Optimal. Leaf size=3 \[ \tan ^{-1}(\sin (x)) \]
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Rubi [A] time = 0.02, antiderivative size = 3, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3190, 203} \[ \tan ^{-1}(\sin (x)) \]
Antiderivative was successfully verified.
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Rule 203
Rule 3190
Rubi steps
\begin {align*} \int \frac {\cos (x)}{1+\sin ^2(x)} \, dx &=\operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sin (x)\right )\\ &=\tan ^{-1}(\sin (x))\\ \end {align*}
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Mathematica [A] time = 0.00, size = 3, normalized size = 1.00 \[ \tan ^{-1}(\sin (x)) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 3, normalized size = 1.00 \[ \arctan \left (\sin \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.94, size = 3, normalized size = 1.00 \[ \arctan \left (\sin \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 4, normalized size = 1.33 \[ \arctan \left (\sin \relax (x )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.12, size = 3, normalized size = 1.00 \[ \arctan \left (\sin \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 3, normalized size = 1.00 \[ \mathrm {atan}\left (\sin \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.23, size = 3, normalized size = 1.00 \[ \operatorname {atan}{\left (\sin {\relax (x )} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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