3.4.2 \(\int \frac {1}{\sqrt {-1+x^2} \sqrt {7-4 \sqrt {3}+x^2}} \, dx\) [302]

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

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 46, normalized size of antiderivative = 1.00, 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 = {432, 430} \begin {gather*} \frac {\sqrt {1-x^2} F\left (\text {ArcSin}(x)\left |-7-4 \sqrt {3}\right .\right )}{\sqrt {7-4 \sqrt {3}} \sqrt {x^2-1}} \end {gather*}

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 430

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

Rule 432

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

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*}

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Mathematica [A]
time = 0.89, size = 48, normalized size = 1.04 \begin {gather*} \frac {\sqrt {1-x^2} F\left (\sin ^{-1}(x)|\frac {1}{-7+4 \sqrt {3}}\right )}{\sqrt {7-4 \sqrt {3}} \sqrt {-1+x^2}} \end {gather*}

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] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (37 ) = 74\).
time = 0.22, size = 117, normalized size = 2.54

method	result	size
default	\(-\frac {i \EllipticF \left (\frac {i x}{-2+\sqrt {3}}, 2 i-i \sqrt {3}\right ) \sqrt {-x^{2}+1}\, \sqrt {-\left (-x^{2}+4 \sqrt {3}-7\right ) \left (-4 \sqrt {3}+7\right )}\, \left (-2+\sqrt {3}\right ) \sqrt {x^{2}-1}\, \sqrt {7+x^{2}-4 \sqrt {3}}}{\left (4 \sqrt {3}-7\right ) \left (-x^{4}+4 x^{2} \sqrt {3}-6 x^{2}-4 \sqrt {3}+7\right )}\)	\(117\)
elliptic	\(-\frac {i \sqrt {-\left (x^{2}-1\right ) \left (-x^{2}+4 \sqrt {3}-7\right )}\, \sqrt {-4 \sqrt {3}+7}\, \sqrt {1-\frac {x^{2}}{4 \sqrt {3}-7}}\, \sqrt {-x^{2}+1}\, \EllipticF \left (\frac {i x}{\sqrt {-4 \sqrt {3}+7}}, 2 i-i \sqrt {3}\right )}{\sqrt {x^{2}-1}\, \sqrt {7+x^{2}-4 \sqrt {3}}\, \sqrt {6 x^{2}-7+x^{4}-4 x^{2} \sqrt {3}+4 \sqrt {3}}}\)	\(128\)

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,method=_RETURNVERBOSE)

[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)*(-4*3^(1/2)+7))^(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*x^2*3^(1/2)-6*x^2-4*3^(1/2)+7)

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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/(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 = 0.00, 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/(x^2-1)^(1/2)/(7+x^2-4*3^(1/2))^(1/2),x, algorithm="fricas")

[Out]

0

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {\left (x - 1\right ) \left (x + 1\right )} \sqrt {x^{2} - 4 \sqrt {3} + 7}}\, dx \end {gather*}

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 = 0.00, 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/(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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{\sqrt {x^2-1}\,\sqrt {x^2-4\,\sqrt {3}+7}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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