Optimal. Leaf size=93 \[ \frac{2 \sqrt{\sec \left (a+b \log \left (c x^n\right )\right )} \sqrt{\cos \left (a+b \log \left (c x^n\right )\right )} \text{EllipticF}\left (\frac{1}{2} \left (a+b \log \left (c x^n\right )\right ),2\right )}{3 b n}+\frac{2 \sin \left (a+b \log \left (c x^n\right )\right ) \sec ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}{3 b n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0610006, antiderivative size = 93, 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, 2641} \[ \frac{2 \sin \left (a+b \log \left (c x^n\right )\right ) \sec ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}{3 b n}+\frac{2 \sqrt{\sec \left (a+b \log \left (c x^n\right )\right )} \sqrt{\cos \left (a+b \log \left (c x^n\right )\right )} F\left (\left .\frac{1}{2} \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{3 b n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3768
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int \frac{\sec ^{\frac{5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \sec ^{\frac{5}{2}}(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac{2 \sec ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{3 b n}+\frac{\operatorname{Subst}\left (\int \sqrt{\sec (a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{3 n}\\ &=\frac{2 \sec ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{3 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 \frac{1}{\sqrt{\cos (a+b x)}} \, dx,x,\log \left (c x^n\right )\right )}{3 n}\\ &=\frac{2 \sqrt{\cos \left (a+b \log \left (c x^n\right )\right )} F\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 )}}{3 b n}+\frac{2 \sec ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{3 b n}\\ \end{align*}
Mathematica [A] time = 0.15626, size = 69, normalized size = 0.74 \[ \frac{2 \sec ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right ) \left (\cos ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right ) \text{EllipticF}\left (\frac{1}{2} \left (a+b \log \left (c x^n\right )\right ),2\right )+\sin \left (a+b \log \left (c x^n\right )\right )\right )}{3 b n} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 2.697, size = 291, normalized size = 3.1 \begin{align*} -{\frac{2}{3\,bn} \left ( -2\,\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 EllipticF} \left ( \cos \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) ,\sqrt{2} \right ) \left ( \sin \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}+\sqrt{ \left ( \sin \left ({\frac{a}{2}}+{\frac{b\ln \left ( c{x}^{n} \right ) }{2}} \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}-1}{\it EllipticF} \left ( \cos \left ({\frac{a}{2}}+{\frac{b\ln \left ( c{x}^{n} \right ) }{2}} \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 ) \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}}{\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 ( 2\, \left ( \cos \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}-1 \right ) ^{-{\frac{3}{2}}} \left ( \sin \left ({\frac{a}{2}}+{\frac{b\ln \left ( c{x}^{n} \right ) }{2}} \right ) \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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{5}{2}}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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{5}{2}}}{x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]