Integrand size = 24, antiderivative size = 42 \[ \int \frac {\arcsin \left (\frac {x}{a}\right )^{3/2}}{\sqrt {a^2-x^2}} \, dx=\frac {2 a \sqrt {1-\frac {x^2}{a^2}} \arcsin \left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {a^2-x^2}} \]
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Time = 0.03 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {4737} \[ \int \frac {\arcsin \left (\frac {x}{a}\right )^{3/2}}{\sqrt {a^2-x^2}} \, dx=\frac {2 a \sqrt {1-\frac {x^2}{a^2}} \arcsin \left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {a^2-x^2}} \]
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Rule 4737
Rubi steps \begin{align*} \text {integral}& = \frac {2 a \sqrt {1-\frac {x^2}{a^2}} \arcsin \left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {a^2-x^2}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00 \[ \int \frac {\arcsin \left (\frac {x}{a}\right )^{3/2}}{\sqrt {a^2-x^2}} \, dx=\frac {2 a \sqrt {1-\frac {x^2}{a^2}} \arcsin \left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {a^2-x^2}} \]
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Time = 0.25 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.90
method | result | size |
default | \(\frac {2 \arcsin \left (\frac {x}{a}\right )^{\frac {5}{2}} \sqrt {\frac {a^{2}-x^{2}}{a^{2}}}\, a}{5 \sqrt {a^{2}-x^{2}}}\) | \(38\) |
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none
Time = 0.26 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.90 \[ \int \frac {\arcsin \left (\frac {x}{a}\right )^{3/2}}{\sqrt {a^2-x^2}} \, dx=\frac {2}{5} \, \sqrt {-\arctan \left (-\frac {x}{\sqrt {a^{2} - x^{2}}}\right )} \arctan \left (-\frac {x}{\sqrt {a^{2} - x^{2}}}\right )^{2} \]
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\[ \int \frac {\arcsin \left (\frac {x}{a}\right )^{3/2}}{\sqrt {a^2-x^2}} \, dx=\int \frac {\operatorname {asin}^{\frac {3}{2}}{\left (\frac {x}{a} \right )}}{\sqrt {- \left (- a + x\right ) \left (a + x\right )}}\, dx \]
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Exception generated. \[ \int \frac {\arcsin \left (\frac {x}{a}\right )^{3/2}}{\sqrt {a^2-x^2}} \, dx=\text {Exception raised: RuntimeError} \]
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none
Time = 0.28 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.36 \[ \int \frac {\arcsin \left (\frac {x}{a}\right )^{3/2}}{\sqrt {a^2-x^2}} \, dx=\frac {2 \, {\left | a \right |} \arcsin \left (\frac {x}{a}\right )^{\frac {5}{2}}}{5 \, a} \]
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Timed out. \[ \int \frac {\arcsin \left (\frac {x}{a}\right )^{3/2}}{\sqrt {a^2-x^2}} \, dx=\int \frac {{\mathrm {asin}\left (\frac {x}{a}\right )}^{3/2}}{\sqrt {a^2-x^2}} \,d x \]
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