3.2.33 \(\int \frac {\sqrt {-a^2+x^2}}{x^4} \, dx\) [133]

Optimal. Leaf size=23 \[ \frac {\left (-a^2+x^2\right )^{3/2}}{3 a^2 x^3} \]

[Out]

1/3*(-a^2+x^2)^(3/2)/a^2/x^3

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Rubi [A]
time = 0.00, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {270} \begin {gather*} \frac {\left (x^2-a^2\right )^{3/2}}{3 a^2 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[-a^2 + x^2]/x^4,x]

[Out]

(-a^2 + x^2)^(3/2)/(3*a^2*x^3)

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 {\sqrt {-a^2+x^2}}{x^4} \, dx &=\frac {\left (-a^2+x^2\right )^{3/2}}{3 a^2 x^3}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 23, normalized size = 1.00 \begin {gather*} \frac {\left (-a^2+x^2\right )^{3/2}}{3 a^2 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-a^2 + x^2]/x^4,x]

[Out]

(-a^2 + x^2)^(3/2)/(3*a^2*x^3)

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in optimal.
time = 2.46, size = 80, normalized size = 3.48 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {I \left (-a^2+x^2\right ) \sqrt {\frac {a^2-x^2}{x^2}}}{3 a^2 x^2},\text {Abs}\left [\frac {a^2}{x^2}\right ]>1\right \}\right \},\frac {\sqrt {1-\frac {a^2}{x^2}}}{3 a^2}-\frac {\sqrt {1-\frac {a^2}{x^2}}}{3 x^2}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[Sqrt[x^2 - a^2]/x^4,x]')

[Out]

Piecewise[{{I / 3 (-a ^ 2 + x ^ 2) Sqrt[(a ^ 2 - x ^ 2) / x ^ 2] / (a ^ 2 x ^ 2), Abs[a ^ 2 / x ^ 2] > 1}}, Sq
rt[1 - a ^ 2 / x ^ 2] / (3 a ^ 2) - Sqrt[1 - a ^ 2 / x ^ 2] / (3 x ^ 2)]

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Maple [A]
time = 0.04, size = 20, normalized size = 0.87

method	result	size
default	\(\frac {\left (-a^{2}+x^{2}\right )^{\frac {3}{2}}}{3 a^{2} x^{3}}\)	\(20\)
gosper	\(-\frac {\left (a -x \right ) \left (a +x \right ) \sqrt {-a^{2}+x^{2}}}{3 x^{3} a^{2}}\)	\(28\)
trager	\(-\frac {\left (a^{2}-x^{2}\right ) \sqrt {-a^{2}+x^{2}}}{3 a^{2} x^{3}}\)	\(29\)
risch	\(\frac {\left (a^{2}-x^{2}\right )^{2}}{3 x^{3} \sqrt {-a^{2}+x^{2}}\, a^{2}}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(-a^2+x^2)^(3/2)/a^2/x^3

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Maxima [A]
time = 0.35, size = 19, normalized size = 0.83 \begin {gather*} \frac {{\left (-a^{2} + x^{2}\right )}^{\frac {3}{2}}}{3 \, a^{2} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(-a^2 + x^2)^(3/2)/(a^2*x^3)

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Fricas [A]
time = 0.32, size = 23, normalized size = 1.00 \begin {gather*} \frac {x^{3} + {\left (-a^{2} + x^{2}\right )}^{\frac {3}{2}}}{3 \, a^{2} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(x^3 + (-a^2 + x^2)^(3/2))/(a^2*x^3)

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Sympy [A]
time = 0.37, size = 76, normalized size = 3.30 \begin {gather*} \begin {cases} - \frac {i \sqrt {\frac {a^{2}}{x^{2}} - 1}}{3 x^{2}} + \frac {i \sqrt {\frac {a^{2}}{x^{2}} - 1}}{3 a^{2}} & \text {for}\: \left |{\frac {a^{2}}{x^{2}}}\right | > 1 \\- \frac {\sqrt {- \frac {a^{2}}{x^{2}} + 1}}{3 x^{2}} + \frac {\sqrt {- \frac {a^{2}}{x^{2}} + 1}}{3 a^{2}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-I*sqrt(a**2/x**2 - 1)/(3*x**2) + I*sqrt(a**2/x**2 - 1)/(3*a**2), Abs(a**2/x**2) > 1), (-sqrt(-a**2
/x**2 + 1)/(3*x**2) + sqrt(-a**2/x**2 + 1)/(3*a**2), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (19) = 38\).
time = 0.00, size = 51, normalized size = 2.22 \begin {gather*} \frac {2 \left (3 \left (\sqrt {-a^{2}+x^{2}}-x\right )^{4}+a^{4}\right )}{3 \left (\left (\sqrt {-a^{2}+x^{2}}-x\right )^{2}+a^{2}\right )^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2+x^2)^(1/2)/x^4,x)

[Out]

2/3*(a^4 + 3*(x - sqrt(-a^2 + x^2))^4)/(a^2 + (x - sqrt(-a^2 + x^2))^2)^3

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Mupad [B]
time = 0.34, size = 19, normalized size = 0.83 \begin {gather*} \frac {{\left (x^2-a^2\right )}^{3/2}}{3\,a^2\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2 - a^2)^(1/2)/x^4,x)

[Out]

(x^2 - a^2)^(3/2)/(3*a^2*x^3)

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