3.241 \(\int \frac{1}{\sqrt{-1+x^2} \sqrt{2+2 x^2}} \, dx\)

Optimal. Leaf size=25 \[ \frac{1}{2} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{x^2-1}}\right ),\frac{1}{2}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0109653, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {253, 222} \[ \frac{1}{2} F\left (\sin ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{x^2-1}}\right )|\frac{1}{2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 253

Int[((a1_.) + (b1_.)*(x_)^(n_))^(p_)*((a2_.) + (b2_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[((a1 + b1*x^n)^FracPa
rt[p]*(a2 + b2*x^n)^FracPart[p])/(a1*a2 + b1*b2*x^(2*n))^FracPart[p], Int[(a1*a2 + b1*b2*x^(2*n))^p, x], x] /;
FreeQ[{a1, b1, a2, b2, n, p}, x] && EqQ[a2*b1 + a1*b2, 0] &&  !IntegerQ[p]

Rule 222

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

Rubi steps

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

Mathematica [C]  time = 0.0095928, size = 46, normalized size = 1.84 \[ \frac{x \sqrt{1-x^4} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};x^4\right )}{\sqrt{x^2-1} \sqrt{2 x^2+2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Sqrt[1 - x^4]*Hypergeometric2F1[1/4, 1/2, 5/4, x^4])/(Sqrt[-1 + x^2]*Sqrt[2 + 2*x^2])

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Maple [C]  time = 0.019, size = 30, normalized size = 1.2 \begin{align*}{-{\frac{i}{2}}{\it EllipticF} \left ( ix,i \right ) \sqrt{2}\sqrt{-{x}^{2}+1}{\frac{1}{\sqrt{{x}^{2}-1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{2 \, x^{2} + 2} \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)/(2*x^2+2)^(1/2),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C]  time = 8.34769, size = 75, normalized size = 3. \begin{align*} \frac{\sqrt{2} i{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{1}{2}, 1, 1 & \frac{3}{4}, \frac{3}{4}, \frac{5}{4} \\\frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4} & 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{x^{4}}} \right )}}{16 \pi ^{\frac{3}{2}}} - \frac{\sqrt{2} i{G_{6, 6}^{3, 5}\left (\begin{matrix} - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4} & 1 \\0, \frac{1}{2}, 0 & - \frac{1}{4}, \frac{1}{4}, \frac{1}{4} \end{matrix} \middle |{\frac{1}{x^{4}}} \right )}}{16 \pi ^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(2)*I*meijerg(((1/2, 1, 1), (3/4, 3/4, 5/4)), ((1/4, 1/2, 3/4, 1, 5/4), (0,)), exp_polar(2*I*pi)/x**4)/(16
*pi**(3/2)) - sqrt(2)*I*meijerg(((-1/4, 0, 1/4, 1/2, 3/4), (1,)), ((0, 1/2, 0), (-1/4, 1/4, 1/4)), x**(-4))/(1
6*pi**(3/2))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{2 \, x^{2} + 2} \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)/(2*x^2+2)^(1/2),x, algorithm="giac")

[Out]

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