3.881 \(\int \frac{1}{x^3 \sqrt{1-x^4}} \, dx\)

Optimal. Leaf size=18 \[ -\frac{\sqrt{1-x^4}}{2 x^2} \]

[Out]

-Sqrt[1 - x^4]/(2*x^2)

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Rubi [A]  time = 0.003119, antiderivative size = 18, 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 = {264} \[ -\frac{\sqrt{1-x^4}}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*Sqrt[1 - x^4]),x]

[Out]

-Sqrt[1 - x^4]/(2*x^2)

Rule 264

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

Rubi steps

\begin{align*} \int \frac{1}{x^3 \sqrt{1-x^4}} \, dx &=-\frac{\sqrt{1-x^4}}{2 x^2}\\ \end{align*}

Mathematica [A]  time = 0.0024418, size = 18, normalized size = 1. \[ -\frac{\sqrt{1-x^4}}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*Sqrt[1 - x^4]),x]

[Out]

-Sqrt[1 - x^4]/(2*x^2)

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Maple [A]  time = 0.002, size = 26, normalized size = 1.4 \begin{align*}{\frac{ \left ( -1+x \right ) \left ( 1+x \right ) \left ({x}^{2}+1 \right ) }{2\,{x}^{2}}{\frac{1}{\sqrt{-{x}^{4}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/(-x^4+1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.0105, size = 19, normalized size = 1.06 \begin{align*} -\frac{\sqrt{-x^{4} + 1}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(-x^4 + 1)/x^2

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Fricas [A]  time = 1.58737, size = 34, normalized size = 1.89 \begin{align*} -\frac{\sqrt{-x^{4} + 1}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(-x^4 + 1)/x^2

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Sympy [A]  time = 0.785154, size = 34, normalized size = 1.89 \begin{align*} \begin{cases} - \frac{i \sqrt{x^{4} - 1}}{2 x^{2}} & \text{for}\: \left |{x^{4}}\right | > 1 \\- \frac{\sqrt{1 - x^{4}}}{2 x^{2}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-I*sqrt(x**4 - 1)/(2*x**2), Abs(x**4) > 1), (-sqrt(1 - x**4)/(2*x**2), True))

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Giac [A]  time = 1.15157, size = 12, normalized size = 0.67 \begin{align*} -\frac{1}{2} \, \sqrt{\frac{1}{x^{4}} - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(1/x^4 - 1)