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

Optimal. Leaf size=72 \[ \frac{1}{3} \sqrt{2 \sqrt{3}-3} \tanh ^{-1}\left (\frac{\left (2 x-\sqrt{3}+1\right )^2}{2 \sqrt{3 \left (2 \sqrt{3}-3\right )} \sqrt{4 x^4+4 \sqrt{3} x^2-1}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.132782, antiderivative size = 72, normalized size of antiderivative = 1., 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 = {1740, 207} \[ \frac{1}{3} \sqrt{2 \sqrt{3}-3} \tanh ^{-1}\left (\frac{\left (2 x-\sqrt{3}+1\right )^2}{2 \sqrt{3 \left (2 \sqrt{3}-3\right )} \sqrt{4 x^4+4 \sqrt{3} x^2-1}}\right ) \]

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]

(Sqrt[-3 + 2*Sqrt[3]]*ArcTanh[(1 - Sqrt[3] + 2*x)^2/(2*Sqrt[3*(-3 + 2*Sqrt[3])]*Sqrt[-1 + 4*Sqrt[3]*x^2 + 4*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 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 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 ) \operatorname{Subst}\left (\int \frac{1}{6 \left (1-\sqrt{3}\right )^4+12 \left (1-\sqrt{3}\right )^3 \left (1+\sqrt{3}\right )+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}} \tanh ^{-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*}

Mathematica [C]  time = 1.63171, size = 623, normalized size = 8.65 \[ \frac{\left (2 x+\sqrt{3}-1\right )^2 \sqrt{\frac{-\frac{4}{2 x+\sqrt{3}-1}+\sqrt{3}+1}{3+\sqrt{3}+i \sqrt{2 \left (2+\sqrt{3}\right )}}} \left (\left (\frac{2 \left (2 i \sqrt{3}-\sqrt{2 \left (2+\sqrt{3}\right )}+\sqrt{6 \left (2+\sqrt{3}\right )}\right )}{2 x+\sqrt{3}-1}+i \left (-1+\sqrt{3}+i \sqrt{2 \left (2+\sqrt{3}\right )}\right )\right ) \sqrt{\sqrt{2 \left (2+\sqrt{3}\right )}+i \left (\frac{8}{2 x+\sqrt{3}-1}-\sqrt{3}+1\right )} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\sqrt{2 \left (2+\sqrt{3}\right )}-i \left (\frac{8}{2 x+\sqrt{3}-1}-\sqrt{3}+1\right )}}{2^{3/4} \sqrt [4]{2+\sqrt{3}}}\right ),\frac{2 i \sqrt{2 \left (2+\sqrt{3}\right )}}{3+\sqrt{3}+i \sqrt{2 \left (2+\sqrt{3}\right )}}\right )+4 \sqrt{3} \sqrt{\frac{2 x^2+\sqrt{3}+2}{\left (2 x+\sqrt{3}-1\right )^2}} \sqrt{\sqrt{2 \left (2+\sqrt{3}\right )}-i \left (\frac{8}{2 x+\sqrt{3}-1}-\sqrt{3}+1\right )} \Pi \left (\frac{2 \sqrt{2 \left (2+\sqrt{3}\right )}}{\sqrt{2 \left (2+\sqrt{3}\right )}+i \left (3+\sqrt{3}\right )};\sin ^{-1}\left (\frac{\sqrt{\sqrt{2 \left (2+\sqrt{3}\right )}-i \left (-\sqrt{3}+1+\frac{8}{2 x+\sqrt{3}-1}\right )}}{2^{3/4} \sqrt [4]{2+\sqrt{3}}}\right )|\frac{2 i \sqrt{2 \left (2+\sqrt{3}\right )}}{3+\sqrt{3}+i \sqrt{2 \left (2+\sqrt{3}\right )}}\right )\right )}{\left (\sqrt{2 \left (2+\sqrt{3}\right )}+i \left (3+\sqrt{3}\right )\right ) \sqrt{8 x^4+8 \sqrt{3} x^2-2} \sqrt{\sqrt{2 \left (2+\sqrt{3}\right )}-i \left (\frac{8}{2 x+\sqrt{3}-1}-\sqrt{3}+1\right )}} \]

Warning: Unable to verify antiderivative.

