Optimal. Leaf size=9 \[ \tan ^{-1}(x)+\log (1-x) \]
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Rubi [A]
time = 0.04, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1600, 1607,
1643, 209} \begin {gather*} \text {ArcTan}(x)+\log (1-x) \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 1600
Rule 1607
Rule 1643
Rubi steps
\begin {align*} \int \frac {-x+x^3}{(-1+x)^2 \left (1+x^2\right )} \, dx &=\int \frac {x+x^2}{(-1+x) \left (1+x^2\right )} \, dx\\ &=\int \frac {x (1+x)}{(-1+x) \left (1+x^2\right )} \, dx\\ &=\int \left (\frac {1}{-1+x}+\frac {1}{1+x^2}\right ) \, dx\\ &=\log (1-x)+\int \frac {1}{1+x^2} \, dx\\ &=\tan ^{-1}(x)+\log (1-x)\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 9, normalized size = 1.00 \begin {gather*} \tan ^{-1}(x)+\log (1-x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 8, normalized size = 0.89
method | result | size |
default | \(\ln \left (-1+x \right )+\arctan \left (x \right )\) | \(8\) |
risch | \(\ln \left (-1+x \right )+\arctan \left (x \right )\) | \(8\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 7, normalized size = 0.78 \begin {gather*} \arctan \left (x\right ) + \log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 7, normalized size = 0.78 \begin {gather*} \arctan \left (x\right ) + \log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 7, normalized size = 0.78 \begin {gather*} \log {\left (x - 1 \right )} + \operatorname {atan}{\left (x \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 28 vs.
\(2 (9) = 18\).
time = 4.32, size = 28, normalized size = 3.11 \begin {gather*} \frac {1}{4} \, \pi - \pi \left \lfloor \frac {\pi + 4 \, \arctan \left (x\right )}{4 \, \pi } + \frac {1}{2} \right \rfloor + \arctan \left (x\right ) + \log \left ({\left | x - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.10, size = 19, normalized size = 2.11 \begin {gather*} \ln \left (x-1\right )-\mathrm {atan}\left (\frac {5}{4\,x+2}-\frac {1}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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