Integrand size = 13, antiderivative size = 25 \[ \int \frac {\cos (x)}{a+b \sin ^2(x)} \, dx=\frac {\arctan \left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}} \]
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Time = 0.03 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3269, 211} \[ \int \frac {\cos (x)}{a+b \sin ^2(x)} \, dx=\frac {\arctan \left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}} \]
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Rule 211
Rule 3269
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sin (x)\right ) \\ & = \frac {\arctan \left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {\cos (x)}{a+b \sin ^2(x)} \, dx=\frac {\arctan \left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}} \]
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Time = 0.37 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.68
method | result | size |
derivativedivides | \(\frac {\arctan \left (\frac {b \sin \left (x \right )}{\sqrt {a b}}\right )}{\sqrt {a b}}\) | \(17\) |
default | \(\frac {\arctan \left (\frac {b \sin \left (x \right )}{\sqrt {a b}}\right )}{\sqrt {a b}}\) | \(17\) |
risch | \(-\frac {\ln \left ({\mathrm e}^{2 i x}-\frac {2 i a \,{\mathrm e}^{i x}}{\sqrt {-a b}}-1\right )}{2 \sqrt {-a b}}+\frac {\ln \left ({\mathrm e}^{2 i x}+\frac {2 i a \,{\mathrm e}^{i x}}{\sqrt {-a b}}-1\right )}{2 \sqrt {-a b}}\) | \(64\) |
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none
Time = 0.29 (sec) , antiderivative size = 78, normalized size of antiderivative = 3.12 \[ \int \frac {\cos (x)}{a+b \sin ^2(x)} \, dx=\left [-\frac {\sqrt {-a b} \log \left (-\frac {b \cos \left (x\right )^{2} + 2 \, \sqrt {-a b} \sin \left (x\right ) + a - b}{b \cos \left (x\right )^{2} - a - b}\right )}{2 \, a b}, \frac {\sqrt {a b} \arctan \left (\frac {\sqrt {a b} \sin \left (x\right )}{a}\right )}{a b}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (24) = 48\).
Time = 0.37 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.64 \[ \int \frac {\cos (x)}{a+b \sin ^2(x)} \, dx=\begin {cases} \frac {\tilde {\infty }}{\sin {\left (x \right )}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {\sin {\left (x \right )}}{a} & \text {for}\: b = 0 \\- \frac {1}{b \sin {\left (x \right )}} & \text {for}\: a = 0 \\\frac {\log {\left (- \sqrt {- \frac {a}{b}} + \sin {\left (x \right )} \right )}}{2 b \sqrt {- \frac {a}{b}}} - \frac {\log {\left (\sqrt {- \frac {a}{b}} + \sin {\left (x \right )} \right )}}{2 b \sqrt {- \frac {a}{b}}} & \text {otherwise} \end {cases} \]
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Time = 0.38 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.64 \[ \int \frac {\cos (x)}{a+b \sin ^2(x)} \, dx=\frac {\arctan \left (\frac {b \sin \left (x\right )}{\sqrt {a b}}\right )}{\sqrt {a b}} \]
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Time = 0.31 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.64 \[ \int \frac {\cos (x)}{a+b \sin ^2(x)} \, dx=\frac {\arctan \left (\frac {b \sin \left (x\right )}{\sqrt {a b}}\right )}{\sqrt {a b}} \]
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Time = 14.32 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.68 \[ \int \frac {\cos (x)}{a+b \sin ^2(x)} \, dx=\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sin \left (x\right )}{\sqrt {a}}\right )}{\sqrt {a}\,\sqrt {b}} \]
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