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 = 0.0404361, antiderivative size = 59, normalized size of antiderivative = 1., 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 3771
Rule 2639
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*}
Mathematica [A] time = 0.108274, 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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Maple [A] time = 1.305, size = 129, normalized size = 2.2 \begin{align*} -{\frac{1}{n\cos \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) b}\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\,{\it EllipticE} \left ( \sqrt{\sin \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) +1},1/2\,\sqrt{2} \right ) -{\it EllipticF} \left ( \sqrt{\sin \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) +1},{\frac{\sqrt{2}}{2}} \right ) \right ){\frac{1}{\sqrt{\sin \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \sqrt{\csc \left (b \log \left (c x^{n}\right ) + a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{x \sqrt{\csc \left (b \log \left (c x^{n}\right ) + a\right )}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \sqrt{\csc{\left (a + b \log{\left (c x^{n} \right )} \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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