3.11.40 \(\int \frac {1}{\sqrt {-1+5 x^2-x^4}} \, dx\) [1040]

Optimal. Leaf size=39 \[ -\frac {F\left (\cos ^{-1}\left (\sqrt {\frac {2}{5+\sqrt {21}}} x\right )|\frac {1}{42} \left (21+5 \sqrt {21}\right )\right )}{\sqrt [4]{21}} \]

[Out]

-1/21*(x^2/(5+21^(1/2)))^(1/2)/x*(5+21^(1/2))^(1/2)*EllipticF((1-2*x^2/(5+21^(1/2)))^(1/2),1/42*(882+210*21^(1
/2))^(1/2))*21^(3/4)

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Rubi [A]
time = 0.05, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1109, 431} \begin {gather*} -\frac {F\left (\text {ArcCos}\left (\sqrt {\frac {2}{5+\sqrt {21}}} x\right )|\frac {1}{42} \left (21+5 \sqrt {21}\right )\right )}{\sqrt [4]{21}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[-1 + 5*x^2 - x^4],x]

[Out]

-(EllipticF[ArcCos[Sqrt[2/(5 + Sqrt[21])]*x], (21 + 5*Sqrt[21])/42]/21^(1/4))

Rule 431

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

Rule 1109

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[2*Sqrt[-c], I
nt[1/(Sqrt[b + q + 2*c*x^2]*Sqrt[-b + q - 2*c*x^2]), x], x]] /; FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0] &&
LtQ[c, 0]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {-1+5 x^2-x^4}} \, dx &=2 \int \frac {1}{\sqrt {5+\sqrt {21}-2 x^2} \sqrt {-5+\sqrt {21}+2 x^2}} \, dx\\ &=-\frac {F\left (\cos ^{-1}\left (\sqrt {\frac {2}{5+\sqrt {21}}} x\right )|\frac {1}{42} \left (21+5 \sqrt {21}\right )\right )}{\sqrt [4]{21}}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(87\) vs. \(2(39)=78\).
time = 10.09, size = 87, normalized size = 2.23 \begin {gather*} \frac {\sqrt {5-\sqrt {21}-2 x^2} \sqrt {2+\left (-5+\sqrt {21}\right ) x^2} F\left (\sin ^{-1}\left (\sqrt {\frac {1}{2} \left (5+\sqrt {21}\right )} x\right )|\frac {23}{2}-\frac {5 \sqrt {21}}{2}\right )}{2 \sqrt {-1+5 x^2-x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[-1 + 5*x^2 - x^4],x]

[Out]

(Sqrt[5 - Sqrt[21] - 2*x^2]*Sqrt[2 + (-5 + Sqrt[21])*x^2]*EllipticF[ArcSin[Sqrt[(5 + Sqrt[21])/2]*x], 23/2 - (
5*Sqrt[21])/2])/(2*Sqrt[-1 + 5*x^2 - x^4])

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Maple [A]
time = 0.06, size = 82, normalized size = 2.10

method	result	size
default	\(\frac {\sqrt {1-\left (\frac {5}{2}-\frac {\sqrt {21}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {5}{2}+\frac {\sqrt {21}}{2}\right ) x^{2}}\, \EllipticF \left (x \left (\frac {\sqrt {7}}{2}-\frac {\sqrt {3}}{2}\right ), \frac {5}{2}+\frac {\sqrt {21}}{2}\right )}{\left (\frac {\sqrt {7}}{2}-\frac {\sqrt {3}}{2}\right ) \sqrt {-x^{4}+5 x^{2}-1}}\)	\(82\)
elliptic	\(\frac {\sqrt {1-\left (\frac {5}{2}-\frac {\sqrt {21}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {5}{2}+\frac {\sqrt {21}}{2}\right ) x^{2}}\, \EllipticF \left (x \left (\frac {\sqrt {7}}{2}-\frac {\sqrt {3}}{2}\right ), \frac {5}{2}+\frac {\sqrt {21}}{2}\right )}{\left (\frac {\sqrt {7}}{2}-\frac {\sqrt {3}}{2}\right ) \sqrt {-x^{4}+5 x^{2}-1}}\)	\(82\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-x^4+5*x^2-1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/(1/2*7^(1/2)-1/2*3^(1/2))*(1-(5/2-1/2*21^(1/2))*x^2)^(1/2)*(1-(5/2+1/2*21^(1/2))*x^2)^(1/2)/(-x^4+5*x^2-1)^(
1/2)*EllipticF(x*(1/2*7^(1/2)-1/2*3^(1/2)),5/2+1/2*21^(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/(-x^4+5*x^2-1)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/sqrt(-x^4 + 5*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^4+5*x^2-1)^(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 {- x^{4} + 5 x^{2} - 1}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x**4+5*x**2-1)**(1/2),x)

[Out]

Integral(1/sqrt(-x**4 + 5*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/(-x^4+5*x^2-1)^(1/2),x, algorithm="giac")

[Out]

integrate(1/sqrt(-x^4 + 5*x^2 - 1), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {1}{\sqrt {-x^4+5\,x^2-1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(5*x^2 - x^4 - 1)^(1/2),x)

[Out]

int(1/(5*x^2 - x^4 - 1)^(1/2), x)

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