Optimal. Leaf size=11 \[ \frac{\tan ^3(x)}{3}+\tan (x) \]
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Rubi [A] time = 0.006882, antiderivative size = 11, 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} \[ \frac{\tan ^3(x)}{3}+\tan (x) \]
Antiderivative was successfully verified.
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Rule 3767
Rubi steps
\begin{align*} \int \sec ^4(x) \, dx &=-\operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (x)\right )\\ &=\tan (x)+\frac{\tan ^3(x)}{3}\\ \end{align*}
Mathematica [A] time = 0.0024551, size = 17, normalized size = 1.55 \[ \frac{2 \tan (x)}{3}+\frac{1}{3} \tan (x) \sec ^2(x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 13, normalized size = 1.2 \begin{align*} - \left ( -{\frac{2}{3}}-{\frac{ \left ( \sec \left ( x \right ) \right ) ^{2}}{3}} \right ) \tan \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.923523, size = 12, normalized size = 1.09 \begin{align*} \frac{1}{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 = 1.96521, size = 51, normalized size = 4.64 \begin{align*} \frac{{\left (2 \, \cos \left (x\right )^{2} + 1\right )} \sin \left (x\right )}{3 \, \cos \left (x\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.061384, size = 19, normalized size = 1.73 \begin{align*} \frac{2 \sin{\left (x \right )}}{3 \cos{\left (x \right )}} + \frac{\sin{\left (x \right )}}{3 \cos ^{3}{\left (x \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07935, size = 12, normalized size = 1.09 \begin{align*} \frac{1}{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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