Integrand size = 23, antiderivative size = 150 \[ \int \frac {(e x)^m}{\sin ^{\frac {3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )} \, dx=\frac {2 (e x)^{1+m} \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{3/2} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-\frac {2 i+2 i m-3 b d n}{4 b d n},-\frac {2 i+2 i m-7 b d n}{4 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{e (2+2 m+3 i b d n) \sin ^{\frac {3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )} \]
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Time = 0.14 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.97, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {4581, 4579, 371} \[ \int \frac {(e x)^m}{\sin ^{\frac {3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )} \, dx=\frac {2 (e x)^{m+1} \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{3/2} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {1}{4} \left (3-\frac {2 i (m+1)}{b d n}\right ),-\frac {2 i m-7 b d n+2 i}{4 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{e (3 i b d n+2 m+2) \sin ^{\frac {3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )} \]
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Rule 371
Rule 4579
Rule 4581
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 \frac {x^{-1+\frac {1+m}{n}}}{\sin ^{\frac {3}{2}}(d (a+b \log (x)))} \, dx,x,c x^n\right )}{e n} \\ & = \frac {\left ((e x)^{1+m} \left (c x^n\right )^{-\frac {3}{2} i b d-\frac {1+m}{n}} \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{3/2}\right ) \text {Subst}\left (\int \frac {x^{-1+\frac {3 i b d}{2}+\frac {1+m}{n}}}{\left (1-e^{2 i a d} x^{2 i b d}\right )^{3/2}} \, dx,x,c x^n\right )}{e n \sin ^{\frac {3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )} \\ & = \frac {2 (e x)^{1+m} \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{3/2} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {1}{4} \left (3-\frac {2 i (1+m)}{b d n}\right ),-\frac {2 i+2 i m-7 b d n}{4 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{e (2+2 m+3 i b d n) \sin ^{\frac {3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(544\) vs. \(2(150)=300\).
Time = 3.99 (sec) , antiderivative size = 544, normalized size of antiderivative = 3.63 \[ \int \frac {(e x)^m}{\sin ^{\frac {3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )} \, dx=\frac {\left (4+8 m+4 m^2+b^2 d^2 n^2\right ) x^{1+i b d n} (e x)^m \sqrt {2-2 e^{2 i a d} \left (c x^n\right )^{2 i b d}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {i \left (1+m+\frac {3}{2} i b d n\right )}{2 b d n},-\frac {2 i+2 i m-7 b d n}{4 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )+\frac {(-2 i-2 i m+3 b d n) x^{1-i b d n} (e x)^m \left (-2 x^{i b d n} \sqrt {-i e^{-i a d} \left (c x^n\right )^{-i b d} \left (-1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )} (b d n \cos (b d n \log (x))-2 (1+m) \sin (b d n \log (x)))+(-2 i-2 i m+b d n) \sqrt {2-2 e^{2 i a d} \left (c x^n\right )^{2 i b d}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {2 i+2 i m+b d n}{4 b d n},-\frac {2 i+2 i m-3 b d n}{4 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \sqrt {\sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}\right )}{\sqrt {\sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}}}{b d n (-2 i-2 i m+3 b d n) \sqrt {-i e^{-i a d} \left (c x^n\right )^{-i b d} \left (-1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )} \left (b d n \cos \left (d \left (a-b n \log (x)+b \log \left (c x^n\right )\right )\right )+2 (1+m) \sin \left (d \left (a-b n \log (x)+b \log \left (c x^n\right )\right )\right )\right )} \]
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\[\int \frac {\left (e x \right )^{m}}{{\sin \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}^{\frac {3}{2}}}d x\]
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Exception generated. \[ \int \frac {(e x)^m}{\sin ^{\frac {3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {(e x)^m}{\sin ^{\frac {3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )} \, dx=\int \frac {\left (e x\right )^{m}}{\sin ^{\frac {3}{2}}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}}\, dx \]
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\[ \int \frac {(e x)^m}{\sin ^{\frac {3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )} \, dx=\int { \frac {\left (e x\right )^{m}}{\sin \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(e x)^m}{\sin ^{\frac {3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(e x)^m}{\sin ^{\frac {3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )} \, dx=\int \frac {{\left (e\,x\right )}^m}{{\sin \left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}^{3/2}} \,d x \]
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