Optimal. Leaf size=19 \[ \frac {\sec ^5(x)}{5}-\frac {2 \sec ^3(x)}{3}+\sec (x) \]
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Rubi [A] time = 0.02, antiderivative size = 19, normalized size of antiderivative = 1.00, 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 194
Rule 2606
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*}
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Mathematica [A] time = 0.01, size = 19, normalized size = 1.00 \[ \frac {\sec ^5(x)}{5}-\frac {2 \sec ^3(x)}{3}+\sec (x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 20, normalized size = 1.05 \[ \frac {15 \, \cos \relax (x)^{4} - 10 \, \cos \relax (x)^{2} + 3}{15 \, \cos \relax (x)^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.03, size = 20, normalized size = 1.05 \[ \frac {15 \, \cos \relax (x)^{4} - 10 \, \cos \relax (x)^{2} + 3}{15 \, \cos \relax (x)^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 48, normalized size = 2.53 \[ \frac {\sin ^{6}\relax (x )}{5 \cos \relax (x )}-\frac {\sin ^{6}\relax (x )}{15 \cos \relax (x )^{3}}+\frac {\left (\sin ^{4}\relax (x )+\frac {4 \left (\sin ^{2}\relax (x )\right )}{3}+\frac {8}{3}\right ) \cos \relax (x )}{5}+\frac {\sin ^{6}\relax (x )}{5 \cos \relax (x )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 20, normalized size = 1.05 \[ \frac {15 \, \cos \relax (x)^{4} - 10 \, \cos \relax (x)^{2} + 3}{15 \, \cos \relax (x)^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.27, size = 17, normalized size = 0.89 \[ \frac {{\cos \relax (x)}^4-\frac {2\,{\cos \relax (x)}^2}{3}+\frac {1}{5}}{{\cos \relax (x)}^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 22, normalized size = 1.16 \[ - \frac {- 15 \cos ^{4}{\relax (x )} + 10 \cos ^{2}{\relax (x )} - 3}{15 \cos ^{5}{\relax (x )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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