3.965 \(\int \frac{1}{x^3 \sqrt{16-x^4}} \, dx\)

Optimal. Leaf size=18 \[ -\frac{\sqrt{16-x^4}}{32 x^2} \]

[Out]

-Sqrt[16 - x^4]/(32*x^2)

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Rubi [A]  time = 0.0032332, 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} \[ -\frac{\sqrt{16-x^4}}{32 x^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*Sqrt[16 - x^4]),x]

[Out]

-Sqrt[16 - x^4]/(32*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{16-x^4}} \, dx &=-\frac{\sqrt{16-x^4}}{32 x^2}\\ \end{align*}

Mathematica [A]  time = 0.0030018, size = 18, normalized size = 1. \[ -\frac{\sqrt{16-x^4}}{32 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*Sqrt[16 - x^4]),x]

[Out]

-Sqrt[16 - x^4]/(32*x^2)

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Maple [A]  time = 0.003, size = 26, normalized size = 1.4 \begin{align*}{\frac{ \left ( -2+x \right ) \left ( 2+x \right ) \left ({x}^{2}+4 \right ) }{32\,{x}^{2}}{\frac{1}{\sqrt{-{x}^{4}+16}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/(-x^4+16)^(1/2),x)

[Out]

1/32/x^2*(-2+x)*(2+x)*(x^2+4)/(-x^4+16)^(1/2)

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Maxima [A]  time = 0.98445, size = 19, normalized size = 1.06 \begin{align*} -\frac{\sqrt{-x^{4} + 16}}{32 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(-x^4+16)^(1/2),x, algorithm="maxima")

[Out]

-1/32*sqrt(-x^4 + 16)/x^2

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Fricas [A]  time = 1.51083, size = 36, normalized size = 2. \begin{align*} -\frac{\sqrt{-x^{4} + 16}}{32 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(-x^4+16)^(1/2),x, algorithm="fricas")

[Out]

-1/32*sqrt(-x^4 + 16)/x^2

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Sympy [A]  time = 0.768701, size = 32, normalized size = 1.78 \begin{align*} \begin{cases} - \frac{\sqrt{-1 + \frac{16}{x^{4}}}}{32} & \text{for}\: \frac{16}{\left |{x^{4}}\right |} > 1 \\- \frac{i \sqrt{1 - \frac{16}{x^{4}}}}{32} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**3/(-x**4+16)**(1/2),x)

[Out]

Piecewise((-sqrt(-1 + 16/x**4)/32, 16/Abs(x**4) > 1), (-I*sqrt(1 - 16/x**4)/32, True))

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Giac [A]  time = 1.16789, size = 15, normalized size = 0.83 \begin{align*} -\frac{1}{32} \, \sqrt{\frac{16}{x^{4}} - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(-x^4+16)^(1/2),x, algorithm="giac")

[Out]

-1/32*sqrt(16/x^4 - 1)