3.10.67 \(\int \frac {1}{x^7 \sqrt {16-x^4}} \, dx\) [967]

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

[Out]

-1/96*(-x^4+16)^(1/2)/x^6-1/768*(-x^4+16)^(1/2)/x^2

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Rubi [A]
time = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, 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 = {277, 270} \begin {gather*} -\frac {\sqrt {16-x^4}}{96 x^6}-\frac {\sqrt {16-x^4}}{768 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^7*Sqrt[16 - x^4]),x]

[Out]

-1/96*Sqrt[16 - x^4]/x^6 - Sqrt[16 - x^4]/(768*x^2)

Rule 270

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]

Rule 277

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]

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*}

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Mathematica [A]
time = 0.09, size = 25, normalized size = 0.68 \begin {gather*} \frac {\left (-8-x^4\right ) \sqrt {16-x^4}}{768 x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^7*Sqrt[16 - x^4]),x]

[Out]

((-8 - x^4)*Sqrt[16 - x^4])/(768*x^6)

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Maple [A]
time = 0.16, size = 30, normalized size = 0.81

method	result	size
trager	\(-\frac {\left (x^{4}+8\right ) \sqrt {-x^{4}+16}}{768 x^{6}}\)	\(20\)
meijerg	\(-\frac {\left (1+\frac {x^{4}}{8}\right ) \sqrt {1-\frac {x^{4}}{16}}}{24 x^{6}}\)	\(22\)
risch	\(\frac {x^{8}-8 x^{4}-128}{768 x^{6} \sqrt {-x^{4}+16}}\)	\(25\)
default	\(\frac {\left (x^{2}-4\right ) \left (x^{2}+4\right ) \left (x^{4}+8\right )}{768 x^{6} \sqrt {-x^{4}+16}}\)	\(30\)
elliptic	\(\frac {\left (x^{2}-4\right ) \left (x^{2}+4\right ) \left (x^{4}+8\right )}{768 x^{6} \sqrt {-x^{4}+16}}\)	\(30\)
gosper	\(\frac {\left (x -2\right ) \left (2+x \right ) \left (x^{2}+4\right ) \left (x^{4}+8\right )}{768 x^{6} \sqrt {-x^{4}+16}}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 29, normalized size = 0.78 \begin {gather*} -\frac {\sqrt {-x^{4} + 16}}{512 \, x^{2}} - \frac {{\left (-x^{4} + 16\right )}^{\frac {3}{2}}}{1536 \, x^{6}} \end {gather*}

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 = 0.35, size = 19, normalized size = 0.51 \begin {gather*} -\frac {{\left (x^{4} + 8\right )} \sqrt {-x^{4} + 16}}{768 \, x^{6}} \end {gather*}

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 [C] Result contains complex when optimal does not.
time = 0.45, size = 66, normalized size = 1.78 \begin {gather*} \begin {cases} - \frac {\sqrt {-1 + \frac {16}{x^{4}}}}{768} - \frac {\sqrt {-1 + \frac {16}{x^{4}}}}{96 x^{4}} & \text {for}\: \frac {1}{\left |{x^{4}}\right |} > \frac {1}{16} \\- \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 {gather*}

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (29) = 58\).
time = 1.13, size = 73, normalized size = 1.97 \begin {gather*} \frac {x^{6} {\left (\frac {9 \, {\left (\sqrt {-x^{4} + 16} - 4\right )}^{2}}{x^{4}} + 1\right )}}{12288 \, {\left (\sqrt {-x^{4} + 16} - 4\right )}^{3}} - \frac {3 \, {\left (\sqrt {-x^{4} + 16} - 4\right )}}{4096 \, x^{2}} - \frac {{\left (\sqrt {-x^{4} + 16} - 4\right )}^{3}}{12288 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12288*x^6*(9*(sqrt(-x^4 + 16) - 4)^2/x^4 + 1)/(sqrt(-x^4 + 16) - 4)^3 - 3/4096*(sqrt(-x^4 + 16) - 4)/x^2 - 1
/12288*(sqrt(-x^4 + 16) - 4)^3/x^6

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Mupad [B]
time = 1.14, size = 19, normalized size = 0.51 \begin {gather*} -\frac {\left (x^4+8\right )\,\sqrt {16-x^4}}{768\,x^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^7*(16 - x^4)^(1/2)),x)

[Out]

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

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