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

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

[Out]

-(x/Sqrt[1 + x^4])

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Rubi [A]  time = 0.0031383, 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, Rules used = {383} \[ -\frac{x}{\sqrt{x^4+1}} \]

Antiderivative was successfully verified.

[In]

Int[(-1 + x^4)/(1 + x^4)^(3/2),x]

[Out]

-(x/Sqrt[1 + x^4])

Rule 383

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*x*(a + b*x^n)^(p + 1))/a, x]
/; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[a*d - b*c*(n*(p + 1) + 1), 0]

Rubi steps

\begin{align*} \int \frac{-1+x^4}{\left (1+x^4\right )^{3/2}} \, dx &=-\frac{x}{\sqrt{1+x^4}}\\ \end{align*}

Mathematica [A]  time = 0.0050631, 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])

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Maple [A]  time = 0.004, size = 11, normalized size = 0.9 \begin{align*} -{x{\frac{1}{\sqrt{{x}^{4}+1}}}} \end{align*}

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)

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Maxima [A]  time = 1.54444, size = 14, normalized size = 1.17 \begin{align*} -\frac{x}{\sqrt{x^{4} + 1}} \end{align*}

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)

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Fricas [A]  time = 1.30916, size = 24, normalized size = 2. \begin{align*} -\frac{x}{\sqrt{x^{4} + 1}} \end{align*}

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)

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Sympy [C]  time = 2.61457, size = 58, normalized size = 4.83 \begin{align*} \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 )} \end{align*}

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))

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Giac [A]  time = 1.06654, size = 14, normalized size = 1.17 \begin{align*} -\frac{x}{\sqrt{x^{4} + 1}} \end{align*}

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)