Optimal. Leaf size=14 \[ \frac {1}{2-x}+\tan ^{-1}(2-x) \]
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Rubi [A]
time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {27, 707, 632,
210} \begin {gather*} \text {ArcTan}(2-x)+\frac {1}{2-x} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 210
Rule 632
Rule 707
Rubi steps
\begin {align*} \int \frac {1}{\left (4-4 x+x^2\right ) \left (5-4 x+x^2\right )} \, dx &=\int \frac {1}{(-2+x)^2 \left (5-4 x+x^2\right )} \, dx\\ &=\frac {1}{2-x}-\int \frac {1}{5-4 x+x^2} \, dx\\ &=\frac {1}{2-x}+2 \text {Subst}\left (\int \frac {1}{-4-x^2} \, dx,x,-4+2 x\right )\\ &=\frac {1}{2-x}+\tan ^{-1}(2-x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 14, normalized size = 1.00 \begin {gather*} -\frac {1}{-2+x}+\tan ^{-1}(2-x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 15, normalized size = 1.07
method | result | size |
default | \(-\frac {1}{-2+x}-\arctan \left (-2+x \right )\) | \(15\) |
risch | \(-\frac {1}{-2+x}-\arctan \left (-2+x \right )\) | \(15\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.27, size = 14, normalized size = 1.00 \begin {gather*} -\frac {1}{x - 2} - \arctan \left (x - 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.74, size = 17, normalized size = 1.21 \begin {gather*} -\frac {{\left (x - 2\right )} \arctan \left (x - 2\right ) + 1}{x - 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 10, normalized size = 0.71 \begin {gather*} - \operatorname {atan}{\left (x - 2 \right )} - \frac {1}{x - 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 14, normalized size = 1.00 \begin {gather*} -\frac {1}{x - 2} - \arctan \left (x - 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.08, size = 14, normalized size = 1.00 \begin {gather*} -\mathrm {atan}\left (x-2\right )-\frac {1}{x-2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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