Optimal. Leaf size=37 \[ \frac {5}{2 (1-x)}+x+2 \tan ^{-1}(x)+\frac {1}{2} \log (1-x)+\frac {3}{4} \log \left (1+x^2\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1643, 649, 209,
266} \begin {gather*} 2 \text {ArcTan}(x)+\frac {3}{4} \log \left (x^2+1\right )+x+\frac {5}{2 (1-x)}+\frac {1}{2} \log (1-x) \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 266
Rule 649
Rule 1643
Rubi steps
\begin {align*} \int \frac {5-4 x+3 x^2+x^4}{(-1+x)^2 \left (1+x^2\right )} \, dx &=\int \left (1+\frac {5}{2 (-1+x)^2}+\frac {1}{2 (-1+x)}+\frac {4+3 x}{2 \left (1+x^2\right )}\right ) \, dx\\ &=\frac {5}{2 (1-x)}+x+\frac {1}{2} \log (1-x)+\frac {1}{2} \int \frac {4+3 x}{1+x^2} \, dx\\ &=\frac {5}{2 (1-x)}+x+\frac {1}{2} \log (1-x)+\frac {3}{2} \int \frac {x}{1+x^2} \, dx+2 \int \frac {1}{1+x^2} \, dx\\ &=\frac {5}{2 (1-x)}+x+2 \tan ^{-1}(x)+\frac {1}{2} \log (1-x)+\frac {3}{4} \log \left (1+x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 33, normalized size = 0.89 \begin {gather*} \frac {5}{2-2 x}+x+2 \tan ^{-1}(x)+\frac {1}{2} \log (-1+x)+\frac {3}{4} \log \left (1+x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 28, normalized size = 0.76
method | result | size |
default | \(x -\frac {5}{2 \left (-1+x \right )}+\frac {\ln \left (-1+x \right )}{2}+\frac {3 \ln \left (x^{2}+1\right )}{4}+2 \arctan \left (x \right )\) | \(28\) |
risch | \(x -\frac {5}{2 \left (-1+x \right )}+\frac {\ln \left (-1+x \right )}{2}+\frac {3 \ln \left (x^{2}+1\right )}{4}+2 \arctan \left (x \right )\) | \(28\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 27, normalized size = 0.73 \begin {gather*} x - \frac {5}{2 \, {\left (x - 1\right )}} + 2 \, \arctan \left (x\right ) + \frac {3}{4} \, \log \left (x^{2} + 1\right ) + \frac {1}{2} \, \log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 44, normalized size = 1.19 \begin {gather*} \frac {4 \, x^{2} + 8 \, {\left (x - 1\right )} \arctan \left (x\right ) + 3 \, {\left (x - 1\right )} \log \left (x^{2} + 1\right ) + 2 \, {\left (x - 1\right )} \log \left (x - 1\right ) - 4 \, x - 10}{4 \, {\left (x - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 29, normalized size = 0.78 \begin {gather*} x + \frac {\log {\left (x - 1 \right )}}{2} + \frac {3 \log {\left (x^{2} + 1 \right )}}{4} + 2 \operatorname {atan}{\left (x \right )} - \frac {5}{2 x - 2} \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. 60 vs.
\(2 (29) = 58\).
time = 3.97, size = 60, normalized size = 1.62 \begin {gather*} \frac {1}{2} \, \pi - 2 \, \pi \left \lfloor \frac {\pi + 4 \, \arctan \left (x\right )}{4 \, \pi } + \frac {1}{2} \right \rfloor + x - \frac {5}{2 \, {\left (x - 1\right )}} + 2 \, \arctan \left (x\right ) + \frac {3}{4} \, \log \left (\frac {2}{x - 1} + \frac {2}{{\left (x - 1\right )}^{2}} + 1\right ) + 2 \, \log \left ({\left | x - 1 \right |}\right ) - 1 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.14, size = 35, normalized size = 0.95 \begin {gather*} x+\frac {\ln \left (x-1\right )}{2}-\frac {5}{2\,\left (x-1\right )}+\ln \left (x-\mathrm {i}\right )\,\left (\frac {3}{4}-\mathrm {i}\right )+\ln \left (x+1{}\mathrm {i}\right )\,\left (\frac {3}{4}+1{}\mathrm {i}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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