3.5.11 \(\int \frac {1+\sqrt {3}+2 x}{(1-\sqrt {3}+2 x) \sqrt {-1-4 \sqrt {3} x^2+4 x^4}} \, dx\) [411]

Optimal. Leaf size=70 \[ -\frac {1}{3} \sqrt {3+2 \sqrt {3}} \tan ^{-1}\left (\frac {\left (1+\sqrt {3}+2 x\right )^2}{2 \sqrt {3 \left (3+2 \sqrt {3}\right )} \sqrt {-1-4 \sqrt {3} x^2+4 x^4}}\right ) \]

[Out]

-1/3*arctan(1/2*(1+2*x+3^(1/2))^2/(9+6*3^(1/2))^(1/2)/(-1+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 = 0.08, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {1754, 209} \begin {gather*} -\frac {1}{3} \sqrt {3+2 \sqrt {3}} \text {ArcTan}\left (\frac {\left (2 x+\sqrt {3}+1\right )^2}{2 \sqrt {3 \left (3+2 \sqrt {3}\right )} \sqrt {4 x^4-4 \sqrt {3} x^2-1}}\right ) \end {gather*}

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]

-1/3*(Sqrt[3 + 2*Sqrt[3]]*ArcTan[(1 + Sqrt[3] + 2*x)^2/(2*Sqrt[3*(3 + 2*Sqrt[3])]*Sqrt[-1 - 4*Sqrt[3]*x^2 + 4*
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*} \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 &=-\left (\left (4 \left (2+\sqrt {3}\right )\right ) \text {Subst}\left (\int \frac {1}{12 \left (1-\sqrt {3}\right ) \left (1+\sqrt {3}\right )^3+6 \left (1+\sqrt {3}\right )^4+2 x^2} \, dx,x,\frac {\left (1+\sqrt {3}+2 x\right )^2}{\sqrt {-1-4 \sqrt {3} x^2+4 x^4}}\right )\right )\\ &=-\frac {1}{3} \sqrt {3+2 \sqrt {3}} \tan ^{-1}\left (\frac {\left (1+\sqrt {3}+2 x\right )^2}{2 \sqrt {3 \left (3+2 \sqrt {3}\right )} \sqrt {-1-4 \sqrt {3} x^2+4 x^4}}\right )\\ \end {align*}

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Mathematica [A]
time = 7.83, size = 81, normalized size = 1.16 \begin {gather*} -\frac {1}{3} \sqrt {3+2 \sqrt {3}} \tan ^{-1}\left (\frac {\sqrt {-9+6 \sqrt {3}} \sqrt {-1-4 \sqrt {3} x^2+4 x^4}}{-1+\left (2-2 \sqrt {3}\right ) x+\left (-4+2 \sqrt {3}\right ) x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 0.39, size = 337, normalized size = 4.81

