Integrand size = 17, antiderivative size = 9 \[ \int \frac {\sec ^2(x)}{\sqrt {1-4 \tan ^2(x)}} \, dx=\frac {1}{2} \arcsin (2 \tan (x)) \]
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Time = 0.06 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3756, 222} \[ \int \frac {\sec ^2(x)}{\sqrt {1-4 \tan ^2(x)}} \, dx=\frac {1}{2} \arcsin (2 \tan (x)) \]
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Rule 222
Rule 3756
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{\sqrt {1-4 x^2}} \, dx,x,\tan (x)\right ) \\ & = \frac {1}{2} \arcsin (2 \tan (x)) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(47\) vs. \(2(9)=18\).
Time = 0.06 (sec) , antiderivative size = 47, normalized size of antiderivative = 5.22 \[ \int \frac {\sec ^2(x)}{\sqrt {1-4 \tan ^2(x)}} \, dx=\frac {\arctan \left (\frac {2 \sin (x)}{\sqrt {1-5 \sin ^2(x)}}\right ) \sqrt {-3+5 \cos (2 x)} \sec (x)}{2 \sqrt {2-8 \tan ^2(x)}} \]
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Time = 0.86 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(\frac {\arcsin \left (2 \tan \left (x \right )\right )}{2}\) | \(8\) |
default | \(\frac {\arcsin \left (2 \tan \left (x \right )\right )}{2}\) | \(8\) |
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Leaf count of result is larger than twice the leaf count of optimal. 45 vs. \(2 (7) = 14\).
Time = 0.26 (sec) , antiderivative size = 45, normalized size of antiderivative = 5.00 \[ \int \frac {\sec ^2(x)}{\sqrt {1-4 \tan ^2(x)}} \, dx=-\frac {1}{4} \, \arctan \left (\frac {{\left (9 \, \cos \left (x\right )^{3} - 8 \, \cos \left (x\right )\right )} \sqrt {\frac {5 \, \cos \left (x\right )^{2} - 4}{\cos \left (x\right )^{2}}}}{4 \, {\left (5 \, \cos \left (x\right )^{2} - 4\right )} \sin \left (x\right )}\right ) \]
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\[ \int \frac {\sec ^2(x)}{\sqrt {1-4 \tan ^2(x)}} \, dx=\int \frac {\sec ^{2}{\left (x \right )}}{\sqrt {- \left (2 \tan {\left (x \right )} - 1\right ) \left (2 \tan {\left (x \right )} + 1\right )}}\, dx \]
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none
Time = 0.33 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.78 \[ \int \frac {\sec ^2(x)}{\sqrt {1-4 \tan ^2(x)}} \, dx=\frac {1}{2} \, \arcsin \left (2 \, \tan \left (x\right )\right ) \]
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none
Time = 0.29 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.78 \[ \int \frac {\sec ^2(x)}{\sqrt {1-4 \tan ^2(x)}} \, dx=\frac {1}{2} \, \arcsin \left (2 \, \tan \left (x\right )\right ) \]
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Timed out. \[ \int \frac {\sec ^2(x)}{\sqrt {1-4 \tan ^2(x)}} \, dx=\int \frac {1}{{\cos \left (x\right )}^2\,\sqrt {1-4\,{\mathrm {tan}\left (x\right )}^2}} \,d x \]
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