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

Optimal. Leaf size=8 \[ \frac{\tan ^{-1}(x)}{\sqrt{2}} \]

[Out]

ArcTan[x]/Sqrt[2]

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Rubi [A]  time = 0.00793174, antiderivative size = 8, 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 \[ \frac{\tan ^{-1}(x)}{\sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

ArcTan[x]/Sqrt[2]

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Rubi in Sympy [A]  time = 2.52358, size = 8, normalized size = 1. \[ \frac{\sqrt{2} \operatorname{atan}{\left (x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(2)*atan(x)/2

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Mathematica [A]  time = 0.00708218, size = 8, normalized size = 1. \[ \frac{\tan ^{-1}(x)}{\sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

ArcTan[x]/Sqrt[2]

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Maple [A]  time = 0.031, size = 8, normalized size = 1. \[{\frac{\arctan \left ( x \right ) \sqrt{2}}{2}} \]

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*arctan(x)*2^(1/2)

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Maxima [A]  time = 1.52895, size = 9, normalized size = 1.12 \[ \frac{1}{2} \, \sqrt{2} \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(2)*arctan(x)

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Fricas [A]  time = 0.244895, size = 46, normalized size = 5.75 \[ -\frac{1}{4} \, \sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{2 \, x^{2} + 2} \sqrt{x^{2} + 1} x}{x^{4} - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 10.5482, size = 8, normalized size = 1. \[ \frac{\sqrt{2} \operatorname{atan}{\left (x \right )}}{2} \]

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)*atan(x)/2

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GIAC/XCAS [A]  time = 0.225048, size = 9, normalized size = 1.12 \[ \frac{1}{2} \, \sqrt{2} \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(2)*arctan(x)