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

Optimal. Leaf size=36 \[ \frac{2 x \sqrt{\frac{\left (1-x^2\right )^2}{x \left (x^2+1\right )}}}{1-x^2} \]

[Out]

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

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Rubi [A]  time = 0.223266, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{2 x \sqrt{\frac{\left (1-x^2\right )^2}{x \left (x^2+1\right )}}}{1-x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ i \int \frac{\sqrt{\frac{\left (x^{2} - 1\right )^{2}}{x \left (x^{2} + 1\right )}}}{- 2 x + 2 i}\, dx + \frac{i \int \frac{\sqrt{\frac{\left (x^{2} - 1\right )^{2}}{x \left (x^{2} + 1\right )}}}{x + i}\, dx}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0270965, size = 29, normalized size = 0.81 \[ -\frac{2 x \sqrt{\frac{\left (x^2-1\right )^2}{x^3+x}}}{x^2-1} \]

Antiderivative was successfully verified.

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

[Out]

(-2*x*Sqrt[(-1 + x^2)^2/(x + x^3)])/(-1 + x^2)

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Maple [A]  time = 0.007, size = 34, normalized size = 0.9 \[ -2\,{\frac{x}{ \left ( -1+x \right ) \left ( 1+x \right ) }\sqrt{{\frac{ \left ({x}^{2}-1 \right ) ^{2}}{x \left ({x}^{2}+1 \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\frac{{\left (x^{2} - 1\right )}^{2}}{{\left (x^{2} + 1\right )} x}}}{x^{2} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.272272, size = 41, normalized size = 1.14 \[ -\frac{2 \, x \sqrt{\frac{x^{4} - 2 \, x^{2} + 1}{x^{3} + x}}}{x^{2} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\frac{\left (x - 1\right )^{2} \left (x + 1\right )^{2}}{x^{3} + x}}}{x^{2} + 1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt((x - 1)**2*(x + 1)**2/(x**3 + x))/(x**2 + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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