3.10.65 \(\int \frac {1}{x^3 \sqrt {16-x^4}} \, dx\) [965]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, 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 = {270} \begin {gather*} -\frac {\sqrt {16-x^4}}{32 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^3*Sqrt[16 - x^4]),x]

[Out]

-1/32*Sqrt[16 - x^4]/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]

Rubi steps

\begin {align*} \int \frac {1}{x^3 \sqrt {16-x^4}} \, dx &=-\frac {\sqrt {16-x^4}}{32 x^2}\\ \end {align*}

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Mathematica [A]
time = 0.08, size = 18, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {16-x^4}}{32 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*Sqrt[16 - x^4]),x]

[Out]

-1/32*Sqrt[16 - x^4]/x^2

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Maple [A]
time = 0.15, size = 25, normalized size = 1.39

method	result	size
trager	\(-\frac {\sqrt {-x^{4}+16}}{32 x^{2}}\)	\(15\)
meijerg	\(-\frac {\sqrt {1-\frac {x^{4}}{16}}}{8 x^{2}}\)	\(15\)
risch	\(\frac {x^{4}-16}{32 x^{2} \sqrt {-x^{4}+16}}\)	\(20\)
default	\(\frac {\left (x^{2}-4\right ) \left (x^{2}+4\right )}{32 x^{2} \sqrt {-x^{4}+16}}\)	\(25\)
elliptic	\(\frac {\left (x^{2}-4\right ) \left (x^{2}+4\right )}{32 x^{2} \sqrt {-x^{4}+16}}\)	\(25\)
gosper	\(\frac {\left (x -2\right ) \left (2+x \right ) \left (x^{2}+4\right )}{32 x^{2} \sqrt {-x^{4}+16}}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 14, normalized size = 0.78 \begin {gather*} -\frac {\sqrt {-x^{4} + 16}}{32 \, x^{2}} \end {gather*}

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 = 0.35, size = 14, normalized size = 0.78 \begin {gather*} -\frac {\sqrt {-x^{4} + 16}}{32 \, x^{2}} \end {gather*}

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 [C] Result contains complex when optimal does not.
time = 0.32, size = 34, normalized size = 1.89 \begin {gather*} \begin {cases} - \frac {i \sqrt {x^{4} - 16}}{32 x^{2}} & \text {for}\: \left |{x^{4}}\right | > 16 \\- \frac {\sqrt {16 - x^{4}}}{32 x^{2}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-I*sqrt(x**4 - 16)/(32*x**2), Abs(x**4) > 16), (-sqrt(16 - x**4)/(32*x**2), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (14) = 28\).
time = 0.85, size = 35, normalized size = 1.94 \begin {gather*} \frac {x^{2}}{64 \, {\left (\sqrt {-x^{4} + 16} - 4\right )}} - \frac {\sqrt {-x^{4} + 16} - 4}{64 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*x^2/(sqrt(-x^4 + 16) - 4) - 1/64*(sqrt(-x^4 + 16) - 4)/x^2

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Mupad [B]
time = 1.14, size = 14, normalized size = 0.78 \begin {gather*} -\frac {\sqrt {16-x^4}}{32\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*(16 - x^4)^(1/2)),x)

[Out]

-(16 - x^4)^(1/2)/(32*x^2)

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