Optimal. Leaf size=8 \[ \frac {\sec ^3(x)}{3} \]
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Rubi [A] time = 0.01, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2606, 30} \[ \frac {\sec ^3(x)}{3} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2606
Rubi steps
\begin {align*} \int \sec ^3(x) \tan (x) \, dx &=\operatorname {Subst}\left (\int x^2 \, dx,x,\sec (x)\right )\\ &=\frac {\sec ^3(x)}{3}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 8, normalized size = 1.00 \[ \frac {\sec ^3(x)}{3} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sec ^3(x) \tan (x) \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.62, size = 6, normalized size = 0.75 \[ \frac {1}{3 \, \cos \relax (x)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.63, size = 6, normalized size = 0.75 \[ \frac {1}{3 \, \cos \relax (x)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 7, normalized size = 0.88
method | result | size |
derivativedivides | \(\frac {\left (\sec ^{3}\relax (x )\right )}{3}\) | \(7\) |
default | \(\frac {\left (\sec ^{3}\relax (x )\right )}{3}\) | \(7\) |
risch | \(\frac {8 \,{\mathrm e}^{3 i x}}{3 \left (1+{\mathrm e}^{2 i x}\right )^{3}}\) | \(17\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 6, normalized size = 0.75 \[ \frac {1}{3 \, \cos \relax (x)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.32, size = 6, normalized size = 0.75 \[ \frac {1}{3\,{\cos \relax (x)}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.07, size = 7, normalized size = 0.88 \[ \frac {1}{3 \cos ^{3}{\relax (x )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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