3.1.62 \(\int \frac {1}{\sqrt {-3-2 x^2+2 x^4}} \, dx\) [62]

Optimal. Leaf size=150 \[ \frac {\sqrt {-3-\left (1-\sqrt {7}\right ) x^2} \sqrt {\frac {3+\left (1+\sqrt {7}\right ) x^2}{3+\left (1-\sqrt {7}\right ) x^2}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{7} x}{\sqrt {-3-\left (1-\sqrt {7}\right ) x^2}}\right )|\frac {1}{14} \left (7-\sqrt {7}\right )\right )}{\sqrt {6} \sqrt [4]{7} \sqrt {\frac {1}{3+\left (1-\sqrt {7}\right ) x^2}} \sqrt {-3-2 x^2+2 x^4}} \]

[Out]

1/42*EllipticF(7^(1/4)*x*2^(1/2)/(-3-x^2*(1-7^(1/2)))^(1/2),1/14*(98-14*7^(1/2))^(1/2))*(-3-x^2*(1-7^(1/2)))^(
1/2)*((3+x^2*(1+7^(1/2)))/(3+x^2*(1-7^(1/2))))^(1/2)*7^(3/4)*6^(1/2)/(2*x^4-2*x^2-3)^(1/2)/(1/(3+x^2*(1-7^(1/2
))))^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 150, 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 = {1112} \begin {gather*} \frac {\sqrt {-\left (\left (1-\sqrt {7}\right ) x^2\right )-3} \sqrt {\frac {\left (1+\sqrt {7}\right ) x^2+3}{\left (1-\sqrt {7}\right ) x^2+3}} F\left (\text {ArcSin}\left (\frac {\sqrt {2} \sqrt [4]{7} x}{\sqrt {-\left (\left (1-\sqrt {7}\right ) x^2\right )-3}}\right )|\frac {1}{14} \left (7-\sqrt {7}\right )\right )}{\sqrt {6} \sqrt [4]{7} \sqrt {\frac {1}{\left (1-\sqrt {7}\right ) x^2+3}} \sqrt {2 x^4-2 x^2-3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[-3 - 2*x^2 + 2*x^4],x]

[Out]

(Sqrt[-3 - (1 - Sqrt[7])*x^2]*Sqrt[(3 + (1 + Sqrt[7])*x^2)/(3 + (1 - Sqrt[7])*x^2)]*EllipticF[ArcSin[(Sqrt[2]*
7^(1/4)*x)/Sqrt[-3 - (1 - Sqrt[7])*x^2]], (7 - Sqrt[7])/14])/(Sqrt[6]*7^(1/4)*Sqrt[(3 + (1 - Sqrt[7])*x^2)^(-1
)]*Sqrt[-3 - 2*x^2 + 2*x^4])

Rule 1112

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

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {-3-2 x^2+2 x^4}} \, dx &=\frac {\sqrt {-3-\left (1-\sqrt {7}\right ) x^2} \sqrt {\frac {3+\left (1+\sqrt {7}\right ) x^2}{3+\left (1-\sqrt {7}\right ) x^2}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{7} x}{\sqrt {-3-\left (1-\sqrt {7}\right ) x^2}}\right )|\frac {1}{14} \left (7-\sqrt {7}\right )\right )}{\sqrt {6} \sqrt [4]{7} \sqrt {\frac {1}{3+\left (1-\sqrt {7}\right ) x^2}} \sqrt {-3-2 x^2+2 x^4}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 10.04, size = 81, normalized size = 0.54 \begin {gather*} -\frac {i \sqrt {3+2 x^2-2 x^4} F\left (i \sinh ^{-1}\left (\sqrt {\frac {2}{-1+\sqrt {7}}} x\right )|\frac {1}{3} \left (-4+\sqrt {7}\right )\right )}{\sqrt {1+\sqrt {7}} \sqrt {-3-2 x^2+2 x^4}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [C] Result contains complex when optimal does not.
time = 0.04, size = 84, normalized size = 0.56

method	result	size
default	\(\frac {3 \sqrt {1-\left (-\frac {1}{3}-\frac {\sqrt {7}}{3}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{3}+\frac {\sqrt {7}}{3}\right ) x^{2}}\, \EllipticF \left (\frac {\sqrt {-3-3 \sqrt {7}}\, x}{3}, \frac {i \sqrt {42}}{6}-\frac {i \sqrt {6}}{6}\right )}{\sqrt {-3-3 \sqrt {7}}\, \sqrt {2 x^{4}-2 x^{2}-3}}\)	\(84\)
elliptic	\(\frac {3 \sqrt {1-\left (-\frac {1}{3}-\frac {\sqrt {7}}{3}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{3}+\frac {\sqrt {7}}{3}\right ) x^{2}}\, \EllipticF \left (\frac {\sqrt {-3-3 \sqrt {7}}\, x}{3}, \frac {i \sqrt {42}}{6}-\frac {i \sqrt {6}}{6}\right )}{\sqrt {-3-3 \sqrt {7}}\, \sqrt {2 x^{4}-2 x^{2}-3}}\)	\(84\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/(-3-3*7^(1/2))^(1/2)*(1-(-1/3-1/3*7^(1/2))*x^2)^(1/2)*(1-(-1/3+1/3*7^(1/2))*x^2)^(1/2)/(2*x^4-2*x^2-3)^(1/2)
*EllipticF(1/3*(-3-3*7^(1/2))^(1/2)*x,1/6*I*42^(1/2)-1/6*I*6^(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^4-2*x^2-3)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/sqrt(2*x^4 - 2*x^2 - 3), 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/(2*x^4-2*x^2-3)^(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 {2 x^{4} - 2 x^{2} - 3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate(1/sqrt(2*x^4 - 2*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^4-2\,x^2-3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(2*x^4 - 2*x^2 - 3)^(1/2),x)

[Out]

int(1/(2*x^4 - 2*x^2 - 3)^(1/2), x)

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