Optimal. Leaf size=31 \[ -\frac {3}{26} \tan ^{-1}\left (\frac {x}{2}\right )+\frac {1}{13} \log (3-x)-\frac {1}{26} \log \left (4+x^2\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {720, 31, 649,
209, 266} \begin {gather*} -\frac {3}{26} \text {ArcTan}\left (\frac {x}{2}\right )-\frac {1}{26} \log \left (x^2+4\right )+\frac {1}{13} \log (3-x) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 209
Rule 266
Rule 649
Rule 720
Rubi steps
\begin {align*} \int \frac {1}{(-3+x) \left (4+x^2\right )} \, dx &=\frac {1}{13} \int \frac {1}{-3+x} \, dx+\frac {1}{13} \int \frac {-3-x}{4+x^2} \, dx\\ &=\frac {1}{13} \log (3-x)-\frac {1}{13} \int \frac {x}{4+x^2} \, dx-\frac {3}{13} \int \frac {1}{4+x^2} \, dx\\ &=-\frac {3}{26} \tan ^{-1}\left (\frac {x}{2}\right )+\frac {1}{13} \log (3-x)-\frac {1}{26} \log \left (4+x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 36, normalized size = 1.16 \begin {gather*} -\frac {3}{26} \tan ^{-1}\left (\frac {x}{2}\right )-\frac {1}{26} \log \left (13+6 (-3+x)+(-3+x)^2\right )+\frac {1}{13} \log (-3+x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.22, size = 22, normalized size = 0.71
method | result | size |
default | \(\frac {\ln \left (x -3\right )}{13}-\frac {\ln \left (x^{2}+4\right )}{26}-\frac {3 \arctan \left (\frac {x}{2}\right )}{26}\) | \(22\) |
risch | \(\frac {\ln \left (x -3\right )}{13}-\frac {\ln \left (9 x^{2}+36\right )}{26}-\frac {3 \arctan \left (\frac {x}{2}\right )}{26}\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 21, normalized size = 0.68 \begin {gather*} -\frac {3}{26} \, \arctan \left (\frac {1}{2} \, x\right ) - \frac {1}{26} \, \log \left (x^{2} + 4\right ) + \frac {1}{13} \, \log \left (x - 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 21, normalized size = 0.68 \begin {gather*} -\frac {3}{26} \, \arctan \left (\frac {1}{2} \, x\right ) - \frac {1}{26} \, \log \left (x^{2} + 4\right ) + \frac {1}{13} \, \log \left (x - 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 22, normalized size = 0.71 \begin {gather*} \frac {\log {\left (x - 3 \right )}}{13} - \frac {\log {\left (x^{2} + 4 \right )}}{26} - \frac {3 \operatorname {atan}{\left (\frac {x}{2} \right )}}{26} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.92, size = 22, normalized size = 0.71 \begin {gather*} -\frac {3}{26} \, \arctan \left (\frac {1}{2} \, x\right ) - \frac {1}{26} \, \log \left (x^{2} + 4\right ) + \frac {1}{13} \, \log \left ({\left | x - 3 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 25, normalized size = 0.81 \begin {gather*} \frac {\ln \left (x-3\right )}{13}+\ln \left (x-2{}\mathrm {i}\right )\,\left (-\frac {1}{26}+\frac {3}{52}{}\mathrm {i}\right )+\ln \left (x+2{}\mathrm {i}\right )\,\left (-\frac {1}{26}-\frac {3}{52}{}\mathrm {i}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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