\(\int \frac {1+\sqrt {3}+x}{(1-\sqrt {3}+x) \sqrt {-4-4 \sqrt {3} x^2+x^4}} \, dx\) [91]

Optimal result

Integrand size = 40, antiderivative size = 63 \[ \int \frac {1+\sqrt {3}+x}{\left (1-\sqrt {3}+x\right ) \sqrt {-4-4 \sqrt {3} x^2+x^4}} \, dx=-\frac {1}{3} \sqrt {3+2 \sqrt {3}} \arctan \left (\frac {\left (1+\sqrt {3}+x\right )^2}{\sqrt {3 \left (3+2 \sqrt {3}\right )} \sqrt {-4-4 \sqrt {3} x^2+x^4}}\right ) \]

[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)

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {1754, 209} \[ \int \frac {1+\sqrt {3}+x}{\left (1-\sqrt {3}+x\right ) \sqrt {-4-4 \sqrt {3} x^2+x^4}} \, dx=-\frac {1}{3} \sqrt {3+2 \sqrt {3}} \arctan \left (\frac {\left (x+\sqrt {3}+1\right )^2}{\sqrt {3 \left (3+2 \sqrt {3}\right )} \sqrt {x^4-4 \sqrt {3} x^2-4}}\right ) \]

[In]

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

[Out]

-1/3*(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])]
)

Rule 209

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

Rule 1754

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] && Eq
Q[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*} \text {integral}& = -\left (\left (4 \left (2+\sqrt {3}\right )\right ) \text {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}} \arctan \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 [A] (verified)

Time = 8.20 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.22 \[ \int \frac {1+\sqrt {3}+x}{\left (1-\sqrt {3}+x\right ) \sqrt {-4-4 \sqrt {3} x^2+x^4}} \, dx=-\frac {1}{3} \sqrt {3+2 \sqrt {3}} \arctan \left (\frac {\sqrt {-9+6 \sqrt {3}} \sqrt {-4-4 \sqrt {3} x^2+x^4}}{-2+\left (2-2 \sqrt {3}\right ) x+\left (-2+\sqrt {3}\right ) x^2}\right ) \]

[In]

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

[Out]

-1/3*(Sqrt[3 + 2*Sqrt[3]]*ArcTan[(Sqrt[-9 + 6*Sqrt[3]]*Sqrt[-4 - 4*Sqrt[3]*x^2 + x^4])/(-2 + (2 - 2*Sqrt[3])*x
+ (-2 + Sqrt[3])*x^2)])

Maple [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3.

Time = 2.31 (sec) , antiderivative size = 311, normalized size of antiderivative = 4.94

method	result	size
elliptic	\(\frac {\sqrt {1-\left (-\frac {\sqrt {3}}{2}-1\right ) x^{2}}\, \sqrt {1-\left (1-\frac {\sqrt {3}}{2}\right ) x^{2}}\, F\left (x \left (\frac {i}{2}+\frac {i \sqrt {3}}{2}\right ), i \sqrt {1-4 \sqrt {3}\, \left (1-\frac {\sqrt {3}}{2}\right )}\right )}{\left (\frac {i}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {-4+x^{4}-4 \sqrt {3}\, x^{2}}}+2 \sqrt {3}\, \left (-\frac {\operatorname {arctanh}\left (\frac {-4 \left (\sqrt {3}-1\right )^{2} \sqrt {3}-8-4 \sqrt {3}\, x^{2}+2 x^{2} \left (\sqrt {3}-1\right )^{2}}{2 \sqrt {\left (\sqrt {3}-1\right )^{4}-4 \left (\sqrt {3}-1\right )^{2} \sqrt {3}-4}\, \sqrt {-4+x^{4}-4 \sqrt {3}\, x^{2}}}\right )}{2 \sqrt {\left (\sqrt {3}-1\right )^{4}-4 \left (\sqrt {3}-1\right )^{2} \sqrt {3}-4}}-\frac {\sqrt {1-\left (-\frac {\sqrt {3}}{2}-1\right ) x^{2}}\, \sqrt {1-\left (1-\frac {\sqrt {3}}{2}\right ) x^{2}}\, \Pi \left (\sqrt {-\frac {\sqrt {3}}{2}-1}\, x , \frac {1}{\left (-\frac {\sqrt {3}}{2}-1\right ) \left (\sqrt {3}-1\right )^{2}}, \frac {\sqrt {1-\frac {\sqrt {3}}{2}}}{\sqrt {-\frac {\sqrt {3}}{2}-1}}\right )}{\sqrt {-\frac {\sqrt {3}}{2}-1}\, \left (\sqrt {3}-1\right ) \sqrt {-4+x^{4}-4 \sqrt {3}\, x^{2}}}\right )\)	\(311\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (45) = 90\).

Time = 0.38 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.78 \[ \int \frac {1+\sqrt {3}+x}{\left (1-\sqrt {3}+x\right ) \sqrt {-4-4 \sqrt {3} x^2+x^4}} \, dx=\frac {1}{6} \, \sqrt {2 \, \sqrt {3} + 3} \arctan \left (-\frac {{\left (9 \, x^{4} - 30 \, x^{3} + 18 \, x^{2} - 2 \, \sqrt {3} {\left (2 \, x^{4} - 10 \, x^{3} + 3 \, x^{2} - 10 \, x + 2\right )} + 24\right )} \sqrt {x^{4} - 4 \, \sqrt {3} x^{2} - 4} \sqrt {2 \, \sqrt {3} + 3}}{11 \, x^{6} - 42 \, x^{5} + 66 \, x^{4} - 176 \, x^{3} - 132 \, x^{2} - 168 \, x - 88}\right ) \]

[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)
)

Sympy [F]

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

[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)

Maxima [F]

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

[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)

Giac [F]

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

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {1+\sqrt {3}+x}{\left (1-\sqrt {3}+x\right ) \sqrt {-4-4 \sqrt {3} x^2+x^4}} \, dx=\int \frac {x+\sqrt {3}+1}{\sqrt {x^4-4\,\sqrt {3}\,x^2-4}\,\left (x-\sqrt {3}+1\right )} \,d x \]

[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)