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 = 0.0336383, antiderivative size = 35, normalized size of antiderivative = 1., 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 1814
Rule 1586
Rule 635
Rule 203
Rule 260
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*}
Mathematica [A] time = 0.0174349, size = 35, normalized size = 1. \[ -\frac{1}{\left (x^2+2\right )^2}+\frac{1}{2} \log \left (x^2+2\right )-\frac{\tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 31, normalized size = 0.9 \begin{align*} - \left ({x}^{2}+2 \right ) ^{-2}+{\frac{\ln \left ({x}^{2}+2 \right ) }{2}}-{\frac{\sqrt{2}}{2}\arctan \left ({\frac{x\sqrt{2}}{2}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.49491, size = 47, normalized size = 1.34 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44246, size = 150, normalized size = 4.29 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.136255, size = 36, normalized size = 1.03 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15804, size = 41, normalized size = 1.17 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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