Optimal. Leaf size=89 \[ \frac{2 \sin \left (a+b \log \left (c x^n\right )\right ) \sqrt{\sec \left (a+b \log \left (c x^n\right )\right )}}{b n}-\frac{2 \sqrt{\sec \left (a+b \log \left (c x^n\right )\right )} \sqrt{\cos \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac{1}{2} \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{b n} \]
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Rubi [A] time = 0.0609058, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {3768, 3771, 2639} \[ \frac{2 \sin \left (a+b \log \left (c x^n\right )\right ) \sqrt{\sec \left (a+b \log \left (c x^n\right )\right )}}{b n}-\frac{2 \sqrt{\sec \left (a+b \log \left (c x^n\right )\right )} \sqrt{\cos \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac{1}{2} \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{b n} \]
Antiderivative was successfully verified.
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Rule 3768
Rule 3771
Rule 2639
Rubi steps
\begin{align*} \int \frac{\sec ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \sec ^{\frac{3}{2}}(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac{2 \sqrt{\sec \left (a+b \log \left (c x^n\right )\right )} \sin \left (a+b \log \left (c x^n\right )\right )}{b n}-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{\sec (a+b x)}} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac{2 \sqrt{\sec \left (a+b \log \left (c x^n\right )\right )} \sin \left (a+b \log \left (c x^n\right )\right )}{b n}-\frac{\left (\sqrt{\cos \left (a+b \log \left (c x^n\right )\right )} \sqrt{\sec \left (a+b \log \left (c x^n\right )\right )}\right ) \operatorname{Subst}\left (\int \sqrt{\cos (a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac{2 \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 ) \sqrt{\sec \left (a+b \log \left (c x^n\right )\right )}}{b n}+\frac{2 \sqrt{\sec \left (a+b \log \left (c x^n\right )\right )} \sin \left (a+b \log \left (c x^n\right )\right )}{b n}\\ \end{align*}
Mathematica [A] time = 0.138382, size = 68, normalized size = 0.76 \[ \frac{2 \sqrt{\sec \left (a+b \log \left (c x^n\right )\right )} \left (\sin \left (a+b \log \left (c x^n\right )\right )-\sqrt{\cos \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac{1}{2} \left (a+b \log \left (c x^n\right )\right )\right |2\right )\right )}{b n} \]
Antiderivative was successfully verified.
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Maple [A] time = 2.643, size = 139, normalized size = 1.6 \begin{align*} -2\,{\frac{\sqrt{2\, \left ( \sin \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}-1}\sqrt{ \left ( \sin \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}}{\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{\sec \left (b \log \left (c x^{n}\right ) + a\right )^{\frac{3}{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{\sec \left (b \log \left (c x^{n}\right ) + a\right )^{\frac{3}{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(-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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