3.967 \(\int \frac{1}{x^7 \sqrt{16-x^4}} \, dx\)

Optimal. Leaf size=37 \[ -\frac{\sqrt{16-x^4}}{768 x^2}-\frac{\sqrt{16-x^4}}{96 x^6} \]

[Out]

-Sqrt[16 - x^4]/(96*x^6) - Sqrt[16 - x^4]/(768*x^2)

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Rubi [A]  time = 0.0076976, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {271, 264} \[ -\frac{\sqrt{16-x^4}}{768 x^2}-\frac{\sqrt{16-x^4}}{96 x^6} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^7*Sqrt[16 - x^4]),x]

[Out]

-Sqrt[16 - x^4]/(96*x^6) - Sqrt[16 - x^4]/(768*x^2)

Rule 271

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

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^7 \sqrt{16-x^4}} \, dx &=-\frac{\sqrt{16-x^4}}{96 x^6}+\frac{1}{24} \int \frac{1}{x^3 \sqrt{16-x^4}} \, dx\\ &=-\frac{\sqrt{16-x^4}}{96 x^6}-\frac{\sqrt{16-x^4}}{768 x^2}\\ \end{align*}

Mathematica [A]  time = 0.0053943, size = 25, normalized size = 0.68 \[ -\frac{\sqrt{1-\frac{x^4}{16}} \left (x^4+8\right )}{192 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^7*Sqrt[16 - x^4]),x]

[Out]

-(Sqrt[1 - x^4/16]*(8 + x^4))/(192*x^6)

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Maple [A]  time = 0.004, size = 31, normalized size = 0.8 \begin{align*}{\frac{ \left ( -2+x \right ) \left ( 2+x \right ) \left ({x}^{2}+4 \right ) \left ({x}^{4}+8 \right ) }{768\,{x}^{6}}{\frac{1}{\sqrt{-{x}^{4}+16}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^7/(-x^4+16)^(1/2),x)

[Out]

1/768*(-2+x)*(2+x)*(x^2+4)*(x^4+8)/x^6/(-x^4+16)^(1/2)

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Maxima [A]  time = 0.997317, size = 39, normalized size = 1.05 \begin{align*} -\frac{\sqrt{-x^{4} + 16}}{512 \, x^{2}} - \frac{{\left (-x^{4} + 16\right )}^{\frac{3}{2}}}{1536 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(-x^4+16)^(1/2),x, algorithm="maxima")

[Out]

-1/512*sqrt(-x^4 + 16)/x^2 - 1/1536*(-x^4 + 16)^(3/2)/x^6

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Fricas [A]  time = 1.46588, size = 51, normalized size = 1.38 \begin{align*} -\frac{{\left (x^{4} + 8\right )} \sqrt{-x^{4} + 16}}{768 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(-x^4+16)^(1/2),x, algorithm="fricas")

[Out]

-1/768*(x^4 + 8)*sqrt(-x^4 + 16)/x^6

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Sympy [A]  time = 1.25166, size = 65, normalized size = 1.76 \begin{align*} \begin{cases} - \frac{\sqrt{-1 + \frac{16}{x^{4}}}}{768} - \frac{\sqrt{-1 + \frac{16}{x^{4}}}}{96 x^{4}} & \text{for}\: \frac{16}{\left |{x^{4}}\right |} > 1 \\- \frac{i \sqrt{1 - \frac{16}{x^{4}}}}{768} - \frac{i \sqrt{1 - \frac{16}{x^{4}}}}{96 x^{4}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**7/(-x**4+16)**(1/2),x)

[Out]

Piecewise((-sqrt(-1 + 16/x**4)/768 - sqrt(-1 + 16/x**4)/(96*x**4), 16/Abs(x**4) > 1), (-I*sqrt(1 - 16/x**4)/76
8 - I*sqrt(1 - 16/x**4)/(96*x**4), True))

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Giac [A]  time = 1.63605, size = 31, normalized size = 0.84 \begin{align*} -\frac{1}{1536} \,{\left (\frac{16}{x^{4}} - 1\right )}^{\frac{3}{2}} - \frac{1}{512} \, \sqrt{\frac{16}{x^{4}} - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(-x^4+16)^(1/2),x, algorithm="giac")

[Out]

-1/1536*(16/x^4 - 1)^(3/2) - 1/512*sqrt(16/x^4 - 1)