Integrand size = 19, antiderivative size = 130 \[ \int x^m \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2 x^{1+m} \sqrt {1-e^{2 i a} \left (c x^n\right )^{2 i b}} \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {2 i+2 i m-b n}{4 b n},-\frac {2 i+2 i m-5 b n}{4 b n},e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{2+2 m+i b n} \]
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Time = 0.11 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {4606, 4604, 371} \[ \int x^m \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2 x^{m+1} \sqrt {1-e^{2 i a} \left (c x^n\right )^{2 i b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {2 i m-b n+2 i}{4 b n},-\frac {2 i m-5 b n+2 i}{4 b n},e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}}{i b n+2 m+2} \]
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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 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}} \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 )}\right ) \text {Subst}\left (\int \frac {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} \\ & = \frac {2 x^{1+m} \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 )} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {2 i+2 i m-b n}{4 b n},-\frac {2 i+2 i m-5 b n}{4 b n},e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{2+2 m+i b n} \\ \end{align*}
Time = 0.69 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.06 \[ \int x^m \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2 e^{-2 i a} x^{1+m} \left (c x^n\right )^{-2 i b} \left (-1+e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \operatorname {Hypergeometric2F1}\left (1,\frac {2 i+2 i m+3 b n}{4 b n},\frac {2 i+2 i m+5 b n}{4 b n},e^{-2 i \left (a+b \log \left (c x^n\right )\right )}\right )}{2+2 m-i b n} \]
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\[\int x^{m} \sqrt {\csc \left (a +b \ln \left (c \,x^{n}\right )\right )}d x\]
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Exception generated. \[ \int x^m \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \, dx=\text {Exception raised: TypeError} \]
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\[ \int x^m \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \, dx=\int x^{m} \sqrt {\csc {\left (a + b \log {\left (c x^{n} \right )} \right )}}\, dx \]
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\[ \int x^m \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \, dx=\int { x^{m} \sqrt {\csc \left (b \log \left (c x^{n}\right ) + a\right )} \,d x } \]
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\[ \int x^m \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \, dx=\int { x^{m} \sqrt {\csc \left (b \log \left (c x^{n}\right ) + a\right )} \,d x } \]
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Timed out. \[ \int x^m \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \, dx=\int x^m\,\sqrt {\frac {1}{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}} \,d x \]
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