Integrand size = 19, antiderivative size = 98 \[ \int \frac {\csc ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {2 \cos \left (a+b \log \left (c x^n\right )\right ) \csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{3 b n}+\frac {2 \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \operatorname {EllipticF}\left (\frac {1}{2} \left (a-\frac {\pi }{2}+b \log \left (c x^n\right )\right ),2\right ) \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}}{3 b n} \]
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Time = 0.07 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3853, 3856, 2720} \[ \int \frac {\csc ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {2 \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )} \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \operatorname {EllipticF}\left (\frac {1}{2} \left (a+b \log \left (c x^n\right )-\frac {\pi }{2}\right ),2\right )}{3 b n}-\frac {2 \cos \left (a+b \log \left (c x^n\right )\right ) \csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{3 b n} \]
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Rule 2720
Rule 3853
Rule 3856
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \csc ^{\frac {5}{2}}(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = -\frac {2 \cos \left (a+b \log \left (c x^n\right )\right ) \csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{3 b n}+\frac {\text {Subst}\left (\int \sqrt {\csc (a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{3 n} \\ & = -\frac {2 \cos \left (a+b \log \left (c x^n\right )\right ) \csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{3 b n}+\frac {\left (\sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {\sin (a+b x)}} \, dx,x,\log \left (c x^n\right )\right )}{3 n} \\ & = -\frac {2 \cos \left (a+b \log \left (c x^n\right )\right ) \csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{3 b n}+\frac {2 \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \operatorname {EllipticF}\left (\frac {1}{2} \left (a-\frac {\pi }{2}+b \log \left (c x^n\right )\right ),2\right ) \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}}{3 b n} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.74 \[ \int \frac {\csc ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {2 \csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \left (\cos \left (a+b \log \left (c x^n\right )\right )+\operatorname {EllipticF}\left (\frac {1}{4} \left (-2 a+\pi -2 b \log \left (c x^n\right )\right ),2\right ) \sin ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )\right )}{3 b n} \]
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Time = 1.45 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.34
method | result | size |
derivativedivides | \(\frac {\sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}\, \sqrt {-2 \sin \left (a +b \ln \left (c \,x^{n}\right )\right )+2}\, \sqrt {-\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \operatorname {EllipticF}\left (\sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}, \frac {\sqrt {2}}{2}\right ) \sin \left (a +b \ln \left (c \,x^{n}\right )\right )-2 {\cos \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{3 n {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}^{\frac {3}{2}} \cos \left (a +b \ln \left (c \,x^{n}\right )\right ) b}\) | \(131\) |
default | \(\frac {\sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}\, \sqrt {-2 \sin \left (a +b \ln \left (c \,x^{n}\right )\right )+2}\, \sqrt {-\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \operatorname {EllipticF}\left (\sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}, \frac {\sqrt {2}}{2}\right ) \sin \left (a +b \ln \left (c \,x^{n}\right )\right )-2 {\cos \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{3 n {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}^{\frac {3}{2}} \cos \left (a +b \ln \left (c \,x^{n}\right )\right ) b}\) | \(131\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.48 \[ \int \frac {\csc ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {-i \, \sqrt {2 i} \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) {\rm weierstrassPInverse}\left (4, 0, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + i \, \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right ) + i \, \sqrt {-2 i} \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) {\rm weierstrassPInverse}\left (4, 0, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) - i \, \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right ) - \frac {2 \, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{\sqrt {\sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}}}{3 \, b n \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )} \]
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Timed out. \[ \int \frac {\csc ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Timed out} \]
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\[ \int \frac {\csc ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int { \frac {\csc \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {5}{2}}}{x} \,d x } \]
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Timed out. \[ \int \frac {\csc ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\csc ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int \frac {{\left (\frac {1}{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}^{5/2}}{x} \,d x \]
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