Optimal. Leaf size=34 \[ \frac {a \sqrt {1-a^2 x^2}}{2 x}-\frac {\text {ArcCos}(a x)}{2 x^2} \]
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Rubi [A]
time = 0.01, 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 = {4724, 270}
\begin {gather*} \frac {a \sqrt {1-a^2 x^2}}{2 x}-\frac {\text {ArcCos}(a x)}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 4724
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 = 0.01, size = 31, normalized size = 0.91 \begin {gather*} \frac {a x \sqrt {1-a^2 x^2}-\text {ArcCos}(a x)}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 38, normalized size = 1.12
method | result | size |
derivativedivides | \(a^{2} \left (-\frac {\arccos \left (a x \right )}{2 a^{2} x^{2}}+\frac {\sqrt {-a^{2} x^{2}+1}}{2 a x}\right )\) | \(38\) |
default | \(a^{2} \left (-\frac {\arccos \left (a x \right )}{2 a^{2} x^{2}}+\frac {\sqrt {-a^{2} x^{2}+1}}{2 a x}\right )\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 28, normalized size = 0.82 \begin {gather*} \frac {\sqrt {-a^{2} x^{2} + 1} a}{2 \, x} - \frac {\arccos \left (a x\right )}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.47, size = 27, normalized size = 0.79 \begin {gather*} \frac {\sqrt {-a^{2} x^{2} + 1} a x - \arccos \left (a x\right )}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.75, size = 53, normalized size = 1.56 \begin {gather*} - \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 68 vs.
\(2 (28) = 56\).
time = 0.45, size = 68, normalized size = 2.00 \begin {gather*} -\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}} \end {gather*}
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 \begin {gather*} \int \frac {\mathrm {acos}\left (a\,x\right )}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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