Integrand size = 6, antiderivative size = 22 \[ \int \frac {\arcsin (x)}{x^2} \, dx=-\frac {\arcsin (x)}{x}-\text {arctanh}\left (\sqrt {1-x^2}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {4723, 272, 65, 212} \[ \int \frac {\arcsin (x)}{x^2} \, dx=-\frac {\arcsin (x)}{x}-\text {arctanh}\left (\sqrt {1-x^2}\right ) \]
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Rule 65
Rule 212
Rule 272
Rule 4723
Rubi steps \begin{align*} \text {integral}& = -\frac {\arcsin (x)}{x}+\int \frac {1}{x \sqrt {1-x^2}} \, dx \\ & = -\frac {\arcsin (x)}{x}+\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {1-x} x} \, dx,x,x^2\right ) \\ & = -\frac {\arcsin (x)}{x}-\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {1-x^2}\right ) \\ & = -\frac {\arcsin (x)}{x}-\text {arctanh}\left (\sqrt {1-x^2}\right ) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {\arcsin (x)}{x^2} \, dx=-\frac {\arcsin (x)}{x}-\text {arctanh}\left (\sqrt {1-x^2}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95
method | result | size |
default | \(-\frac {\arcsin \left (x \right )}{x}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-x^{2}+1}}\right )\) | \(21\) |
parts | \(-\frac {\arcsin \left (x \right )}{x}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-x^{2}+1}}\right )\) | \(21\) |
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Time = 0.27 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.77 \[ \int \frac {\arcsin (x)}{x^2} \, dx=-\frac {x \log \left (\sqrt {-x^{2} + 1} + 1\right ) - x \log \left (\sqrt {-x^{2} + 1} - 1\right ) + 2 \, \arcsin \left (x\right )}{2 \, x} \]
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Time = 0.98 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {\arcsin (x)}{x^2} \, dx=\begin {cases} - \operatorname {acosh}{\left (\frac {1}{x} \right )} & \text {for}\: \frac {1}{\left |{x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{x} \right )} & \text {otherwise} \end {cases} - \frac {\operatorname {asin}{\left (x \right )}}{x} \]
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Time = 0.29 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.50 \[ \int \frac {\arcsin (x)}{x^2} \, dx=-\frac {\arcsin \left (x\right )}{x} - \log \left (\frac {2 \, \sqrt {-x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) \]
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Time = 0.27 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.73 \[ \int \frac {\arcsin (x)}{x^2} \, dx=-\frac {\arcsin \left (x\right )}{x} - \frac {1}{2} \, \log \left (\sqrt {-x^{2} + 1} + 1\right ) + \frac {1}{2} \, \log \left (-\sqrt {-x^{2} + 1} + 1\right ) \]
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Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {\arcsin (x)}{x^2} \, dx=-\mathrm {atanh}\left (\frac {1}{\sqrt {1-x^2}}\right )-\frac {\mathrm {asin}\left (x\right )}{x} \]
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