Integrand size = 19, antiderivative size = 123 \[ \int (e x)^m \csc \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {2 e^{i a d} (e x)^{1+m} \left (c x^n\right )^{i b d} \operatorname {Hypergeometric2F1}\left (1,-\frac {i+i m-b d n}{2 b d n},-\frac {i (1+m)-3 b d n}{2 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{e (i (1+m)-b d n)} \]
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Time = 0.09 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.96, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {4606, 4602, 371} \[ \int (e x)^m \csc \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {2 e^{i a d} (e x)^{m+1} \left (c x^n\right )^{i b d} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} \left (1-\frac {i (m+1)}{b d n}\right ),-\frac {i (m+1)-3 b d n}{2 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{e (-b d n+i (m+1))} \]
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Rule 371
Rule 4602
Rule 4606
Rubi steps \begin{align*} \text {integral}& = \frac {\left ((e x)^{1+m} \left (c x^n\right )^{-\frac {1+m}{n}}\right ) \text {Subst}\left (\int x^{-1+\frac {1+m}{n}} \csc (d (a+b \log (x))) \, dx,x,c x^n\right )}{e n} \\ & = -\frac {\left (2 i e^{i a d} (e x)^{1+m} \left (c x^n\right )^{-\frac {1+m}{n}}\right ) \text {Subst}\left (\int \frac {x^{-1+i b d+\frac {1+m}{n}}}{1-e^{2 i a d} x^{2 i b d}} \, dx,x,c x^n\right )}{e n} \\ & = \frac {2 e^{i a d} (e x)^{1+m} \left (c x^n\right )^{i b d} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} \left (1-\frac {i (1+m)}{b d n}\right ),-\frac {i (1+m)-3 b d n}{2 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{i (e+e m)-b d e n} \\ \end{align*}
Time = 0.29 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.47 \[ \int (e x)^m \csc \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {2 x^{1+i b d n} (e x)^m \operatorname {Hypergeometric2F1}\left (1,\frac {-i-i m+b d n}{2 b d n},-\frac {i (1+m+3 i b d n)}{2 b d n},x^{2 i b d n} \left (\cos \left (2 d \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right )+i \sin \left (2 d \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right )\right )\right ) \left (-i \cos \left (d \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right )+\sin \left (d \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right )\right )}{1+m+i b d n} \]
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\[\int \left (e x \right )^{m} \csc \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )d x\]
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\[ \int (e x)^m \csc \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { \left (e x\right )^{m} \csc \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \]
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\[ \int (e x)^m \csc \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int \left (e x\right )^{m} \csc {\left (a d + b d \log {\left (c x^{n} \right )} \right )}\, dx \]
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\[ \int (e x)^m \csc \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { \left (e x\right )^{m} \csc \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \]
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\[ \int (e x)^m \csc \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int { \left (e x\right )^{m} \csc \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) \,d x } \]
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Timed out. \[ \int (e x)^m \csc \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int \frac {{\left (e\,x\right )}^m}{\sin \left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )} \,d x \]
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