Optimal. Leaf size=25 \[ \frac{\sec ^7(x)}{7}-\frac{2 \sec ^5(x)}{5}+\frac{\sec ^3(x)}{3} \]
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Rubi [A] time = 0.0291992, antiderivative size = 25, 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, 270} \[ \frac{\sec ^7(x)}{7}-\frac{2 \sec ^5(x)}{5}+\frac{\sec ^3(x)}{3} \]
Antiderivative was successfully verified.
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Rule 2606
Rule 270
Rubi steps
\begin{align*} \int \sec ^3(x) \tan ^5(x) \, dx &=\operatorname{Subst}\left (\int x^2 \left (-1+x^2\right )^2 \, dx,x,\sec (x)\right )\\ &=\operatorname{Subst}\left (\int \left (x^2-2 x^4+x^6\right ) \, dx,x,\sec (x)\right )\\ &=\frac{\sec ^3(x)}{3}-\frac{2 \sec ^5(x)}{5}+\frac{\sec ^7(x)}{7}\\ \end{align*}
Mathematica [A] time = 0.011756, size = 25, normalized size = 1. \[ \frac{\sec ^7(x)}{7}-\frac{2 \sec ^5(x)}{5}+\frac{\sec ^3(x)}{3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.012, size = 58, normalized size = 2.3 \begin{align*}{\frac{ \left ( \sin \left ( x \right ) \right ) ^{6}}{7\, \left ( \cos \left ( x \right ) \right ) ^{7}}}+{\frac{ \left ( \sin \left ( x \right ) \right ) ^{6}}{35\, \left ( \cos \left ( x \right ) \right ) ^{5}}}-{\frac{ \left ( \sin \left ( x \right ) \right ) ^{6}}{105\, \left ( \cos \left ( x \right ) \right ) ^{3}}}+{\frac{ \left ( \sin \left ( x \right ) \right ) ^{6}}{35\,\cos \left ( x \right ) }}+{\frac{\cos \left ( x \right ) }{35} \left ({\frac{8}{3}}+ \left ( \sin \left ( x \right ) \right ) ^{4}+{\frac{4\, \left ( \sin \left ( x \right ) \right ) ^{2}}{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.926329, size = 27, normalized size = 1.08 \begin{align*} \frac{35 \, \cos \left (x\right )^{4} - 42 \, \cos \left (x\right )^{2} + 15}{105 \, \cos \left (x\right )^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.05767, size = 66, normalized size = 2.64 \begin{align*} \frac{35 \, \cos \left (x\right )^{4} - 42 \, \cos \left (x\right )^{2} + 15}{105 \, \cos \left (x\right )^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.114735, size = 20, normalized size = 0.8 \begin{align*} \frac{35 \cos ^{4}{\left (x \right )} - 42 \cos ^{2}{\left (x \right )} + 15}{105 \cos ^{7}{\left (x \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06472, size = 27, normalized size = 1.08 \begin{align*} \frac{35 \, \cos \left (x\right )^{4} - 42 \, \cos \left (x\right )^{2} + 15}{105 \, \cos \left (x\right )^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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