Optimal. Leaf size=13 \[ \frac {\tanh ^{-1}(\sin (2 a x))}{2 a} \]
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Rubi [A] time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3770} \[ \frac {\tanh ^{-1}(\sin (2 a x))}{2 a} \]
Antiderivative was successfully verified.
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Rule 3770
Rubi steps
\begin {align*} \int \sec (2 a x) \, dx &=\frac {\tanh ^{-1}(\sin (2 a x))}{2 a}\\ \end {align*}
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Mathematica [B] time = 0.01, size = 37, normalized size = 2.85 \[ \frac {\log (\sin (a x)+\cos (a x))}{2 a}-\frac {\log (\cos (a x)-\sin (a x))}{2 a} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sec (2 a x) \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 1.16, size = 26, normalized size = 2.00 \[ \frac {\log \left (\sin \left (2 \, a x\right ) + 1\right ) - \log \left (-\sin \left (2 \, a x\right ) + 1\right )}{4 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 18, normalized size = 1.38
method | result | size |
derivativedivides | \(\frac {\ln \left (\sec \left (2 a x \right )+\tan \left (2 a x \right )\right )}{2 a}\) | \(18\) |
default | \(\frac {\ln \left (\sec \left (2 a x \right )+\tan \left (2 a x \right )\right )}{2 a}\) | \(18\) |
norman | \(-\frac {\ln \left (\tan \left (a x \right )-1\right )}{2 a}+\frac {\ln \left (\tan \left (a x \right )+1\right )}{2 a}\) | \(26\) |
risch | \(\frac {\ln \left ({\mathrm e}^{2 i a x}+i\right )}{2 a}-\frac {\ln \left ({\mathrm e}^{2 i a x}-i\right )}{2 a}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 17, normalized size = 1.31 \[ \frac {\log \left (\sec \left (2 \, a x\right ) + \tan \left (2 \, a x\right )\right )}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.23, size = 11, normalized size = 0.85 \[ \frac {\mathrm {atanh}\left (\sin \left (2\,a\,x\right )\right )}{2\,a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 29, normalized size = 2.23 \[ \begin {cases} \frac {- \frac {\log {\left (\sin {\left (2 a x \right )} - 1 \right )}}{2} + \frac {\log {\left (\sin {\left (2 a x \right )} + 1 \right )}}{2}}{2 a} & \text {for}\: 2 a \neq 0 \\x & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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