3.43 \(\int \frac{\left (-1+x^2\right )^3}{\left (1+x^2\right )^4} \, dx\)

Optimal. Leaf size=34 \[ -\frac{x \left (1-x^2\right )^2}{3 \left (x^2+1\right )^3}-\frac{2 x}{3 \left (x^2+1\right )} \]

[Out]

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

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Rubi [A]  time = 0.0259221, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{x \left (1-x^2\right )^2}{3 \left (x^2+1\right )^3}-\frac{2 x}{3 \left (x^2+1\right )} \]

Antiderivative was successfully verified.

[In]  Int[(-1 + x^2)^3/(1 + x^2)^4,x]

[Out]

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

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Rubi in Sympy [A]  time = 8.89048, size = 27, normalized size = 0.79 \[ - \frac{x \left (- x^{2} + 1\right )^{2}}{3 \left (x^{2} + 1\right )^{3}} - \frac{2 x}{3 \left (x^{2} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**2-1)**3/(x**2+1)**4,x)

[Out]

-x*(-x**2 + 1)**2/(3*(x**2 + 1)**3) - 2*x/(3*(x**2 + 1))

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Mathematica [A]  time = 0.0118829, size = 24, normalized size = 0.71 \[ -\frac{x \left (3 x^4+2 x^2+3\right )}{3 \left (x^2+1\right )^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(-1 + x^2)^3/(1 + x^2)^4,x]

[Out]

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

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Maple [A]  time = 0.01, size = 23, normalized size = 0.7 \[{\frac{1}{ \left ({x}^{2}+1 \right ) ^{3}} \left ( -{x}^{5}-{\frac{2\,{x}^{3}}{3}}-x \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^2-1)^3/(x^2+1)^4,x)

[Out]

(-x^5-2/3*x^3-x)/(x^2+1)^3

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Maxima [A]  time = 1.3427, size = 45, normalized size = 1.32 \[ -\frac{3 \, x^{5} + 2 \, x^{3} + 3 \, x}{3 \,{\left (x^{6} + 3 \, x^{4} + 3 \, x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 - 1)^3/(x^2 + 1)^4,x, algorithm="maxima")

[Out]

-1/3*(3*x^5 + 2*x^3 + 3*x)/(x^6 + 3*x^4 + 3*x^2 + 1)

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Fricas [A]  time = 0.194033, size = 45, normalized size = 1.32 \[ -\frac{3 \, x^{5} + 2 \, x^{3} + 3 \, x}{3 \,{\left (x^{6} + 3 \, x^{4} + 3 \, x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 - 1)^3/(x^2 + 1)^4,x, algorithm="fricas")

[Out]

-1/3*(3*x^5 + 2*x^3 + 3*x)/(x^6 + 3*x^4 + 3*x^2 + 1)

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Sympy [A]  time = 0.319225, size = 31, normalized size = 0.91 \[ - \frac{3 x^{5} + 2 x^{3} + 3 x}{3 x^{6} + 9 x^{4} + 9 x^{2} + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**2-1)**3/(x**2+1)**4,x)

[Out]

-(3*x**5 + 2*x**3 + 3*x)/(3*x**6 + 9*x**4 + 9*x**2 + 3)

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GIAC/XCAS [A]  time = 0.234433, size = 27, normalized size = 0.79 \[ -\frac{3 \,{\left (x + \frac{1}{x}\right )}^{2} - 4}{3 \,{\left (x + \frac{1}{x}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 - 1)^3/(x^2 + 1)^4,x, algorithm="giac")

[Out]

-1/3*(3*(x + 1/x)^2 - 4)/(x + 1/x)^3