Integrand size = 13, antiderivative size = 86 \[ \int x \csc \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {2 e^{i a} x^2 \left (c x^n\right )^{i b} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} \left (1-\frac {2 i}{b n}\right ),\frac {1}{2} \left (3-\frac {2 i}{b n}\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{2 i-b n} \]
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Time = 0.07 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {4606, 4602, 371} \[ \int x \csc \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {2 e^{i a} x^2 \left (c x^n\right )^{i b} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} \left (1-\frac {2 i}{b n}\right ),\frac {1}{2} \left (3-\frac {2 i}{b n}\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{-b n+2 i} \]
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Rule 371
Rule 4602
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 (a+b \log (x)) \, dx,x,c x^n\right )}{n} \\ & = -\frac {\left (2 i e^{i a} x^2 \left (c x^n\right )^{-2/n}\right ) \text {Subst}\left (\int \frac {x^{-1+i b+\frac {2}{n}}}{1-e^{2 i a} x^{2 i b}} \, dx,x,c x^n\right )}{n} \\ & = \frac {2 e^{i a} x^2 \left (c x^n\right )^{i b} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} \left (1-\frac {2 i}{b n}\right ),\frac {1}{2} \left (3-\frac {2 i}{b n}\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{2 i-b n} \\ \end{align*}
Time = 1.11 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.91 \[ \int x \csc \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {2 e^{i a} x^2 \left (c x^n\right )^{i b} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2}-\frac {i}{b n},\frac {3}{2}-\frac {i}{b n},e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )}{-2 i+b n} \]
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\[\int x \csc \left (a +b \ln \left (c \,x^{n}\right )\right )d x\]
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\[ \int x \csc \left (a+b \log \left (c x^n\right )\right ) \, dx=\int { x \csc \left (b \log \left (c x^{n}\right ) + a\right ) \,d x } \]
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\[ \int x \csc \left (a+b \log \left (c x^n\right )\right ) \, dx=\int x \csc {\left (a + b \log {\left (c x^{n} \right )} \right )}\, dx \]
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\[ \int x \csc \left (a+b \log \left (c x^n\right )\right ) \, dx=\int { x \csc \left (b \log \left (c x^{n}\right ) + a\right ) \,d x } \]
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\[ \int x \csc \left (a+b \log \left (c x^n\right )\right ) \, dx=\int { x \csc \left (b \log \left (c x^{n}\right ) + a\right ) \,d x } \]
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Timed out. \[ \int x \csc \left (a+b \log \left (c x^n\right )\right ) \, dx=\int \frac {x}{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )} \,d x \]
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