Optimal. Leaf size=51 \[ -\frac{2-x}{216 \left (x^2-4 x+13\right )}-\frac{2-x}{36 \left (x^2-4 x+13\right )^2}+\frac{1}{648} \tan ^{-1}\left (\frac{x-2}{3}\right ) \]
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Rubi [A] time = 0.0140963, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {614, 618, 204} \[ -\frac{2-x}{216 \left (x^2-4 x+13\right )}-\frac{2-x}{36 \left (x^2-4 x+13\right )^2}+\frac{1}{648} \tan ^{-1}\left (\frac{x-2}{3}\right ) \]
Antiderivative was successfully verified.
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Rule 614
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{\left (13-4 x+x^2\right )^3} \, dx &=-\frac{2-x}{36 \left (13-4 x+x^2\right )^2}+\frac{1}{12} \int \frac{1}{\left (13-4 x+x^2\right )^2} \, dx\\ &=-\frac{2-x}{36 \left (13-4 x+x^2\right )^2}-\frac{2-x}{216 \left (13-4 x+x^2\right )}+\frac{1}{216} \int \frac{1}{13-4 x+x^2} \, dx\\ &=-\frac{2-x}{36 \left (13-4 x+x^2\right )^2}-\frac{2-x}{216 \left (13-4 x+x^2\right )}-\frac{1}{108} \operatorname{Subst}\left (\int \frac{1}{-36-x^2} \, dx,x,-4+2 x\right )\\ &=-\frac{2-x}{36 \left (13-4 x+x^2\right )^2}-\frac{2-x}{216 \left (13-4 x+x^2\right )}+\frac{1}{648} \tan ^{-1}\left (\frac{1}{3} (-2+x)\right )\\ \end{align*}
Mathematica [A] time = 0.016078, size = 36, normalized size = 0.71 \[ \frac{1}{648} \left (\frac{3 (x-2) \left (x^2-4 x+19\right )}{\left (x^2-4 x+13\right )^2}+\tan ^{-1}\left (\frac{x-2}{3}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 44, normalized size = 0.9 \begin{align*}{\frac{2\,x-4}{72\, \left ({x}^{2}-4\,x+13 \right ) ^{2}}}+{\frac{2\,x-4}{432\,{x}^{2}-1728\,x+5616}}+{\frac{1}{648}\arctan \left ( -{\frac{2}{3}}+{\frac{x}{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.41552, size = 59, normalized size = 1.16 \begin{align*} \frac{x^{3} - 6 \, x^{2} + 27 \, x - 38}{216 \,{\left (x^{4} - 8 \, x^{3} + 42 \, x^{2} - 104 \, x + 169\right )}} + \frac{1}{648} \, \arctan \left (\frac{1}{3} \, x - \frac{2}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.1216, size = 180, normalized size = 3.53 \begin{align*} \frac{3 \, x^{3} - 18 \, x^{2} +{\left (x^{4} - 8 \, x^{3} + 42 \, x^{2} - 104 \, x + 169\right )} \arctan \left (\frac{1}{3} \, x - \frac{2}{3}\right ) + 81 \, x - 114}{648 \,{\left (x^{4} - 8 \, x^{3} + 42 \, x^{2} - 104 \, x + 169\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.152098, size = 42, normalized size = 0.82 \begin{align*} \frac{x^{3} - 6 x^{2} + 27 x - 38}{216 x^{4} - 1728 x^{3} + 9072 x^{2} - 22464 x + 36504} + \frac{\operatorname{atan}{\left (\frac{x}{3} - \frac{2}{3} \right )}}{648} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05459, size = 46, normalized size = 0.9 \begin{align*} \frac{x^{3} - 6 \, x^{2} + 27 \, x - 38}{216 \,{\left (x^{2} - 4 \, x + 13\right )}^{2}} + \frac{1}{648} \, \arctan \left (\frac{1}{3} \, x - \frac{2}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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