Optimal. Leaf size=15 \[ \frac {\tanh ^{-1}\left (\sqrt {2} \sin (x)\right )}{\sqrt {2}} \]
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Rubi [A] time = 0.01, antiderivative size = 15, 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 = {4356, 206} \[ \frac {\tanh ^{-1}\left (\sqrt {2} \sin (x)\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 4356
Rubi steps
\begin {align*} \int \cos (x) \sec (2 x) \, dx &=\operatorname {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\sin (x)\right )\\ &=\frac {\tanh ^{-1}\left (\sqrt {2} \sin (x)\right )}{\sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 15, normalized size = 1.00 \[ \frac {\tanh ^{-1}\left (\sqrt {2} \sin (x)\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.13, size = 33, normalized size = 2.20 \[ \frac {1}{4} \, \sqrt {2} \log \left (-\frac {2 \, \cos \relax (x)^{2} - 2 \, \sqrt {2} \sin \relax (x) - 3}{2 \, \cos \relax (x)^{2} - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.13, size = 31, normalized size = 2.07 \[ \frac {1}{4} \, \sqrt {2} \log \left ({\left | \frac {1}{2} \, \sqrt {2} + \sin \relax (x) \right |}\right ) - \frac {1}{4} \, \sqrt {2} \log \left ({\left | -\frac {1}{2} \, \sqrt {2} + \sin \relax (x) \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 13, normalized size = 0.87 \[ \frac {\arctanh \left (\sin \relax (x ) \sqrt {2}\right ) \sqrt {2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.08, size = 137, normalized size = 9.13 \[ \frac {1}{8} \, \sqrt {2} \log \left (2 \, \cos \relax (x)^{2} + 2 \, \sin \relax (x)^{2} + 2 \, \sqrt {2} \cos \relax (x) + 2 \, \sqrt {2} \sin \relax (x) + 2\right ) - \frac {1}{8} \, \sqrt {2} \log \left (2 \, \cos \relax (x)^{2} + 2 \, \sin \relax (x)^{2} + 2 \, \sqrt {2} \cos \relax (x) - 2 \, \sqrt {2} \sin \relax (x) + 2\right ) + \frac {1}{8} \, \sqrt {2} \log \left (2 \, \cos \relax (x)^{2} + 2 \, \sin \relax (x)^{2} - 2 \, \sqrt {2} \cos \relax (x) + 2 \, \sqrt {2} \sin \relax (x) + 2\right ) - \frac {1}{8} \, \sqrt {2} \log \left (2 \, \cos \relax (x)^{2} + 2 \, \sin \relax (x)^{2} - 2 \, \sqrt {2} \cos \relax (x) - 2 \, \sqrt {2} \sin \relax (x) + 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 12, normalized size = 0.80 \[ \frac {\sqrt {2}\,\mathrm {atanh}\left (\sqrt {2}\,\sin \relax (x)\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos {\relax (x )} \sec {\left (2 x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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