Integrand size = 28, antiderivative size = 28 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x (a+b \arcsin (c x))^2} \, dx=-\frac {\left (1-c^2 x^2\right )^2}{b c x (a+b \arcsin (c x))}-\frac {9 \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )}{4 b^2}-\frac {3 \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{4 b^2}-\frac {9 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{4 b^2}-\frac {3 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{4 b^2}-\frac {\text {Int}\left (\frac {1-c^2 x^2}{x^2 (a+b \arcsin (c x))},x\right )}{b c} \]
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Not integrable
Time = 0.24 (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 = {} \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x (a+b \arcsin (c x))^2} \, dx=\int \frac {\left (1-c^2 x^2\right )^{3/2}}{x (a+b \arcsin (c x))^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = -\frac {\left (1-c^2 x^2\right )^2}{b c x (a+b \arcsin (c x))}-\frac {\int \frac {1-c^2 x^2}{x^2 (a+b \arcsin (c x))} \, dx}{b c}-\frac {(3 c) \int \frac {1-c^2 x^2}{a+b \arcsin (c x)} \, dx}{b} \\ & = -\frac {\left (1-c^2 x^2\right )^2}{b c x (a+b \arcsin (c x))}-\frac {3 \text {Subst}\left (\int \frac {\cos ^3\left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{b^2}-\frac {\int \frac {1-c^2 x^2}{x^2 (a+b \arcsin (c x))} \, dx}{b c} \\ & = -\frac {\left (1-c^2 x^2\right )^2}{b c x (a+b \arcsin (c x))}-\frac {3 \text {Subst}\left (\int \left (\frac {\cos \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{4 x}+\frac {3 \cos \left (\frac {a}{b}-\frac {x}{b}\right )}{4 x}\right ) \, dx,x,a+b \arcsin (c x)\right )}{b^2}-\frac {\int \frac {1-c^2 x^2}{x^2 (a+b \arcsin (c x))} \, dx}{b c} \\ & = -\frac {\left (1-c^2 x^2\right )^2}{b c x (a+b \arcsin (c x))}-\frac {3 \text {Subst}\left (\int \frac {\cos \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{4 b^2}-\frac {9 \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{4 b^2}-\frac {\int \frac {1-c^2 x^2}{x^2 (a+b \arcsin (c x))} \, dx}{b c} \\ & = -\frac {\left (1-c^2 x^2\right )^2}{b c x (a+b \arcsin (c x))}-\frac {\int \frac {1-c^2 x^2}{x^2 (a+b \arcsin (c x))} \, dx}{b c}-\frac {\left (9 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{4 b^2}-\frac {\left (3 \cos \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {3 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{4 b^2}-\frac {\left (9 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{4 b^2}-\frac {\left (3 \sin \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {3 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{4 b^2} \\ & = -\frac {\left (1-c^2 x^2\right )^2}{b c x (a+b \arcsin (c x))}-\frac {9 \cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )}{4 b^2}-\frac {3 \cos \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{4 b^2}-\frac {9 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{4 b^2}-\frac {3 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{4 b^2}-\frac {\int \frac {1-c^2 x^2}{x^2 (a+b \arcsin (c x))} \, dx}{b c} \\ \end{align*}
Not integrable
Time = 8.85 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x (a+b \arcsin (c x))^2} \, dx=\int \frac {\left (1-c^2 x^2\right )^{3/2}}{x (a+b \arcsin (c x))^2} \, dx \]
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Not integrable
Time = 0.15 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93
\[\int \frac {\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}{x \left (a +b \arcsin \left (c x \right )\right )^{2}}d x\]
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Not integrable
Time = 0.25 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.54 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x (a+b \arcsin (c x))^2} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x} \,d x } \]
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Not integrable
Time = 3.02 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x (a+b \arcsin (c x))^2} \, dx=\int \frac {\left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}{x \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}\, dx \]
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Not integrable
Time = 0.92 (sec) , antiderivative size = 146, normalized size of antiderivative = 5.21 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x (a+b \arcsin (c x))^2} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x} \,d x } \]
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Exception generated. \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x (a+b \arcsin (c x))^2} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 0.15 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x (a+b \arcsin (c x))^2} \, dx=\int \frac {{\left (1-c^2\,x^2\right )}^{3/2}}{x\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2} \,d x \]
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