Integrand size = 16, antiderivative size = 18 \[ \int \frac {\sec ^2(x)}{a-a \sin ^2(x)} \, dx=\frac {\tan (x)}{a}+\frac {\tan ^3(x)}{3 a} \]
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Time = 0.05 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3254, 3852} \[ \int \frac {\sec ^2(x)}{a-a \sin ^2(x)} \, dx=\frac {\tan ^3(x)}{3 a}+\frac {\tan (x)}{a} \]
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Rule 3254
Rule 3852
Rubi steps \begin{align*} \text {integral}& = \frac {\int \sec ^4(x) \, dx}{a} \\ & = -\frac {\text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (x)\right )}{a} \\ & = \frac {\tan (x)}{a}+\frac {\tan ^3(x)}{3 a} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83 \[ \int \frac {\sec ^2(x)}{a-a \sin ^2(x)} \, dx=\frac {\tan (x)+\frac {\tan ^3(x)}{3}}{a} \]
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Time = 0.39 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78
method | result | size |
default | \(\frac {\frac {\left (\tan ^{3}\left (x \right )\right )}{3}+\tan \left (x \right )}{a}\) | \(14\) |
parallelrisch | \(\frac {\tan \left (x \right ) \left (2+\sec ^{2}\left (x \right )\right )}{3 a}\) | \(14\) |
risch | \(\frac {4 i \left (3 \,{\mathrm e}^{2 i x}+1\right )}{3 \left ({\mathrm e}^{2 i x}+1\right )^{3} a}\) | \(25\) |
norman | \(\frac {-\frac {2 \tan \left (\frac {x}{2}\right )}{a}+\frac {4 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{3 a}-\frac {2 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{a}}{\left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )^{3}}\) | \(44\) |
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none
Time = 0.27 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {\sec ^2(x)}{a-a \sin ^2(x)} \, dx=\frac {{\left (2 \, \cos \left (x\right )^{2} + 1\right )} \sin \left (x\right )}{3 \, a \cos \left (x\right )^{3}} \]
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\[ \int \frac {\sec ^2(x)}{a-a \sin ^2(x)} \, dx=- \frac {\int \frac {\sec ^{2}{\left (x \right )}}{\sin ^{2}{\left (x \right )} - 1}\, dx}{a} \]
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none
Time = 0.31 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {\sec ^2(x)}{a-a \sin ^2(x)} \, dx=\frac {\tan \left (x\right )^{3} + 3 \, \tan \left (x\right )}{3 \, a} \]
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none
Time = 0.31 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {\sec ^2(x)}{a-a \sin ^2(x)} \, dx=\frac {\tan \left (x\right )^{3} + 3 \, \tan \left (x\right )}{3 \, a} \]
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Time = 13.68 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.72 \[ \int \frac {\sec ^2(x)}{a-a \sin ^2(x)} \, dx=\frac {\mathrm {tan}\left (x\right )\,\left ({\mathrm {tan}\left (x\right )}^2+3\right )}{3\,a} \]
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