Optimal. Leaf size=17 \[ \frac {\tan ^5(x)}{5}+\frac {\tan ^3(x)}{3} \]
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Rubi [A] time = 0.02, antiderivative size = 17, normalized size of antiderivative = 1.00, 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 14
Rule 2607
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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 0.43, size = 20, normalized size = 1.18 \[ -\frac {{\left (2 \, \cos \relax (x)^{4} + \cos \relax (x)^{2} - 3\right )} \sin \relax (x)}{15 \, \cos \relax (x)^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.77, size = 13, normalized size = 0.76 \[ \frac {1}{5} \, \tan \relax (x)^{5} + \frac {1}{3} \, \tan \relax (x)^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 22, normalized size = 1.29 \[ \frac {2 \left (\sin ^{3}\relax (x )\right )}{15 \cos \relax (x )^{3}}+\frac {\sin ^{3}\relax (x )}{5 \cos \relax (x )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 13, normalized size = 0.76 \[ \frac {1}{5} \, \tan \relax (x)^{5} + \frac {1}{3} \, \tan \relax (x)^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.17, size = 13, normalized size = 0.76 \[ \frac {{\mathrm {tan}\relax (x)}^5}{5}+\frac {{\mathrm {tan}\relax (x)}^3}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.07, size = 29, normalized size = 1.71 \[ - \frac {2 \sin {\relax (x )}}{15 \cos {\relax (x )}} - \frac {\sin {\relax (x )}}{15 \cos ^{3}{\relax (x )}} + \frac {\sin {\relax (x )}}{5 \cos ^{5}{\relax (x )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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