Integrand size = 15, antiderivative size = 109 \[ \int \frac {1}{\sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}} \, dx=\frac {2 x \sqrt {1+e^{2 i a} \left (c x^n\right )^{2 i b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} \left (1-\frac {2 i}{b n}\right ),\frac {1}{4} \left (5-\frac {2 i}{b n}\right ),-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2+i b n) \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}} \]
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Time = 0.07 (sec) , antiderivative size = 109, 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 = {4572, 4580, 371} \[ \int \frac {1}{\sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}} \, dx=\frac {2 x \sqrt {1+e^{2 i a} \left (c x^n\right )^{2 i b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} \left (1-\frac {2 i}{b n}\right ),\frac {1}{4} \left (5-\frac {2 i}{b n}\right ),-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2+i b n) \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}} \]
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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 \frac {x^{-1+\frac {1}{n}}}{\sqrt {\cos (a+b \log (x))}} \, dx,x,c x^n\right )}{n} \\ & = \frac {\left (x \left (c x^n\right )^{-\frac {i b}{2}-\frac {1}{n}} \sqrt {1+e^{2 i a} \left (c x^n\right )^{2 i b}}\right ) \text {Subst}\left (\int \frac {x^{-1+\frac {i b}{2}+\frac {1}{n}}}{\sqrt {1+e^{2 i a} x^{2 i b}}} \, dx,x,c x^n\right )}{n \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}} \\ & = \frac {2 x \sqrt {1+e^{2 i a} \left (c x^n\right )^{2 i b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} \left (1-\frac {2 i}{b n}\right ),\frac {1}{4} \left (5-\frac {2 i}{b n}\right ),-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2+i b n) \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}} \\ \end{align*}
Time = 0.38 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.23 \[ \int \frac {1}{\sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}} \, dx=-\frac {2 i \sqrt {2} \sqrt {1+e^{2 i \left (a+b \log \left (c x^n\right )\right )}} x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4}-\frac {i}{2 b n},\frac {5}{4}-\frac {i}{2 b n},-e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {e^{-i \left (a+b \log \left (c x^n\right )\right )} \left (1+e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )} (-2 i+b n)} \]
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\[\int \frac {1}{\sqrt {\cos \left (a +b \ln \left (c \,x^{n}\right )\right )}}d x\]
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Exception generated. \[ \int \frac {1}{\sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {1}{\sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}} \, dx=\int \frac {1}{\sqrt {\cos {\left (a + b \log {\left (c x^{n} \right )} \right )}}}\, dx \]
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\[ \int \frac {1}{\sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}} \, dx=\int { \frac {1}{\sqrt {\cos \left (b \log \left (c x^{n}\right ) + a\right )}} \,d x } \]
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\[ \int \frac {1}{\sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}} \, dx=\int { \frac {1}{\sqrt {\cos \left (b \log \left (c x^{n}\right ) + a\right )}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}} \, dx=\int \frac {1}{\sqrt {\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}} \,d x \]
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