Integrand size = 9, antiderivative size = 9 \[ \int \cos ^2(x) \sec (3 x) \, dx=\frac {1}{2} \text {arctanh}(2 \sin (x)) \]
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Time = 0.02 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {212} \[ \int \cos ^2(x) \sec (3 x) \, dx=\frac {1}{2} \text {arctanh}(2 \sin (x)) \]
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Rule 212
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{1-4 x^2} \, dx,x,\sin (x)\right ) \\ & = \frac {1}{2} \text {arctanh}(2 \sin (x)) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00 \[ \int \cos ^2(x) \sec (3 x) \, dx=\frac {1}{2} \text {arctanh}(2 \sin (x)) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(19\) vs. \(2(7)=14\).
Time = 0.90 (sec) , antiderivative size = 20, normalized size of antiderivative = 2.22
method | result | size |
default | \(\frac {\ln \left (1+2 \sin \left (x \right )\right )}{4}-\frac {\ln \left (2 \sin \left (x \right )-1\right )}{4}\) | \(20\) |
risch | \(-\frac {\ln \left (-i {\mathrm e}^{i x}+{\mathrm e}^{2 i x}-1\right )}{4}+\frac {\ln \left (i {\mathrm e}^{i x}+{\mathrm e}^{2 i x}-1\right )}{4}\) | \(38\) |
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Leaf count of result is larger than twice the leaf count of optimal. 19 vs. \(2 (7) = 14\).
Time = 0.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 2.11 \[ \int \cos ^2(x) \sec (3 x) \, dx=\frac {1}{4} \, \log \left (2 \, \sin \left (x\right ) + 1\right ) - \frac {1}{4} \, \log \left (-2 \, \sin \left (x\right ) + 1\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (7) = 14\).
Time = 1.80 (sec) , antiderivative size = 76, normalized size of antiderivative = 8.44 \[ \int \cos ^2(x) \sec (3 x) \, dx=- \frac {\log {\left (\sin {\left (3 x \right )} - 1 \right )}}{12} + \frac {\log {\left (\sin {\left (3 x \right )} + 1 \right )}}{12} - \frac {\log {\left (\tan {\left (\frac {x}{2} \right )} - 1 \right )}}{6} + \frac {\log {\left (\tan {\left (\frac {x}{2} \right )} + 1 \right )}}{6} - \frac {\log {\left (\tan ^{2}{\left (\frac {x}{2} \right )} - 4 \tan {\left (\frac {x}{2} \right )} + 1 \right )}}{12} + \frac {\log {\left (\tan ^{2}{\left (\frac {x}{2} \right )} + 4 \tan {\left (\frac {x}{2} \right )} + 1 \right )}}{12} \]
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\[ \int \cos ^2(x) \sec (3 x) \, dx=\int { \frac {\cos \left (x\right )^{2}}{\cos \left (3 \, x\right )} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 21 vs. \(2 (7) = 14\).
Time = 0.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 2.33 \[ \int \cos ^2(x) \sec (3 x) \, dx=\frac {1}{4} \, \log \left ({\left | 2 \, \sin \left (x\right ) + 1 \right |}\right ) - \frac {1}{4} \, \log \left ({\left | 2 \, \sin \left (x\right ) - 1 \right |}\right ) \]
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Time = 0.40 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.78 \[ \int \cos ^2(x) \sec (3 x) \, dx=\frac {\mathrm {atanh}\left (2\,\sin \left (x\right )\right )}{2} \]
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