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

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

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Rubi [A]  time = 0.13, antiderivative size = 72, 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 = {1740, 207} \begin {gather*} \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 ) \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]

(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 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])

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}+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*}

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Mathematica [C]  time = 1.73, size = 623, normalized size = 8.65 \begin {gather*} \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 (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 )+\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 )} F\left (\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 )}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 12.22, size = 81, normalized size = 1.12 \begin {gather*} \frac {1}{3} \sqrt {2 \sqrt {3}-3} \tanh ^{-1}\left (\frac {\sqrt {9+6 \sqrt {3}} \sqrt {4 x^4+4 \sqrt {3} x^2-1}}{\left (4+2 \sqrt {3}\right ) x^2+\left (-2-2 \sqrt {3}\right ) x+1}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(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[(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)])/3

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fricas [B]  time = 1.29, size = 328, normalized size = 4.56 \begin {gather*} \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 {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]

1/12*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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \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]

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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maple [C]  time = 0.16, size = 336, normalized size = 4.67 \begin {gather*} \frac {\sqrt {-\left (2 \sqrt {3}-4\right ) x^{2}+1}\, \sqrt {-\left (4+2 \sqrt {3}\right ) x^{2}+1}\, \EllipticF \left (\left (i \sqrt {3}-i\right ) x , i \sqrt {1+\sqrt {3}\, \left (4+2 \sqrt {3}\right )}\right )}{\left (i \sqrt {3}-i\right ) \sqrt {4 x^{4}+4 \sqrt {3}\, x^{2}-1}}-2 \sqrt {3}\, \left (-\frac {\sqrt {-\left (2 \sqrt {3}-4\right ) x^{2}+1}\, \sqrt {-\left (4+2 \sqrt {3}\right ) x^{2}+1}\, \EllipticPi \left (\sqrt {2 \sqrt {3}-4}\, x , \frac {1}{\left (2 \sqrt {3}-4\right ) \left (-\frac {1}{2}-\frac {\sqrt {3}}{2}\right )^{2}}, \frac {\sqrt {4+2 \sqrt {3}}}{\sqrt {2 \sqrt {3}-4}}\right )}{2 \sqrt {2 \sqrt {3}-4}\, \left (-\frac {1}{2}-\frac {\sqrt {3}}{2}\right ) \sqrt {4 x^{4}+4 \sqrt {3}\, x^{2}-1}}-\frac {\arctanh \left (\frac {4 \sqrt {3}\, x^{2}+8 \left (-\frac {1}{2}-\frac {\sqrt {3}}{2}\right )^{2} x^{2}+4 \sqrt {3}\, \left (-\frac {1}{2}-\frac {\sqrt {3}}{2}\right )^{2}-2}{2 \sqrt {4 \left (-\frac {1}{2}-\frac {\sqrt {3}}{2}\right )^{4}+4 \sqrt {3}\, \left (-\frac {1}{2}-\frac {\sqrt {3}}{2}\right )^{2}-1}\, \sqrt {4 x^{4}+4 \sqrt {3}\, x^{2}-1}}\right )}{4 \sqrt {4 \left (-\frac {1}{2}-\frac {\sqrt {3}}{2}\right )^{4}+4 \sqrt {3}\, \left (-\frac {1}{2}-\frac {\sqrt {3}}{2}\right )^{2}-1}}\right ) \end {gather*}

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*3^(1/2)*x^2)^(1/2),x)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \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]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {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 - sqrt(3) + 1)/((2*x + 1 + sqrt(3))*sqrt(4*x**4 + 4*sqrt(3)*x**2 - 1)), x)

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