Integrand size = 19, antiderivative size = 24 \[ \int \frac {1}{x \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}} \, dx=\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} \left (a+b \log \left (c x^n\right )\right ),2\right )}{b n} \]
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Time = 0.03 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2720} \[ \int \frac {1}{x \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}} \, dx=\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} \left (a+b \log \left (c x^n\right )\right ),2\right )}{b n} \]
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Rule 2720
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\sqrt {\cos (a+b x)}} \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = \frac {2 \operatorname {EllipticF}\left (\frac {1}{2} \left (a+b \log \left (c x^n\right )\right ),2\right )}{b n} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}} \, dx=\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} \left (a+b \log \left (c x^n\right )\right ),2\right )}{b n} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.18 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08
method | result | size |
derivativedivides | \(\frac {2 \,\operatorname {InverseJacobiAM}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}, \sqrt {2}\right )}{n b}\) | \(26\) |
default | \(\frac {2 \,\operatorname {InverseJacobiAM}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}, \sqrt {2}\right )}{n b}\) | \(26\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 78, normalized size of antiderivative = 3.25 \[ \int \frac {1}{x \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}} \, dx=\frac {-i \, \sqrt {2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + i \, \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right ) + i \, \sqrt {2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) - i \, \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )}{b n} \]
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\[ \int \frac {1}{x \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}} \, dx=\int \frac {1}{x \sqrt {\cos {\left (a + b \log {\left (c x^{n} \right )} \right )}}}\, dx \]
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\[ \int \frac {1}{x \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}} \, dx=\int { \frac {1}{x \sqrt {\cos \left (b \log \left (c x^{n}\right ) + a\right )}} \,d x } \]
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\[ \int \frac {1}{x \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}} \, dx=\int { \frac {1}{x \sqrt {\cos \left (b \log \left (c x^{n}\right ) + a\right )}} \,d x } \]
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Time = 26.37 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}} \, dx=\frac {2\,\mathrm {F}\left (\frac {a}{2}+\frac {b\,\ln \left (c\,x^n\right )}{2}\middle |2\right )}{b\,n} \]
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