Integrand size = 12, antiderivative size = 86 \[ \int x^2 \sqrt {\arcsin (a x)} \, dx=\frac {1}{3} x^3 \sqrt {\arcsin (a x)}-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{4 a^3}+\frac {\sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )}{12 a^3} \]
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Time = 0.11 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {4725, 4809, 3393, 3386, 3432} \[ \int x^2 \sqrt {\arcsin (a x)} \, dx=-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{4 a^3}+\frac {\sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )}{12 a^3}+\frac {1}{3} x^3 \sqrt {\arcsin (a x)} \]
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Rule 3386
Rule 3393
Rule 3432
Rule 4725
Rule 4809
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x^3 \sqrt {\arcsin (a x)}-\frac {1}{6} a \int \frac {x^3}{\sqrt {1-a^2 x^2} \sqrt {\arcsin (a x)}} \, dx \\ & = \frac {1}{3} x^3 \sqrt {\arcsin (a x)}-\frac {\text {Subst}\left (\int \frac {\sin ^3(x)}{\sqrt {x}} \, dx,x,\arcsin (a x)\right )}{6 a^3} \\ & = \frac {1}{3} x^3 \sqrt {\arcsin (a x)}-\frac {\text {Subst}\left (\int \left (\frac {3 \sin (x)}{4 \sqrt {x}}-\frac {\sin (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\arcsin (a x)\right )}{6 a^3} \\ & = \frac {1}{3} x^3 \sqrt {\arcsin (a x)}+\frac {\text {Subst}\left (\int \frac {\sin (3 x)}{\sqrt {x}} \, dx,x,\arcsin (a x)\right )}{24 a^3}-\frac {\text {Subst}\left (\int \frac {\sin (x)}{\sqrt {x}} \, dx,x,\arcsin (a x)\right )}{8 a^3} \\ & = \frac {1}{3} x^3 \sqrt {\arcsin (a x)}+\frac {\text {Subst}\left (\int \sin \left (3 x^2\right ) \, dx,x,\sqrt {\arcsin (a x)}\right )}{12 a^3}-\frac {\text {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {\arcsin (a x)}\right )}{4 a^3} \\ & = \frac {1}{3} x^3 \sqrt {\arcsin (a x)}-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arcsin (a x)}\right )}{4 a^3}+\frac {\sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arcsin (a x)}\right )}{12 a^3} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.03 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.47 \[ \int x^2 \sqrt {\arcsin (a x)} \, dx=\frac {9 \sqrt {-i \arcsin (a x)} \Gamma \left (\frac {3}{2},-i \arcsin (a x)\right )+9 \sqrt {i \arcsin (a x)} \Gamma \left (\frac {3}{2},i \arcsin (a x)\right )-\sqrt {3} \left (\sqrt {-i \arcsin (a x)} \Gamma \left (\frac {3}{2},-3 i \arcsin (a x)\right )+\sqrt {i \arcsin (a x)} \Gamma \left (\frac {3}{2},3 i \arcsin (a x)\right )\right )}{72 a^3 \sqrt {\arcsin (a x)}} \]
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Time = 0.04 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.12
method | result | size |
default | \(-\frac {-\operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {3}\, \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }+9 \,\operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }-18 a x \arcsin \left (a x \right )+6 \arcsin \left (a x \right ) \sin \left (3 \arcsin \left (a x \right )\right )}{72 a^{3} \sqrt {\arcsin \left (a x \right )}}\) | \(96\) |
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Exception generated. \[ \int x^2 \sqrt {\arcsin (a x)} \, dx=\text {Exception raised: TypeError} \]
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\[ \int x^2 \sqrt {\arcsin (a x)} \, dx=\int x^{2} \sqrt {\operatorname {asin}{\left (a x \right )}}\, dx \]
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Exception generated. \[ \int x^2 \sqrt {\arcsin (a x)} \, dx=\text {Exception raised: RuntimeError} \]
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Result contains complex when optimal does not.
Time = 0.33 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.92 \[ \int x^2 \sqrt {\arcsin (a x)} \, dx=\frac {\left (i - 1\right ) \, \sqrt {6} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {6} \sqrt {\arcsin \left (a x\right )}\right )}{288 \, a^{3}} - \frac {\left (i + 1\right ) \, \sqrt {6} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {6} \sqrt {\arcsin \left (a x\right )}\right )}{288 \, a^{3}} - \frac {\left (i - 1\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arcsin \left (a x\right )}\right )}{32 \, a^{3}} + \frac {\left (i + 1\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arcsin \left (a x\right )}\right )}{32 \, a^{3}} + \frac {i \, \sqrt {\arcsin \left (a x\right )} e^{\left (3 i \, \arcsin \left (a x\right )\right )}}{24 \, a^{3}} - \frac {i \, \sqrt {\arcsin \left (a x\right )} e^{\left (i \, \arcsin \left (a x\right )\right )}}{8 \, a^{3}} + \frac {i \, \sqrt {\arcsin \left (a x\right )} e^{\left (-i \, \arcsin \left (a x\right )\right )}}{8 \, a^{3}} - \frac {i \, \sqrt {\arcsin \left (a x\right )} e^{\left (-3 i \, \arcsin \left (a x\right )\right )}}{24 \, a^{3}} \]
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Timed out. \[ \int x^2 \sqrt {\arcsin (a x)} \, dx=\int x^2\,\sqrt {\mathrm {asin}\left (a\,x\right )} \,d x \]
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