Optimal. Leaf size=22 \[ \frac{1}{3} \sqrt{5 \cos ^2(3 x)+1} \sec (3 x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0909895, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {264} \[ \frac{1}{3} \sqrt{5 \cos ^2(3 x)+1} \sec (3 x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 264
Rubi steps
\begin{align*} \int \frac{\sec (3 x) \tan (3 x)}{\sqrt{1+5 \cos ^2(3 x)}} \, dx &=-\left (\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1+5 x^2}} \, dx,x,\cos (3 x)\right )\right )\\ &=\frac{1}{3} \sqrt{1+5 \cos ^2(3 x)} \sec (3 x)\\ \end{align*}
Mathematica [A] time = 0.0330922, size = 22, normalized size = 1. \[ \frac{1}{3} \sqrt{5 \cos ^2(3 x)+1} \sec (3 x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.032, size = 34, normalized size = 1.6 \begin{align*}{\frac{ \left ( \sec \left ( 3\,x \right ) \right ) ^{2}+5}{3\,\sec \left ( 3\,x \right ) }{\frac{1}{\sqrt{{\frac{ \left ( \sec \left ( 3\,x \right ) \right ) ^{2}+5}{ \left ( \sec \left ( 3\,x \right ) \right ) ^{2}}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.43293, size = 27, normalized size = 1.23 \begin{align*} \frac{\sqrt{5 \, \cos \left (3 \, x\right )^{2} + 1}}{3 \, \cos \left (3 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.44364, size = 50, normalized size = 2.27 \begin{align*} \frac{\sqrt{5 \, \cos \left (3 \, x\right )^{2} + 1}}{3 \, \cos \left (3 \, 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{\tan{\left (3 x \right )} \sec{\left (3 x \right )}}{\sqrt{5 \cos ^{2}{\left (3 x \right )} + 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.24747, size = 159, normalized size = 7.23 \begin{align*} \frac{2 \, \sqrt{2}{\left (\sqrt{3} \tan \left (\frac{3}{2} \, x\right )^{2} + \sqrt{3} - \sqrt{3 \, \tan \left (\frac{3}{2} \, x\right )^{4} - 4 \, \tan \left (\frac{3}{2} \, x\right )^{2} + 3}\right )}}{3 \,{\left ({\left (\sqrt{3} \tan \left (\frac{3}{2} \, x\right )^{2} - \sqrt{3 \, \tan \left (\frac{3}{2} \, x\right )^{4} - 4 \, \tan \left (\frac{3}{2} \, x\right )^{2} + 3}\right )}^{2} - 2 \, \sqrt{3}{\left (\sqrt{3} \tan \left (\frac{3}{2} \, x\right )^{2} - \sqrt{3 \, \tan \left (\frac{3}{2} \, x\right )^{4} - 4 \, \tan \left (\frac{3}{2} \, x\right )^{2} + 3}\right )} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]