Integrand size = 11, antiderivative size = 11 \[ \int \frac {1}{a \sec (x)+b \tan (x)} \, dx=\frac {\log (a+b \sin (x))}{b} \]
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Time = 0.04 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3238, 2747, 31} \[ \int \frac {1}{a \sec (x)+b \tan (x)} \, dx=\frac {\log (a+b \sin (x))}{b} \]
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Rule 31
Rule 2747
Rule 3238
Rubi steps \begin{align*} \text {integral}& = \int \frac {\cos (x)}{a+b \sin (x)} \, dx \\ & = \frac {\text {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \sin (x)\right )}{b} \\ & = \frac {\log (a+b \sin (x))}{b} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {1}{a \sec (x)+b \tan (x)} \, dx=\frac {\log (a+b \sin (x))}{b} \]
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Time = 0.55 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09
method | result | size |
default | \(\frac {\ln \left (a +b \sin \left (x \right )\right )}{b}\) | \(12\) |
risch | \(-\frac {i x}{b}+\frac {\ln \left ({\mathrm e}^{2 i x}+\frac {2 i a \,{\mathrm e}^{i x}}{b}-1\right )}{b}\) | \(33\) |
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none
Time = 0.28 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {1}{a \sec (x)+b \tan (x)} \, dx=\frac {\log \left (b \sin \left (x\right ) + a\right )}{b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 32 vs. \(2 (8) = 16\).
Time = 0.13 (sec) , antiderivative size = 32, normalized size of antiderivative = 2.91 \[ \int \frac {1}{a \sec (x)+b \tan (x)} \, dx=\begin {cases} \frac {\log {\left (\frac {a \sec {\left (x \right )}}{b} + \tan {\left (x \right )} \right )}}{b} - \frac {\log {\left (\tan ^{2}{\left (x \right )} + 1 \right )}}{2 b} & \text {for}\: b \neq 0 \\\frac {\tan {\left (x \right )}}{a \sec {\left (x \right )}} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (11) = 22\).
Time = 0.30 (sec) , antiderivative size = 50, normalized size of antiderivative = 4.55 \[ \int \frac {1}{a \sec (x)+b \tan (x)} \, dx=\frac {\log \left (a + \frac {2 \, b \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {a \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}\right )}{b} - \frac {\log \left (\frac {\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right )}{b} \]
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none
Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09 \[ \int \frac {1}{a \sec (x)+b \tan (x)} \, dx=\frac {\log \left ({\left | b \sin \left (x\right ) + a \right |}\right )}{b} \]
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Time = 31.00 (sec) , antiderivative size = 55, normalized size of antiderivative = 5.00 \[ \int \frac {1}{a \sec (x)+b \tan (x)} \, dx=\frac {2\,\mathrm {atanh}\left (\frac {b\,\left (2\,a^3\,\sin \left (x\right )+\frac {5\,a^2\,b}{2}-b^3-\frac {a^2\,b\,\cos \left (2\,x\right )}{2}\right )}{{\left (2\,a^2+\sin \left (x\right )\,a\,b-b^2\right )}^2}\right )}{b} \]
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