Integrand size = 15, antiderivative size = 106 \[ \int x \csc ^p\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x^2 \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^p \csc ^p\left (a+b \log \left (c x^n\right )\right ) \operatorname {Hypergeometric2F1}\left (p,\frac {1}{2} \left (-\frac {2 i}{b n}+p\right ),\frac {1}{2} \left (2-\frac {2 i}{b n}+p\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{2+i b n p} \]
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Time = 0.10 (sec) , antiderivative size = 106, 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, 4604, 371} \[ \int x \csc ^p\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x^2 \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^p \operatorname {Hypergeometric2F1}\left (p,\frac {1}{2} \left (p-\frac {2 i}{b n}\right ),\frac {1}{2} \left (p-\frac {2 i}{b n}+2\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \csc ^p\left (a+b \log \left (c x^n\right )\right )}{2+i b n p} \]
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Rule 371
Rule 4604
Rule 4606
Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^2 \left (c x^n\right )^{-2/n}\right ) \text {Subst}\left (\int x^{-1+\frac {2}{n}} \csc ^p(a+b \log (x)) \, dx,x,c x^n\right )}{n} \\ & = \frac {\left (x^2 \left (c x^n\right )^{-\frac {2}{n}-i b p} \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^p \csc ^p\left (a+b \log \left (c x^n\right )\right )\right ) \text {Subst}\left (\int x^{-1+\frac {2}{n}+i b p} \left (1-e^{2 i a} x^{2 i b}\right )^{-p} \, dx,x,c x^n\right )}{n} \\ & = \frac {x^2 \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^p \csc ^p\left (a+b \log \left (c x^n\right )\right ) \operatorname {Hypergeometric2F1}\left (p,\frac {1}{2} \left (-\frac {2 i}{b n}+p\right ),\frac {1}{2} \left (2-\frac {2 i}{b n}+p\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{2+i b n p} \\ \end{align*}
Time = 0.95 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.34 \[ \int x \csc ^p\left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {i x^2 \left (2-2 e^{2 i a} \left (c x^n\right )^{2 i b}\right )^p \left (\frac {i e^{i a} \left (c x^n\right )^{i b}}{-1+e^{2 i a} \left (c x^n\right )^{2 i b}}\right )^p \operatorname {Hypergeometric2F1}\left (-\frac {i}{b n}+\frac {p}{2},p,1-\frac {i}{b n}+\frac {p}{2},e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{-2 i+b n p} \]
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\[\int x {\csc \left (a +b \ln \left (c \,x^{n}\right )\right )}^{p}d x\]
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\[ \int x \csc ^p\left (a+b \log \left (c x^n\right )\right ) \, dx=\int { x \csc \left (b \log \left (c x^{n}\right ) + a\right )^{p} \,d x } \]
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\[ \int x \csc ^p\left (a+b \log \left (c x^n\right )\right ) \, dx=\int x \csc ^{p}{\left (a + b \log {\left (c x^{n} \right )} \right )}\, dx \]
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\[ \int x \csc ^p\left (a+b \log \left (c x^n\right )\right ) \, dx=\int { x \csc \left (b \log \left (c x^{n}\right ) + a\right )^{p} \,d x } \]
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\[ \int x \csc ^p\left (a+b \log \left (c x^n\right )\right ) \, dx=\int { x \csc \left (b \log \left (c x^{n}\right ) + a\right )^{p} \,d x } \]
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Timed out. \[ \int x \csc ^p\left (a+b \log \left (c x^n\right )\right ) \, dx=\int x\,{\left (\frac {1}{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}^p \,d x \]
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