Integrand size = 14, antiderivative size = 48 \[ \int x^{-1+n} \csc ^{-1}\left (a+b x^n\right ) \, dx=\frac {\left (a+b x^n\right ) \csc ^{-1}\left (a+b x^n\right )}{b n}+\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{\left (a+b x^n\right )^2}}\right )}{b n} \]
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Time = 0.05 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6847, 5359, 379, 272, 65, 212} \[ \int x^{-1+n} \csc ^{-1}\left (a+b x^n\right ) \, dx=\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{\left (a+b x^n\right )^2}}\right )}{b n}+\frac {\left (a+b x^n\right ) \csc ^{-1}\left (a+b x^n\right )}{b n} \]
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Rule 65
Rule 212
Rule 272
Rule 379
Rule 5359
Rule 6847
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \csc ^{-1}(a+b x) \, dx,x,x^n\right )}{n} \\ & = \frac {\left (a+b x^n\right ) \csc ^{-1}\left (a+b x^n\right )}{b n}+\frac {\text {Subst}\left (\int \frac {1}{(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}} \, dx,x,x^n\right )}{n} \\ & = \frac {\left (a+b x^n\right ) \csc ^{-1}\left (a+b x^n\right )}{b n}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {1}{x^2}} x} \, dx,x,a+b x^n\right )}{b n} \\ & = \frac {\left (a+b x^n\right ) \csc ^{-1}\left (a+b x^n\right )}{b n}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {1-x} x} \, dx,x,\frac {1}{\left (a+b x^n\right )^2}\right )}{2 b n} \\ & = \frac {\left (a+b x^n\right ) \csc ^{-1}\left (a+b x^n\right )}{b n}+\frac {\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {1-\frac {1}{\left (a+b x^n\right )^2}}\right )}{b n} \\ & = \frac {\left (a+b x^n\right ) \csc ^{-1}\left (a+b x^n\right )}{b n}+\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{\left (a+b x^n\right )^2}}\right )}{b n} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(130\) vs. \(2(48)=96\).
Time = 0.20 (sec) , antiderivative size = 130, normalized size of antiderivative = 2.71 \[ \int x^{-1+n} \csc ^{-1}\left (a+b x^n\right ) \, dx=\frac {\left (a+b x^n\right ) \csc ^{-1}\left (a+b x^n\right )}{b n}+\frac {\sqrt {-1+\left (a+b x^n\right )^2} \left (-\log \left (1-\frac {a+b x^n}{\sqrt {-1+\left (a+b x^n\right )^2}}\right )+\log \left (1+\frac {a+b x^n}{\sqrt {-1+\left (a+b x^n\right )^2}}\right )\right )}{2 b n \left (a+b x^n\right ) \sqrt {1-\frac {1}{\left (a+b x^n\right )^2}}} \]
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\[\int x^{-1+n} \operatorname {arccsc}\left (a +b \,x^{n}\right )d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (46) = 92\).
Time = 0.30 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.96 \[ \int x^{-1+n} \csc ^{-1}\left (a+b x^n\right ) \, dx=\frac {b x^{n} \operatorname {arccsc}\left (b x^{n} + a\right ) - 2 \, a \arctan \left (-b x^{n} - a + \sqrt {b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2} - 1}\right ) - \log \left (-b x^{n} - a + \sqrt {b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2} - 1}\right )}{b n} \]
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Timed out. \[ \int x^{-1+n} \csc ^{-1}\left (a+b x^n\right ) \, dx=\text {Timed out} \]
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none
Time = 0.18 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.38 \[ \int x^{-1+n} \csc ^{-1}\left (a+b x^n\right ) \, dx=\frac {2 \, {\left (b x^{n} + a\right )} \operatorname {arccsc}\left (b x^{n} + a\right ) + \log \left (\sqrt {-\frac {1}{{\left (b x^{n} + a\right )}^{2}} + 1} + 1\right ) - \log \left (-\sqrt {-\frac {1}{{\left (b x^{n} + a\right )}^{2}} + 1} + 1\right )}{2 \, b n} \]
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Time = 0.33 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.54 \[ \int x^{-1+n} \csc ^{-1}\left (a+b x^n\right ) \, dx=\frac {b {\left (\frac {2 \, {\left (b x^{n} + a\right )} \arcsin \left (\frac {1}{b x^{n} + a}\right )}{b^{2}} + \frac {\log \left (\sqrt {-\frac {1}{{\left (b x^{n} + a\right )}^{2}} + 1} + 1\right ) - \log \left (-\sqrt {-\frac {1}{{\left (b x^{n} + a\right )}^{2}} + 1} + 1\right )}{b^{2}}\right )}}{2 \, n} \]
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Time = 0.89 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.88 \[ \int x^{-1+n} \csc ^{-1}\left (a+b x^n\right ) \, dx=\frac {\mathrm {atanh}\left (\frac {1}{\sqrt {1-\frac {1}{{\left (a+b\,x^n\right )}^2}}}\right )+\mathrm {asin}\left (\frac {1}{a+b\,x^n}\right )\,\left (a+b\,x^n\right )}{b\,n} \]
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