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

Optimal. Leaf size=14 \[ -\frac{2 x}{\sqrt{x \left (x^2+1\right )}} \]

[Out]

(-2*x)/Sqrt[x*(1 + x^2)]

_______________________________________________________________________________________

Rubi [A]  time = 0.246295, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{2 x}{\sqrt{x \left (x^2+1\right )}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*x)/Sqrt[x*(1 + x^2)]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 50.0794, size = 235, normalized size = 16.79 \[ - \frac{\sqrt{2} \sqrt{x} \left (1 + i\right ) \left (x^{2} + 1\right ) \sqrt{- i x + 1} E\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x} \left (1 + i\right )}{2} \right )}\middle | -1\right )}{2 \left (x + i\right ) \sqrt{x^{3} + x} \sqrt{i x + 1}} - \frac{\sqrt{2} \sqrt{x} \left (1 - i\right ) \left (x^{2} + 1\right ) \sqrt{i x + 1} E\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x} \left (-1 + i\right )}{2} \right )}\middle | -1\right )}{4 \left (- \frac{x}{2} + \frac{i}{2}\right ) \sqrt{x^{3} + x} \sqrt{- i x + 1}} - \frac{i x \left (x^{2} + 1\right )}{\left (x + i\right ) \sqrt{x^{3} + x}} - \frac{2 i x \left (x^{2} + 1\right )}{\left (- 2 x + 2 i\right ) \sqrt{x^{3} + x}} + \frac{\sqrt{\frac{x^{2} + 1}{\left (x + 1\right )^{2}}} \left (x + 1\right ) \sqrt{x^{3} + x} F\left (2 \operatorname{atan}{\left (\sqrt{x} \right )}\middle | \frac{1}{2}\right )}{\sqrt{x} \left (x^{2} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(2)*sqrt(x)*(1 + I)*(x**2 + 1)*sqrt(-I*x + 1)*elliptic_e(asin(sqrt(2)*sqrt(
x)*(1 + I)/2), -1)/(2*(x + I)*sqrt(x**3 + x)*sqrt(I*x + 1)) - sqrt(2)*sqrt(x)*(1
- I)*(x**2 + 1)*sqrt(I*x + 1)*elliptic_e(asin(sqrt(2)*sqrt(x)*(-1 + I)/2), -1)/
(4*(-x/2 + I/2)*sqrt(x**3 + x)*sqrt(-I*x + 1)) - I*x*(x**2 + 1)/((x + I)*sqrt(x*
*3 + x)) - 2*I*x*(x**2 + 1)/((-2*x + 2*I)*sqrt(x**3 + x)) + sqrt((x**2 + 1)/(x +
1)**2)*(x + 1)*sqrt(x**3 + x)*elliptic_f(2*atan(sqrt(x)), 1/2)/(sqrt(x)*(x**2 +
1))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0196121, size = 12, normalized size = 0.86 \[ -\frac{2 x}{\sqrt{x^3+x}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*x)/Sqrt[x + x^3]

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 13, normalized size = 0.9 \[ -2\,{\frac{x}{\sqrt{x \left ({x}^{2}+1 \right ) }}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.273522, size = 22, normalized size = 1.57 \[ -\frac{2 \, \sqrt{x^{3} + x}}{x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(x^3 + x)/(x^2 + 1)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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