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

Optimal. Leaf size=12 \[ -\frac{x}{\sqrt{x^4+1}} \]

[Out]

-(x/Sqrt[1 + x^4])

_______________________________________________________________________________________

Rubi [A]  time = 0.00872242, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ -\frac{x}{\sqrt{x^4+1}} \]

Antiderivative was successfully verified.

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

[Out]

-(x/Sqrt[1 + x^4])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 2.94961, size = 10, normalized size = 0.83 \[ - \frac{x}{\sqrt{x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x/sqrt(x**4 + 1)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0143957, size = 12, normalized size = 1. \[ -\frac{x}{\sqrt{x^4+1}} \]

Antiderivative was successfully verified.

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

[Out]

-(x/Sqrt[1 + x^4])

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 11, normalized size = 0.9 \[ -{x{\frac{1}{\sqrt{{x}^{4}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^4-1)/(x^4+1)^(3/2),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 5.83016, size = 14, normalized size = 1.17 \[ -\frac{x}{\sqrt{x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x/sqrt(x^4 + 1)

_______________________________________________________________________________________

Fricas [A]  time = 0.215414, size = 14, normalized size = 1.17 \[ -\frac{x}{\sqrt{x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x/sqrt(x^4 + 1)

_______________________________________________________________________________________

Sympy [A]  time = 3.7857, size = 58, normalized size = 4.83 \[ \frac{x^{5} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{5}{4}, \frac{3}{2} \\ \frac{9}{4} \end{matrix}\middle |{x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac{9}{4}\right )} - \frac{x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{3}{2} \\ \frac{5}{4} \end{matrix}\middle |{x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac{5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**5*gamma(5/4)*hyper((5/4, 3/2), (9/4,), x**4*exp_polar(I*pi))/(4*gamma(9/4)) -
x*gamma(1/4)*hyper((1/4, 3/2), (5/4,), x**4*exp_polar(I*pi))/(4*gamma(5/4))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.210607, size = 14, normalized size = 1.17 \[ -\frac{x}{\sqrt{x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x/sqrt(x^4 + 1)