Integrand size = 19, antiderivative size = 72 \[ \int \frac {\sqrt {\sinh \left (a+b \log \left (c x^n\right )\right )}}{x} \, dx=-\frac {2 i E\left (\left .\frac {1}{2} \left (i a-\frac {\pi }{2}+i b \log \left (c x^n\right )\right )\right |2\right ) \sqrt {\sinh \left (a+b \log \left (c x^n\right )\right )}}{b n \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )}} \]
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Time = 0.03 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2721, 2719} \[ \int \frac {\sqrt {\sinh \left (a+b \log \left (c x^n\right )\right )}}{x} \, dx=-\frac {2 i \sqrt {\sinh \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac {1}{2} \left (i a+i b \log \left (c x^n\right )-\frac {\pi }{2}\right )\right |2\right )}{b n \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )}} \]
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Rule 2719
Rule 2721
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \sqrt {\sinh (a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = \frac {\sqrt {\sinh \left (a+b \log \left (c x^n\right )\right )} \text {Subst}\left (\int \sqrt {i \sinh (a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{n \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )}} \\ & = -\frac {2 i E\left (\left .\frac {1}{2} \left (i a-\frac {\pi }{2}+i b \log \left (c x^n\right )\right )\right |2\right ) \sqrt {\sinh \left (a+b \log \left (c x^n\right )\right )}}{b n \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {\sinh \left (a+b \log \left (c x^n\right )\right )}}{x} \, dx=\frac {2 E\left (\left .\frac {1}{2} \left (\frac {\pi }{2}-i \left (a+b \log \left (c x^n\right )\right )\right )\right |2\right ) \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )}}{b n \sqrt {\sinh \left (a+b \log \left (c x^n\right )\right )}} \]
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Time = 0.80 (sec) , antiderivative size = 146, normalized size of antiderivative = 2.03
method | result | size |
derivativedivides | \(\frac {\sqrt {-i \left (\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )+i\right )}\, \sqrt {2}\, \sqrt {-i \left (-\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )+i\right )}\, \sqrt {i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \left (2 \operatorname {EllipticE}\left (\sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}, \frac {\sqrt {2}}{2}\right )-\operatorname {EllipticF}\left (\sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}, \frac {\sqrt {2}}{2}\right )\right )}{n \cosh \left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, b}\) | \(146\) |
default | \(\frac {\sqrt {-i \left (\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )+i\right )}\, \sqrt {2}\, \sqrt {-i \left (-\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )+i\right )}\, \sqrt {i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \left (2 \operatorname {EllipticE}\left (\sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}, \frac {\sqrt {2}}{2}\right )-\operatorname {EllipticF}\left (\sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}, \frac {\sqrt {2}}{2}\right )\right )}{n \cosh \left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, b}\) | \(146\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.81 \[ \int \frac {\sqrt {\sinh \left (a+b \log \left (c x^n\right )\right )}}{x} \, dx=-\frac {2 \, {\left (\sqrt {2} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )\right ) + \sqrt {\sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}\right )}}{b n} \]
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\[ \int \frac {\sqrt {\sinh \left (a+b \log \left (c x^n\right )\right )}}{x} \, dx=\int \frac {\sqrt {\sinh {\left (a + b \log {\left (c x^{n} \right )} \right )}}}{x}\, dx \]
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\[ \int \frac {\sqrt {\sinh \left (a+b \log \left (c x^n\right )\right )}}{x} \, dx=\int { \frac {\sqrt {\sinh \left (b \log \left (c x^{n}\right ) + a\right )}}{x} \,d x } \]
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\[ \int \frac {\sqrt {\sinh \left (a+b \log \left (c x^n\right )\right )}}{x} \, dx=\int { \frac {\sqrt {\sinh \left (b \log \left (c x^{n}\right ) + a\right )}}{x} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {\sinh \left (a+b \log \left (c x^n\right )\right )}}{x} \, dx=\int \frac {\sqrt {\mathrm {sinh}\left (a+b\,\ln \left (c\,x^n\right )\right )}}{x} \,d x \]
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