Optimal. Leaf size=17 \[ \frac{\sec ^5(x)}{5}-\frac{\sec ^3(x)}{3} \]
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Rubi [A] time = 0.0252672, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2606, 14} \[ \frac{\sec ^5(x)}{5}-\frac{\sec ^3(x)}{3} \]
Antiderivative was successfully verified.
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Rule 2606
Rule 14
Rubi steps
\begin{align*} \int \sec ^3(x) \tan ^3(x) \, dx &=\operatorname{Subst}\left (\int x^2 \left (-1+x^2\right ) \, dx,x,\sec (x)\right )\\ &=\operatorname{Subst}\left (\int \left (-x^2+x^4\right ) \, dx,x,\sec (x)\right )\\ &=-\frac{1}{3} \sec ^3(x)+\frac{\sec ^5(x)}{5}\\ \end{align*}
Mathematica [A] time = 0.0161254, size = 17, normalized size = 1. \[ \frac{\sec ^5(x)}{5}-\frac{\sec ^3(x)}{3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0., size = 42, normalized size = 2.5 \begin{align*}{\frac{ \left ( \sin \left ( x \right ) \right ) ^{4}}{5\, \left ( \cos \left ( x \right ) \right ) ^{5}}}+{\frac{ \left ( \sin \left ( x \right ) \right ) ^{4}}{15\, \left ( \cos \left ( x \right ) \right ) ^{3}}}-{\frac{ \left ( \sin \left ( x \right ) \right ) ^{4}}{15\,\cos \left ( x \right ) }}-{\frac{ \left ( 2+ \left ( \sin \left ( x \right ) \right ) ^{2} \right ) \cos \left ( x \right ) }{15}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.944727, size = 19, normalized size = 1.12 \begin{align*} -\frac{5 \, \cos \left (x\right )^{2} - 3}{15 \, \cos \left (x\right )^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.85028, size = 45, normalized size = 2.65 \begin{align*} -\frac{5 \, \cos \left (x\right )^{2} - 3}{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 = 0.096019, size = 15, normalized size = 0.88 \begin{align*} - \frac{5 \cos ^{2}{\left (x \right )} - 3}{15 \cos ^{5}{\left (x \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06395, size = 19, normalized size = 1.12 \begin{align*} -\frac{5 \, \cos \left (x\right )^{2} - 3}{15 \, \cos \left (x\right )^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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