3.1.79 \(\int \frac {1}{\sqrt {3+8 x^2+2 x^4}} \, dx\) [79]

Optimal. Leaf size=110 \[ \frac {\sqrt {\frac {3+\left (4-\sqrt {10}\right ) x^2}{3+\left (4+\sqrt {10}\right ) x^2}} \left (3+\left (4+\sqrt {10}\right ) x^2\right ) F\left (\tan ^{-1}\left (\sqrt {\frac {1}{3} \left (4+\sqrt {10}\right )} x\right )|-\frac {2}{3} \left (5-2 \sqrt {10}\right )\right )}{\sqrt {3 \left (4+\sqrt {10}\right )} \sqrt {3+8 x^2+2 x^4}} \]

[Out]

(1/(9+x^2*(12+3*10^(1/2))))^(1/2)*(9+x^2*(12+3*10^(1/2)))^(1/2)*EllipticF(x*(12+3*10^(1/2))^(1/2)/(9+x^2*(12+3
*10^(1/2)))^(1/2),1/3*(-30+12*10^(1/2))^(1/2))*(3+x^2*(10^(1/2)+4))*((3+x^2*(4-10^(1/2)))/(3+x^2*(10^(1/2)+4))
)^(1/2)/(2*x^4+8*x^2+3)^(1/2)/(12+3*10^(1/2))^(1/2)

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Rubi [A]
time = 0.05, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {1113} \begin {gather*} \frac {\sqrt {\frac {\left (4-\sqrt {10}\right ) x^2+3}{\left (4+\sqrt {10}\right ) x^2+3}} \left (\left (4+\sqrt {10}\right ) x^2+3\right ) F\left (\text {ArcTan}\left (\sqrt {\frac {1}{3} \left (4+\sqrt {10}\right )} x\right )|-\frac {2}{3} \left (5-2 \sqrt {10}\right )\right )}{\sqrt {3 \left (4+\sqrt {10}\right )} \sqrt {2 x^4+8 x^2+3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[3 + 8*x^2 + 2*x^4],x]

[Out]

(Sqrt[(3 + (4 - Sqrt[10])*x^2)/(3 + (4 + Sqrt[10])*x^2)]*(3 + (4 + Sqrt[10])*x^2)*EllipticF[ArcTan[Sqrt[(4 + S
qrt[10])/3]*x], (-2*(5 - 2*Sqrt[10]))/3])/(Sqrt[3*(4 + Sqrt[10])]*Sqrt[3 + 8*x^2 + 2*x^4])

Rule 1113

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(2*a + (b + q
)*x^2)*(Sqrt[(2*a + (b - q)*x^2)/(2*a + (b + q)*x^2)]/(2*a*Rt[(b + q)/(2*a), 2]*Sqrt[a + b*x^2 + c*x^4]))*Elli
pticF[ArcTan[Rt[(b + q)/(2*a), 2]*x], 2*(q/(b + q))], x] /; PosQ[(b + q)/a] &&  !(PosQ[(b - q)/a] && SimplerSq
rtQ[(b - q)/(2*a), (b + q)/(2*a)])] /; FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {3+8 x^2+2 x^4}} \, dx &=\frac {\sqrt {\frac {3+\left (4-\sqrt {10}\right ) x^2}{3+\left (4+\sqrt {10}\right ) x^2}} \left (3+\left (4+\sqrt {10}\right ) x^2\right ) F\left (\tan ^{-1}\left (\sqrt {\frac {1}{3} \left (4+\sqrt {10}\right )} x\right )|-\frac {2}{3} \left (5-2 \sqrt {10}\right )\right )}{\sqrt {3 \left (4+\sqrt {10}\right )} \sqrt {3+8 x^2+2 x^4}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 10.07, size = 98, normalized size = 0.89 \begin {gather*} -\frac {i \sqrt {\frac {-4+\sqrt {10}-2 x^2}{-4+\sqrt {10}}} \sqrt {4+\sqrt {10}+2 x^2} F\left (i \sinh ^{-1}\left (\sqrt {\frac {2}{4+\sqrt {10}}} x\right )|\frac {13}{3}+\frac {4 \sqrt {10}}{3}\right )}{\sqrt {6+16 x^2+4 x^4}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[1/Sqrt[3 + 8*x^2 + 2*x^4],x]

