Optimal. Leaf size=13 \[ -\frac {1}{2 \left ((x+2)^2+1\right )} \]
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Rubi [A] time = 0.01, antiderivative size = 13, 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 = {372, 261} \[ -\frac {1}{2 \left ((x+2)^2+1\right )} \]
Antiderivative was successfully verified.
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Rule 261
Rule 372
Rubi steps
\begin {align*} \int \frac {2+x}{\left (1+(2+x)^2\right )^2} \, dx &=\operatorname {Subst}\left (\int \frac {x}{\left (1+x^2\right )^2} \, dx,x,2+x\right )\\ &=-\frac {1}{2 \left (1+(2+x)^2\right )}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 13, normalized size = 1.00 \[ -\frac {1}{2 \left ((x+2)^2+1\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 12, normalized size = 0.92 \[ -\frac {1}{2 \, {\left (x^{2} + 4 \, x + 5\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 12, normalized size = 0.92 \[ -\frac {1}{2 \, {\left (x^{2} + 4 \, x + 5\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 13, normalized size = 1.00 \[ -\frac {1}{2 \left (x^{2}+4 x +5\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 11, normalized size = 0.85 \[ -\frac {1}{2 \, {\left ({\left (x + 2\right )}^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.14, size = 13, normalized size = 1.00 \[ -\frac {1}{2\,\left ({\left (x+2\right )}^2+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 12, normalized size = 0.92 \[ - \frac {1}{2 x^{2} + 8 x + 10} \]
Verification of antiderivative is not currently implemented for this CAS.
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