Integrand size = 14, antiderivative size = 66 \[ \int (d x)^m \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {(d x)^{1+m} \left (a+b \csc ^{-1}(c x)\right )}{d (1+m)}+\frac {b (d x)^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {m}{2},1-\frac {m}{2},\frac {1}{c^2 x^2}\right )}{c m (1+m)} \]
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Time = 0.03 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {5329, 346, 371} \[ \int (d x)^m \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {(d x)^{m+1} \left (a+b \csc ^{-1}(c x)\right )}{d (m+1)}+\frac {b (d x)^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {m}{2},1-\frac {m}{2},\frac {1}{c^2 x^2}\right )}{c m (m+1)} \]
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Rule 346
Rule 371
Rule 5329
Rubi steps \begin{align*} \text {integral}& = \frac {(d x)^{1+m} \left (a+b \csc ^{-1}(c x)\right )}{d (1+m)}+\frac {(b d) \int \frac {(d x)^{-1+m}}{\sqrt {1-\frac {1}{c^2 x^2}}} \, dx}{c (1+m)} \\ & = \frac {(d x)^{1+m} \left (a+b \csc ^{-1}(c x)\right )}{d (1+m)}-\frac {\left (b \left (\frac {1}{x}\right )^m (d x)^m\right ) \text {Subst}\left (\int \frac {x^{-1-m}}{\sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{c (1+m)} \\ & = \frac {(d x)^{1+m} \left (a+b \csc ^{-1}(c x)\right )}{d (1+m)}+\frac {b (d x)^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {m}{2},1-\frac {m}{2},\frac {1}{c^2 x^2}\right )}{c m (1+m)} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.26 \[ \int (d x)^m \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {(d x)^m \left ((1+m) x \left (a+b \csc ^{-1}(c x)\right )+\frac {b \sqrt {1-c^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},c^2 x^2\right )}{c \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{(1+m)^2} \]
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\[\int \left (d x \right )^{m} \left (a +b \,\operatorname {arccsc}\left (c x \right )\right )d x\]
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\[ \int (d x)^m \left (a+b \csc ^{-1}(c x)\right ) \, dx=\int { {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} \left (d x\right )^{m} \,d x } \]
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\[ \int (d x)^m \left (a+b \csc ^{-1}(c x)\right ) \, dx=\int \left (d x\right )^{m} \left (a + b \operatorname {acsc}{\left (c x \right )}\right )\, dx \]
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\[ \int (d x)^m \left (a+b \csc ^{-1}(c x)\right ) \, dx=\int { {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} \left (d x\right )^{m} \,d x } \]
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\[ \int (d x)^m \left (a+b \csc ^{-1}(c x)\right ) \, dx=\int { {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} \left (d x\right )^{m} \,d x } \]
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Timed out. \[ \int (d x)^m \left (a+b \csc ^{-1}(c x)\right ) \, dx=\int {\left (d\,x\right )}^m\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right ) \,d x \]
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