Integrand size = 15, antiderivative size = 87 \[ \int \frac {\sec \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=-\frac {2 e^{i a} \left (c x^n\right )^{i b} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} \left (1+\frac {i}{b n}\right ),\frac {1}{2} \left (3+\frac {i}{b n}\right ),-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(1-i b n) x} \]
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Time = 0.07 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4605, 4601, 371} \[ \int \frac {\sec \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=-\frac {2 e^{i a} \left (c x^n\right )^{i b} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} \left (1+\frac {i}{b n}\right ),\frac {1}{2} \left (3+\frac {i}{b n}\right ),-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{x (1-i b n)} \]
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Rule 371
Rule 4601
Rule 4605
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c x^n\right )^{\frac {1}{n}} \text {Subst}\left (\int x^{-1-\frac {1}{n}} \sec (a+b \log (x)) \, dx,x,c x^n\right )}{n x} \\ & = \frac {\left (2 e^{i a} \left (c x^n\right )^{\frac {1}{n}}\right ) \text {Subst}\left (\int \frac {x^{-1+i b-\frac {1}{n}}}{1+e^{2 i a} x^{2 i b}} \, dx,x,c x^n\right )}{n x} \\ & = -\frac {2 e^{i a} \left (c x^n\right )^{i b} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} \left (1+\frac {i}{b n}\right ),\frac {1}{2} \left (3+\frac {i}{b n}\right ),-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(1-i b n) x} \\ \end{align*}
Time = 0.54 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.98 \[ \int \frac {\sec \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\frac {2 e^{i a} \left (c x^n\right )^{i b} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2}+\frac {i}{2 b n},\frac {3}{2}+\frac {i}{2 b n},-e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )}{(-1+i b n) x} \]
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\[\int \frac {\sec \left (a +b \ln \left (c \,x^{n}\right )\right )}{x^{2}}d x\]
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\[ \int \frac {\sec \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\int { \frac {\sec \left (b \log \left (c x^{n}\right ) + a\right )}{x^{2}} \,d x } \]
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\[ \int \frac {\sec \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\int \frac {\sec {\left (a + b \log {\left (c x^{n} \right )} \right )}}{x^{2}}\, dx \]
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\[ \int \frac {\sec \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\int { \frac {\sec \left (b \log \left (c x^{n}\right ) + a\right )}{x^{2}} \,d x } \]
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\[ \int \frac {\sec \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\int { \frac {\sec \left (b \log \left (c x^{n}\right ) + a\right )}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\sec \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\int \frac {1}{x^2\,\cos \left (a+b\,\ln \left (c\,x^n\right )\right )} \,d x \]
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