Optimal. Leaf size=24 \[ \frac{2 x+3}{2 \left (x^2+4 x+5\right )}+\tan ^{-1}(x+2) \]
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Rubi [A] time = 0.0085993, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {638, 618, 204} \[ \frac{2 x+3}{2 \left (x^2+4 x+5\right )}+\tan ^{-1}(x+2) \]
Antiderivative was successfully verified.
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Rule 638
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{4+x}{\left (5+4 x+x^2\right )^2} \, dx &=\frac{3+2 x}{2 \left (5+4 x+x^2\right )}+\int \frac{1}{5+4 x+x^2} \, dx\\ &=\frac{3+2 x}{2 \left (5+4 x+x^2\right )}-2 \operatorname{Subst}\left (\int \frac{1}{-4-x^2} \, dx,x,4+2 x\right )\\ &=\frac{3+2 x}{2 \left (5+4 x+x^2\right )}+\tan ^{-1}(2+x)\\ \end{align*}
Mathematica [A] time = 0.0108832, size = 24, normalized size = 1. \[ \frac{2 x+3}{2 \left (x^2+4 x+5\right )}+\tan ^{-1}(x+2) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 23, normalized size = 1. \begin{align*}{\frac{4\,x+6}{4\,{x}^{2}+16\,x+20}}+\arctan \left ( 2+x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.6477, size = 30, normalized size = 1.25 \begin{align*} \frac{2 \, x + 3}{2 \,{\left (x^{2} + 4 \, x + 5\right )}} + \arctan \left (x + 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57199, size = 88, normalized size = 3.67 \begin{align*} \frac{2 \,{\left (x^{2} + 4 \, x + 5\right )} \arctan \left (x + 2\right ) + 2 \, x + 3}{2 \,{\left (x^{2} + 4 \, x + 5\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.193603, size = 19, normalized size = 0.79 \begin{align*} \frac{2 x + 3}{2 x^{2} + 8 x + 10} + \operatorname{atan}{\left (x + 2 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25279, size = 30, normalized size = 1.25 \begin{align*} \frac{2 \, x + 3}{2 \,{\left (x^{2} + 4 \, x + 5\right )}} + \arctan \left (x + 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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