Integrand size = 13, antiderivative size = 112 \[ \int \cos ^p\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{-p} \cos ^p\left (a+b \log \left (c x^n\right )\right ) \operatorname {Hypergeometric2F1}\left (-p,-\frac {i+b n p}{2 b n},\frac {1}{2} \left (2-\frac {i}{b n}-p\right ),-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{1-i b n p} \]
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Time = 0.08 (sec) , antiderivative size = 112, 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 = {4572, 4580, 371} \[ \int \cos ^p\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,-\frac {b n p+i}{2 b n},\frac {1}{2} \left (-p-\frac {i}{b n}+2\right ),-e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \cos ^p\left (a+b \log \left (c x^n\right )\right )}{1-i b n p} \]
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Rule 371
Rule 4572
Rule 4580
Rubi steps \begin{align*} \text {integral}& = \frac {\left (x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int x^{-1+\frac {1}{n}} \cos ^p(a+b \log (x)) \, dx,x,c x^n\right )}{n} \\ & = \frac {\left (x \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} \cos ^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} \\ & = \frac {x \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{-p} \cos ^p\left (a+b \log \left (c x^n\right )\right ) \operatorname {Hypergeometric2F1}\left (-p,-\frac {i+b n p}{2 b n},\frac {1}{2} \left (2-\frac {i}{b n}-p\right ),-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{1-i b n p} \\ \end{align*}
Time = 0.58 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.28 \[ \int \cos ^p\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {i x \left (e^{-i a} \left (c x^n\right )^{-i b}+e^{i a} \left (c x^n\right )^{i b}\right )^p \left (2+2 e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,-\frac {i+b n p}{2 b n},1-\frac {i}{2 b n}-\frac {p}{2},-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{i+b n p} \]
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\[\int {\cos \left (a +b \ln \left (c \,x^{n}\right )\right )}^{p}d x\]
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\[ \int \cos ^p\left (a+b \log \left (c x^n\right )\right ) \, dx=\int { \cos \left (b \log \left (c x^{n}\right ) + a\right )^{p} \,d x } \]
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\[ \int \cos ^p\left (a+b \log \left (c x^n\right )\right ) \, dx=\int \cos ^{p}{\left (a + b \log {\left (c x^{n} \right )} \right )}\, dx \]
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\[ \int \cos ^p\left (a+b \log \left (c x^n\right )\right ) \, dx=\int { \cos \left (b \log \left (c x^{n}\right ) + a\right )^{p} \,d x } \]
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\[ \int \cos ^p\left (a+b \log \left (c x^n\right )\right ) \, dx=\int { \cos \left (b \log \left (c x^{n}\right ) + a\right )^{p} \,d x } \]
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Timed out. \[ \int \cos ^p\left (a+b \log \left (c x^n\right )\right ) \, dx=\int {\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}^p \,d x \]
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