrt{3}}}} \left (-x+\sqrt{3}+1\right )^2 \left (\left (\frac{2 \left (2 i \sqrt{3} \sqrt{i \left (\sqrt{3}+1-\frac{8}{-x+\sqrt{3}+1}\right )+\sqrt{4-2 \sqrt{3}}}+\sqrt{6} \sqrt{2 \sqrt{4-2 \sqrt{3}}-\sqrt{12-6 \sqrt{3}}+i \sqrt{3}-i+\frac{8 i \left (-2+\sqrt{3}\right )}{-x+\sqrt{3}+1}}+\sqrt{-\frac{2 i \left (\left (-1+\sqrt{3}\right ) x-8 \sqrt{3}+14\right )}{-x+\sqrt{3}+1}+4 \sqrt{4-2 \sqrt{3}}-2 \sqrt{12-6 \sqrt{3}}}\right )}{x-\sqrt{3}-1}+i \sqrt{3} \sqrt{i \left (\sqrt{3}+1-\frac{8}{-x+\sqrt{3}+1}\right )+\sqrt{4-2 \sqrt{3}}}+i \sqrt{i \left (\sqrt{3}+1-\frac{8}{-x+\sqrt{3}+1}\right )+\sqrt{4-2 \sqrt{3}}}+\sqrt{-\frac{2 i \left (\left (-1+\sqrt{3}\right ) x-8 \sqrt{3}+14\right )}{-x+\sqrt{3}+1}+4 \sqrt{4-2 \sqrt{3}}-2 \sqrt{12-6 \sqrt{3}}}\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\sqrt{4-2 \sqrt{3}}-i \left (\sqrt{3}+1-\frac{8}{-x+\sqrt{3}+1}\right )}}{2^{3/4} \sqrt [4]{2-\sqrt{3}}}\right ),\frac{2 \sqrt{4-2 \sqrt{3}}}{\sqrt{4-2 \sqrt{3}}+i \left (-3+\sqrt{3}\right )}\right )+2 \sqrt{6} \sqrt{\sqrt{4-2 \sqrt{3}}-i \left (\sqrt{3}+1-\frac{8}{-x+\sqrt{3}+1}\right )} \sqrt{\frac{x^2-2 \sqrt{3}+4}{\left (-x+\sqrt{3}+1\right )^2}} \Pi \left (\frac{2 \sqrt{4-2 \sqrt{3}}}{\sqrt{4-2 \sqrt{3}}-i \left (-3+\sqrt{3}\right )};\sin ^{-1}\left (\frac{\sqrt{\sqrt{4-2 \sqrt{3}}-i \left (\sqrt{3}+1-\frac{8}{-x+\sqrt{3}+1}\right )}}{2^{3/4} \sqrt [4]{2-\sqrt{3}}}\right )|\frac{2 \sqrt{4-2 \sqrt{3}}}{\sqrt{4-2 \sqrt{3}}+i \left (-3+\sqrt{3}\right )}\right )\right )}{\left (\sqrt{4-2 \sqrt{3}}-i \left (-3+\sqrt{3}\right )\right ) \sqrt{\sqrt{4-2 \sqrt{3}}-i \left (\sqrt{3}+1-\frac{8}{-x+\sqrt{3}+1}\right )} \sqrt{x^4-4 \sqrt{3} x^2-4}} \]

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]

