Optimal. Leaf size=9 \[ \frac{1}{2} \sin ^{-1}(2 \tan (x)) \]
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Rubi [A] time = 0.0468873, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {3675, 216} \[ \frac{1}{2} \sin ^{-1}(2 \tan (x)) \]
Antiderivative was successfully verified.
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Rule 3675
Rule 216
Rubi steps
\begin{align*} \int \frac{\sec ^2(x)}{\sqrt{1-4 \tan ^2(x)}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\sqrt{1-4 x^2}} \, dx,x,\tan (x)\right )\\ &=\frac{1}{2} \sin ^{-1}(2 \tan (x))\\ \end{align*}
Mathematica [B] time = 0.0622612, size = 47, normalized size = 5.22 \[ \frac{\sqrt{5 \cos (2 x)-3} \sec (x) \tan ^{-1}\left (\frac{2 \sin (x)}{\sqrt{1-5 \sin ^2(x)}}\right )}{2 \sqrt{2-8 \tan ^2(x)}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.361, size = 172, normalized size = 19.1 \begin{align*}{\frac{\sqrt{2} \left ( \sin \left ( x \right ) \right ) ^{2}}{\sqrt{9+4\,\sqrt{5}}\cos \left ( x \right ) \left ( -1+\cos \left ( x \right ) \right ) }\sqrt{{\frac{2\,\cos \left ( x \right ) \sqrt{5}-2\,\sqrt{5}+5\,\cos \left ( x \right ) -4}{1+\cos \left ( x \right ) }}}\sqrt{-2\,{\frac{2\,\cos \left ( x \right ) \sqrt{5}-2\,\sqrt{5}-5\,\cos \left ( x \right ) +4}{1+\cos \left ( x \right ) }}} \left ( 2\,{\it EllipticPi} \left ({\frac{\sqrt{9+4\,\sqrt{5}} \left ( -1+\cos \left ( x \right ) \right ) }{\sin \left ( x \right ) }}, \left ( 9+4\,\sqrt{5} \right ) ^{-1},{\frac{\sqrt{9-4\,\sqrt{5}}}{\sqrt{9+4\,\sqrt{5}}}} \right ) -{\it EllipticF} \left ({\frac{ \left ( -1+\cos \left ( x \right ) \right ) \left ( \sqrt{5}+2 \right ) }{\sin \left ( x \right ) }},9-4\,\sqrt{5} \right ) \right ){\frac{1}{\sqrt{{\frac{5\, \left ( \cos \left ( x \right ) \right ) ^{2}-4}{ \left ( \cos \left ( x \right ) \right ) ^{2}}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.48008, size = 9, normalized size = 1. \begin{align*} \frac{1}{2} \, \arcsin \left (2 \, \tan \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.24892, size = 135, normalized size = 15. \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15862, size = 9, normalized size = 1. \begin{align*} \frac{1}{2} \, \arcsin \left (2 \, \tan \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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