Optimal. Leaf size=17 \[ \frac{\tan ^5(x)}{5}+\frac{\tan ^3(x)}{3} \]
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Rubi [A] time = 0.0232369, 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 = {2607, 14} \[ \frac{\tan ^5(x)}{5}+\frac{\tan ^3(x)}{3} \]
Antiderivative was successfully verified.
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Rule 2607
Rule 14
Rubi steps
\begin{align*} \int \sec ^4(x) \tan ^2(x) \, dx &=\operatorname{Subst}\left (\int x^2 \left (1+x^2\right ) \, dx,x,\tan (x)\right )\\ &=\operatorname{Subst}\left (\int \left (x^2+x^4\right ) \, dx,x,\tan (x)\right )\\ &=\frac{\tan ^3(x)}{3}+\frac{\tan ^5(x)}{5}\\ \end{align*}
Mathematica [A] time = 0.0157197, size = 27, normalized size = 1.59 \[ -\frac{2 \tan (x)}{15}+\frac{1}{5} \tan (x) \sec ^4(x)-\frac{1}{15} \tan (x) \sec ^2(x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 22, normalized size = 1.3 \begin{align*}{\frac{ \left ( \sin \left ( x \right ) \right ) ^{3}}{5\, \left ( \cos \left ( x \right ) \right ) ^{5}}}+{\frac{2\, \left ( \sin \left ( x \right ) \right ) ^{3}}{15\, \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 = 0.923833, size = 18, normalized size = 1.06 \begin{align*} \frac{1}{5} \, \tan \left (x\right )^{5} + \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 [A] time = 2.22829, size = 69, normalized size = 4.06 \begin{align*} -\frac{{\left (2 \, \cos \left (x\right )^{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 [B] time = 0.06336, size = 29, normalized size = 1.71 \begin{align*} - \frac{2 \sin{\left (x \right )}}{15 \cos{\left (x \right )}} - \frac{\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 = 1.06102, size = 18, normalized size = 1.06 \begin{align*} \frac{1}{5} \, \tan \left (x\right )^{5} + \frac{1}{3} \, \tan \left (x\right )^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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