Optimal. Leaf size=68 \[ \frac{2 \text{EllipticF}\left (\frac{1}{2} \left (a+b \log \left (c x^n\right )-\frac{\pi }{2}\right ),2\right )}{3 b n}-\frac{2 \cos \left (a+b \log \left (c x^n\right )\right )}{3 b n \sin ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )} \]
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Rubi [A] time = 0.0421785, antiderivative size = 68, 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 = {2636, 2641} \[ \frac{2 F\left (\left .\frac{1}{2} \left (a+b \log \left (c x^n\right )-\frac{\pi }{2}\right )\right |2\right )}{3 b n}-\frac{2 \cos \left (a+b \log \left (c x^n\right )\right )}{3 b n \sin ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2641
Rubi steps
\begin{align*} \int \frac{1}{x \sin ^{\frac{5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\sin ^{\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 )}{3 b n \sin ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}+\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{\sin (a+b x)}} \, dx,x,\log \left (c x^n\right )\right )}{3 n}\\ &=\frac{2 F\left (\left .\frac{1}{2} \left (a-\frac{\pi }{2}+b \log \left (c x^n\right )\right )\right |2\right )}{3 b n}-\frac{2 \cos \left (a+b \log \left (c x^n\right )\right )}{3 b n \sin ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}\\ \end{align*}
Mathematica [A] time = 0.214424, size = 61, normalized size = 0.9 \[ \frac{2 \left (\text{EllipticF}\left (\frac{1}{4} \left (2 a+2 b \log \left (c x^n\right )-\pi \right ),2\right )-\frac{\cos \left (a+b \log \left (c x^n\right )\right )}{\sin ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}\right )}{3 b n} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.235, size = 131, normalized size = 1.9 \begin{align*}{\frac{1}{3\,bn\cos \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) } \left ( \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 ) }{\it 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\, \left ( \cos \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2} \right ) \left ( \sin \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{-{\frac{3}{2}}}} \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 \sin \left (b \log \left (c x^{n}\right ) + a\right )^{\frac{5}{2}}}\,{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}{{\left (x \cos \left (b \log \left (c x^{n}\right ) + a\right )^{2} - x\right )} \sqrt{\sin \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(-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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \sin \left (b \log \left (c x^{n}\right ) + a\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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