Integrand size = 24, antiderivative size = 42 \[ \int \frac {1}{\sqrt {c-a^2 c x^2} \arcsin (a x)^{3/2}} \, dx=-\frac {2 \sqrt {1-a^2 x^2}}{a \sqrt {c-a^2 c x^2} \sqrt {\arcsin (a x)}} \]
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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} \[ \int \frac {1}{\sqrt {c-a^2 c x^2} \arcsin (a x)^{3/2}} \, dx=-\frac {2 \sqrt {1-a^2 x^2}}{a \sqrt {\arcsin (a x)} \sqrt {c-a^2 c x^2}} \]
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Rule 4737
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {1-a^2 x^2}}{a \sqrt {c-a^2 c x^2} \sqrt {\arcsin (a x)}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {c-a^2 c x^2} \arcsin (a x)^{3/2}} \, dx=-\frac {2 \sqrt {1-a^2 x^2}}{a \sqrt {c-a^2 c x^2} \sqrt {\arcsin (a x)}} \]
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Time = 0.14 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.90
method | result | size |
default | \(-\frac {2 \sqrt {-a^{2} x^{2}+1}}{\sqrt {\arcsin \left (a x \right )}\, a \sqrt {-c \left (a^{2} x^{2}-1\right )}}\) | \(38\) |
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Time = 0.25 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.14 \[ \int \frac {1}{\sqrt {c-a^2 c x^2} \arcsin (a x)^{3/2}} \, dx=\frac {2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{{\left (a^{3} c x^{2} - a c\right )} \sqrt {\arcsin \left (a x\right )}} \]
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\[ \int \frac {1}{\sqrt {c-a^2 c x^2} \arcsin (a x)^{3/2}} \, dx=\int \frac {1}{\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \operatorname {asin}^{\frac {3}{2}}{\left (a x \right )}}\, dx \]
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Exception generated. \[ \int \frac {1}{\sqrt {c-a^2 c x^2} \arcsin (a x)^{3/2}} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {1}{\sqrt {c-a^2 c x^2} \arcsin (a x)^{3/2}} \, dx=\int { \frac {1}{\sqrt {-a^{2} c x^{2} + c} \arcsin \left (a x\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {c-a^2 c x^2} \arcsin (a x)^{3/2}} \, dx=\int \frac {1}{{\mathrm {asin}\left (a\,x\right )}^{3/2}\,\sqrt {c-a^2\,c\,x^2}} \,d x \]
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