Optimal. Leaf size=19 \[ \frac{\tan ^5(x)}{5}+\frac{2 \tan ^3(x)}{3}+\tan (x) \]
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Rubi [A] time = 0.0088159, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3767} \[ \frac{\tan ^5(x)}{5}+\frac{2 \tan ^3(x)}{3}+\tan (x) \]
Antiderivative was successfully verified.
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Rule 3767
Rubi steps
\begin{align*} \int \sec ^6(x) \, dx &=-\operatorname{Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-\tan (x)\right )\\ &=\tan (x)+\frac{2 \tan ^3(x)}{3}+\frac{\tan ^5(x)}{5}\\ \end{align*}
Mathematica [A] time = 0.0027936, size = 27, normalized size = 1.42 \[ \frac{8 \tan (x)}{15}+\frac{1}{5} \tan (x) \sec ^4(x)+\frac{4}{15} \tan (x) \sec ^2(x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 19, normalized size = 1. \begin{align*} - \left ( -{\frac{8}{15}}-{\frac{ \left ( \sec \left ( x \right ) \right ) ^{4}}{5}}-{\frac{4\, \left ( \sec \left ( x \right ) \right ) ^{2}}{15}} \right ) \tan \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.926333, size = 20, normalized size = 1.05 \begin{align*} \frac{1}{5} \, \tan \left (x\right )^{5} + \frac{2}{3} \, \tan \left (x\right )^{3} + \tan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.94805, size = 70, normalized size = 3.68 \begin{align*} \frac{{\left (8 \, \cos \left (x\right )^{4} + 4 \, \cos \left (x\right )^{2} + 3\right )} \sin \left (x\right )}{15 \, \cos \left (x\right )^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.062246, size = 31, normalized size = 1.63 \begin{align*} \frac{8 \sin{\left (x \right )}}{15 \cos{\left (x \right )}} + \frac{4 \sin{\left (x \right )}}{15 \cos ^{3}{\left (x \right )}} + \frac{\sin{\left (x \right )}}{5 \cos ^{5}{\left (x \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05634, size = 20, normalized size = 1.05 \begin{align*} \frac{1}{5} \, \tan \left (x\right )^{5} + \frac{2}{3} \, \tan \left (x\right )^{3} + \tan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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