Optimal. Leaf size=12 \[ \frac {x}{2}+\frac {\tan (x)}{2} \]
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Rubi [A] time = 0.05, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {388, 203} \[ \frac {x}{2}+\frac {\tan (x)}{2} \]
Antiderivative was successfully verified.
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Rule 203
Rule 388
Rubi steps
\begin {align*} \int \frac {1+\cos ^2(x)}{1+\cos (2 x)} \, dx &=\operatorname {Subst}\left (\int \frac {2+x^2}{2+2 x^2} \, dx,x,\tan (x)\right )\\ &=\frac {\tan (x)}{2}+\operatorname {Subst}\left (\int \frac {1}{2+2 x^2} \, dx,x,\tan (x)\right )\\ &=\frac {x}{2}+\frac {\tan (x)}{2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 12, normalized size = 1.00 \[ \frac {x}{2}+\frac {\tan (x)}{2} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.09, size = 13, normalized size = 1.08 \[ \frac {x \cos \relax (x) + \sin \relax (x)}{2 \, \cos \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 8, normalized size = 0.67 \[ \frac {1}{2} \, x + \frac {1}{2} \, \tan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 9, normalized size = 0.75 \[ \frac {x}{2}+\frac {\tan \relax (x )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.31, size = 18, normalized size = 1.50 \[ \frac {1}{2} \, x + \frac {\sin \left (2 \, x\right )}{2 \, {\left (\cos \left (2 \, x\right ) + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.93, size = 8, normalized size = 0.67 \[ \frac {x}{2}+\frac {\mathrm {tan}\relax (x)}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.41, size = 7, normalized size = 0.58 \[ \frac {x}{2} + \frac {\tan {\relax (x )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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