Integrand size = 16, antiderivative size = 16 \[ \int \frac {(d x)^m}{\left (a+b \csc ^{-1}(c x)\right )^2} \, dx=\text {Int}\left (\frac {(d x)^m}{\left (a+b \csc ^{-1}(c x)\right )^2},x\right ) \]
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Not integrable
Time = 0.02 (sec) , antiderivative size = 16, 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 = {} \[ \int \frac {(d x)^m}{\left (a+b \csc ^{-1}(c x)\right )^2} \, dx=\int \frac {(d x)^m}{\left (a+b \csc ^{-1}(c x)\right )^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {(d x)^m}{\left (a+b \csc ^{-1}(c x)\right )^2} \, dx \\ \end{align*}
Not integrable
Time = 1.39 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {(d x)^m}{\left (a+b \csc ^{-1}(c x)\right )^2} \, dx=\int \frac {(d x)^m}{\left (a+b \csc ^{-1}(c x)\right )^2} \, dx \]
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Not integrable
Time = 2.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00
\[\int \frac {\left (d x \right )^{m}}{\left (a +b \,\operatorname {arccsc}\left (c x \right )\right )^{2}}d x\]
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Not integrable
Time = 0.27 (sec) , antiderivative size = 32, normalized size of antiderivative = 2.00 \[ \int \frac {(d x)^m}{\left (a+b \csc ^{-1}(c x)\right )^2} \, dx=\int { \frac {\left (d x\right )^{m}}{{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}^{2}} \,d x } \]
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Not integrable
Time = 5.31 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int \frac {(d x)^m}{\left (a+b \csc ^{-1}(c x)\right )^2} \, dx=\int \frac {\left (d x\right )^{m}}{\left (a + b \operatorname {acsc}{\left (c x \right )}\right )^{2}}\, dx \]
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Not integrable
Time = 1.74 (sec) , antiderivative size = 684, normalized size of antiderivative = 42.75 \[ \int \frac {(d x)^m}{\left (a+b \csc ^{-1}(c x)\right )^2} \, dx=\int { \frac {\left (d x\right )^{m}}{{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}^{2}} \,d x } \]
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Not integrable
Time = 1.73 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {(d x)^m}{\left (a+b \csc ^{-1}(c x)\right )^2} \, dx=\int { \frac {\left (d x\right )^{m}}{{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}^{2}} \,d x } \]
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Not integrable
Time = 0.82 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.38 \[ \int \frac {(d x)^m}{\left (a+b \csc ^{-1}(c x)\right )^2} \, dx=\int \frac {{\left (d\,x\right )}^m}{{\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}^2} \,d x \]
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