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

Optimal. Leaf size=68 \[ \frac{\sqrt{-\frac{x^2}{-(a+1) x^2-a+1}} \sqrt{(a+1) x^2+a-1} \tan ^{-1}\left (\frac{\sqrt{(a+1) x^2+a-1}}{\sqrt{2}}\right )}{\sqrt{2} x} \]

[Out]

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

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Rubi [A]  time = 0.335117, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{\sqrt{-\frac{x^2}{-(a+1) x^2-a+1}} \sqrt{(a+1) x^2+a-1} \tan ^{-1}\left (\frac{\sqrt{(a+1) x^2+a-1}}{\sqrt{2}}\right )}{\sqrt{2} x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 17.7705, size = 60, normalized size = 0.88 \[ \frac{\sqrt{2} \sqrt{\frac{x^{2}}{a + x^{2} \left (a + 1\right ) - 1}} \sqrt{a + x^{2} \left (a + 1\right ) - 1} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{a + x^{2} \left (a + 1\right ) - 1}}{2} \right )}}{2 x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(2)*sqrt(x**2/(a + x**2*(a + 1) - 1))*sqrt(a + x**2*(a + 1) - 1)*atan(sqrt(2
)*sqrt(a + x**2*(a + 1) - 1)/2)/(2*x)

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Mathematica [A]  time = 0.0714029, size = 67, normalized size = 0.99 \[ \frac{\sqrt{a x^2+a+x^2-1} \sqrt{\frac{x^2}{(a+1) x^2+a-1}} \tan ^{-1}\left (\frac{\sqrt{a \left (x^2+1\right )+x^2-1}}{\sqrt{2}}\right )}{\sqrt{2} x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.04, size = 60, normalized size = 0.9 \[{\frac{\sqrt{2}}{2\,x}\sqrt{{\frac{{x}^{2}}{a{x}^{2}+{x}^{2}+a-1}}}\sqrt{a{x}^{2}+{x}^{2}+a-1}\arctan \left ({\frac{\sqrt{2}}{2}\sqrt{a{x}^{2}+{x}^{2}+a-1}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(x^2/(a*x^2+x^2+a-1))^(1/2)/x*(a*x^2+x^2+a-1)^(1/2)*2^(1/2)*arctan(1/2*(a*x^
2+x^2+a-1)^(1/2)*2^(1/2))

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Maxima [A]  time = 0.817198, size = 32, normalized size = 0.47 \[ \frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} \sqrt{a x^{2} + x^{2} + a - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.280264, size = 57, normalized size = 0.84 \[ \frac{1}{4} \, \sqrt{2} \arctan \left (\frac{\sqrt{2}{\left ({\left (a + 1\right )} x^{2} + a - 3\right )} \sqrt{\frac{x^{2}}{{\left (a + 1\right )} x^{2} + a - 1}}}{4 \, x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*sqrt(2)*arctan(1/4*sqrt(2)*((a + 1)*x^2 + a - 3)*sqrt(x^2/((a + 1)*x^2 + a -
1))/x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.275341, size = 82, normalized size = 1.21 \[ \frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} \sqrt{a x^{2} + x^{2} + a - 1}\right ){\rm sign}\left (a x^{2} + x^{2} + a - 1\right ){\rm sign}\left (x\right ) - \frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} \sqrt{a - 1}\right ){\rm sign}\left (a - 1\right ){\rm sign}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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