Integrand size = 15, antiderivative size = 86 \[ \int x^2 \csc \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {2 e^{i a} x^3 \left (c x^n\right )^{i b} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} \left (1-\frac {3 i}{b n}\right ),\frac {3}{2} \left (1-\frac {i}{b n}\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{3 i-b n} \]
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Time = 0.08 (sec) , antiderivative size = 86, 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 = {4606, 4602, 371} \[ \int x^2 \csc \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {2 e^{i a} x^3 \left (c x^n\right )^{i b} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} \left (1-\frac {3 i}{b n}\right ),\frac {3}{2} \left (1-\frac {i}{b n}\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{-b n+3 i} \]
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Rule 371
Rule 4602
Rule 4606
Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^3 \left (c x^n\right )^{-3/n}\right ) \text {Subst}\left (\int x^{-1+\frac {3}{n}} \csc (a+b \log (x)) \, dx,x,c x^n\right )}{n} \\ & = -\frac {\left (2 i e^{i a} x^3 \left (c x^n\right )^{-3/n}\right ) \text {Subst}\left (\int \frac {x^{-1+i b+\frac {3}{n}}}{1-e^{2 i a} x^{2 i b}} \, dx,x,c x^n\right )}{n} \\ & = \frac {2 e^{i a} x^3 \left (c x^n\right )^{i b} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} \left (1-\frac {3 i}{b n}\right ),\frac {3}{2} \left (1-\frac {i}{b n}\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{3 i-b n} \\ \end{align*}
Time = 1.15 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.95 \[ \int x^2 \csc \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {2 e^{i a} x^3 \left (c x^n\right )^{i b} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2}-\frac {3 i}{2 b n},\frac {3}{2}-\frac {3 i}{2 b n},e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )}{-3 i+b n} \]
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\[\int x^{2} \csc \left (a +b \ln \left (c \,x^{n}\right )\right )d x\]
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\[ \int x^2 \csc \left (a+b \log \left (c x^n\right )\right ) \, dx=\int { x^{2} \csc \left (b \log \left (c x^{n}\right ) + a\right ) \,d x } \]
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\[ \int x^2 \csc \left (a+b \log \left (c x^n\right )\right ) \, dx=\int x^{2} \csc {\left (a + b \log {\left (c x^{n} \right )} \right )}\, dx \]
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\[ \int x^2 \csc \left (a+b \log \left (c x^n\right )\right ) \, dx=\int { x^{2} \csc \left (b \log \left (c x^{n}\right ) + a\right ) \,d x } \]
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\[ \int x^2 \csc \left (a+b \log \left (c x^n\right )\right ) \, dx=\int { x^{2} \csc \left (b \log \left (c x^{n}\right ) + a\right ) \,d x } \]
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Timed out. \[ \int x^2 \csc \left (a+b \log \left (c x^n\right )\right ) \, dx=\int \frac {x^2}{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )} \,d x \]
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