Optimal. Leaf size=59 \[ \frac {2 \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )} \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac {1}{2} \left (a+b \log \left (c x^n\right )-\frac {\pi }{2}\right )\right |2\right )}{b n} \]
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Rubi [A] time = 0.04, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3771, 2639} \[ \frac {2 \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )} \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac {1}{2} \left (a+b \log \left (c x^n\right )-\frac {\pi }{2}\right )\right |2\right )}{b n} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 3771
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {\csc (a+b x)}} \, dx,x,\log \left (c x^n\right )\right )}{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 ) \operatorname {Subst}\left (\int \sqrt {\sin (a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac {2 \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b \log \left (c x^n\right )\right )\right |2\right ) \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}}{b n}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 58, normalized size = 0.98 \[ -\frac {2 \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )} \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac {1}{4} \left (-2 a-2 b \log \left (c x^n\right )+\pi \right )\right |2\right )}{b n} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.81, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{x \sqrt {\csc \left (b \log \left (c x^{n}\right ) + a\right )}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \sqrt {\csc \left (b \log \left (c x^{n}\right ) + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 129, normalized size = 2.19 \[ -\frac {\sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}\, \sqrt {-2 \sin \left (a +b \ln \left (c \,x^{n}\right )\right )+2}\, \sqrt {-\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \left (2 \EllipticE \left (\sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}, \frac {\sqrt {2}}{2}\right )-\EllipticF \left (\sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}, \frac {\sqrt {2}}{2}\right )\right )}{n \cos \left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}\, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \sqrt {\csc \left (b \log \left (c x^{n}\right ) + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{x\,\sqrt {\frac {1}{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \sqrt {\csc {\left (a + b \log {\left (c x^{n} \right )} \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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