Optimal. Leaf size=38 \[ \frac {1-x}{x^2-4 x+5}+\frac {5}{2} \log \left (x^2-4 x+5\right )-2 \tan ^{-1}(2-x) \]
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Rubi [A] time = 0.03, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {1660, 634, 618, 204, 628} \[ \frac {1-x}{x^2-4 x+5}+\frac {5}{2} \log \left (x^2-4 x+5\right )-2 \tan ^{-1}(2-x) \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 1660
Rubi steps
\begin {align*} \int \frac {-41+55 x-27 x^2+5 x^3}{\left (5-4 x+x^2\right )^2} \, dx &=\frac {1-x}{5-4 x+x^2}+\frac {1}{4} \int \frac {-32+20 x}{5-4 x+x^2} \, dx\\ &=\frac {1-x}{5-4 x+x^2}+2 \int \frac {1}{5-4 x+x^2} \, dx+\frac {5}{2} \int \frac {-4+2 x}{5-4 x+x^2} \, dx\\ &=\frac {1-x}{5-4 x+x^2}+\frac {5}{2} \log \left (5-4 x+x^2\right )-4 \operatorname {Subst}\left (\int \frac {1}{-4-x^2} \, dx,x,-4+2 x\right )\\ &=\frac {1-x}{5-4 x+x^2}-2 \tan ^{-1}(2-x)+\frac {5}{2} \log \left (5-4 x+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 38, normalized size = 1.00 \[ \frac {1-x}{x^2-4 x+5}+\frac {5}{2} \log \left (x^2-4 x+5\right )-2 \tan ^{-1}(2-x) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.02, size = 38, normalized size = 1.00 \[ \frac {1-x}{x^2-4 x+5}+\frac {5}{2} \log \left (x^2-4 x+5\right )-2 \tan ^{-1}(2-x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 50, normalized size = 1.32 \[ \frac {4 \, {\left (x^{2} - 4 \, x + 5\right )} \arctan \left (x - 2\right ) + 5 \, {\left (x^{2} - 4 \, x + 5\right )} \log \left (x^{2} - 4 \, x + 5\right ) - 2 \, x + 2}{2 \, {\left (x^{2} - 4 \, x + 5\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.92, size = 33, normalized size = 0.87 \[ -\frac {x - 1}{x^{2} - 4 \, x + 5} + 2 \, \arctan \left (x - 2\right ) + \frac {5}{2} \, \log \left (x^{2} - 4 \, x + 5\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.40, size = 35, normalized size = 0.92
method | result | size |
default | \(\frac {1-x}{x^{2}-4 x +5}+2 \arctan \left (-2+x \right )+\frac {5 \ln \left (x^{2}-4 x +5\right )}{2}\) | \(35\) |
risch | \(\frac {1-x}{x^{2}-4 x +5}+2 \arctan \left (-2+x \right )+\frac {5 \ln \left (x^{2}-4 x +5\right )}{2}\) | \(35\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 33, normalized size = 0.87 \[ -\frac {x - 1}{x^{2} - 4 \, x + 5} + 2 \, \arctan \left (x - 2\right ) + \frac {5}{2} \, \log \left (x^{2} - 4 \, x + 5\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.20, size = 41, normalized size = 1.08 \[ 2\,\mathrm {atan}\left (x-2\right )+\frac {5\,\ln \left (x^2-4\,x+5\right )}{2}-\frac {x}{x^2-4\,x+5}+\frac {1}{x^2-4\,x+5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 31, normalized size = 0.82 \[ \frac {1 - x}{x^{2} - 4 x + 5} + \frac {5 \log {\left (x^{2} - 4 x + 5 \right )}}{2} + 2 \operatorname {atan}{\left (x - 2 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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