3.226 \(\int \frac{1}{\sqrt{1+x^2} \sqrt{2+4 x^2}} \, dx\)

Optimal. Leaf size=49 \[ \frac{\sqrt{2 x^2+1} F\left (\left .\tan ^{-1}(x)\right |-1\right )}{\sqrt{2} \sqrt{x^2+1} \sqrt{\frac{2 x^2+1}{x^2+1}}} \]

[Out]

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

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Rubi [A]  time = 0.0333253, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ \frac{\sqrt{2 x^2+1} F\left (\left .\tan ^{-1}(x)\right |-1\right )}{\sqrt{2} \sqrt{x^2+1} \sqrt{\frac{2 x^2+1}{x^2+1}}} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[1 + x^2]*Sqrt[2 + 4*x^2]),x]

[Out]

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

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Rubi in Sympy [A]  time = 5.19043, size = 46, normalized size = 0.94 \[ \frac{\sqrt{2} \sqrt{4 x^{2} + 2} F\left (\operatorname{atan}{\left (x \right )}\middle | -1\right )}{2 \sqrt{\frac{4 x^{2} + 2}{x^{2} + 1}} \sqrt{x^{2} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(x**2+1)**(1/2)/(4*x**2+2)**(1/2),x)

[Out]

sqrt(2)*sqrt(4*x**2 + 2)*elliptic_f(atan(x), -1)/(2*sqrt((4*x**2 + 2)/(x**2 + 1)
)*sqrt(x**2 + 1))

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Mathematica [C]  time = 0.0415428, size = 17, normalized size = 0.35 \[ -\frac{i F\left (\left .i \sinh ^{-1}(x)\right |2\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[1 + x^2]*Sqrt[2 + 4*x^2]),x]

[Out]

((-I)*EllipticF[I*ArcSinh[x], 2])/Sqrt[2]

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Maple [A]  time = 0.057, size = 15, normalized size = 0.3 \[ -{\frac{i}{2}}{\it EllipticF} \left ( ix,\sqrt{2} \right ) \sqrt{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^2+1)^(1/2)/(4*x^2+2)^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{4 \, x^{2} + 2} \sqrt{x^{2} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(4*x^2 + 2)*sqrt(x^2 + 1)),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(4*x^2 + 2)*sqrt(x^2 + 1)),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\sqrt{2} \int \frac{1}{\sqrt{x^{2} + 1} \sqrt{2 x^{2} + 1}}\, dx}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x**2+1)**(1/2)/(4*x**2+2)**(1/2),x)

[Out]

sqrt(2)*Integral(1/(sqrt(x**2 + 1)*sqrt(2*x**2 + 1)), x)/2

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{4 \, x^{2} + 2} \sqrt{x^{2} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(4*x^2 + 2)*sqrt(x^2 + 1)),x, algorithm="giac")

[Out]

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