Integrand size = 7, antiderivative size = 5 \[ \int \frac {1}{\sec (x)+\tan (x)} \, dx=\log (1+\sin (x)) \]
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Time = 0.03 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3238, 2746, 31} \[ \int \frac {1}{\sec (x)+\tan (x)} \, dx=\log (\sin (x)+1) \]
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Rule 31
Rule 2746
Rule 3238
Rubi steps \begin{align*} \text {integral}& = \int \frac {\cos (x)}{1+\sin (x)} \, dx \\ & = \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,\sin (x)\right ) \\ & = \log (1+\sin (x)) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(16\) vs. \(2(5)=10\).
Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 3.20 \[ \int \frac {1}{\sec (x)+\tan (x)} \, dx=2 \log \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right ) \]
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Time = 0.35 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.20
method | result | size |
default | \(\ln \left (1+\sin \left (x \right )\right )\) | \(6\) |
risch | \(-i x +2 \ln \left (i+{\mathrm e}^{i x}\right )\) | \(17\) |
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none
Time = 0.25 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sec (x)+\tan (x)} \, dx=\log \left (\sin \left (x\right ) + 1\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 17 vs. \(2 (5) = 10\).
Time = 0.06 (sec) , antiderivative size = 17, normalized size of antiderivative = 3.40 \[ \int \frac {1}{\sec (x)+\tan (x)} \, dx=\log {\left (\tan {\left (x \right )} + \sec {\left (x \right )} \right )} - \frac {\log {\left (\tan ^{2}{\left (x \right )} + 1 \right )}}{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (5) = 10\).
Time = 0.22 (sec) , antiderivative size = 31, normalized size of antiderivative = 6.20 \[ \int \frac {1}{\sec (x)+\tan (x)} \, dx=2 \, \log \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1} + 1\right ) - \log \left (\frac {\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 22 vs. \(2 (5) = 10\).
Time = 0.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 4.40 \[ \int \frac {1}{\sec (x)+\tan (x)} \, dx=-\log \left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right ) + 2 \, \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) + 1 \right |}\right ) \]
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Time = 30.21 (sec) , antiderivative size = 21, normalized size of antiderivative = 4.20 \[ \int \frac {1}{\sec (x)+\tan (x)} \, dx=2\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )-\ln \left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right ) \]
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