3.302 \(\int \frac{1}{\sqrt{-1+x^2} \sqrt{7-4 \sqrt{3}+x^2}} \, dx\)

Optimal. Leaf size=46 \[ \frac{\sqrt{1-x^2} \text{EllipticF}\left (\sin ^{-1}(x),-7-4 \sqrt{3}\right )}{\sqrt{7-4 \sqrt{3}} \sqrt{x^2-1}} \]

[Out]

(Sqrt[1 - x^2]*EllipticF[ArcSin[x], -7 - 4*Sqrt[3]])/(Sqrt[7 - 4*Sqrt[3]]*Sqrt[-1 + x^2])

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Rubi [A]  time = 0.0407638, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {421, 419} \[ \frac{\sqrt{1-x^2} F\left (\sin ^{-1}(x)|-7-4 \sqrt{3}\right )}{\sqrt{7-4 \sqrt{3}} \sqrt{x^2-1}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[-1 + x^2]*Sqrt[7 - 4*Sqrt[3] + x^2]),x]

[Out]

(Sqrt[1 - x^2]*EllipticF[ArcSin[x], -7 - 4*Sqrt[3]])/(Sqrt[7 - 4*Sqrt[3]]*Sqrt[-1 + x^2])

Rule 421

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 + (d*x^2)/c]/Sqrt[c + d*
x^2], Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + (d*x^2)/c]), x], x] /; FreeQ[{a, b, c, d}, x] &&  !GtQ[c, 0]

Rule 419

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1*EllipticF[ArcSin[Rt[-(d/c),
2]*x], (b*c)/(a*d)])/(Sqrt[a]*Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &
& GtQ[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-(b/a), -(d/c)])

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{-1+x^2} \sqrt{7-4 \sqrt{3}+x^2}} \, dx &=\frac{\sqrt{1-x^2} \int \frac{1}{\sqrt{1-x^2} \sqrt{7-4 \sqrt{3}+x^2}} \, dx}{\sqrt{-1+x^2}}\\ &=\frac{\sqrt{1-x^2} F\left (\sin ^{-1}(x)|-7-4 \sqrt{3}\right )}{\sqrt{7-4 \sqrt{3}} \sqrt{-1+x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0817452, size = 48, normalized size = 1.04 \[ \frac{\sqrt{1-x^2} \text{EllipticF}\left (\sin ^{-1}(x),\frac{1}{4 \sqrt{3}-7}\right )}{\sqrt{7-4 \sqrt{3}} \sqrt{x^2-1}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[-1 + x^2]*Sqrt[7 - 4*Sqrt[3] + x^2]),x]

[Out]

(Sqrt[1 - x^2]*EllipticF[ArcSin[x], (-7 + 4*Sqrt[3])^(-1)])/(Sqrt[7 - 4*Sqrt[3]]*Sqrt[-1 + x^2])

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Maple [B]  time = 0.106, size = 117, normalized size = 2.5 \begin{align*}{\frac{-i \left ( -2+\sqrt{3} \right ) }{ \left ( 4\,\sqrt{3}-7 \right ) \left ( -{x}^{4}+4\,\sqrt{3}{x}^{2}-6\,{x}^{2}-4\,\sqrt{3}+7 \right ) }{\it EllipticF} \left ({\frac{ix}{-2+\sqrt{3}}},2\,i-i\sqrt{3} \right ) \sqrt{-{x}^{2}+1}\sqrt{- \left ( -{x}^{2}+4\,\sqrt{3}-7 \right ) \left ( 7-4\,\sqrt{3} \right ) }\sqrt{{x}^{2}-1}\sqrt{7+{x}^{2}-4\,\sqrt{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2-1)^(1/2)/(7+x^2-4*3^(1/2))^(1/2),x)

[Out]

-I*EllipticF(I*x/(-2+3^(1/2)),2*I-I*3^(1/2))*(-x^2+1)^(1/2)*(-(-x^2+4*3^(1/2)-7)*(7-4*3^(1/2)))^(1/2)/(4*3^(1/
2)-7)*(-2+3^(1/2))*(x^2-1)^(1/2)*(7+x^2-4*3^(1/2))^(1/2)/(-x^4+4*3^(1/2)*x^2-6*x^2-4*3^(1/2)+7)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{x^{2} - 4 \, \sqrt{3} + 7} \sqrt{x^{2} - 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^2-1)^(1/2)/(7+x^2-4*3^(1/2))^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(x^2 - 4*sqrt(3) + 7)*sqrt(x^2 - 1)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (x^{2} + 4 \, \sqrt{3} + 7\right )} \sqrt{x^{2} - 4 \, \sqrt{3} + 7} \sqrt{x^{2} - 1}}{x^{6} + 13 \, x^{4} - 13 \, x^{2} - 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^2-1)^(1/2)/(7+x^2-4*3^(1/2))^(1/2),x, algorithm="fricas")

[Out]

integral((x^2 + 4*sqrt(3) + 7)*sqrt(x^2 - 4*sqrt(3) + 7)*sqrt(x^2 - 1)/(x^6 + 13*x^4 - 13*x^2 - 1), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{\left (x - 1\right ) \left (x + 1\right )} \sqrt{x^{2} - 4 \sqrt{3} + 7}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x**2-1)**(1/2)/(7+x**2-4*3**(1/2))**(1/2),x)

[Out]

Integral(1/(sqrt((x - 1)*(x + 1))*sqrt(x**2 - 4*sqrt(3) + 7)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{x^{2} - 4 \, \sqrt{3} + 7} \sqrt{x^{2} - 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^2-1)^(1/2)/(7+x^2-4*3^(1/2))^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(x^2 - 4*sqrt(3) + 7)*sqrt(x^2 - 1)), x)