Optimal. Leaf size=11 \[ \frac {1}{2} \tanh ^{-1}(2 \sin (x) \cos (x)) \]
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Rubi [A] time = 0.03, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3675, 206} \[ \frac {1}{2} \tanh ^{-1}(2 \sin (x) \cos (x)) \]
Antiderivative was successfully verified.
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Rule 206
Rule 3675
Rubi steps
\begin {align*} \int \frac {\sec ^2(x)}{1-\tan ^2(x)} \, dx &=\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tan (x)\right )\\ &=\frac {1}{2} \tanh ^{-1}(2 \cos (x) \sin (x))\\ \end {align*}
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Mathematica [B] time = 0.01, size = 23, normalized size = 2.09 \[ \frac {1}{2} \log (\sin (x)+\cos (x))-\frac {1}{2} \log (\cos (x)-\sin (x)) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.57, size = 23, normalized size = 2.09 \[ \frac {1}{4} \, \log \left (2 \, \cos \relax (x) \sin \relax (x) + 1\right ) - \frac {1}{4} \, \log \left (-2 \, \cos \relax (x) \sin \relax (x) + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 17, normalized size = 1.55 \[ \frac {1}{2} \, \log \left ({\left | \tan \relax (x) + 1 \right |}\right ) - \frac {1}{2} \, \log \left ({\left | \tan \relax (x) - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 4, normalized size = 0.36 \[ \arctanh \left (\tan \relax (x )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 15, normalized size = 1.36 \[ \frac {1}{2} \, \log \left (\tan \relax (x) + 1\right ) - \frac {1}{2} \, \log \left (\tan \relax (x) - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.08, size = 3, normalized size = 0.27 \[ \mathrm {atanh}\left (\mathrm {tan}\relax (x)\right ) \]
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 \[ - \int \frac {\sec ^{2}{\relax (x )}}{\tan ^{2}{\relax (x )} - 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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