Optimal. Leaf size=42 \[ \frac {x+4}{4 \left (x^2+2\right )}+\frac {1}{2} \log \left (x^2+2\right )+\frac {5 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{4 \sqrt {2}} \]
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Rubi [A] time = 0.02, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1814, 635, 203, 260} \[ \frac {x+4}{4 \left (x^2+2\right )}+\frac {1}{2} \log \left (x^2+2\right )+\frac {5 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{4 \sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 203
Rule 260
Rule 635
Rule 1814
Rubi steps
\begin {align*} \int \frac {3+x^2+x^3}{\left (2+x^2\right )^2} \, dx &=\frac {4+x}{4 \left (2+x^2\right )}-\frac {1}{4} \int \frac {-5-4 x}{2+x^2} \, dx\\ &=\frac {4+x}{4 \left (2+x^2\right )}+\frac {5}{4} \int \frac {1}{2+x^2} \, dx+\int \frac {x}{2+x^2} \, dx\\ &=\frac {4+x}{4 \left (2+x^2\right )}+\frac {5 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{4 \sqrt {2}}+\frac {1}{2} \log \left (2+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 42, normalized size = 1.00 \[ \frac {x+4}{4 \left (x^2+2\right )}+\frac {1}{2} \log \left (x^2+2\right )+\frac {5 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{4 \sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 44, normalized size = 1.05 \[ \frac {5 \, \sqrt {2} {\left (x^{2} + 2\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) + 4 \, {\left (x^{2} + 2\right )} \log \left (x^{2} + 2\right ) + 2 \, x + 8}{8 \, {\left (x^{2} + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 33, normalized size = 0.79 \[ \frac {5}{8} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) + \frac {x + 4}{4 \, {\left (x^{2} + 2\right )}} + \frac {1}{2} \, \log \left (x^{2} + 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 35, normalized size = 0.83 \[ \frac {5 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{2}\right )}{8}+\frac {\ln \left (x^{2}+2\right )}{2}+\frac {\frac {x}{4}+1}{x^{2}+2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.36, size = 33, normalized size = 0.79 \[ \frac {5}{8} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) + \frac {x + 4}{4 \, {\left (x^{2} + 2\right )}} + \frac {1}{2} \, \log \left (x^{2} + 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.19, size = 39, normalized size = 0.93 \[ \frac {\ln \left (x^2+2\right )}{2}+\frac {5\,\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x}{2}\right )}{8}+\frac {x}{4\,\left (x^2+2\right )}+\frac {1}{x^2+2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 36, normalized size = 0.86 \[ \frac {x + 4}{4 x^{2} + 8} + \frac {\log {\left (x^{2} + 2 \right )}}{2} + \frac {5 \sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} x}{2} \right )}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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