Integrand size = 15, antiderivative size = 103 \[ \int x^m \sec \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {2 e^{i a} x^{1+m} \left (c x^n\right )^{i b} \operatorname {Hypergeometric2F1}\left (1,-\frac {i+i m-b n}{2 b n},-\frac {i (1+m)-3 b n}{2 b n},-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{1+m+i b n} \]
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Time = 0.08 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.96, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4605, 4601, 371} \[ \int x^m \sec \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {2 e^{i a} x^{m+1} \left (c x^n\right )^{i b} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} \left (1-\frac {i (m+1)}{b n}\right ),-\frac {i (m+1)-3 b n}{2 b n},-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{i b n+m+1} \]
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Rule 371
Rule 4601
Rule 4605
Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{1+m} \left (c x^n\right )^{-\frac {1+m}{n}}\right ) \text {Subst}\left (\int x^{-1+\frac {1+m}{n}} \sec (a+b \log (x)) \, dx,x,c x^n\right )}{n} \\ & = \frac {\left (2 e^{i a} x^{1+m} \left (c x^n\right )^{-\frac {1+m}{n}}\right ) \text {Subst}\left (\int \frac {x^{-1+i b+\frac {1+m}{n}}}{1+e^{2 i a} x^{2 i b}} \, dx,x,c x^n\right )}{n} \\ & = \frac {2 e^{i a} x^{1+m} \left (c x^n\right )^{i b} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} \left (1-\frac {i (1+m)}{b n}\right ),-\frac {i (1+m)-3 b n}{2 b n},-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{1+m+i b n} \\ \end{align*}
Time = 1.38 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.96 \[ \int x^m \sec \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {2 e^{i a} x^{1+m} \left (c x^n\right )^{i b} \operatorname {Hypergeometric2F1}\left (1,\frac {-i-i m+b n}{2 b n},-\frac {i (1+m+3 i b n)}{2 b n},-e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )}{1+m+i b n} \]
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\[\int x^{m} \sec \left (a +b \ln \left (c \,x^{n}\right )\right )d x\]
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\[ \int x^m \sec \left (a+b \log \left (c x^n\right )\right ) \, dx=\int { x^{m} \sec \left (b \log \left (c x^{n}\right ) + a\right ) \,d x } \]
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\[ \int x^m \sec \left (a+b \log \left (c x^n\right )\right ) \, dx=\int x^{m} \sec {\left (a + b \log {\left (c x^{n} \right )} \right )}\, dx \]
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\[ \int x^m \sec \left (a+b \log \left (c x^n\right )\right ) \, dx=\int { x^{m} \sec \left (b \log \left (c x^{n}\right ) + a\right ) \,d x } \]
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\[ \int x^m \sec \left (a+b \log \left (c x^n\right )\right ) \, dx=\int { x^{m} \sec \left (b \log \left (c x^{n}\right ) + a\right ) \,d x } \]
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Timed out. \[ \int x^m \sec \left (a+b \log \left (c x^n\right )\right ) \, dx=\int \frac {x^m}{\cos \left (a+b\,\ln \left (c\,x^n\right )\right )} \,d x \]
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