Integrand size = 28, antiderivative size = 28 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x (a+b \arcsin (c x))} \, dx=\frac {5 \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{4 b}+\frac {\operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {3 a}{b}\right )}{4 b}-\frac {5 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{4 b}-\frac {\cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{4 b}+\text {Int}\left (\frac {1}{x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))},x\right ) \]
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Not integrable
Time = 0.44 (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 = {} \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x (a+b \arcsin (c x))} \, dx=\int \frac {\left (1-c^2 x^2\right )^{3/2}}{x (a+b \arcsin (c x))} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}-\frac {2 c^2 x}{\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}+\frac {c^4 x^3}{\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}\right ) \, dx \\ & = -\left (\left (2 c^2\right ) \int \frac {x}{\sqrt {1-c^2 x^2} (a+b \arcsin (c x))} \, dx\right )+c^4 \int \frac {x^3}{\sqrt {1-c^2 x^2} (a+b \arcsin (c x))} \, dx+\int \frac {1}{x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))} \, dx \\ & = -\frac {\text {Subst}\left (\int \frac {\sin ^3\left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{b}+\frac {2 \text {Subst}\left (\int \frac {\sin \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{b}+\int \frac {1}{x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))} \, dx \\ & = -\frac {\text {Subst}\left (\int \left (-\frac {\sin \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{4 x}+\frac {3 \sin \left (\frac {a}{b}-\frac {x}{b}\right )}{4 x}\right ) \, dx,x,a+b \arcsin (c x)\right )}{b}-\frac {\left (2 \cos \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 )}{b}+\frac {\left (2 \sin \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 )}{b}+\int \frac {1}{x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))} \, dx \\ & = \frac {2 \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{b}-\frac {2 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{b}+\frac {\text {Subst}\left (\int \frac {\sin \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{4 b}-\frac {3 \text {Subst}\left (\int \frac {\sin \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{4 b}+\int \frac {1}{x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))} \, dx \\ & = \frac {2 \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{b}-\frac {2 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{b}+\frac {\left (3 \cos \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}-\frac {\cos \left (\frac {3 a}{b}\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {3 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{4 b}-\frac {\left (3 \sin \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}+\frac {\sin \left (\frac {3 a}{b}\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {3 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{4 b}+\int \frac {1}{x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))} \, dx \\ & = \frac {5 \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{4 b}+\frac {\operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {3 a}{b}\right )}{4 b}-\frac {5 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{4 b}-\frac {\cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{4 b}+\int \frac {1}{x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))} \, dx \\ \end{align*}
Not integrable
Time = 3.35 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x (a+b \arcsin (c x))} \, dx=\int \frac {\left (1-c^2 x^2\right )^{3/2}}{x (a+b \arcsin (c x))} \, dx \]
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Not integrable
Time = 0.09 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93
\[\int \frac {\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}{x \left (a +b \arcsin \left (c x \right )\right )}d x\]
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Not integrable
Time = 0.24 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x (a+b \arcsin (c x))} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (b \arcsin \left (c x\right ) + a\right )} x} \,d x } \]
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Not integrable
Time = 2.15 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x (a+b \arcsin (c x))} \, dx=\int \frac {\left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}{x \left (a + b \operatorname {asin}{\left (c x \right )}\right )}\, dx \]
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Not integrable
Time = 0.41 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x (a+b \arcsin (c x))} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (b \arcsin \left (c x\right ) + a\right )} x} \,d x } \]
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Exception generated. \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x (a+b \arcsin (c x))} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 0.15 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x (a+b \arcsin (c x))} \, dx=\int \frac {{\left (1-c^2\,x^2\right )}^{3/2}}{x\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )} \,d x \]
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