Integrand size = 20, antiderivative size = 19 \[ \int \frac {\sin (2 x)}{a^2-b^2 \sin ^2(x)} \, dx=-\frac {\log \left (a^2-b^2 \sin ^2(x)\right )}{b^2} \]
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Time = 0.04 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {12, 266} \[ \int \frac {\sin (2 x)}{a^2-b^2 \sin ^2(x)} \, dx=-\frac {\log \left (a^2-b^2 \sin ^2(x)\right )}{b^2} \]
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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 x^2} \, dx,x,\sin (x)\right ) \\ & = 2 \text {Subst}\left (\int \frac {x}{a^2-b^2 x^2} \, dx,x,\sin (x)\right ) \\ & = -\frac {\log \left (a^2-b^2 \sin ^2(x)\right )}{b^2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {\sin (2 x)}{a^2-b^2 \sin ^2(x)} \, dx=-\frac {\log \left (a^2-b^2 \sin ^2(x)\right )}{b^2} \]
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Time = 6.18 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05
method | result | size |
derivativedivides | \(-\frac {\ln \left (a^{2}-b^{2} \left (\sin ^{2}\left (x \right )\right )\right )}{b^{2}}\) | \(20\) |
default | \(-\frac {\ln \left (a^{2}-b^{2} \left (\sin ^{2}\left (x \right )\right )\right )}{b^{2}}\) | \(20\) |
risch | \(\frac {2 i x}{b^{2}}-\frac {\ln \left ({\mathrm e}^{4 i x}+\frac {2 \left (2 a^{2}-b^{2}\right ) {\mathrm e}^{2 i x}}{b^{2}}+1\right )}{b^{2}}\) | \(43\) |
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Time = 0.27 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.21 \[ \int \frac {\sin (2 x)}{a^2-b^2 \sin ^2(x)} \, dx=-\frac {\log \left (b^{2} \cos \left (x\right )^{2} + a^{2} - b^{2}\right )}{b^{2}} \]
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Time = 0.91 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.79 \[ \int \frac {\sin (2 x)}{a^2-b^2 \sin ^2(x)} \, dx=2 \left (\begin {cases} - \frac {\cos ^{2}{\left (x \right )}}{2 a^{2}} & \text {for}\: b^{2} = 0 \\- \frac {\log {\left (a^{2} - b^{2} \sin ^{2}{\left (x \right )} \right )}}{2 b^{2}} & \text {otherwise} \end {cases}\right ) \]
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none
Time = 0.23 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {\sin (2 x)}{a^2-b^2 \sin ^2(x)} \, dx=-\frac {\log \left (b^{2} \sin \left (x\right )^{2} - a^{2}\right )}{b^{2}} \]
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Time = 0.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \frac {\sin (2 x)}{a^2-b^2 \sin ^2(x)} \, dx=-\frac {\log \left ({\left | b^{2} \sin \left (x\right )^{2} - a^{2} \right |}\right )}{b^{2}} \]
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Time = 0.59 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.53 \[ \int \frac {\sin (2 x)}{a^2-b^2 \sin ^2(x)} \, dx=\frac {\mathrm {atan}\left (\frac {b^2\,{\sin \left (x\right )}^2}{a^2\,{\cos \left (x\right )}^2\,2{}\mathrm {i}+a^2\,{\sin \left (x\right )}^2\,2{}\mathrm {i}-b^2\,{\sin \left (x\right )}^2\,1{}\mathrm {i}}\right )\,2{}\mathrm {i}}{b^2} \]
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