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 = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {2 x+3}{2 \left (x^2+4 x+5\right )}+\tan ^{-1}(x+2) \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 638
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*}
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Mathematica [A] time = 0.01, size = 24, normalized size = 1.00 \begin {gather*} \frac {2 x+3}{2 \left (x^2+4 x+5\right )}+\tan ^{-1}(x+2) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {4+x}{\left (5+4 x+x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.43, size = 31, normalized size = 1.29 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 22, normalized size = 0.92 \begin {gather*} \frac {2 \, x + 3}{2 \, {\left (x^{2} + 4 \, x + 5\right )}} + \arctan \left (x + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 23, normalized size = 0.96 \begin {gather*} \arctan \left (x +2\right )+\frac {4 x +6}{4 x^{2}+16 x +20} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.18, size = 22, normalized size = 0.92 \begin {gather*} \frac {2 \, x + 3}{2 \, {\left (x^{2} + 4 \, x + 5\right )}} + \arctan \left (x + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 19, normalized size = 0.79 \begin {gather*} \mathrm {atan}\left (x+2\right )+\frac {x+\frac {3}{2}}{x^2+4\,x+5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 19, normalized size = 0.79 \begin {gather*} \frac {2 x + 3}{2 x^{2} + 8 x + 10} + \operatorname {atan}{\left (x + 2 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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