Optimal. Leaf size=35 \[ -\frac {1}{\left (2+x^2\right )^2}-\frac {\tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{\sqrt {2}}+\frac {1}{2} \log \left (2+x^2\right ) \]
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Rubi [A]
time = 0.02, 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 = {1828, 1600,
649, 209, 266} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {x}{\sqrt {2}}\right )}{\sqrt {2}}-\frac {1}{\left (x^2+2\right )^2}+\frac {1}{2} \log \left (x^2+2\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 266
Rule 649
Rule 1600
Rule 1828
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 = 0.01, size = 35, normalized size = 1.00 \begin {gather*} -\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 {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.19, size = 31, normalized size = 0.89
method | result | size |
default | \(-\frac {1}{\left (x^{2}+2\right )^{2}}+\frac {\ln \left (x^{2}+2\right )}{2}-\frac {\arctan \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}}{2}\) | \(31\) |
risch | \(-\frac {1}{\left (x^{2}+2\right )^{2}}+\frac {\ln \left (x^{2}+2\right )}{2}-\frac {\arctan \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}}{2}\) | \(31\) |
meijerg | \(-\frac {\sqrt {2}\, \left (\frac {\sqrt {2}\, x \left (\frac {3 x^{2}}{2}+5\right )}{4 \left (1+\frac {x^{2}}{2}\right )^{2}}+\frac {3 \arctan \left (\frac {\sqrt {2}\, x}{2}\right )}{2}\right )}{8}-\frac {x^{2} \left (\frac {9 x^{2}}{2}+6\right )}{24 \left (1+\frac {x^{2}}{2}\right )^{2}}+\frac {\ln \left (1+\frac {x^{2}}{2}\right )}{2}-\frac {\sqrt {2}\, \left (-\frac {x \sqrt {2}\, \left (\frac {25 x^{2}}{2}+15\right )}{20 \left (1+\frac {x^{2}}{2}\right )^{2}}+\frac {3 \arctan \left (\frac {\sqrt {2}\, x}{2}\right )}{2}\right )}{8}+\frac {x^{4}}{8 \left (1+\frac {x^{2}}{2}\right )^{2}}-\frac {\sqrt {2}\, \left (-\frac {x \sqrt {2}\, \left (-\frac {3 x^{2}}{2}+3\right )}{12 \left (1+\frac {x^{2}}{2}\right )^{2}}+\frac {\arctan \left (\frac {\sqrt {2}\, x}{2}\right )}{2}\right )}{4}+\frac {x^{2} \left (\frac {x^{2}}{2}+2\right )}{4 \left (1+\frac {x^{2}}{2}\right )^{2}}\) | \(179\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 35, normalized size = 1.00 \begin {gather*} -\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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 55, normalized size = 1.57 \begin {gather*} -\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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 36, normalized size = 1.03 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.36, size = 30, normalized size = 0.86 \begin {gather*} -\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 ) \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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