Optimal. Leaf size=23 \[ -\frac {1}{2} \tan ^{-1}\left (\frac {x}{2}\right )+\log (x)+\frac {1}{2} \log \left (4+x^2\right ) \]
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Rubi [A]
time = 0.04, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1607, 1816,
649, 209, 266} \begin {gather*} -\frac {1}{2} \text {ArcTan}\left (\frac {x}{2}\right )+\frac {1}{2} \log \left (x^2+4\right )+\log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 266
Rule 649
Rule 1607
Rule 1816
Rubi steps
\begin {align*} \int \frac {4-x+2 x^2}{4 x+x^3} \, dx &=\int \frac {4-x+2 x^2}{x \left (4+x^2\right )} \, dx\\ &=\int \left (\frac {1}{x}+\frac {-1+x}{4+x^2}\right ) \, dx\\ &=\log (x)+\int \frac {-1+x}{4+x^2} \, dx\\ &=\log (x)-\int \frac {1}{4+x^2} \, dx+\int \frac {x}{4+x^2} \, dx\\ &=-\frac {1}{2} \tan ^{-1}\left (\frac {x}{2}\right )+\log (x)+\frac {1}{2} \log \left (4+x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 23, normalized size = 1.00 \begin {gather*} -\frac {1}{2} \tan ^{-1}\left (\frac {x}{2}\right )+\log (x)+\frac {1}{2} \log \left (4+x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 18, normalized size = 0.78
method | result | size |
default | \(-\frac {\arctan \left (\frac {x}{2}\right )}{2}+\ln \left (x \right )+\frac {\ln \left (x^{2}+4\right )}{2}\) | \(18\) |
risch | \(-\frac {\arctan \left (\frac {x}{2}\right )}{2}+\ln \left (x \right )+\frac {\ln \left (x^{2}+4\right )}{2}\) | \(18\) |
meijerg | \(\ln \left (x \right )-\ln \left (2\right )+\frac {\ln \left (1+\frac {x^{2}}{4}\right )}{2}-\frac {\arctan \left (\frac {x}{2}\right )}{2}\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.61, size = 17, normalized size = 0.74 \begin {gather*} -\frac {1}{2} \, \arctan \left (\frac {1}{2} \, x\right ) + \frac {1}{2} \, \log \left (x^{2} + 4\right ) + \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 17, normalized size = 0.74 \begin {gather*} -\frac {1}{2} \, \arctan \left (\frac {1}{2} \, x\right ) + \frac {1}{2} \, \log \left (x^{2} + 4\right ) + \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 17, normalized size = 0.74 \begin {gather*} \log {\left (x \right )} + \frac {\log {\left (x^{2} + 4 \right )}}{2} - \frac {\operatorname {atan}{\left (\frac {x}{2} \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.84, size = 18, normalized size = 0.78 \begin {gather*} -\frac {1}{2} \, \arctan \left (\frac {1}{2} \, x\right ) + \frac {1}{2} \, \log \left (x^{2} + 4\right ) + \log \left ({\left | x \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 21, normalized size = 0.91 \begin {gather*} \ln \left (x\right )+\ln \left (x-2{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {1}{4}{}\mathrm {i}\right )+\ln \left (x+2{}\mathrm {i}\right )\,\left (\frac {1}{2}-\frac {1}{4}{}\mathrm {i}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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