method	result	size
elliptic	\(\frac {\sqrt {1-\left (-4-2 \sqrt {3}\right ) x^{2}}\, \sqrt {1-\left (-2 \sqrt {3}+4\right ) x^{2}}\, \EllipticF \left (x \left (i+i \sqrt {3}\right ), i \sqrt {1-\sqrt {3}\, \left (-2 \sqrt {3}+4\right )}\right )}{\left (i+i \sqrt {3}\right ) \sqrt {-1+4 x^{4}-4 x^{2} \sqrt {3}}}+\sqrt {3}\, \left (-\frac {\arctanh \left (\frac {-4 \sqrt {3}\, \left (\frac {\sqrt {3}}{2}-\frac {1}{2}\right )^{2}-2-4 x^{2} \sqrt {3}+8 x^{2} \left (\frac {\sqrt {3}}{2}-\frac {1}{2}\right )^{2}}{2 \sqrt {4 \left (\frac {\sqrt {3}}{2}-\frac {1}{2}\right )^{4}-4 \sqrt {3}\, \left (\frac {\sqrt {3}}{2}-\frac {1}{2}\right )^{2}-1}\, \sqrt {-1+4 x^{4}-4 x^{2} \sqrt {3}}}\right )}{2 \sqrt {4 \left (\frac {\sqrt {3}}{2}-\frac {1}{2}\right )^{4}-4 \sqrt {3}\, \left (\frac {\sqrt {3}}{2}-\frac {1}{2}\right )^{2}-1}}-\frac {\sqrt {1-\left (-4-2 \sqrt {3}\right ) x^{2}}\, \sqrt {1-\left (-2 \sqrt {3}+4\right ) x^{2}}\, \EllipticPi \left (\sqrt {-4-2 \sqrt {3}}\, x , \frac {1}{\left (-4-2 \sqrt {3}\right ) \left (\frac {\sqrt {3}}{2}-\frac {1}{2}\right )^{2}}, \frac {\sqrt {-2 \sqrt {3}+4}}{\sqrt {-4-2 \sqrt {3}}}\right )}{\sqrt {-4-2 \sqrt {3}}\, \left (\frac {\sqrt {3}}{2}-\frac {1}{2}\right ) \sqrt {-1+4 x^{4}-4 x^{2} \sqrt {3}}}\right )\)	\(336\)
default	\(\frac {\sqrt {1-\left (-4-2 \sqrt {3}\right ) x^{2}}\, \sqrt {1-\left (-2 \sqrt {3}+4\right ) x^{2}}\, \EllipticF \left (x \left (i+i \sqrt {3}\right ), i \sqrt {1-\sqrt {3}\, \left (-2 \sqrt {3}+4\right )}\right )}{\left (i+i \sqrt {3}\right ) \sqrt {-1+4 x^{4}-4 x^{2} \sqrt {3}}}+2 \sqrt {3}\, \left (-\frac {\arctanh \left (\frac {-4 \sqrt {3}\, \left (\frac {\sqrt {3}}{2}-\frac {1}{2}\right )^{2}-2-4 x^{2} \sqrt {3}+8 x^{2} \left (\frac {\sqrt {3}}{2}-\frac {1}{2}\right )^{2}}{2 \sqrt {4 \left (\frac {\sqrt {3}}{2}-\frac {1}{2}\right )^{4}-4 \sqrt {3}\, \left (\frac {\sqrt {3}}{2}-\frac {1}{2}\right )^{2}-1}\, \sqrt {-1+4 x^{4}-4 x^{2} \sqrt {3}}}\right )}{4 \sqrt {4 \left (\frac {\sqrt {3}}{2}-\frac {1}{2}\right )^{4}-4 \sqrt {3}\, \left (\frac {\sqrt {3}}{2}-\frac {1}{2}\right )^{2}-1}}-\frac {\sqrt {1-\left (-4-2 \sqrt {3}\right ) x^{2}}\, \sqrt {1-\left (-2 \sqrt {3}+4\right ) x^{2}}\, \EllipticPi \left (\sqrt {-4-2 \sqrt {3}}\, x , \frac {1}{\left (-4-2 \sqrt {3}\right ) \left (\frac {\sqrt {3}}{2}-\frac {1}{2}\right )^{2}}, \frac {\sqrt {-2 \sqrt {3}+4}}{\sqrt {-4-2 \sqrt {3}}}\right )}{2 \sqrt {-4-2 \sqrt {3}}\, \left (\frac {\sqrt {3}}{2}-\frac {1}{2}\right ) \sqrt {-1+4 x^{4}-4 x^{2} \sqrt {3}}}\right )\)	\(337\)

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,method=_RETURNVERBOSE)

[Out]

1/(I+I*3^(1/2))*(1-(-4-2*3^(1/2))*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)*Ellip
ticF(x*(I+I*3^(1/2)),I*(1-3^(1/2)*(-2*3^(1/2)+4))^(1/2))+2*3^(1/2)*(-1/4/(4*(1/2*3^(1/2)-1/2)^4-4*3^(1/2)*(1/2
*3^(1/2)-1/2)^2-1)^(1/2)*arctanh(1/2*(-4*3^(1/2)*(1/2*3^(1/2)-1/2)^2-2-4*x^2*3^(1/2)+8*x^2*(1/2*3^(1/2)-1/2)^2
)/(4*(1/2*3^(1/2)-1/2)^4-4*3^(1/2)*(1/2*3^(1/2)-1/2)^2-1)^(1/2)/(-1+4*x^4-4*x^2*3^(1/2))^(1/2))-1/2/(-4-2*3^(1
/2))^(1/2)/(1/2*3^(1/2)-1/2)*(1-(-4-2*3^(1/2))*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)*EllipticPi((-4-2*3^(1/2))^(1/2)*x,1/(-4-2*3^(1/2))/(1/2*3^(1/2)-1/2)^2,(-2*3^(1/2)+4)^(1/2)/(-4-2*3^(1
/2))^(1/2)))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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, 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 [B] Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (50) = 100\).
time = 0.52, size = 114, normalized size = 1.63 \begin {gather*} \frac {1}{6} \, \sqrt {2 \, \sqrt {3} + 3} \arctan \left (-\frac {{\left (36 \, x^{4} - 60 \, x^{3} + 18 \, x^{2} - \sqrt {3} {\left (16 \, x^{4} - 40 \, x^{3} + 6 \, x^{2} - 10 \, x + 1\right )} + 6\right )} \sqrt {4 \, x^{4} - 4 \, \sqrt {3} x^{2} - 1} \sqrt {2 \, \sqrt {3} + 3}}{88 \, x^{6} - 168 \, x^{5} + 132 \, x^{4} - 176 \, x^{3} - 66 \, x^{2} - 42 \, x - 11}\right ) \end {gather*}

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, algorithm="fricas")

[Out]

1/6*sqrt(2*sqrt(3) + 3)*arctan(-(36*x^4 - 60*x^3 + 18*x^2 - sqrt(3)*(16*x^4 - 40*x^3 + 6*x^2 - 10*x + 1) + 6)*
sqrt(4*x^4 - 4*sqrt(3)*x^2 - 1)*sqrt(2*sqrt(3) + 3)/(88*x^6 - 168*x^5 + 132*x^4 - 176*x^3 - 66*x^2 - 42*x - 11
))

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

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]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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, 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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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