[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 + Sqrt[3] + 2*x)^2*Sqrt[(1 + Sqrt[3] - 4/(-1 + Sqrt[3] + 2*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] + 2*x))*Sqrt[Sqrt[2*(2 + Sqrt[3])] + I*(1 - Sqrt[3] + 8/(-1 + Sqrt[3] + 2*x))]*EllipticF[Arc
Sin[Sqrt[Sqrt[2*(2 + Sqrt[3])] - I*(1 - Sqrt[3] + 8/(-1 + Sqrt[3] + 2*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])])] + 4*Sqrt[3]*Sqrt[(2 + Sqrt[3] + 2*x^2)/(-1
+ Sqrt[3] + 2*x)^2]*Sqrt[Sqrt[2*(2 + Sqrt[3])] - I*(1 - Sqrt[3] + 8/(-1 + Sqrt[3] + 2*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] + 2*x))]/(2^(3/4)*(2 + Sqrt[3])^(1/4))], ((2*I)*Sqrt[2*(2 + Sqrt[3])])/(3 + Sqrt[3] + I*S
qrt[2*(2 + Sqrt[3])])]))/((Sqrt[2*(2 + Sqrt[3])] + I*(3 + Sqrt[3]))*Sqrt[-2 + 8*Sqrt[3]*x^2 + 8*x^4]*Sqrt[Sqrt
[2*(2 + Sqrt[3])] - I*(1 - Sqrt[3] + 8/(-1 + Sqrt[3] + 2*x))])

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Maple [C]  time = 0.143, size = 336, normalized size = 4.7 \begin{align*}{\frac{{\it EllipticF} \left ( x \left ( i\sqrt{3}-i \right ) ,i\sqrt{1+\sqrt{3} \left ( 2\,\sqrt{3}+4 \right ) } \right ) }{i\sqrt{3}-i}\sqrt{1- \left ( 2\,\sqrt{3}-4 \right ){x}^{2}}\sqrt{1- \left ( 2\,\sqrt{3}+4 \right ){x}^{2}}{\frac{1}{\sqrt{-1+4\,{x}^{4}+4\,{x}^{2}\sqrt{3}}}}}-2\,\sqrt{3} \left ( -1/4\,{\frac{1}{\sqrt{4\, \left ( -1/2-1/2\,\sqrt{3} \right ) ^{4}+4\,\sqrt{3} \left ( -1/2-1/2\,\sqrt{3} \right ) ^{2}-1}}{\it Artanh} \left ( 1/2\,{\frac{4\,\sqrt{3} \left ( -1/2-1/2\,\sqrt{3} \right ) ^{2}-2+4\,{x}^{2}\sqrt{3}+8\,{x}^{2} \left ( -1/2-1/2\,\sqrt{3} \right ) ^{2}}{\sqrt{4\, \left ( -1/2-1/2\,\sqrt{3} \right ) ^{4}+4\,\sqrt{3} \left ( -1/2-1/2\,\sqrt{3} \right ) ^{2}-1}\sqrt{-1+4\,{x}^{4}+4\,{x}^{2}\sqrt{3}}}} \right ) }-1/2\,{\frac{\sqrt{1- \left ( 2\,\sqrt{3}-4 \right ){x}^{2}}\sqrt{1- \left ( 2\,\sqrt{3}+4 \right ){x}^{2}}}{\sqrt{2\,\sqrt{3}-4} \left ( -1/2-1/2\,\sqrt{3} \right ) \sqrt{-1+4\,{x}^{4}+4\,{x}^{2}\sqrt{3}}}{\it EllipticPi} \left ( \sqrt{2\,\sqrt{3}-4}x,{\frac{1}{ \left ( 2\,\sqrt{3}-4 \right ) \left ( -1/2-1/2\,\sqrt{3} \right ) ^{2}}},{\frac{\sqrt{2\,\sqrt{3}+4}}{\sqrt{2\,\sqrt{3}-4}}} \right ) } \right ) \end{align*}

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)

[Out]

