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]

-1/3*arctan((1+x+3^(1/2))^2/(9+6*3^(1/2))^(1/2)/(-4+x^4-4*3^(1/2)*x^2)^(1/2))*(3+2*3^(1/2))^(1/2)

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Rubi [A]  time = 0.13, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, 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 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])

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]

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*}

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Mathematica [C]  time = 7.99, size = 876, normalized size = 13.90 \[ -\frac {\sqrt {2} \sqrt {\frac {\sqrt {3}-1-\frac {4}{-x+\sqrt {3}+1}}{-3+\sqrt {3}-i \sqrt {4-2 \sqrt {3}}}} \left (-x+\sqrt {3}+1\right )^2 \left (\left (\frac {2 \left (2 i \sqrt {3} \sqrt {i \left (\sqrt {3}+1-\frac {8}{-x+\sqrt {3}+1}\right )+\sqrt {4-2 \sqrt {3}}}+\sqrt {6} \sqrt {2 \sqrt {4-2 \sqrt {3}}-\sqrt {12-6 \sqrt {3}}+i \sqrt {3}-i+\frac {8 i \left (-2+\sqrt {3}\right )}{-x+\sqrt {3}+1}}+\sqrt {-\frac {2 i \left (\left (-1+\sqrt {3}\right ) x-8 \sqrt {3}+14\right )}{-x+\sqrt {3}+1}+4 \sqrt {4-2 \sqrt {3}}-2 \sqrt {12-6 \sqrt {3}}}\right )}{x-\sqrt {3}-1}+i \sqrt {3} \sqrt {i \left (\sqrt {3}+1-\frac {8}{-x+\sqrt {3}+1}\right )+\sqrt {4-2 \sqrt {3}}}+i \sqrt {i \left (\sqrt {3}+1-\frac {8}{-x+\sqrt {3}+1}\right )+\sqrt {4-2 \sqrt {3}}}+\sqrt {-\frac {2 i \left (\left (-1+\sqrt {3}\right ) x-8 \sqrt {3}+14\right )}{-x+\sqrt {3}+1}+4 \sqrt {4-2 \sqrt {3}}-2 \sqrt {12-6 \sqrt {3}}}\right ) \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt {\sqrt {4-2 \sqrt {3}}-i \left (\sqrt {3}+1-\frac {8}{-x+\sqrt {3}+1}\right )}}{2^{3/4} \sqrt [4]{2-\sqrt {3}}}\right ),\frac {2 \sqrt {4-2 \sqrt {3}}}{\sqrt {4-2 \sqrt {3}}+i \left (-3+\sqrt {3}\right )}\right )+2 \sqrt {6} \sqrt {\sqrt {4-2 \sqrt {3}}-i \left (\sqrt {3}+1-\frac {8}{-x+\sqrt {3}+1}\right )} \sqrt {\frac {x^2-2 \sqrt {3}+4}{\left (-x+\sqrt {3}+1\right )^2}} \Pi \left (\frac {2 \sqrt {4-2 \sqrt {3}}}{\sqrt {4-2 \sqrt {3}}-i \left (-3+\sqrt {3}\right )};\sin ^{-1}\left (\frac {\sqrt {\sqrt {4-2 \sqrt {3}}-i \left (\sqrt {3}+1-\frac {8}{-x+\sqrt {3}+1}\right )}}{2^{3/4} \sqrt [4]{2-\sqrt {3}}}\right )|\frac {2 \sqrt {4-2 \sqrt {3}}}{\sqrt {4-2 \sqrt {3}}+i \left (-3+\sqrt {3}\right )}\right )\right )}{\left (\sqrt {4-2 \sqrt {3}}-i \left (-3+\sqrt {3}\right )\right ) \sqrt {\sqrt {4-2 \sqrt {3}}-i \left (\sqrt {3}+1-\frac {8}{-x+\sqrt {3}+1}\right )} \sqrt {x^4-4 \sqrt {3} x^2-4}} \]

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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fricas [B]  time = 0.91, size = 112, normalized size = 1.78 \[ \frac {1}{6} \, \sqrt {2 \, \sqrt {3} + 3} \arctan \left (-\frac {{\left (9 \, x^{4} - 30 \, x^{3} + 18 \, x^{2} - 2 \, \sqrt {3} {\left (2 \, x^{4} - 10 \, x^{3} + 3 \, x^{2} - 10 \, x + 2\right )} + 24\right )} \sqrt {x^{4} - 4 \, \sqrt {3} x^{2} - 4} \sqrt {2 \, \sqrt {3} + 3}}{11 \, x^{6} - 42 \, x^{5} + 66 \, x^{4} - 176 \, x^{3} - 132 \, x^{2} - 168 \, x - 88}\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x + \sqrt {3} + 1}{\sqrt {x^{4} - 4 \, \sqrt {3} x^{2} - 4} {\left (x - \sqrt {3} + 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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)

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maple [C]  time = 0.16, size = 311, normalized size = 4.94 \[ \frac {\sqrt {-\left (-1-\frac {\sqrt {3}}{2}\right ) x^{2}+1}\, \sqrt {-\left (1-\frac {\sqrt {3}}{2}\right ) x^{2}+1}\, \EllipticF \left (\left (\frac {i}{2}+\frac {i \sqrt {3}}{2}\right ) x , i \sqrt {1-4 \sqrt {3}\, \left (1-\frac {\sqrt {3}}{2}\right )}\right )}{\left (\frac {i}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x^{4}-4 \sqrt {3}\, x^{2}-4}}+2 \sqrt {3}\, \left (-\frac {\sqrt {-\left (-1-\frac {\sqrt {3}}{2}\right ) x^{2}+1}\, \sqrt {-\left (1-\frac {\sqrt {3}}{2}\right ) x^{2}+1}\, \EllipticPi \left (\sqrt {-1-\frac {\sqrt {3}}{2}}\, x , \frac {1}{\left (-1-\frac {\sqrt {3}}{2}\right ) \left (\sqrt {3}-1\right )^{2}}, \frac {\sqrt {1-\frac {\sqrt {3}}{2}}}{\sqrt {-1-\frac {\sqrt {3}}{2}}}\right )}{\sqrt {-1-\frac {\sqrt {3}}{2}}\, \left (\sqrt {3}-1\right ) \sqrt {x^{4}-4 \sqrt {3}\, x^{2}-4}}-\frac {\arctanh \left (\frac {-4 \sqrt {3}\, x^{2}+2 \left (\sqrt {3}-1\right )^{2} x^{2}-4 \sqrt {3}\, \left (\sqrt {3}-1\right )^{2}-8}{2 \sqrt {\left (\sqrt {3}-1\right )^{4}-4 \sqrt {3}\, \left (\sqrt {3}-1\right )^{2}-4}\, \sqrt {x^{4}-4 \sqrt {3}\, x^{2}-4}}\right )}{2 \sqrt {\left (\sqrt {3}-1\right )^{4}-4 \sqrt {3}\, \left (\sqrt {3}-1\right )^{2}-4}}\right ) \]

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+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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x + \sqrt {3} + 1}{\sqrt {x^{4} - 4 \, \sqrt {3} x^{2} - 4} {\left (x - \sqrt {3} + 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x+\sqrt {3}+1}{\sqrt {x^4-4\,\sqrt {3}\,x^2-4}\,\left (x-\sqrt {3}+1\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x + 1 + \sqrt {3}}{\left (x - \sqrt {3} + 1\right ) \sqrt {x^{4} - 4 \sqrt {3} x^{2} - 4}}\, dx \]

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

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