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 = 0.01, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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 204
Rule 614
Rule 618
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*}
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Mathematica [A] time = 0.02, 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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IntegrateAlgebraic [A] time = 0.02, size = 40, normalized size = 0.78 \[ \frac {(x-2) \left (x^2-4 x+19\right )}{216 \left (x^2-4 x+13\right )^2}-\frac {1}{648} \tan ^{-1}\left (\frac {2}{3}-\frac {x}{3}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.03, size = 62, normalized size = 1.22 \[ \frac {3 \, x^{3} - 18 \, x^{2} + {\left (x^{4} - 8 \, x^{3} + 42 \, x^{2} - 104 \, x + 169\right )} \arctan \left (\frac {1}{3} \, x - \frac {2}{3}\right ) + 81 \, x - 114}{648 \, {\left (x^{4} - 8 \, x^{3} + 42 \, x^{2} - 104 \, x + 169\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.85, size = 34, normalized size = 0.67 \[ \frac {x^{3} - 6 \, x^{2} + 27 \, x - 38}{216 \, {\left (x^{2} - 4 \, x + 13\right )}^{2}} + \frac {1}{648} \, \arctan \left (\frac {1}{3} \, x - \frac {2}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.37, size = 36, normalized size = 0.71
method | result | size |
risch | \(\frac {\frac {1}{216} x^{3}-\frac {1}{36} x^{2}+\frac {1}{8} x -\frac {19}{108}}{\left (x^{2}-4 x +13\right )^{2}}+\frac {\arctan \left (-\frac {2}{3}+\frac {x}{3}\right )}{648}\) | \(36\) |
default | \(\frac {2 x -4}{72 \left (x^{2}-4 x +13\right )^{2}}+\frac {2 x -4}{432 x^{2}-1728 x +5616}+\frac {\arctan \left (-\frac {2}{3}+\frac {x}{3}\right )}{648}\) | \(44\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 44, normalized size = 0.86 \[ \frac {x^{3} - 6 \, x^{2} + 27 \, x - 38}{216 \, {\left (x^{4} - 8 \, x^{3} + 42 \, x^{2} - 104 \, x + 169\right )}} + \frac {1}{648} \, \arctan \left (\frac {1}{3} \, x - \frac {2}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.20, size = 39, normalized size = 0.76 \[ \frac {\mathrm {atan}\left (\frac {x}{3}-\frac {2}{3}\right )}{648}+6\,\left (x-2\right )\,\left (\frac {1}{1296\,\left (x^2-4\,x+13\right )}+\frac {1}{216\,{\left (x^2-4\,x+13\right )}^2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 42, normalized size = 0.82 \[ \frac {x^{3} - 6 x^{2} + 27 x - 38}{216 x^{4} - 1728 x^{3} + 9072 x^{2} - 22464 x + 36504} + \frac {\operatorname {atan}{\left (\frac {x}{3} - \frac {2}{3} \right )}}{648} \]
Verification of antiderivative is not currently implemented for this CAS.
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