Integrand size = 8, antiderivative size = 27 \[ \int \frac {\arccos (a x)}{x^2} \, dx=-\frac {\arccos (a x)}{x}+a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4724, 272, 65, 214} \[ \int \frac {\arccos (a x)}{x^2} \, dx=a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {\arccos (a x)}{x} \]
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Rule 65
Rule 214
Rule 272
Rule 4724
Rubi steps \begin{align*} \text {integral}& = -\frac {\arccos (a x)}{x}-a \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {\arccos (a x)}{x}-\frac {1}{2} a \text {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right ) \\ & = -\frac {\arccos (a x)}{x}+\frac {\text {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2}} \, dx,x,\sqrt {1-a^2 x^2}\right )}{a} \\ & = -\frac {\arccos (a x)}{x}+a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.26 \[ \int \frac {\arccos (a x)}{x^2} \, dx=-\frac {\arccos (a x)}{x}-a \log (x)+a \log \left (1+\sqrt {1-a^2 x^2}\right ) \]
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Time = 0.08 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96
method | result | size |
parts | \(-\frac {\arccos \left (a x \right )}{x}+a \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\) | \(26\) |
derivativedivides | \(a \left (-\frac {\arccos \left (a x \right )}{a x}+\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )\) | \(29\) |
default | \(a \left (-\frac {\arccos \left (a x \right )}{a x}+\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )\) | \(29\) |
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Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (25) = 50\).
Time = 0.26 (sec) , antiderivative size = 82, normalized size of antiderivative = 3.04 \[ \int \frac {\arccos (a x)}{x^2} \, dx=\frac {a x \log \left (\sqrt {-a^{2} x^{2} + 1} + 1\right ) - a x \log \left (\sqrt {-a^{2} x^{2} + 1} - 1\right ) + 2 \, {\left (x - 1\right )} \arccos \left (a x\right ) - 2 \, x \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} a x}{a^{2} x^{2} - 1}\right )}{2 \, x} \]
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Result contains complex when optimal does not.
Time = 0.98 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.26 \[ \int \frac {\arccos (a x)}{x^2} \, dx=- a \left (\begin {cases} - \operatorname {acosh}{\left (\frac {1}{a x} \right )} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{a x} \right )} & \text {otherwise} \end {cases}\right ) - \frac {\operatorname {acos}{\left (a x \right )}}{x} \]
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none
Time = 0.30 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.41 \[ \int \frac {\arccos (a x)}{x^2} \, dx=a \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - \frac {\arccos \left (a x\right )}{x} \]
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none
Time = 0.28 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.78 \[ \int \frac {\arccos (a x)}{x^2} \, dx=\frac {1}{2} \, a {\left (\log \left (\sqrt {-a^{2} x^{2} + 1} + 1\right ) - \log \left (-\sqrt {-a^{2} x^{2} + 1} + 1\right )\right )} - \frac {\arccos \left (a x\right )}{x} \]
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Time = 0.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {\arccos (a x)}{x^2} \, dx=a\,\mathrm {atanh}\left (\frac {1}{\sqrt {1-a^2\,x^2}}\right )-\frac {\mathrm {acos}\left (a\,x\right )}{x} \]
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