Optimal. Leaf size=8 \[ \frac{\tan ^3(x)}{3} \]
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Rubi [A] time = 0.021766, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2607, 30} \[ \frac{\tan ^3(x)}{3} \]
Antiderivative was successfully verified.
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Rule 2607
Rule 30
Rubi steps
\begin{align*} \int \sec ^2(x) \tan ^2(x) \, dx &=\operatorname{Subst}\left (\int x^2 \, dx,x,\tan (x)\right )\\ &=\frac{\tan ^3(x)}{3}\\ \end{align*}
Mathematica [A] time = 0.0027176, size = 8, normalized size = 1. \[ \frac{\tan ^3(x)}{3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 11, normalized size = 1.4 \begin{align*}{\frac{ \left ( \sin \left ( x \right ) \right ) ^{3}}{3\, \left ( \cos \left ( x \right ) \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.932575, size = 8, normalized size = 1. \begin{align*} \frac{1}{3} \, \tan \left (x\right )^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.74863, size = 50, normalized size = 6.25 \begin{align*} -\frac{{\left (\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 = 0.062806, size = 17, normalized size = 2.12 \begin{align*} - \frac{\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 = 1.07314, size = 8, normalized size = 1. \begin{align*} \frac{1}{3} \, \tan \left (x\right )^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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