Integrand size = 8, antiderivative size = 44 \[ \int \sqrt {\arcsin (a x)} \, dx=x \sqrt {\arcsin (a x)}-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{a} \]
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Time = 0.05 (sec) , antiderivative size = 44, 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 = {4715, 4809, 3386, 3432} \[ \int \sqrt {\arcsin (a x)} \, dx=x \sqrt {\arcsin (a x)}-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{a} \]
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Rule 3386
Rule 3432
Rule 4715
Rule 4809
Rubi steps \begin{align*} \text {integral}& = x \sqrt {\arcsin (a x)}-\frac {1}{2} a \int \frac {x}{\sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}} \, dx \\ & = x \sqrt {\arcsin (a x)}-\frac {\text {Subst}\left (\int \frac {\sin (x)}{\sqrt {x}} \, dx,x,\arcsin (a x)\right )}{2 a} \\ & = x \sqrt {\arcsin (a x)}-\frac {\text {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {\arcsin (a x)}\right )}{a} \\ & = x \sqrt {\arcsin (a x)}-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{a} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.03 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.50 \[ \int \sqrt {\arcsin (a x)} \, dx=\frac {\sqrt {-i \arcsin (a x)} \Gamma \left (\frac {3}{2},-i \arcsin (a x)\right )+\sqrt {i \arcsin (a x)} \Gamma \left (\frac {3}{2},i \arcsin (a x)\right )}{2 a \sqrt {\arcsin (a x)}} \]
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Time = 0.04 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.11
method | result | size |
default | \(\frac {-\operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }+2 a x \arcsin \left (a x \right )}{2 a \sqrt {\arcsin \left (a x \right )}}\) | \(49\) |
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Exception generated. \[ \int \sqrt {\arcsin (a x)} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \sqrt {\arcsin (a x)} \, dx=\int \sqrt {\operatorname {asin}{\left (a x \right )}}\, dx \]
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Exception generated. \[ \int \sqrt {\arcsin (a x)} \, dx=\text {Exception raised: RuntimeError} \]
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Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.89 \[ \int \sqrt {\arcsin (a x)} \, dx=-\frac {\left (i - 1\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arcsin \left (a x\right )}\right )}{8 \, a} + \frac {\left (i + 1\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arcsin \left (a x\right )}\right )}{8 \, a} - \frac {i \, \sqrt {\arcsin \left (a x\right )} e^{\left (i \, \arcsin \left (a x\right )\right )}}{2 \, a} + \frac {i \, \sqrt {\arcsin \left (a x\right )} e^{\left (-i \, \arcsin \left (a x\right )\right )}}{2 \, a} \]
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Timed out. \[ \int \sqrt {\arcsin (a x)} \, dx=\int \sqrt {\mathrm {asin}\left (a\,x\right )} \,d x \]
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