Optimal. Leaf size=9 \[ \frac {1}{2} \tanh ^{-1}(2 \sin (x)) \]
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Rubi [A] time = 0.02, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {206} \[ \frac {1}{2} \tanh ^{-1}(2 \sin (x)) \]
Antiderivative was successfully verified.
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Rule 206
Rubi steps
\begin {align*} \int \cos ^2(x) \sec (3 x) \, dx &=\operatorname {Subst}\left (\int \frac {1}{1-4 x^2} \, dx,x,\sin (x)\right )\\ &=\frac {1}{2} \tanh ^{-1}(2 \sin (x))\\ \end {align*}
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Mathematica [A] time = 0.01, size = 9, normalized size = 1.00 \[ \frac {1}{2} \tanh ^{-1}(2 \sin (x)) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos ^2(x) \sec (3 x) \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 1.03, size = 19, normalized size = 2.11 \[ \frac {1}{4} \, \log \left (2 \, \sin \relax (x) + 1\right ) - \frac {1}{4} \, \log \left (-2 \, \sin \relax (x) + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.65, size = 21, normalized size = 2.33 \[ \frac {1}{4} \, \log \left ({\left | 2 \, \sin \relax (x) + 1 \right |}\right ) - \frac {1}{4} \, \log \left ({\left | 2 \, \sin \relax (x) - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.15, size = 20, normalized size = 2.22
method | result | size |
default | \(-\frac {\ln \left (2 \sin \relax (x )-1\right )}{4}+\frac {\ln \left (1+2 \sin \relax (x )\right )}{4}\) | \(20\) |
risch | \(\frac {\ln \left (i {\mathrm e}^{i x}+{\mathrm e}^{2 i x}-1\right )}{4}-\frac {\ln \left (-i {\mathrm e}^{i x}+{\mathrm e}^{2 i x}-1\right )}{4}\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \relax (x)^{2}}{\cos \left (3 \, x\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.36, size = 7, normalized size = 0.78 \[ \frac {\mathrm {atanh}\left (2\,\sin \relax (x)\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 5.46, size = 76, normalized size = 8.44 \[ - \frac {\log {\left (\sin {\left (3 x \right )} - 1 \right )}}{12} + \frac {\log {\left (\sin {\left (3 x \right )} + 1 \right )}}{12} - \frac {\log {\left (\tan {\left (\frac {x}{2} \right )} - 1 \right )}}{6} + \frac {\log {\left (\tan {\left (\frac {x}{2} \right )} + 1 \right )}}{6} - \frac {\log {\left (\tan ^{2}{\left (\frac {x}{2} \right )} - 4 \tan {\left (\frac {x}{2} \right )} + 1 \right )}}{12} + \frac {\log {\left (\tan ^{2}{\left (\frac {x}{2} \right )} + 4 \tan {\left (\frac {x}{2} \right )} + 1 \right )}}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
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