Optimal. Leaf size=59 \[ \frac{2 \sin \left (a+b \log \left (c x^n\right )\right )}{b n \sqrt{\cos \left (a+b \log \left (c x^n\right )\right )}}-\frac{2 E\left (\left .\frac{1}{2} \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{b n} \]
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Rubi [A] time = 0.0410599, 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 = {2636, 2639} \[ \frac{2 \sin \left (a+b \log \left (c x^n\right )\right )}{b n \sqrt{\cos \left (a+b \log \left (c x^n\right )\right )}}-\frac{2 E\left (\left .\frac{1}{2} \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{b n} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2639
Rubi steps
\begin{align*} \int \frac{1}{x \cos ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\cos ^{\frac{3}{2}}(a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac{2 \sin \left (a+b \log \left (c x^n\right )\right )}{b n \sqrt{\cos \left (a+b \log \left (c x^n\right )\right )}}-\frac{\operatorname{Subst}\left (\int \sqrt{\cos (a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac{2 E\left (\left .\frac{1}{2} \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{b n}+\frac{2 \sin \left (a+b \log \left (c x^n\right )\right )}{b n \sqrt{\cos \left (a+b \log \left (c x^n\right )\right )}}\\ \end{align*}
Mathematica [A] time = 0.152792, size = 54, normalized size = 0.92 \[ \frac{2 \left (\frac{\sin \left (a+b \log \left (c x^n\right )\right )}{\sqrt{\cos \left (a+b \log \left (c x^n\right )\right )}}-E\left (\left .\frac{1}{2} \left (a+b \log \left (c x^n\right )\right )\right |2\right )\right )}{b n} \]
Antiderivative was successfully verified.
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Maple [A] time = 2.377, size = 139, normalized size = 2.4 \begin{align*} -2\,{\frac{\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 ) }{n\sin \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \sqrt{2\, \left ( \cos \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}-1}b}} \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 \cos \left (b \log \left (c x^{n}\right ) + a\right )^{\frac{3}{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}{x \cos \left (b \log \left (c x^{n}\right ) + a\right )^{\frac{3}{2}}}, 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 \cos \left (b \log \left (c x^{n}\right ) + a\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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