Integrand size = 18, antiderivative size = 15 \[ \int \frac {\cos (x)}{a^2-b^2 \sin ^2(x)} \, dx=\frac {\text {arctanh}\left (\frac {b \sin (x)}{a}\right )}{a b} \]
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Time = 0.02 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {3269, 214} \[ \int \frac {\cos (x)}{a^2-b^2 \sin ^2(x)} \, dx=\frac {\text {arctanh}\left (\frac {b \sin (x)}{a}\right )}{a b} \]
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Rule 214
Rule 3269
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{a^2-b^2 x^2} \, dx,x,\sin (x)\right ) \\ & = \frac {\text {arctanh}\left (\frac {b \sin (x)}{a}\right )}{a b} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {\cos (x)}{a^2-b^2 \sin ^2(x)} \, dx=\frac {\text {arctanh}\left (\frac {b \sin (x)}{a}\right )}{a b} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(32\) vs. \(2(15)=30\).
Time = 0.50 (sec) , antiderivative size = 33, normalized size of antiderivative = 2.20
method | result | size |
derivativedivides | \(\frac {\ln \left (a +b \sin \left (x \right )\right )}{2 b a}-\frac {\ln \left (-b \sin \left (x \right )+a \right )}{2 b a}\) | \(33\) |
default | \(\frac {\ln \left (a +b \sin \left (x \right )\right )}{2 b a}-\frac {\ln \left (-b \sin \left (x \right )+a \right )}{2 b a}\) | \(33\) |
parallelrisch | \(\frac {-\ln \left (\frac {-b \sin \left (x \right )+a}{\cos \left (x \right )+1}\right )+\ln \left (\frac {a +b \sin \left (x \right )}{\cos \left (x \right )+1}\right )}{2 a b}\) | \(41\) |
norman | \(-\frac {\ln \left (a \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-2 b \tan \left (\frac {x}{2}\right )+a \right )}{2 b a}+\frac {\ln \left (a \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+2 b \tan \left (\frac {x}{2}\right )+a \right )}{2 b a}\) | \(54\) |
risch | \(\frac {\ln \left ({\mathrm e}^{2 i x}+\frac {2 i a \,{\mathrm e}^{i x}}{b}-1\right )}{2 b a}-\frac {\ln \left ({\mathrm e}^{2 i x}-\frac {2 i a \,{\mathrm e}^{i x}}{b}-1\right )}{2 b a}\) | \(58\) |
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none
Time = 0.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.73 \[ \int \frac {\cos (x)}{a^2-b^2 \sin ^2(x)} \, dx=\frac {\log \left (b \sin \left (x\right ) + a\right ) - \log \left (-b \sin \left (x\right ) + a\right )}{2 \, a b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (10) = 20\).
Time = 0.25 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.93 \[ \int \frac {\cos (x)}{a^2-b^2 \sin ^2(x)} \, dx=\begin {cases} \frac {\tilde {\infty }}{\sin {\left (x \right )}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {1}{b^{2} \sin {\left (x \right )}} & \text {for}\: a = 0 \\\frac {\sin {\left (x \right )}}{a^{2}} & \text {for}\: b = 0 \\- \frac {\log {\left (- \frac {a}{b} + \sin {\left (x \right )} \right )}}{2 a b} + \frac {\log {\left (\frac {a}{b} + \sin {\left (x \right )} \right )}}{2 a b} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (15) = 30\).
Time = 0.21 (sec) , antiderivative size = 33, normalized size of antiderivative = 2.20 \[ \int \frac {\cos (x)}{a^2-b^2 \sin ^2(x)} \, dx=\frac {\log \left (b \sin \left (x\right ) + a\right )}{2 \, a b} - \frac {\log \left (b \sin \left (x\right ) - a\right )}{2 \, a b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (15) = 30\).
Time = 0.28 (sec) , antiderivative size = 35, normalized size of antiderivative = 2.33 \[ \int \frac {\cos (x)}{a^2-b^2 \sin ^2(x)} \, dx=\frac {\log \left ({\left | b \sin \left (x\right ) + a \right |}\right )}{2 \, a b} - \frac {\log \left ({\left | b \sin \left (x\right ) - a \right |}\right )}{2 \, a b} \]
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Time = 0.20 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {\cos (x)}{a^2-b^2 \sin ^2(x)} \, dx=\frac {\mathrm {atanh}\left (\frac {b\,\sin \left (x\right )}{a}\right )}{a\,b} \]
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