Integrand size = 20, antiderivative size = 20 \[ \int \frac {1}{\left (c-a^2 c x^2\right )^2 \arcsin (a x)^2} \, dx=-\frac {1}{a c^2 \left (1-a^2 x^2\right )^{3/2} \arcsin (a x)}+\frac {3 a \text {Int}\left (\frac {x}{\left (1-a^2 x^2\right )^{5/2} \arcsin (a x)},x\right )}{c^2} \]
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Not integrable
Time = 0.07 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{\left (c-a^2 c x^2\right )^2 \arcsin (a x)^2} \, dx=\int \frac {1}{\left (c-a^2 c x^2\right )^2 \arcsin (a x)^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = -\frac {1}{a c^2 \left (1-a^2 x^2\right )^{3/2} \arcsin (a x)}+\frac {(3 a) \int \frac {x}{\left (1-a^2 x^2\right )^{5/2} \arcsin (a x)} \, dx}{c^2} \\ \end{align*}
Not integrable
Time = 13.43 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\left (c-a^2 c x^2\right )^2 \arcsin (a x)^2} \, dx=\int \frac {1}{\left (c-a^2 c x^2\right )^2 \arcsin (a x)^2} \, dx \]
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Not integrable
Time = 0.49 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00
\[\int \frac {1}{\left (-a^{2} c \,x^{2}+c \right )^{2} \arcsin \left (a x \right )^{2}}d x\]
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Not integrable
Time = 0.24 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.80 \[ \int \frac {1}{\left (c-a^2 c x^2\right )^2 \arcsin (a x)^2} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} - c\right )}^{2} \arcsin \left (a x\right )^{2}} \,d x } \]
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Not integrable
Time = 1.60 (sec) , antiderivative size = 41, normalized size of antiderivative = 2.05 \[ \int \frac {1}{\left (c-a^2 c x^2\right )^2 \arcsin (a x)^2} \, dx=\frac {\int \frac {1}{a^{4} x^{4} \operatorname {asin}^{2}{\left (a x \right )} - 2 a^{2} x^{2} \operatorname {asin}^{2}{\left (a x \right )} + \operatorname {asin}^{2}{\left (a x \right )}}\, dx}{c^{2}} \]
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Not integrable
Time = 0.67 (sec) , antiderivative size = 202, normalized size of antiderivative = 10.10 \[ \int \frac {1}{\left (c-a^2 c x^2\right )^2 \arcsin (a x)^2} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} - c\right )}^{2} \arcsin \left (a x\right )^{2}} \,d x } \]
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Not integrable
Time = 1.38 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15 \[ \int \frac {1}{\left (c-a^2 c x^2\right )^2 \arcsin (a x)^2} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} - c\right )}^{2} \arcsin \left (a x\right )^{2}} \,d x } \]
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Not integrable
Time = 0.15 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\left (c-a^2 c x^2\right )^2 \arcsin (a x)^2} \, dx=\int \frac {1}{{\mathrm {asin}\left (a\,x\right )}^2\,{\left (c-a^2\,c\,x^2\right )}^2} \,d x \]
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