Integrand size = 15, antiderivative size = 109 \[ \int \csc ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {2 x \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{5/2} \csc ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right ) \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {1}{4} \left (5-\frac {2 i}{b n}\right ),\frac {1}{4} \left (9-\frac {2 i}{b n}\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{2+5 i b n} \]
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Time = 0.09 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4600, 4604, 371} \[ \int \csc ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {2 x \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{5/2} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {1}{4} \left (5-\frac {2 i}{b n}\right ),\frac {1}{4} \left (9-\frac {2 i}{b n}\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \csc ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{2+5 i b n} \]
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Rule 371
Rule 4600
Rule 4604
Rubi steps \begin{align*} \text {integral}& = \frac {\left (x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int x^{-1+\frac {1}{n}} \csc ^{\frac {5}{2}}(a+b \log (x)) \, dx,x,c x^n\right )}{n} \\ & = \frac {\left (x \left (c x^n\right )^{-\frac {5 i b}{2}-\frac {1}{n}} \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{5/2} \csc ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )\right ) \text {Subst}\left (\int \frac {x^{-1+\frac {5 i b}{2}+\frac {1}{n}}}{\left (1-e^{2 i a} x^{2 i b}\right )^{5/2}} \, dx,x,c x^n\right )}{n} \\ & = \frac {2 x \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{5/2} \csc ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right ) \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {1}{4} \left (5-\frac {2 i}{b n}\right ),\frac {1}{4} \left (9-\frac {2 i}{b n}\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{2+5 i b n} \\ \end{align*}
Time = 1.17 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.60 \[ \int \csc ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {2 e^{-2 i \left (a-b n \log (x)+b \log \left (c x^n\right )\right )} x^{1-2 i b n} \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \left (-e^{2 i a} \left (c x^n\right )^{2 i b} \left (2+b n \cot \left (a+b \log \left (c x^n\right )\right )\right )+(2+i b n) \left (-1+e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {3}{4}+\frac {i}{2 b n},\frac {5}{4}+\frac {i}{2 b n},e^{-2 i \left (a+b \log \left (c x^n\right )\right )}\right )\right )}{3 b^2 n^2} \]
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\[\int {\csc \left (a +b \ln \left (c \,x^{n}\right )\right )}^{\frac {5}{2}}d x\]
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Exception generated. \[ \int \csc ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \csc ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Timed out} \]
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\[ \int \csc ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx=\int { \csc \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {5}{2}} \,d x } \]
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Timed out. \[ \int \csc ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Timed out} \]
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Timed out. \[ \int \csc ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx=\int {\left (\frac {1}{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}^{5/2} \,d x \]
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