[Out]

((-I)*Sqrt[(-4 + Sqrt[10] - 2*x^2)/(-4 + Sqrt[10])]*Sqrt[4 + Sqrt[10] + 2*x^2]*EllipticF[I*ArcSinh[Sqrt[2/(4 +
Sqrt[10])]*x], 13/3 + (4*Sqrt[10])/3])/Sqrt[6 + 16*x^2 + 4*x^4]

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Maple [A]
time = 0.07, size = 82, normalized size = 0.75

method	result	size
default	\(\frac {3 \sqrt {1-\left (-\frac {4}{3}+\frac {\sqrt {10}}{3}\right ) x^{2}}\, \sqrt {1-\left (-\frac {4}{3}-\frac {\sqrt {10}}{3}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {-12+3 \sqrt {10}}}{3}, \frac {2 \sqrt {6}}{3}+\frac {\sqrt {15}}{3}\right )}{\sqrt {-12+3 \sqrt {10}}\, \sqrt {2 x^{4}+8 x^{2}+3}}\)	\(82\)
elliptic	\(\frac {3 \sqrt {1-\left (-\frac {4}{3}+\frac {\sqrt {10}}{3}\right ) x^{2}}\, \sqrt {1-\left (-\frac {4}{3}-\frac {\sqrt {10}}{3}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {-12+3 \sqrt {10}}}{3}, \frac {2 \sqrt {6}}{3}+\frac {\sqrt {15}}{3}\right )}{\sqrt {-12+3 \sqrt {10}}\, \sqrt {2 x^{4}+8 x^{2}+3}}\)	\(82\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(2*x^4+8*x^2+3)^(1/2),x,method=_RETURNVERBOSE)

[Out]

3/(-12+3*10^(1/2))^(1/2)*(1-(-4/3+1/3*10^(1/2))*x^2)^(1/2)*(1-(-4/3-1/3*10^(1/2))*x^2)^(1/2)/(2*x^4+8*x^2+3)^(
1/2)*EllipticF(1/3*x*(-12+3*10^(1/2))^(1/2),2/3*6^(1/2)+1/3*15^(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/(2*x^4+8*x^2+3)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/sqrt(2*x^4 + 8*x^2 + 3), x)

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Fricas [A]
time = 0.09, size = 35, normalized size = 0.32 \begin {gather*} -\frac {1}{6} \, {\left (\sqrt {10} + 4\right )} \sqrt {\sqrt {10} - 4} {\rm ellipticF}\left (\frac {1}{3} \, \sqrt {3} x \sqrt {\sqrt {10} - 4}, \frac {4}{3} \, \sqrt {10} + \frac {13}{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*x^4+8*x^2+3)^(1/2),x, algorithm="fricas")

[Out]

-1/6*(sqrt(10) + 4)*sqrt(sqrt(10) - 4)*ellipticF(1/3*sqrt(3)*x*sqrt(sqrt(10) - 4), 4/3*sqrt(10) + 13/3)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {2 x^{4} + 8 x^{2} + 3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*x**4+8*x**2+3)**(1/2),x)

[Out]

Integral(1/sqrt(2*x**4 + 8*x**2 + 3), 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/(2*x^4+8*x^2+3)^(1/2),x, algorithm="giac")

[Out]

integrate(1/sqrt(2*x^4 + 8*x^2 + 3), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\sqrt {2\,x^4+8\,x^2+3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(8*x^2 + 2*x^4 + 3)^(1/2),x)

[Out]

int(1/(8*x^2 + 2*x^4 + 3)^(1/2), x)

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