Optimal. Leaf size=5 \[ -\tan ^{-1}(\cos (x)) \]
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Rubi [A] time = 0.03, antiderivative size = 5, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {4339, 203} \[ -\tan ^{-1}(\cos (x)) \]
Antiderivative was successfully verified.
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Rule 203
Rule 4339
Rubi steps
\begin {align*} \int \frac {\sec (x) \tan (x)}{1+\sec ^2(x)} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\cos (x)\right )\\ &=-\tan ^{-1}(\cos (x))\\ \end {align*}
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Mathematica [A] time = 0.02, size = 5, normalized size = 1.00 \[ -\tan ^{-1}(\cos (x)) \]
Antiderivative was successfully verified.
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fricas [A] time = 4.94, size = 5, normalized size = 1.00 \[ -\arctan \left (\cos \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 5, normalized size = 1.00 \[ -\arctan \left (\cos \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 4, normalized size = 0.80 \[ \arctan \left (\sec \relax (x )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 5, normalized size = 1.00 \[ -\arctan \left (\cos \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.95, size = 7, normalized size = 1.40 \[ \mathrm {atan}\left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 3, normalized size = 0.60 \[ \operatorname {atan}{\left (\sec {\relax (x )} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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