Integrand size = 8, antiderivative size = 34 \[ \int \frac {\arccos (a x)}{x^3} \, dx=\frac {a \sqrt {1-a^2 x^2}}{2 x}-\frac {\arccos (a x)}{2 x^2} \]
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Time = 0.01 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4724, 270} \[ \int \frac {\arccos (a x)}{x^3} \, dx=\frac {a \sqrt {1-a^2 x^2}}{2 x}-\frac {\arccos (a x)}{2 x^2} \]
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Rule 270
Rule 4724
Rubi steps \begin{align*} \text {integral}& = -\frac {\arccos (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 {\arccos (a x)}{2 x^2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.91 \[ \int \frac {\arccos (a x)}{x^3} \, dx=\frac {a x \sqrt {1-a^2 x^2}-\arccos (a x)}{2 x^2} \]
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Time = 0.07 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.85
method | result | size |
parts | \(-\frac {\arccos \left (a x \right )}{2 x^{2}}+\frac {a \sqrt {-a^{2} x^{2}+1}}{2 x}\) | \(29\) |
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 | \(a^{2} \left (-\frac {\arccos \left (a x \right )}{2 a^{2} x^{2}}+\frac {\sqrt {-a^{2} x^{2}+1}}{2 a x}\right )\) | \(38\) |
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none
Time = 0.24 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.79 \[ \int \frac {\arccos (a x)}{x^3} \, dx=\frac {\sqrt {-a^{2} x^{2} + 1} a x - \arccos \left (a x\right )}{2 \, x^{2}} \]
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Result contains complex when optimal does not.
Time = 0.72 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.56 \[ \int \frac {\arccos (a x)}{x^3} \, dx=- \frac {a \left (\begin {cases} - \frac {i \sqrt {a^{2} x^{2} - 1}}{x} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{x} & \text {otherwise} \end {cases}\right )}{2} - \frac {\operatorname {acos}{\left (a x \right )}}{2 x^{2}} \]
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none
Time = 0.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.82 \[ \int \frac {\arccos (a x)}{x^3} \, dx=\frac {\sqrt {-a^{2} x^{2} + 1} a}{2 \, x} - \frac {\arccos \left (a x\right )}{2 \, x^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (28) = 56\).
Time = 0.29 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.00 \[ \int \frac {\arccos (a x)}{x^3} \, dx=-\frac {1}{4} \, {\left (\frac {a^{4} x}{{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} {\left | a \right |}} - \frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{x {\left | a \right |}}\right )} a - \frac {\arccos \left (a x\right )}{2 \, x^{2}} \]
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Timed out. \[ \int \frac {\arccos (a x)}{x^3} \, dx=\int \frac {\mathrm {acos}\left (a\,x\right )}{x^3} \,d x \]
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