Optimal. Leaf size=33 \[ \frac{2 x \sqrt{\frac{\left (1-x^2\right )^2}{x^3+x}}}{1-x^2} \]
[Out]
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Rubi [A] time = 0.303391, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{2 x \sqrt{\frac{\left (1-x^2\right )^2}{x^3+x}}}{1-x^2} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[(-1 + x^2)^2/(x + x^3)]/(1 + x^2),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ i \int \frac{\sqrt{\frac{\left (x^{2} - 1\right )^{2}}{x^{3} + x}}}{- 2 x + 2 i}\, dx + \frac{i \int \frac{\sqrt{\frac{\left (x^{2} - 1\right )^{2}}{x^{3} + x}}}{x + i}\, dx}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(((x**2-1)**2/(x**3+x))**(1/2)/(x**2+1),x)
[Out]
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Mathematica [A] time = 0.0123657, size = 29, normalized size = 0.88 \[ -\frac{2 x \sqrt{\frac{\left (x^2-1\right )^2}{x^3+x}}}{x^2-1} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[(-1 + x^2)^2/(x + x^3)]/(1 + x^2),x]
[Out]
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Maple [A] time = 0.007, size = 34, normalized size = 1. \[ -2\,{\frac{x}{ \left ( -1+x \right ) \left ( 1+x \right ) }\sqrt{{\frac{ \left ({x}^{2}-1 \right ) ^{2}}{x \left ({x}^{2}+1 \right ) }}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(((x^2-1)^2/(x^3+x))^(1/2)/(x^2+1),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\frac{{\left (x^{2} - 1\right )}^{2}}{x^{3} + x}}}{x^{2} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((x^2 - 1)^2/(x^3 + x))/(x^2 + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.277066, size = 41, normalized size = 1.24 \[ -\frac{2 \, x \sqrt{\frac{x^{4} - 2 \, x^{2} + 1}{x^{3} + x}}}{x^{2} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((x^2 - 1)^2/(x^3 + x))/(x^2 + 1),x, algorithm="fricas")
[Out]
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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**3+x))**(1/2)/(x**2+1),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\frac{{\left (x^{2} - 1\right )}^{2}}{x^{3} + x}}}{x^{2} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((x^2 - 1)^2/(x^3 + x))/(x^2 + 1),x, algorithm="giac")
[Out]