Optimal. Leaf size=35 \[ -\frac {1}{\left (x^2+2\right )^2}+\frac {1}{2} \log \left (x^2+2\right )-\frac {\tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{\sqrt {2}} \]
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Rubi [A] time = 0.03, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {1814, 1586, 635, 203, 260} \[ -\frac {1}{\left (x^2+2\right )^2}+\frac {1}{2} \log \left (x^2+2\right )-\frac {\tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 203
Rule 260
Rule 635
Rule 1586
Rule 1814
Rubi steps
\begin {align*} \int \frac {-4+8 x-4 x^2+4 x^3-x^4+x^5}{\left (2+x^2\right )^3} \, dx &=-\frac {1}{\left (2+x^2\right )^2}-\frac {1}{8} \int \frac {16-16 x+8 x^2-8 x^3}{\left (2+x^2\right )^2} \, dx\\ &=-\frac {1}{\left (2+x^2\right )^2}-\frac {1}{8} \int \frac {8-8 x}{2+x^2} \, dx\\ &=-\frac {1}{\left (2+x^2\right )^2}-\int \frac {1}{2+x^2} \, dx+\int \frac {x}{2+x^2} \, dx\\ &=-\frac {1}{\left (2+x^2\right )^2}-\frac {\tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{\sqrt {2}}+\frac {1}{2} \log \left (2+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 35, normalized size = 1.00 \[ -\frac {1}{\left (x^2+2\right )^2}+\frac {1}{2} \log \left (x^2+2\right )-\frac {\tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 55, normalized size = 1.57 \[ -\frac {\sqrt {2} {\left (x^{4} + 4 \, x^{2} + 4\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) - {\left (x^{4} + 4 \, x^{2} + 4\right )} \log \left (x^{2} + 2\right ) + 2}{2 \, {\left (x^{4} + 4 \, x^{2} + 4\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 30, normalized size = 0.86 \[ -\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) - \frac {1}{{\left (x^{2} + 2\right )}^{2}} + \frac {1}{2} \, \log \left (x^{2} + 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 31, normalized size = 0.89 \[ -\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{2}\right )}{2}+\frac {\ln \left (x^{2}+2\right )}{2}-\frac {1}{\left (x^{2}+2\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.22, size = 35, normalized size = 1.00 \[ -\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) - \frac {1}{x^{4} + 4 \, x^{2} + 4} + \frac {1}{2} \, \log \left (x^{2} + 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.12, size = 35, normalized size = 1.00 \[ \frac {\ln \left (x^2+2\right )}{2}-\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x}{2}\right )}{2}-\frac {1}{x^4+4\,x^2+4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 36, normalized size = 1.03 \[ \frac {\log {\left (x^{2} + 2 \right )}}{2} - \frac {\sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} x}{2} \right )}}{2} - \frac {1}{x^{4} + 4 x^{2} + 4} \]
Verification of antiderivative is not currently implemented for this CAS.
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