Optimal. Leaf size=38 \[ -\frac {\sqrt {1-\frac {x^2}{a^2}}}{2 a x}-\frac {\sin ^{-1}\left (\frac {x}{a}\right )}{2 x^2} \]
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Rubi [A] time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5265, 4627, 264} \[ -\frac {\sqrt {1-\frac {x^2}{a^2}}}{2 a x}-\frac {\sin ^{-1}\left (\frac {x}{a}\right )}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 264
Rule 4627
Rule 5265
Rubi steps
\begin {align*} \int \frac {\csc ^{-1}\left (\frac {a}{x}\right )}{x^3} \, dx &=\int \frac {\sin ^{-1}\left (\frac {x}{a}\right )}{x^3} \, dx\\ &=-\frac {\sin ^{-1}\left (\frac {x}{a}\right )}{2 x^2}+\frac {\int \frac {1}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx}{2 a}\\ &=-\frac {\sqrt {1-\frac {x^2}{a^2}}}{2 a x}-\frac {\sin ^{-1}\left (\frac {x}{a}\right )}{2 x^2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 35, normalized size = 0.92 \[ -\frac {x \sqrt {1-\frac {x^2}{a^2}}+a \csc ^{-1}\left (\frac {a}{x}\right )}{2 a x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 38, normalized size = 1.00 \[ -\frac {a^{2} \operatorname {arccsc}\left (\frac {a}{x}\right ) + x^{2} \sqrt {\frac {a^{2} - x^{2}}{x^{2}}}}{2 \, a^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 61, normalized size = 1.61 \[ -\frac {a {\left (\frac {a + \sqrt {a^{2} - x^{2}}}{a^{2} x} - \frac {x}{{\left (a + \sqrt {a^{2} - x^{2}}\right )} a^{2}}\right )}}{4 \, {\left | a \right |}} - \frac {\arcsin \left (\frac {x}{a}\right )}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 54, normalized size = 1.42 \[ -\frac {\frac {a^{2} \mathrm {arccsc}\left (\frac {a}{x}\right )}{2 x^{2}}+\frac {x \left (-1+\frac {a^{2}}{x^{2}}\right )}{2 \sqrt {\frac {\left (-1+\frac {a^{2}}{x^{2}}\right ) x^{2}}{a^{2}}}\, a}}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 32, normalized size = 0.84 \[ -\frac {\operatorname {arccsc}\left (\frac {a}{x}\right )}{2 \, x^{2}} - \frac {\sqrt {-\frac {x^{2}}{a^{2}} + 1}}{2 \, a x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\mathrm {asin}\left (\frac {x}{a}\right )}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.16, size = 51, normalized size = 1.34 \[ - \frac {\operatorname {acsc}{\left (\frac {a}{x} \right )}}{2 x^{2}} + \frac {\begin {cases} - \frac {\sqrt {\frac {a^{2}}{x^{2}} - 1}}{a} & \text {for}\: \left |{\frac {a^{2}}{x^{2}}}\right | > 1 \\- \frac {i \sqrt {- \frac {a^{2}}{x^{2}} + 1}}{a} & \text {otherwise} \end {cases}}{2 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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