Optimal. Leaf size=14 \[ \frac {x}{2}+\frac {1}{2} \sin (x) \cos (x) \]
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Rubi [A] time = 0.02, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {3657, 2635, 8} \[ \frac {x}{2}+\frac {1}{2} \sin (x) \cos (x) \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 3657
Rubi steps
\begin {align*} \int \frac {1}{1+\tan ^2(x)} \, dx &=\int \cos ^2(x) \, dx\\ &=\frac {1}{2} \cos (x) \sin (x)+\frac {\int 1 \, dx}{2}\\ &=\frac {x}{2}+\frac {1}{2} \cos (x) \sin (x)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 14, normalized size = 1.00 \[ \frac {x}{2}+\frac {1}{4} \sin (2 x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 20, normalized size = 1.43 \[ \frac {x \tan \relax (x)^{2} + x + \tan \relax (x)}{2 \, {\left (\tan \relax (x)^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.92, size = 16, normalized size = 1.14 \[ \frac {1}{2} \, x + \frac {\tan \relax (x)}{2 \, {\left (\tan \relax (x)^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 17, normalized size = 1.21 \[ \frac {x}{2}+\frac {\tan \relax (x )}{2 \left (\tan ^{2}\relax (x )\right )+2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 16, normalized size = 1.14 \[ \frac {1}{2} \, x + \frac {\tan \relax (x)}{2 \, {\left (\tan \relax (x)^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.18, size = 10, normalized size = 0.71 \[ \frac {x}{2}+\frac {\sin \left (2\,x\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.39, size = 36, normalized size = 2.57 \[ \frac {x \tan ^{2}{\relax (x )}}{2 \tan ^{2}{\relax (x )} + 2} + \frac {x}{2 \tan ^{2}{\relax (x )} + 2} + \frac {\tan {\relax (x )}}{2 \tan ^{2}{\relax (x )} + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
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