1/(I*3^(1/2)-I)*(1-(2*3^(1/2)-4)*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)*Ellipti
cF(x*(I*3^(1/2)-I),I*(1+3^(1/2)*(2*3^(1/2)+4))^(1/2))-2*3^(1/2)*(-1/4/(4*(-1/2-1/2*3^(1/2))^4+4*3^(1/2)*(-1/2-
1/2*3^(1/2))^2-1)^(1/2)*arctanh(1/2*(4*3^(1/2)*(-1/2-1/2*3^(1/2))^2-2+4*x^2*3^(1/2)+8*x^2*(-1/2-1/2*3^(1/2))^2
)/(4*(-1/2-1/2*3^(1/2))^4+4*3^(1/2)*(-1/2-1/2*3^(1/2))^2-1)^(1/2)/(-1+4*x^4+4*x^2*3^(1/2))^(1/2))-1/2/(2*3^(1/
2)-4)^(1/2)/(-1/2-1/2*3^(1/2))*(1-(2*3^(1/2)-4)*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((2*3^(1/2)-4)^(1/2)*x,1/(2*3^(1/2)-4)/(-1/2-1/2*3^(1/2))^2,(2*3^(1/2)+4)^(1/2)/(2*3^(1/2)-4
)^(1/2)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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{align*}

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]  time = 2.70544, size = 950, normalized size = 13.19 \begin{align*} \frac{1}{12} \, \sqrt{2 \, \sqrt{3} - 3} \log \left (-\frac{2368 \, x^{12} - 6528 \, x^{11} + 12864 \, x^{10} - 19264 \, x^{9} + 14832 \, x^{8} - 10944 \, x^{7} + 6432 \, x^{6} + 5472 \, x^{5} + 3708 \, x^{4} + 2408 \, x^{3} + 804 \, x^{2} +{\left (1728 \, x^{10} - 4800 \, x^{9} + 8208 \, x^{8} - 8928 \, x^{7} + 6048 \, x^{6} - 3024 \, x^{5} - 504 \, x^{4} - 504 \, x^{3} - 324 \, x^{2} + 2 \, \sqrt{3}{\left (496 \, x^{10} - 1408 \, x^{9} + 2304 \, x^{8} - 2640 \, x^{7} + 1848 \, x^{6} - 504 \, x^{5} + 336 \, x^{4} + 204 \, x^{3} + 63 \, x^{2} + 26 \, x + 4\right )} - 72 \, x - 15\right )} \sqrt{4 \, x^{4} + 4 \, \sqrt{3} x^{2} - 1} \sqrt{2 \, \sqrt{3} - 3} + 3 \, \sqrt{3}{\left (448 \, x^{12} - 1280 \, x^{11} + 2560 \, x^{10} - 3200 \, x^{9} + 3696 \, x^{8} - 1920 \, x^{7} - 960 \, x^{5} - 924 \, x^{4} - 400 \, x^{3} - 160 \, x^{2} - 40 \, x - 7\right )} + 204 \, x + 37}{64 \, x^{12} + 384 \, x^{11} + 768 \, x^{10} + 320 \, x^{9} - 720 \, x^{8} - 576 \, x^{7} + 384 \, x^{6} + 288 \, x^{5} - 180 \, x^{4} - 40 \, x^{3} + 48 \, x^{2} - 12 \, x + 1}\right ) \end{align*}

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]

1/12*sqrt(2*sqrt(3) - 3)*log(-(2368*x^12 - 6528*x^11 + 12864*x^10 - 19264*x^9 + 14832*x^8 - 10944*x^7 + 6432*x
^6 + 5472*x^5 + 3708*x^4 + 2408*x^3 + 804*x^2 + (1728*x^10 - 4800*x^9 + 8208*x^8 - 8928*x^7 + 6048*x^6 - 3024*
x^5 - 504*x^4 - 504*x^3 - 324*x^2 + 2*sqrt(3)*(496*x^10 - 1408*x^9 + 2304*x^8 - 2640*x^7 + 1848*x^6 - 504*x^5
+ 336*x^4 + 204*x^3 + 63*x^2 + 26*x + 4) - 72*x - 15)*sqrt(4*x^4 + 4*sqrt(3)*x^2 - 1)*sqrt(2*sqrt(3) - 3) + 3*
sqrt(3)*(448*x^12 - 1280*x^11 + 2560*x^10 - 3200*x^9 + 3696*x^8 - 1920*x^7 - 960*x^5 - 924*x^4 - 400*x^3 - 160
*x^2 - 40*x - 7) + 204*x + 37)/(64*x^12 + 384*x^11 + 768*x^10 + 320*x^9 - 720*x^8 - 576*x^7 + 384*x^6 + 288*x^
5 - 180*x^4 - 40*x^3 + 48*x^2 - 12*x + 1))

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

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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{align*}

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)