Integrand size = 17, antiderivative size = 115 \[ \int \frac {\sin ^p\left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=-\frac {\left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (\frac {i}{b n}-p\right ),-p,\frac {1}{2} \left (2+\frac {i}{b n}-p\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sin ^p\left (a+b \log \left (c x^n\right )\right )}{(1+i b n p) x} \]
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Time = 0.12 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {4581, 4579, 371} \[ \int \frac {\sin ^p\left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=-\frac {\left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (\frac {i}{b n}-p\right ),-p,\frac {1}{2} \left (-p+\frac {i}{b n}+2\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sin ^p\left (a+b \log \left (c x^n\right )\right )}{x (1+i b n p)} \]
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Rule 371
Rule 4579
Rule 4581
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c x^n\right )^{\frac {1}{n}} \text {Subst}\left (\int x^{-1-\frac {1}{n}} \sin ^p(a+b \log (x)) \, dx,x,c x^n\right )}{n x} \\ & = \frac {\left (\left (c x^n\right )^{\frac {1}{n}+i b p} \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{-p} \sin ^p\left (a+b \log \left (c x^n\right )\right )\right ) \text {Subst}\left (\int x^{-1-\frac {1}{n}-i b p} \left (1-e^{2 i a} x^{2 i b}\right )^p \, dx,x,c x^n\right )}{n x} \\ & = -\frac {\left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (\frac {i}{b n}-p\right ),-p,\frac {1}{2} \left (2+\frac {i}{b n}-p\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sin ^p\left (a+b \log \left (c x^n\right )\right )}{(1+i b n p) x} \\ \end{align*}
Time = 0.56 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.27 \[ \int \frac {\sin ^p\left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\frac {\left (2-2 e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{-p} \left (-i e^{-i a} \left (c x^n\right )^{-i b} \left (-1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )\right )^p \operatorname {Hypergeometric2F1}\left (\frac {i}{2 b n}-\frac {p}{2},-p,1+\frac {i}{2 b n}-\frac {p}{2},e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(-1-i b n p) x} \]
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\[\int \frac {{\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}^{p}}{x^{2}}d x\]
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\[ \int \frac {\sin ^p\left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\int { \frac {\sin \left (b \log \left (c x^{n}\right ) + a\right )^{p}}{x^{2}} \,d x } \]
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\[ \int \frac {\sin ^p\left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\int \frac {\sin ^{p}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x^{2}}\, dx \]
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\[ \int \frac {\sin ^p\left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\int { \frac {\sin \left (b \log \left (c x^{n}\right ) + a\right )^{p}}{x^{2}} \,d x } \]
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\[ \int \frac {\sin ^p\left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\int { \frac {\sin \left (b \log \left (c x^{n}\right ) + a\right )^{p}}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\sin ^p\left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\int \frac {{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}^p}{x^2} \,d x \]
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