Optimal. Leaf size=63 \[ \frac{6 E\left (\left .\frac{1}{2} \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{5 b n}+\frac{2 \sin \left (a+b \log \left (c x^n\right )\right ) \cos ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}{5 b n} \]
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Rubi [A] time = 0.0423989, antiderivative size = 63, 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 = {2635, 2639} \[ \frac{6 E\left (\left .\frac{1}{2} \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{5 b n}+\frac{2 \sin \left (a+b \log \left (c x^n\right )\right ) \cos ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}{5 b n} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2639
Rubi steps
\begin{align*} \int \frac{\cos ^{\frac{5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \cos ^{\frac{5}{2}}(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac{2 \cos ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{5 b n}+\frac{3 \operatorname{Subst}\left (\int \sqrt{\cos (a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{5 n}\\ &=\frac{6 E\left (\left .\frac{1}{2} \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{5 b n}+\frac{2 \cos ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{5 b n}\\ \end{align*}
Mathematica [A] time = 0.132875, size = 58, normalized size = 0.92 \[ \frac{6 E\left (\left .\frac{1}{2} \left (a+b \log \left (c x^n\right )\right )\right |2\right )+\sin \left (2 \left (a+b \log \left (c x^n\right )\right )\right ) \sqrt{\cos \left (a+b \log \left (c x^n\right )\right )}}{5 b n} \]
Antiderivative was successfully verified.
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Maple [B] time = 2.823, size = 280, normalized size = 4.4 \begin{align*} -{\frac{2}{5\,bn}\sqrt{ \left ( 2\, \left ( \cos \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}-1 \right ) \left ( \sin \left ({\frac{a}{2}}+{\frac{b\ln \left ( c{x}^{n} \right ) }{2}} \right ) \right ) ^{2}} \left ( -8\,\cos \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \left ( \sin \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{6}+8\,\cos \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \left ( \sin \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{4}-3\,\sqrt{ \left ( \sin \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}-1}{\it EllipticE} \left ( \cos \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) ,\sqrt{2} \right ) -2\, \left ( \sin \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}\cos \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ){\frac{1}{\sqrt{-2\, \left ( \sin \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{4}+ \left ( \sin \left ({\frac{a}{2}}+{\frac{b\ln \left ( c{x}^{n} \right ) }{2}} \right ) \right ) ^{2}}}} \left ( \sin \left ({\frac{a}{2}}+{\frac{b\ln \left ( c{x}^{n} \right ) }{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{2\, \left ( \cos \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}-1}}}} \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{\cos \left (b \log \left (c x^{n}\right ) + a\right )^{\frac{5}{2}}}{x}\,{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{\cos \left (b \log \left (c x^{n}\right ) + a\right )^{\frac{5}{2}}}{x}, 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{\cos \left (b \log \left (c x^{n}\right ) + a\right )^{\frac{5}{2}}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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