Integrand size = 15, antiderivative size = 110 \[ \int \frac {1}{\csc ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2 x \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1}{4} \left (-5-\frac {2 i}{b n}\right ),-\frac {2 i+b n}{4 b n},e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2-5 i b n) \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{5/2} \csc ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \]
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Time = 0.09 (sec) , antiderivative size = 110, 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} \[ \int \frac {1}{\csc ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2 x \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1}{4} \left (-5-\frac {2 i}{b n}\right ),-\frac {b n+2 i}{4 b n},e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2-5 i b n) \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{5/2} \csc ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \]
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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 \frac {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}}\right ) \text {Subst}\left (\int 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 \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 )} \\ & = \frac {2 x \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1}{4} \left (-5-\frac {2 i}{b n}\right ),-\frac {2 i+b n}{4 b n},e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2-5 i b 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 )} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(579\) vs. \(2(110)=220\).
Time = 7.48 (sec) , antiderivative size = 579, normalized size of antiderivative = 5.26 \[ \int \frac {1}{\csc ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {x \left (-\frac {60 b^3 e^{i a} n^3 \left (c x^n\right )^{i b} \sqrt {2-2 e^{2 i a} \left (c x^n\right )^{2 i b}} \sqrt {\frac {i e^{i a} \left (c x^n\right )^{i b}}{-1+e^{2 i a} \left (c x^n\right )^{2 i b}}} \left ((2 i+b n) x^{2 i b n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4}-\frac {i}{2 b n},\frac {7}{4}-\frac {i}{2 b n},e^{2 i a} \left (c x^n\right )^{2 i b}\right )+(-2 i+3 b n) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {2 i+b n}{4 b n},\frac {3}{4}-\frac {i}{2 b n},e^{2 i a} \left (c x^n\right )^{2 i b}\right )\right )}{(2 i+b n) (-2 i+3 b n) \left ((2 i+b n) x^{2 i b n}+e^{2 i a} (-2 i+b n) \left (c x^n\right )^{2 i b}\right )}+\frac {4 b n \cos \left (a-b n \log (x)+b \log \left (c x^n\right )\right )-12 b n \cos \left (a+b n \log (x)+b \log \left (c x^n\right )\right )+8 b n \cos \left (b n \log (x)-3 \left (a+b \log \left (c x^n\right )\right )\right )+8 \sin \left (a-b n \log (x)+b \log \left (c x^n\right )\right )+60 b^2 n^2 \sin \left (a-b n \log (x)+b \log \left (c x^n\right )\right )+4 \sin \left (a+b n \log (x)+b \log \left (c x^n\right )\right )-5 b^2 n^2 \sin \left (a+b n \log (x)+b \log \left (c x^n\right )\right )-4 \sin \left (3 a-b n \log (x)+3 b \log \left (c x^n\right )\right )-5 b^2 n^2 \sin \left (3 a-b n \log (x)+3 b \log \left (c x^n\right )\right )}{\sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \left (b n \cos \left (a-b n \log (x)+b \log \left (c x^n\right )\right )+2 \sin \left (a-b n \log (x)+b \log \left (c x^n\right )\right )\right )}\right )}{2 \left (4+25 b^2 n^2\right )} \]
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\[\int \frac {1}{{\csc \left (a +b \ln \left (c \,x^{n}\right )\right )}^{\frac {5}{2}}}d x\]
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Exception generated. \[ \int \frac {1}{\csc ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {1}{\csc ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int \frac {1}{\csc ^{\frac {5}{2}}{\left (a + b \log {\left (c x^{n} \right )} \right )}}\, dx \]
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\[ \int \frac {1}{\csc ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int { \frac {1}{\csc \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {1}{\csc ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int { \frac {1}{\csc \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\csc ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int \frac {1}{{\left (\frac {1}{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}^{5/2}} \,d x \]
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