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 = {2686, 30}
\begin {gather*} \frac {\sec ^3(x)}{3} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2686
Rubi steps
\begin {align*} \int \sec ^3(x) \tan (x) \, dx &=\text {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 \begin {gather*} \frac {\sec ^3(x)}{3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 7, normalized size = 0.88
method | result | size |
derivativedivides | \(\frac {\left (\sec ^{3}\left (x \right )\right )}{3}\) | \(7\) |
default | \(\frac {\left (\sec ^{3}\left (x \right )\right )}{3}\) | \(7\) |
risch | \(\frac {8 \,{\mathrm e}^{3 i x}}{3 \left ({\mathrm e}^{2 i x}+1\right )^{3}}\) | \(17\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.96, size = 6, normalized size = 0.75 \begin {gather*} \frac {1}{3 \, \cos \left (x\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.09, size = 6, normalized size = 0.75 \begin {gather*} \frac {1}{3 \, \cos \left (x\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 7, normalized size = 0.88 \begin {gather*} \frac {1}{3 \cos ^{3}{\left (x \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.60, size = 6, normalized size = 0.75 \begin {gather*} \frac {1}{3 \, \cos \left (x\right )^{3}} \end {gather*}
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 \begin {gather*} \frac {1}{3\,{\cos \left (x\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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