Integrand size = 19, antiderivative size = 29 \[ \int \frac {1}{x \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}} \, dx=\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} \left (a-\frac {\pi }{2}+b \log \left (c x^n\right )\right ),2\right )}{b n} \]
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Time = 0.03 (sec) , antiderivative size = 29, 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} \[ \int \frac {1}{x \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}} \, dx=\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} \left (a+b \log \left (c x^n\right )-\frac {\pi }{2}\right ),2\right )}{b n} \]
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Rule 2720
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\sqrt {\sin (a+b x)}} \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = \frac {2 \operatorname {EllipticF}\left (\frac {1}{2} \left (a-\frac {\pi }{2}+b \log \left (c x^n\right )\right ),2\right )}{b n} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.10 \[ \int \frac {1}{x \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}} \, dx=-\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} \left (-a+\frac {\pi }{2}-b \log \left (c x^n\right )\right ),2\right )}{b n} \]
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Time = 0.87 (sec) , antiderivative size = 102, normalized size of antiderivative = 3.52
method | result | size |
derivativedivides | \(\frac {\sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}\, \sqrt {-2 \sin \left (a +b \ln \left (c \,x^{n}\right )\right )+2}\, \sqrt {-\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \operatorname {EllipticF}\left (\sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}, \frac {\sqrt {2}}{2}\right )}{n \cos \left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}\, b}\) | \(102\) |
default | \(\frac {\sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}\, \sqrt {-2 \sin \left (a +b \ln \left (c \,x^{n}\right )\right )+2}\, \sqrt {-\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \operatorname {EllipticF}\left (\sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}, \frac {\sqrt {2}}{2}\right )}{n \cos \left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}\, b}\) | \(102\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.83 \[ \int \frac {1}{x \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}} \, dx=\frac {\sqrt {2} \sqrt {-i} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + i \, \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right ) + \sqrt {2} \sqrt {i} {\rm weierstrassPInverse}\left (4, 0, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) - i \, \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )}{b n} \]
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\[ \int \frac {1}{x \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}} \, dx=\int \frac {1}{x \sqrt {\sin {\left (a + b \log {\left (c x^{n} \right )} \right )}}}\, dx \]
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\[ \int \frac {1}{x \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}} \, dx=\int { \frac {1}{x \sqrt {\sin \left (b \log \left (c x^{n}\right ) + a\right )}} \,d x } \]
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\[ \int \frac {1}{x \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}} \, dx=\int { \frac {1}{x \sqrt {\sin \left (b \log \left (c x^{n}\right ) + a\right )}} \,d x } \]
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Time = 25.90 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \frac {1}{x \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}} \, dx=-\frac {2\,\mathrm {F}\left (\frac {\pi }{4}-\frac {a}{2}-\frac {b\,\ln \left (c\,x^n\right )}{2}\middle |2\right )}{b\,n} \]
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