Integrand size = 11, antiderivative size = 17 \[ \int \frac {1}{(b \cos (x)+a \sin (x))^2} \, dx=\frac {\sin (x)}{b (b \cos (x)+a \sin (x))} \]
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Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {3154} \[ \int \frac {1}{(b \cos (x)+a \sin (x))^2} \, dx=\frac {\sin (x)}{b (a \sin (x)+b \cos (x))} \]
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Rule 3154
Rubi steps \begin{align*} \text {integral}& = \frac {\sin (x)}{b (b \cos (x)+a \sin (x))} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(b \cos (x)+a \sin (x))^2} \, dx=\frac {\sin (x)}{b (b \cos (x)+a \sin (x))} \]
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Time = 0.30 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82
method | result | size |
default | \(-\frac {1}{a \left (a \tan \left (x \right )+b \right )}\) | \(14\) |
parallelrisch | \(\frac {\sin \left (x \right )}{b \left (b \cos \left (x \right )+a \sin \left (x \right )\right )}\) | \(18\) |
norman | \(\frac {-\frac {1}{a}+\frac {\tan ^{2}\left (\frac {x}{2}\right )}{a}}{-b \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+2 a \tan \left (\frac {x}{2}\right )+b}\) | \(38\) |
risch | \(-\frac {2 i}{\left (a \,{\mathrm e}^{2 i x}+i b \,{\mathrm e}^{2 i x}-a +i b \right ) \left (i b +a \right )}\) | \(38\) |
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Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (17) = 34\).
Time = 0.26 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.29 \[ \int \frac {1}{(b \cos (x)+a \sin (x))^2} \, dx=-\frac {a \cos \left (x\right ) - b \sin \left (x\right )}{{\left (a^{2} b + b^{3}\right )} \cos \left (x\right ) + {\left (a^{3} + a b^{2}\right )} \sin \left (x\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 602 vs. \(2 (14) = 28\).
Time = 134.57 (sec) , antiderivative size = 602, normalized size of antiderivative = 35.41 \[ \int \frac {1}{(b \cos (x)+a \sin (x))^2} \, dx=\begin {cases} \frac {\tilde {\infty } \tan {\left (\frac {x}{2} \right )}}{\tan ^{2}{\left (\frac {x}{2} \right )} - 1} & \text {for}\: a = 0 \wedge b = 0 \\\frac {x \tan ^{4}{\left (\frac {x}{2} \right )}}{2 b^{2} \sin ^{2}{\left (x \right )} \tan ^{4}{\left (\frac {x}{2} \right )} - 4 b^{2} \sin ^{2}{\left (x \right )} \tan ^{2}{\left (\frac {x}{2} \right )} + 2 b^{2} \sin ^{2}{\left (x \right )} + 8 b^{2} \sin {\left (x \right )} \cos {\left (x \right )} \tan ^{3}{\left (\frac {x}{2} \right )} - 8 b^{2} \sin {\left (x \right )} \cos {\left (x \right )} \tan {\left (\frac {x}{2} \right )} + 8 b^{2} \cos ^{2}{\left (x \right )} \tan ^{2}{\left (\frac {x}{2} \right )}} + \frac {2 x \tan ^{2}{\left (\frac {x}{2} \right )}}{2 b^{2} \sin ^{2}{\left (x \right )} \tan ^{4}{\left (\frac {x}{2} \right )} - 4 b^{2} \sin ^{2}{\left (x \right )} \tan ^{2}{\left (\frac {x}{2} \right )} + 2 b^{2} \sin ^{2}{\left (x \right )} + 8 b^{2} \sin {\left (x \right )} \cos {\left (x \right )} \tan ^{3}{\left (\frac {x}{2} \right )} - 8 b^{2} \sin {\left (x \right )} \cos {\left (x \right )} \tan {\left (\frac {x}{2} \right )} + 8 b^{2} \cos ^{2}{\left (x \right )} \tan ^{2}{\left (\frac {x}{2} \right )}} + \frac {x}{2 b^{2} \sin ^{2}{\left (x \right )} \tan ^{4}{\left (\frac {x}{2} \right )} - 4 b^{2} \sin ^{2}{\left (x \right )} \tan ^{2}{\left (\frac {x}{2} \right )} + 2 b^{2} \sin ^{2}{\left (x \right )} + 8 b^{2} \sin {\left (x \right )} \cos {\left (x \right )} \tan ^{3}{\left (\frac {x}{2} \right )} - 8 b^{2} \sin {\left (x \right )} \cos {\left (x \right )} \tan {\left (\frac {x}{2} \right )} + 8 b^{2} \cos ^{2}{\left (x \right )} \tan ^{2}{\left (\frac {x}{2} \right )}} + \frac {2 \tan ^{3}{\left (\frac {x}{2} \right )}}{2 b^{2} \sin ^{2}{\left (x \right )} \tan ^{4}{\left (\frac {x}{2} \right )} - 4 b^{2} \sin ^{2}{\left (x \right )} \tan ^{2}{\left (\frac {x}{2} \right )} + 2 b^{2} \sin ^{2}{\left (x \right )} + 8 b^{2} \sin {\left (x \right )} \cos {\left (x \right )} \tan ^{3}{\left (\frac {x}{2} \right )} - 8 b^{2} \sin {\left (x \right )} \cos {\left (x \right )} \tan {\left (\frac {x}{2} \right )} + 8 b^{2} \cos ^{2}{\left (x \right )} \tan ^{2}{\left (\frac {x}{2} \right )}} - \frac {2 \tan {\left (\frac {x}{2} \right )}}{2 b^{2} \sin ^{2}{\left (x \right )} \tan ^{4}{\left (\frac {x}{2} \right )} - 4 b^{2} \sin ^{2}{\left (x \right )} \tan ^{2}{\left (\frac {x}{2} \right )} + 2 b^{2} \sin ^{2}{\left (x \right )} + 8 b^{2} \sin {\left (x \right )} \cos {\left (x \right )} \tan ^{3}{\left (\frac {x}{2} \right )} - 8 b^{2} \sin {\left (x \right )} \cos {\left (x \right )} \tan {\left (\frac {x}{2} \right )} + 8 b^{2} \cos ^{2}{\left (x \right )} \tan ^{2}{\left (\frac {x}{2} \right )}} & \text {for}\: a = \frac {b \left (\tan {\left (\frac {x}{2} \right )} - 1\right ) \left (\tan {\left (\frac {x}{2} \right )} + 1\right )}{2 \tan {\left (\frac {x}{2} \right )}} \\\frac {\frac {\tan {\left (\frac {x}{2} \right )}}{2} - \frac {1}{2 \tan {\left (\frac {x}{2} \right )}}}{a^{2}} & \text {for}\: b = 0 \\\frac {2 \tan {\left (\frac {x}{2} \right )}}{2 a b \tan {\left (\frac {x}{2} \right )} - b^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + b^{2}} & \text {otherwise} \end {cases} \]
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Time = 0.18 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int \frac {1}{(b \cos (x)+a \sin (x))^2} \, dx=-\frac {1}{a^{2} \tan \left (x\right ) + a b} \]
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Time = 0.28 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \frac {1}{(b \cos (x)+a \sin (x))^2} \, dx=-\frac {1}{{\left (a \tan \left (x\right ) + b\right )} a} \]
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Time = 0.65 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.71 \[ \int \frac {1}{(b \cos (x)+a \sin (x))^2} \, dx=\frac {2\,\mathrm {tan}\left (\frac {x}{2}\right )}{b\,\left (-b\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+2\,a\,\mathrm {tan}\left (\frac {x}{2}\right )+b\right )} \]
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