Optimal. Leaf size=9 \[ \frac {1}{2} \sin ^{-1}(2 \tan (x)) \]
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Rubi [A] time = 0.05, antiderivative size = 9, normalized size of antiderivative = 1.00, 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 216
Rule 3675
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*}
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Mathematica [B] time = 0.06, 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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fricas [B] time = 0.83, size = 45, normalized size = 5.00 \[ -\frac {1}{4} \, \arctan \left (\frac {{\left (9 \, \cos \relax (x)^{3} - 8 \, \cos \relax (x)\right )} \sqrt {\frac {5 \, \cos \relax (x)^{2} - 4}{\cos \relax (x)^{2}}}}{4 \, {\left (5 \, \cos \relax (x)^{2} - 4\right )} \sin \relax (x)}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 7, normalized size = 0.78 \[ \frac {1}{2} \, \arcsin \left (2 \, \tan \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.69, size = 171, normalized size = 19.00 \[ -\frac {\sqrt {2}\, \sqrt {\frac {2 \cos \relax (x ) \sqrt {5}+5 \cos \relax (x )-2 \sqrt {5}-4}{1+\cos \relax (x )}}\, \sqrt {-\frac {2 \left (2 \cos \relax (x ) \sqrt {5}-5 \cos \relax (x )-2 \sqrt {5}+4\right )}{1+\cos \relax (x )}}\, \left (\EllipticF \left (\frac {\left (-1+\cos \relax (x )\right ) \left (\sqrt {5}+2\right )}{\sin \relax (x )}, 9-4 \sqrt {5}\right )-2 \EllipticPi \left (\frac {\sqrt {9+4 \sqrt {5}}\, \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}, \frac {1}{9+4 \sqrt {5}}, \frac {\sqrt {9-4 \sqrt {5}}}{\sqrt {9+4 \sqrt {5}}}\right )\right ) \left (\sin ^{2}\relax (x )\right )}{\sqrt {\frac {5 \left (\cos ^{2}\relax (x )\right )-4}{\cos \relax (x )^{2}}}\, \cos \relax (x ) \left (-1+\cos \relax (x )\right ) \sqrt {9+4 \sqrt {5}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 7, normalized size = 0.78 \[ \frac {1}{2} \, \arcsin \left (2 \, \tan \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.11 \[ \int \frac {1}{{\cos \relax (x)}^2\,\sqrt {1-4\,{\mathrm {tan}\relax (x)}^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec ^{2}{\relax (x )}}{\sqrt {- \left (2 \tan {\relax (x )} - 1\right ) \left (2 \tan {\relax (x )} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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