Integrand size = 19, antiderivative size = 111 \[ \int \frac {\sin ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=-\frac {2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4} \left (-3+\frac {4 i}{b n}\right ),\frac {1}{4} \left (1+\frac {4 i}{b n}\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sin ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{(4+3 i b n) x^2 \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2}} \]
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Time = 0.10 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {4581, 4579, 371} \[ \int \frac {\sin ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=-\frac {2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4} \left (\frac {4 i}{b n}-3\right ),\frac {1}{4} \left (1+\frac {4 i}{b n}\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sin ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x^2 (4+3 i b n) \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2}} \]
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Rule 371
Rule 4579
Rule 4581
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c x^n\right )^{2/n} \text {Subst}\left (\int x^{-1-\frac {2}{n}} \sin ^{\frac {3}{2}}(a+b \log (x)) \, dx,x,c x^n\right )}{n x^2} \\ & = \frac {\left (\left (c x^n\right )^{\frac {3 i b}{2}+\frac {2}{n}} \sin ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )\right ) \text {Subst}\left (\int x^{-1-\frac {3 i b}{2}-\frac {2}{n}} \left (1-e^{2 i a} x^{2 i b}\right )^{3/2} \, dx,x,c x^n\right )}{n x^2 \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2}} \\ & = -\frac {2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4} \left (-3+\frac {4 i}{b n}\right ),\frac {1}{4} \left (1+\frac {4 i}{b n}\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sin ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{(4+3 i b n) x^2 \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2}} \\ \end{align*}
Time = 0.93 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.95 \[ \int \frac {\sin ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\frac {6 b^2 \sqrt {2-2 e^{2 i \left (a+b \log \left (c x^n\right )\right )}} n^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4}+\frac {i}{b n},\frac {5}{4}+\frac {i}{b n},e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {-i e^{-i \left (a+b \log \left (c x^n\right )\right )} \left (-1+e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )} (4+3 i b n) (4 i+b n) (4 i+3 b n) x^2}-\frac {2 \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )} \left (3 b n \cos \left (a+b \log \left (c x^n\right )\right )+4 \sin \left (a+b \log \left (c x^n\right )\right )\right )}{\left (16+9 b^2 n^2\right ) x^2} \]
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\[\int \frac {{\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}^{\frac {3}{2}}}{x^{3}}d x\]
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Exception generated. \[ \int \frac {\sin ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {\sin ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\int \frac {\sin ^{\frac {3}{2}}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x^{3}}\, dx \]
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\[ \int \frac {\sin ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\int { \frac {\sin \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}}}{x^{3}} \,d x } \]
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\[ \int \frac {\sin ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\int { \frac {\sin \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}}}{x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\sin ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\int \frac {{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}^{3/2}}{x^3} \,d x \]
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