Optimal. Leaf size=8 \[ \frac {\tan ^3(x)}{3} \]
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Rubi [A] time = 0.02, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2607, 30} \[ \frac {\tan ^3(x)}{3} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2607
Rubi steps
\begin {align*} \int \sec ^2(x) \tan ^2(x) \, dx &=\operatorname {Subst}\left (\int x^2 \, dx,x,\tan (x)\right )\\ &=\frac {\tan ^3(x)}{3}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 8, normalized size = 1.00 \[ \frac {\tan ^3(x)}{3} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sec ^2(x) \tan ^2(x) \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 0.89, size = 14, normalized size = 1.75 \[ -\frac {{\left (\cos \relax (x)^{2} - 1\right )} \sin \relax (x)}{3 \, \cos \relax (x)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.62, size = 6, normalized size = 0.75 \[ \frac {1}{3} \, \tan \relax (x)^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 11, normalized size = 1.38
method | result | size |
default | \(\frac {\sin ^{3}\relax (x )}{3 \cos \relax (x )^{3}}\) | \(11\) |
risch | \(-\frac {2 i \left (3 \,{\mathrm e}^{4 i x}+1\right )}{3 \left (1+{\mathrm e}^{2 i x}\right )^{3}}\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 6, normalized size = 0.75 \[ \frac {1}{3} \, \tan \relax (x)^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 6, normalized size = 0.75 \[ \frac {{\mathrm {tan}\relax (x)}^3}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.07, size = 17, normalized size = 2.12 \[ - \frac {\sin {\relax (x )}}{3 \cos {\relax (x )}} + \frac {\sin {\relax (x )}}{3 \cos ^{3}{\relax (x )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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