Integrand size = 23, antiderivative size = 23 \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{x^3} \, dx=\text {Int}\left (\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{x^3},x\right ) \]
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Not integrable
Time = 0.09 (sec) , antiderivative size = 23, 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 = {} \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{x^3} \, dx=\int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{x^3} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{x^3} \, dx \\ \end{align*}
Not integrable
Time = 11.89 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{x^3} \, dx=\int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{x^3} \, dx \]
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Not integrable
Time = 2.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91
\[\int \frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arccsc}\left (c x \right )\right )}{x^{3}}d x\]
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Not integrable
Time = 0.27 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.74 \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{x^3} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
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Not integrable
Time = 63.27 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{x^3} \, dx=\int \frac {\left (a + b \operatorname {acsc}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{\frac {3}{2}}}{x^{3}}\, dx \]
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Exception generated. \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{x^3} \, dx=\text {Exception raised: ValueError} \]
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Not integrable
Time = 0.36 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{x^3} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
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Not integrable
Time = 1.49 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17 \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{x^3} \, dx=\int \frac {{\left (e\,x^2+d\right )}^{3/2}\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{x^3} \,d x \]
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