Integrand size = 28, antiderivative size = 28 \[ \int \frac {\sqrt {1-c^2 x^2}}{x (a+b \arcsin (c x))^2} \, dx=-\frac {1-c^2 x^2}{b c x (a+b \arcsin (c x))}-\frac {\cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )}{b^2}-\frac {\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{b^2}-\frac {\text {Int}\left (\frac {1}{x^2 (a+b \arcsin (c x))},x\right )}{b c} \]
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Not integrable
Time = 0.14 (sec) , antiderivative size = 28, 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 = {} \[ \int \frac {\sqrt {1-c^2 x^2}}{x (a+b \arcsin (c x))^2} \, dx=\int \frac {\sqrt {1-c^2 x^2}}{x (a+b \arcsin (c x))^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = -\frac {1-c^2 x^2}{b c x (a+b \arcsin (c x))}-\frac {\int \frac {1}{x^2 (a+b \arcsin (c x))} \, dx}{b c}-\frac {c \int \frac {1}{a+b \arcsin (c x)} \, dx}{b} \\ & = -\frac {1-c^2 x^2}{b c x (a+b \arcsin (c x))}-\frac {\text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{b^2}-\frac {\int \frac {1}{x^2 (a+b \arcsin (c x))} \, dx}{b c} \\ & = -\frac {1-c^2 x^2}{b c x (a+b \arcsin (c x))}-\frac {\int \frac {1}{x^2 (a+b \arcsin (c x))} \, dx}{b c}-\frac {\cos \left (\frac {a}{b}\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{b^2}-\frac {\sin \left (\frac {a}{b}\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{b^2} \\ & = -\frac {1-c^2 x^2}{b c x (a+b \arcsin (c x))}-\frac {\cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )}{b^2}-\frac {\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{b^2}-\frac {\int \frac {1}{x^2 (a+b \arcsin (c x))} \, dx}{b c} \\ \end{align*}
Not integrable
Time = 10.62 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\sqrt {1-c^2 x^2}}{x (a+b \arcsin (c x))^2} \, dx=\int \frac {\sqrt {1-c^2 x^2}}{x (a+b \arcsin (c x))^2} \, dx \]
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Not integrable
Time = 0.39 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93
\[\int \frac {\sqrt {-c^{2} x^{2}+1}}{x \left (a +b \arcsin \left (c x \right )\right )^{2}}d x\]
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Not integrable
Time = 0.25 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.54 \[ \int \frac {\sqrt {1-c^2 x^2}}{x (a+b \arcsin (c x))^2} \, dx=\int { \frac {\sqrt {-c^{2} x^{2} + 1}}{{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x} \,d x } \]
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Not integrable
Time = 1.09 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt {1-c^2 x^2}}{x (a+b \arcsin (c x))^2} \, dx=\int \frac {\sqrt {- \left (c x - 1\right ) \left (c x + 1\right )}}{x \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}\, dx \]
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Not integrable
Time = 0.77 (sec) , antiderivative size = 128, normalized size of antiderivative = 4.57 \[ \int \frac {\sqrt {1-c^2 x^2}}{x (a+b \arcsin (c x))^2} \, dx=\int { \frac {\sqrt {-c^{2} x^{2} + 1}}{{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x} \,d x } \]
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Exception generated. \[ \int \frac {\sqrt {1-c^2 x^2}}{x (a+b \arcsin (c x))^2} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 0.16 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1-c^2 x^2}}{x (a+b \arcsin (c x))^2} \, dx=\int \frac {\sqrt {1-c^2\,x^2}}{x\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2} \,d x \]
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