Optimal. Leaf size=34 \[ \frac {a \sqrt {1-a^2 x^2}}{2 x}-\frac {\cos ^{-1}(a x)}{2 x^2} \]
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Rubi [A] time = 0.02, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4628, 264} \[ \frac {a \sqrt {1-a^2 x^2}}{2 x}-\frac {\cos ^{-1}(a x)}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 264
Rule 4628
Rubi steps
\begin {align*} \int \frac {\cos ^{-1}(a x)}{x^3} \, dx &=-\frac {\cos ^{-1}(a x)}{2 x^2}-\frac {1}{2} a \int \frac {1}{x^2 \sqrt {1-a^2 x^2}} \, dx\\ &=\frac {a \sqrt {1-a^2 x^2}}{2 x}-\frac {\cos ^{-1}(a x)}{2 x^2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 31, normalized size = 0.91 \[ \frac {a x \sqrt {1-a^2 x^2}-\cos ^{-1}(a x)}{2 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 27, normalized size = 0.79 \[ \frac {\sqrt {-a^{2} x^{2} + 1} a x - \arccos \left (a x\right )}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 68, normalized size = 2.00 \[ -\frac {1}{4} \, {\left (\frac {a^{4} x}{{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} {\left | a \right |}} - \frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{x {\left | a \right |}}\right )} a - \frac {\arccos \left (a x\right )}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 38, normalized size = 1.12 \[ a^{2} \left (-\frac {\arccos \left (a x \right )}{2 a^{2} x^{2}}+\frac {\sqrt {-a^{2} x^{2}+1}}{2 a x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 28, normalized size = 0.82 \[ \frac {\sqrt {-a^{2} x^{2} + 1} a}{2 \, x} - \frac {\arccos \left (a x\right )}{2 \, x^{2}} \]
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 \[ \int \frac {\mathrm {acos}\left (a\,x\right )}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.12, size = 53, normalized size = 1.56 \[ - \frac {a \left (\begin {cases} - \frac {i \sqrt {a^{2} x^{2} - 1}}{x} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{x} & \text {otherwise} \end {cases}\right )}{2} - \frac {\operatorname {acos}{\left (a x \right )}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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