-((Sqrt[2]*Sqrt[(-1 + Sqrt[3] - 4/(1 + Sqrt[3] - x))/(-3 + Sqrt[3] - I*Sqrt[4 - 2*Sqrt[3]])]*(1 + Sqrt[3] - x)
^2*((I*Sqrt[Sqrt[4 - 2*Sqrt[3]] + I*(1 + Sqrt[3] - 8/(1 + Sqrt[3] - x))] + I*Sqrt[3]*Sqrt[Sqrt[4 - 2*Sqrt[3]]
+ I*(1 + Sqrt[3] - 8/(1 + Sqrt[3] - x))] + Sqrt[-2*Sqrt[12 - 6*Sqrt[3]] + 4*Sqrt[4 - 2*Sqrt[3]] - ((2*I)*(14 -
8*Sqrt[3] + (-1 + Sqrt[3])*x))/(1 + Sqrt[3] - x)] + (2*((2*I)*Sqrt[3]*Sqrt[Sqrt[4 - 2*Sqrt[3]] + I*(1 + Sqrt[
3] - 8/(1 + Sqrt[3] - x))] + Sqrt[6]*Sqrt[-I + I*Sqrt[3] - Sqrt[12 - 6*Sqrt[3]] + 2*Sqrt[4 - 2*Sqrt[3]] + ((8*
I)*(-2 + Sqrt[3]))/(1 + Sqrt[3] - x)] + Sqrt[-2*Sqrt[12 - 6*Sqrt[3]] + 4*Sqrt[4 - 2*Sqrt[3]] - ((2*I)*(14 - 8*
Sqrt[3] + (-1 + Sqrt[3])*x))/(1 + Sqrt[3] - x)]))/(-1 - Sqrt[3] + x))*EllipticF[ArcSin[Sqrt[Sqrt[4 - 2*Sqrt[3]
] - I*(1 + Sqrt[3] - 8/(1 + Sqrt[3] - x))]/(2^(3/4)*(2 - Sqrt[3])^(1/4))], (2*Sqrt[4 - 2*Sqrt[3]])/(Sqrt[4 - 2
*Sqrt[3]] + I*(-3 + Sqrt[3]))] + 2*Sqrt[6]*Sqrt[Sqrt[4 - 2*Sqrt[3]] - I*(1 + Sqrt[3] - 8/(1 + Sqrt[3] - x))]*S
qrt[(4 - 2*Sqrt[3] + x^2)/(1 + Sqrt[3] - x)^2]*EllipticPi[(2*Sqrt[4 - 2*Sqrt[3]])/(Sqrt[4 - 2*Sqrt[3]] - I*(-3
+ Sqrt[3])), ArcSin[Sqrt[Sqrt[4 - 2*Sqrt[3]] - I*(1 + Sqrt[3] - 8/(1 + Sqrt[3] - x))]/(2^(3/4)*(2 - Sqrt[3])^
(1/4))], (2*Sqrt[4 - 2*Sqrt[3]])/(Sqrt[4 - 2*Sqrt[3]] + I*(-3 + Sqrt[3]))]))/((Sqrt[4 - 2*Sqrt[3]] - I*(-3 + S
qrt[3]))*Sqrt[Sqrt[4 - 2*Sqrt[3]] - I*(1 + Sqrt[3] - 8/(1 + Sqrt[3] - x))]*Sqrt[-4 - 4*Sqrt[3]*x^2 + x^4]))

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Maple [C]  time = 0.129, size = 311, normalized size = 4.9 \begin{align*}{\frac{{\it EllipticF} \left ( x \left ({\frac{i}{2}}+{\frac{i}{2}}\sqrt{3} \right ) ,i\sqrt{1-4\,\sqrt{3} \left ( 1-1/2\,\sqrt{3} \right ) } \right ) }{{\frac{i}{2}}+{\frac{i}{2}}\sqrt{3}}\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 ( \sqrt{3}-1 \right ) ^{4}-4\,\sqrt{3} \left ( \sqrt{3}-1 \right ) ^{2}-4}}{\it Artanh} \left ( 1/2\,{\frac{-4\,\sqrt{3} \left ( \sqrt{3}-1 \right ) ^{2}-8-4\,\sqrt{3}{x}^{2}+2\,{x}^{2} \left ( \sqrt{3}-1 \right ) ^{2}}{\sqrt{ \left ( \sqrt{3}-1 \right ) ^{4}-4\,\sqrt{3} \left ( \sqrt{3}-1 \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 ( \sqrt{3}-1 \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 ( \sqrt{3}-1 \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+1/2*I*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)*EllipticF(x*(1/2*I+1/2*I*3^(1/2)),I*(1-4*3^(1/2)*(1-1/2*3^(1/2)))^(1/2))+2*3^(1/2)*(-1/2/((3^(1/2)-1)^4-4*
3^(1/2)*(3^(1/2)-1)^2-4)^(1/2)*arctanh(1/2*(-4*3^(1/2)*(3^(1/2)-1)^2-8-4*3^(1/2)*x^2+2*x^2*(3^(1/2)-1)^2)/((3^
(1/2)-1)^4-4*3^(1/2)*(3^(1/2)-1)^2-4)^(1/2)/(-4+x^4-4*3^(1/2)*x^2)^(1/2))-1/(-1-1/2*3^(1/2))^(1/2)/(3^(1/2)-1)
*(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))/(3^(1/2)-1)^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. \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)

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Fricas [B]  time = 2.83386, size = 301, normalized size = 4.78 \begin{align*} \frac{1}{6} \, \sqrt{2 \, \sqrt{3} + 3} \arctan \left (-\frac{{\left (9 \, x^{4} - 30 \, x^{3} + 18 \, x^{2} - 2 \, \sqrt{3}{\left (2 \, x^{4} - 10 \, x^{3} + 3 \, x^{2} - 10 \, x + 2\right )} + 24\right )} \sqrt{x^{4} - 4 \, \sqrt{3} x^{2} - 4} \sqrt{2 \, \sqrt{3} + 3}}{11 \, x^{6} - 42 \, x^{5} + 66 \, x^{4} - 176 \, x^{3} - 132 \, x^{2} - 168 \, x - 88}\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]

1/6*sqrt(2*sqrt(3) + 3)*arctan(-(9*x^4 - 30*x^3 + 18*x^2 - 2*sqrt(3)*(2*x^4 - 10*x^3 + 3*x^2 - 10*x + 2) + 24)
*sqrt(x^4 - 4*sqrt(3)*x^2 - 4)*sqrt(2*sqrt(3) + 3)/(11*x^6 - 42*x^5 + 66*x^4 - 176*x^3 - 132*x^2 - 168*x - 88)
)

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

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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)