Optimal. Leaf size=19 \[ \frac{\sec ^5(x)}{5}-\frac{2 \sec ^3(x)}{3}+\sec (x) \]
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Rubi [A] time = 0.0163057, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2606, 194} \[ \frac{\sec ^5(x)}{5}-\frac{2 \sec ^3(x)}{3}+\sec (x) \]
Antiderivative was successfully verified.
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Rule 2606
Rule 194
Rubi steps
\begin{align*} \int \sec (x) \tan ^5(x) \, dx &=\operatorname{Subst}\left (\int \left (-1+x^2\right )^2 \, dx,x,\sec (x)\right )\\ &=\operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\sec (x)\right )\\ &=\sec (x)-\frac{2 \sec ^3(x)}{3}+\frac{\sec ^5(x)}{5}\\ \end{align*}
Mathematica [A] time = 0.0097963, size = 19, normalized size = 1. \[ \frac{\sec ^5(x)}{5}-\frac{2 \sec ^3(x)}{3}+\sec (x) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.008, size = 48, normalized size = 2.5 \begin{align*}{\frac{ \left ( \sin \left ( x \right ) \right ) ^{6}}{5\, \left ( \cos \left ( x \right ) \right ) ^{5}}}-{\frac{ \left ( \sin \left ( x \right ) \right ) ^{6}}{15\, \left ( \cos \left ( x \right ) \right ) ^{3}}}+{\frac{ \left ( \sin \left ( x \right ) \right ) ^{6}}{5\,\cos \left ( x \right ) }}+{\frac{\cos \left ( x \right ) }{5} \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 = 0.937615, size = 27, normalized size = 1.42 \begin{align*} \frac{15 \, \cos \left (x\right )^{4} - 10 \, \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 = 2.02845, size = 63, normalized size = 3.32 \begin{align*} \frac{15 \, \cos \left (x\right )^{4} - 10 \, \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 = 0.103827, size = 20, normalized size = 1.05 \begin{align*} \frac{15 \cos ^{4}{\left (x \right )} - 10 \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 = 1.05896, size = 27, normalized size = 1.42 \begin{align*} \frac{15 \, \cos \left (x\right )^{4} - 10 \, \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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