Integrand size = 8, antiderivative size = 28 \[ \int \frac {\arcsin (a x)}{x^2} \, dx=-\frac {\arcsin (a x)}{x}-a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 28, 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 = {4723, 272, 65, 214} \[ \int \frac {\arcsin (a x)}{x^2} \, dx=-a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {\arcsin (a x)}{x} \]
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Rule 65
Rule 214
Rule 272
Rule 4723
Rubi steps \begin{align*} \text {integral}& = -\frac {\arcsin (a x)}{x}+a \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {\arcsin (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 {\arcsin (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 {\arcsin (a x)}{x}-a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\arcsin (a x)}{x^2} \, dx=-\frac {\arcsin (a x)}{x}-a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96
method | result | size |
parts | \(-\frac {\arcsin \left (a x \right )}{x}-a \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\) | \(27\) |
derivativedivides | \(a \left (-\frac {\arcsin \left (a x \right )}{a x}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )\) | \(31\) |
default | \(a \left (-\frac {\arcsin \left (a x \right )}{a x}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )\) | \(31\) |
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Time = 0.28 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.75 \[ \int \frac {\arcsin (a x)}{x^2} \, dx=-\frac {a x \log \left (\sqrt {-a^{2} x^{2} + 1} + 1\right ) - a x \log \left (\sqrt {-a^{2} x^{2} + 1} - 1\right ) + 2 \, \arcsin \left (a x\right )}{2 \, x} \]
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Result contains complex when optimal does not.
Time = 0.82 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14 \[ \int \frac {\arcsin (a x)}{x^2} \, dx=a \left (\begin {cases} - \operatorname {acosh}{\left (\frac {1}{a x} \right )} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{a x} \right )} & \text {otherwise} \end {cases}\right ) - \frac {\operatorname {asin}{\left (a x \right )}}{x} \]
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Time = 0.27 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.39 \[ \int \frac {\arcsin (a x)}{x^2} \, dx=-a \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - \frac {\arcsin \left (a x\right )}{x} \]
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Time = 0.27 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.71 \[ \int \frac {\arcsin (a x)}{x^2} \, dx=-\frac {1}{2} \, a {\left (\log \left (\sqrt {-a^{2} x^{2} + 1} + 1\right ) - \log \left (-\sqrt {-a^{2} x^{2} + 1} + 1\right )\right )} - \frac {\arcsin \left (a x\right )}{x} \]
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Time = 0.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {\arcsin (a x)}{x^2} \, dx=-\frac {\mathrm {asin}\left (a\,x\right )}{x}-a\,\mathrm {atanh}\left (\frac {1}{\sqrt {1-a^2\,x^2}}\right ) \]
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