Integrand size = 19, antiderivative size = 129 \[ \int \frac {x^m}{\sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}} \, dx=\frac {2 x^{1+m} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {2 i+2 i m+b n}{4 b n},-\frac {2 i+2 i m-3 b n}{4 b n},e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2+2 m-i b n) \sqrt {1-e^{2 i a} \left (c x^n\right )^{2 i b}} \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}} \]
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Time = 0.11 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.98, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {4606, 4604, 371} \[ \int \frac {x^m}{\sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}} \, dx=\frac {2 x^{m+1} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4} \left (-\frac {2 i (m+1)}{b n}-1\right ),-\frac {2 i m-3 b n+2 i}{4 b n},e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(-i b n+2 m+2) \sqrt {1-e^{2 i a} \left (c x^n\right )^{2 i b}} \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}} \]
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Rule 371
Rule 4604
Rule 4606
Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{1+m} \left (c x^n\right )^{-\frac {1+m}{n}}\right ) \text {Subst}\left (\int \frac {x^{-1+\frac {1+m}{n}}}{\sqrt {\csc (a+b \log (x))}} \, dx,x,c x^n\right )}{n} \\ & = \frac {\left (x^{1+m} \left (c x^n\right )^{\frac {i b}{2}-\frac {1+m}{n}}\right ) \text {Subst}\left (\int x^{-1-\frac {i b}{2}+\frac {1+m}{n}} \sqrt {1-e^{2 i a} x^{2 i b}} \, dx,x,c x^n\right )}{n \sqrt {1-e^{2 i a} \left (c x^n\right )^{2 i b}} \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}} \\ & = \frac {2 x^{1+m} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4} \left (-1-\frac {2 i (1+m)}{b n}\right ),-\frac {2 i+2 i m-3 b n}{4 b n},e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2+2 m-i b n) \sqrt {1-e^{2 i a} \left (c x^n\right )^{2 i b}} \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(441\) vs. \(2(129)=258\).
Time = 5.48 (sec) , antiderivative size = 441, normalized size of antiderivative = 3.42 \[ \int \frac {x^m}{\sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}} \, dx=-\frac {2 b e^{i a} n x^{1+m} \left (c x^n\right )^{i b} \sqrt {2-2 e^{2 i a} \left (c x^n\right )^{2 i b}} \sqrt {\frac {i e^{i a} \left (c x^n\right )^{i b}}{-1+e^{2 i a} \left (c x^n\right )^{2 i b}}} \left ((2 i+2 i m+b n) x^{2 i b n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {i \left (1+m+\frac {3 i b n}{2}\right )}{2 b n},-\frac {2 i+2 i m-7 b n}{4 b n},e^{2 i a} \left (c x^n\right )^{2 i b}\right )+(-2 i-2 i m+3 b n) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {2 i+2 i m+b n}{4 b n},-\frac {2 i+2 i m-3 b n}{4 b n},e^{2 i a} \left (c x^n\right )^{2 i b}\right )\right )}{(2+2 m-i b n) (2+2 m+3 i b n) \left ((2 i+2 i m+b n) x^{2 i b n}+e^{2 i a} (-2 i-2 i m+b n) \left (c x^n\right )^{2 i b}\right )}+\frac {2 x^{1+m} \sin \left (a-b n \log (x)+b \log \left (c x^n\right )\right )}{\sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \left (b n \cos \left (a-b n \log (x)+b \log \left (c x^n\right )\right )+2 (1+m) \sin \left (a-b n \log (x)+b \log \left (c x^n\right )\right )\right )} \]
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\[\int \frac {x^{m}}{\sqrt {\csc \left (a +b \ln \left (c \,x^{n}\right )\right )}}d x\]
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Exception generated. \[ \int \frac {x^m}{\sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {x^m}{\sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}} \, dx=\int \frac {x^{m}}{\sqrt {\csc {\left (a + b \log {\left (c x^{n} \right )} \right )}}}\, dx \]
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\[ \int \frac {x^m}{\sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}} \, dx=\int { \frac {x^{m}}{\sqrt {\csc \left (b \log \left (c x^{n}\right ) + a\right )}} \,d x } \]
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\[ \int \frac {x^m}{\sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}} \, dx=\int { \frac {x^{m}}{\sqrt {\csc \left (b \log \left (c x^{n}\right ) + a\right )}} \,d x } \]
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Timed out. \[ \int \frac {x^m}{\sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}} \, dx=\int \frac {x^m}{\sqrt {\frac {1}{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}}} \,d x \]
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