Integrand size = 15, antiderivative size = 110 \[ \int \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2 x \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {2 i+b n}{4 b n},\frac {1}{4} \left (3-\frac {2 i}{b n}\right ),-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2-i b n) \sqrt {1+e^{2 i a} \left (c x^n\right )^{2 i b}}} \]
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Time = 0.09 (sec) , antiderivative size = 110, 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} \[ \int \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2 x \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {b n+2 i}{4 b n},\frac {1}{4} \left (3-\frac {2 i}{b n}\right ),-e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}}{(2-i b n) \sqrt {1+e^{2 i a} \left (c x^n\right )^{2 i b}}} \]
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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}} \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 {\cos \left (a+b \log \left (c x^n\right )\right )}\right ) \text {Subst}\left (\int 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 {1+e^{2 i a} \left (c x^n\right )^{2 i b}}} \\ & = \frac {2 x \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {2 i+b n}{4 b n},\frac {1}{4} \left (3-\frac {2 i}{b n}\right ),-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2-i b n) \sqrt {1+e^{2 i a} \left (c x^n\right )^{2 i b}}} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(377\) vs. \(2(110)=220\).
Time = 3.78 (sec) , antiderivative size = 377, normalized size of antiderivative = 3.43 \[ \int \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2 b e^{i a} n x \left (c x^n\right )^{i b} \sqrt {2+2 e^{2 i a} \left (c x^n\right )^{2 i b}} \left ((2 i+b n) x^{2 i b n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4}-\frac {i}{2 b n},\frac {7}{4}-\frac {i}{2 b n},-e^{2 i a} \left (c x^n\right )^{2 i b}\right )+(-2 i+3 b n) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {2 i+b n}{4 b n},\frac {3}{4}-\frac {i}{2 b n},-e^{2 i a} \left (c x^n\right )^{2 i b}\right )\right )}{(2 i+b n) (-2 i+3 b n) \sqrt {e^{-i a} \left (c x^n\right )^{-i b}+e^{i a} \left (c x^n\right )^{i b}} \left ((-2+i b n) x^{2 i b n}-i e^{2 i a} (-2 i+b n) \left (c x^n\right )^{2 i b}\right )}-\frac {2 x \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} \cos \left (a-b n \log (x)+b \log \left (c x^n\right )\right )}{-2 \cos \left (a-b n \log (x)+b \log \left (c x^n\right )\right )+b n \sin \left (a-b n \log (x)+b \log \left (c x^n\right )\right )} \]
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\[\int \sqrt {\cos \left (a +b \ln \left (c \,x^{n}\right )\right )}d x\]
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Exception generated. \[ \int \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} \, dx=\int \sqrt {\cos {\left (a + b \log {\left (c x^{n} \right )} \right )}}\, dx \]
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\[ \int \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} \, dx=\int { \sqrt {\cos \left (b \log \left (c x^{n}\right ) + a\right )} \,d x } \]
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\[ \int \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} \, dx=\int { \sqrt {\cos \left (b \log \left (c x^{n}\right ) + a\right )} \,d x } \]
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Timed out. \[ \int \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} \, dx=\int \sqrt {\cos \left (a+b\,\ln \left (c\,x^n\right )\right )} \,d x \]
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