Integrand size = 19, antiderivative size = 18 \[ \int \frac {\sin (2 x)}{a^2+b^2 \cos ^2(x)} \, dx=-\frac {\log \left (a^2+b^2 \cos ^2(x)\right )}{b^2} \]
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Time = 0.04 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {12, 266} \[ \int \frac {\sin (2 x)}{a^2+b^2 \cos ^2(x)} \, dx=-\frac {\log \left (a^2-b^2 \sin ^2(x)+b^2\right )}{b^2} \]
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Rule 12
Rule 266
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {2 x}{a^2+b^2-b^2 x^2} \, dx,x,\sin (x)\right ) \\ & = 2 \text {Subst}\left (\int \frac {x}{a^2+b^2-b^2 x^2} \, dx,x,\sin (x)\right ) \\ & = -\frac {\log \left (a^2+b^2-b^2 \sin ^2(x)\right )}{b^2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {\sin (2 x)}{a^2+b^2 \cos ^2(x)} \, dx=-\frac {\log \left (a^2+b^2-b^2 \sin ^2(x)\right )}{b^2} \]
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Time = 10.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06
method | result | size |
derivativedivides | \(-\frac {\ln \left (a^{2}+b^{2} \left (\cos ^{2}\left (x \right )\right )\right )}{b^{2}}\) | \(19\) |
default | \(-\frac {\ln \left (a^{2}+b^{2} \left (\cos ^{2}\left (x \right )\right )\right )}{b^{2}}\) | \(19\) |
risch | \(\frac {2 i x}{b^{2}}-\frac {\ln \left ({\mathrm e}^{4 i x}+\frac {2 \left (2 a^{2}+b^{2}\right ) {\mathrm e}^{2 i x}}{b^{2}}+1\right )}{b^{2}}\) | \(41\) |
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Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {\sin (2 x)}{a^2+b^2 \cos ^2(x)} \, dx=-\frac {\log \left (b^{2} \cos \left (x\right )^{2} + a^{2}\right )}{b^{2}} \]
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Time = 0.87 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.89 \[ \int \frac {\sin (2 x)}{a^2+b^2 \cos ^2(x)} \, dx=2 \left (\begin {cases} - \frac {\cos ^{2}{\left (x \right )}}{2 a^{2}} & \text {for}\: b^{2} = 0 \\- \frac {\log {\left (a^{2} + b^{2} \cos ^{2}{\left (x \right )} \right )}}{2 b^{2}} & \text {otherwise} \end {cases}\right ) \]
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Time = 0.21 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {\sin (2 x)}{a^2+b^2 \cos ^2(x)} \, dx=-\frac {\log \left (b^{2} \cos \left (x\right )^{2} + a^{2}\right )}{b^{2}} \]
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Time = 0.28 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {\sin (2 x)}{a^2+b^2 \cos ^2(x)} \, dx=-\frac {\log \left (b^{2} \cos \left (x\right )^{2} + a^{2}\right )}{b^{2}} \]
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Time = 0.46 (sec) , antiderivative size = 58, normalized size of antiderivative = 3.22 \[ \int \frac {\sin (2 x)}{a^2+b^2 \cos ^2(x)} \, dx=\frac {2\,\mathrm {atanh}\left (\frac {b^2}{2\,a^2+b^2\,{\cos \left (x\right )}^2+b^2}-\frac {b^2\,{\cos \left (x\right )}^2}{2\,a^2+b^2\,{\cos \left (x\right )}^2+b^2}\right )}{b^2} \]
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