Optimal. Leaf size=24 \[ \frac{2 \text{EllipticF}\left (\frac{1}{2} \left (a+b \log \left (c x^n\right )\right ),2\right )}{b n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0279455, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {2641} \[ \frac{2 F\left (\left .\frac{1}{2} \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{b n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2641
Rubi steps
\begin{align*} \int \frac{1}{x \sqrt{\cos \left (a+b \log \left (c x^n\right )\right )}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{\cos (a+b x)}} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac{2 F\left (\left .\frac{1}{2} \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{b n}\\ \end{align*}
Mathematica [A] time = 0.0853128, size = 24, normalized size = 1. \[ \frac{2 \text{EllipticF}\left (\frac{1}{2} \left (a+b \log \left (c x^n\right )\right ),2\right )}{b n} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.037, size = 26, normalized size = 1.1 \begin{align*} 2\,{\frac{{\it InverseJacobiAM} \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) ,\sqrt{2} \right ) }{bn}} \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{1}{x \sqrt{\cos \left (b \log \left (c x^{n}\right ) + a\right )}}\,{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{1}{x \sqrt{\cos \left (b \log \left (c x^{n}\right ) + a\right )}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \sqrt{\cos{\left (a + b \log{\left (c x^{n} \right )} \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \sqrt{\cos \left (b \log \left (c x^{n}\right ) + a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]