3.132 \(\int \frac{1}{x^3 \sqrt{-16+x^2}} \, dx\)

Optimal. Leaf size=35 \[ \frac{\sqrt{x^2-16}}{32 x^2}+\frac{1}{128} \tan ^{-1}\left (\frac{\sqrt{x^2-16}}{4}\right ) \]

[Out]

Sqrt[-16 + x^2]/(32*x^2) + ArcTan[Sqrt[-16 + x^2]/4]/128

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Rubi [A]  time = 0.0339582, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{\sqrt{x^2-16}}{32 x^2}+\frac{1}{128} \tan ^{-1}\left (\frac{\sqrt{x^2-16}}{4}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*Sqrt[-16 + x^2]),x]

[Out]

Sqrt[-16 + x^2]/(32*x^2) + ArcTan[Sqrt[-16 + x^2]/4]/128

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Rubi in Sympy [A]  time = 2.16624, size = 26, normalized size = 0.74 \[ \frac{\operatorname{atan}{\left (\frac{\sqrt{x^{2} - 16}}{4} \right )}}{128} + \frac{\sqrt{x^{2} - 16}}{32 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**3/(x**2-16)**(1/2),x)

[Out]

atan(sqrt(x**2 - 16)/4)/128 + sqrt(x**2 - 16)/(32*x**2)

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Mathematica [A]  time = 0.0206181, size = 33, normalized size = 0.94 \[ \frac{\sqrt{x^2-16}}{32 x^2}-\frac{1}{128} \tan ^{-1}\left (\frac{4}{\sqrt{x^2-16}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^3*Sqrt[-16 + x^2]),x]

[Out]

Sqrt[-16 + x^2]/(32*x^2) - ArcTan[4/Sqrt[-16 + x^2]]/128

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Maple [A]  time = 0.01, size = 26, normalized size = 0.7 \[{\frac{1}{32\,{x}^{2}}\sqrt{{x}^{2}-16}}-{\frac{1}{128}\arctan \left ( 4\,{\frac{1}{\sqrt{{x}^{2}-16}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(x^2-16)^(1/2),x)

[Out]

1/32*(x^2-16)^(1/2)/x^2-1/128*arctan(4/(x^2-16)^(1/2))

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Maxima [A]  time = 1.50128, size = 30, normalized size = 0.86 \[ \frac{\sqrt{x^{2} - 16}}{32 \, x^{2}} - \frac{1}{128} \, \arcsin \left (\frac{4}{{\left | x \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^2 - 16)*x^3),x, algorithm="maxima")

[Out]

1/32*sqrt(x^2 - 16)/x^2 - 1/128*arcsin(4/abs(x))

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Fricas [A]  time = 0.20376, size = 115, normalized size = 3.29 \[ -\frac{2 \, x^{3} -{\left (x^{4} - \sqrt{x^{2} - 16} x^{3} - 8 \, x^{2}\right )} \arctan \left (-\frac{1}{4} \, x + \frac{1}{4} \, \sqrt{x^{2} - 16}\right ) - 2 \,{\left (x^{2} - 8\right )} \sqrt{x^{2} - 16} - 32 \, x}{64 \,{\left (x^{4} - \sqrt{x^{2} - 16} x^{3} - 8 \, x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^2 - 16)*x^3),x, algorithm="fricas")

[Out]

-1/64*(2*x^3 - (x^4 - sqrt(x^2 - 16)*x^3 - 8*x^2)*arctan(-1/4*x + 1/4*sqrt(x^2 -
16)) - 2*(x^2 - 8)*sqrt(x^2 - 16) - 32*x)/(x^4 - sqrt(x^2 - 16)*x^3 - 8*x^2)

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Sympy [A]  time = 4.0036, size = 68, normalized size = 1.94 \[ \begin{cases} \frac{i \operatorname{acosh}{\left (\frac{4}{x} \right )}}{128} + \frac{i \sqrt{-1 + \frac{16}{x^{2}}}}{32 x} & \text{for}\: 16 \left |{\frac{1}{x^{2}}}\right | > 1 \\- \frac{\operatorname{asin}{\left (\frac{4}{x} \right )}}{128} + \frac{1}{32 x \sqrt{1 - \frac{16}{x^{2}}}} - \frac{1}{2 x^{3} \sqrt{1 - \frac{16}{x^{2}}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/(x**2-16)**(1/2),x)

[Out]

Piecewise((I*acosh(4/x)/128 + I*sqrt(-1 + 16/x**2)/(32*x), 16*Abs(x**(-2)) > 1),
(-asin(4/x)/128 + 1/(32*x*sqrt(1 - 16/x**2)) - 1/(2*x**3*sqrt(1 - 16/x**2)), Tr
ue))

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GIAC/XCAS [A]  time = 0.207506, size = 34, normalized size = 0.97 \[ \frac{\sqrt{x^{2} - 16}}{32 \, x^{2}} + \frac{1}{128} \, \arctan \left (\frac{1}{4} \, \sqrt{x^{2} - 16}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^2 - 16)*x^3),x, algorithm="giac")

[Out]

1/32*sqrt(x^2 - 16)/x^2 + 1/128*arctan(1/4*sqrt(x^2 - 16))