Integrand size = 20, antiderivative size = 20 \[ \int \frac {1}{\left (c-a^2 c x^2\right ) \arcsin (a x)^2} \, dx=-\frac {1}{a c \sqrt {1-a^2 x^2} \arcsin (a x)}+\frac {a \text {Int}\left (\frac {x}{\left (1-a^2 x^2\right )^{3/2} \arcsin (a x)},x\right )}{c} \]
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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 = {} \[ \int \frac {1}{\left (c-a^2 c x^2\right ) \arcsin (a x)^2} \, dx=\int \frac {1}{\left (c-a^2 c x^2\right ) \arcsin (a x)^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = -\frac {1}{a c \sqrt {1-a^2 x^2} \arcsin (a x)}+\frac {a \int \frac {x}{\left (1-a^2 x^2\right )^{3/2} \arcsin (a x)} \, dx}{c} \\ \end{align*}
Not integrable
Time = 4.27 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\left (c-a^2 c x^2\right ) \arcsin (a x)^2} \, dx=\int \frac {1}{\left (c-a^2 c x^2\right ) \arcsin (a x)^2} \, dx \]
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Not integrable
Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00
\[\int \frac {1}{\left (-a^{2} c \,x^{2}+c \right ) \arcsin \left (a x \right )^{2}}d x\]
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Not integrable
Time = 0.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int \frac {1}{\left (c-a^2 c x^2\right ) \arcsin (a x)^2} \, dx=\int { -\frac {1}{{\left (a^{2} c x^{2} - c\right )} \arcsin \left (a x\right )^{2}} \,d x } \]
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Not integrable
Time = 0.93 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.30 \[ \int \frac {1}{\left (c-a^2 c x^2\right ) \arcsin (a x)^2} \, dx=- \frac {\int \frac {1}{a^{2} x^{2} \operatorname {asin}^{2}{\left (a x \right )} - \operatorname {asin}^{2}{\left (a x \right )}}\, dx}{c} \]
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Not integrable
Time = 0.61 (sec) , antiderivative size = 153, normalized size of antiderivative = 7.65 \[ \int \frac {1}{\left (c-a^2 c x^2\right ) \arcsin (a x)^2} \, dx=\int { -\frac {1}{{\left (a^{2} c x^{2} - c\right )} \arcsin \left (a x\right )^{2}} \,d x } \]
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Not integrable
Time = 0.51 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int \frac {1}{\left (c-a^2 c x^2\right ) \arcsin (a x)^2} \, dx=\int { -\frac {1}{{\left (a^{2} c x^{2} - c\right )} \arcsin \left (a x\right )^{2}} \,d x } \]
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Not integrable
Time = 0.15 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\left (c-a^2 c x^2\right ) \arcsin (a x)^2} \, dx=\int \frac {1}{{\mathrm {asin}\left (a\,x\right )}^2\,\left (c-a^2\,c\,x^2\right )} \,d x \]
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