Optimal. Leaf size=32 \[ \frac{1}{4 \left (x^4+2 x^2+2\right )}+\frac{1}{4} \log \left (x^4+2 x^2+2\right ) \]
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Rubi [A] time = 0.0263224, antiderivative size = 39, normalized size of antiderivative = 1.22, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {1247, 686, 628} \[ \frac{1}{4} \log \left (x^4+2 x^2+2\right )-\frac{\left (x^2+1\right )^2}{4 \left (x^4+2 x^2+2\right )} \]
Antiderivative was successfully verified.
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Rule 1247
Rule 686
Rule 628
Rubi steps
\begin{align*} \int \frac{x \left (1+x^2\right )^3}{\left (2+2 x^2+x^4\right )^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(1+x)^3}{\left (2+2 x+x^2\right )^2} \, dx,x,x^2\right )\\ &=-\frac{\left (1+x^2\right )^2}{4 \left (2+2 x^2+x^4\right )}+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1+x}{2+2 x+x^2} \, dx,x,x^2\right )\\ &=-\frac{\left (1+x^2\right )^2}{4 \left (2+2 x^2+x^4\right )}+\frac{1}{4} \log \left (2+2 x^2+x^4\right )\\ \end{align*}
Mathematica [A] time = 0.0127078, size = 26, normalized size = 0.81 \[ \frac{1}{4} \left (\frac{1}{\left (x^2+1\right )^2+1}+\log \left (\left (x^2+1\right )^2+1\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 29, normalized size = 0.9 \begin{align*}{\frac{1}{4\,{x}^{4}+8\,{x}^{2}+8}}+{\frac{\ln \left ({x}^{4}+2\,{x}^{2}+2 \right ) }{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.949625, size = 38, normalized size = 1.19 \begin{align*} \frac{1}{4 \,{\left (x^{4} + 2 \, x^{2} + 2\right )}} + \frac{1}{4} \, \log \left (x^{4} + 2 \, x^{2} + 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.9647, size = 92, normalized size = 2.88 \begin{align*} \frac{{\left (x^{4} + 2 \, x^{2} + 2\right )} \log \left (x^{4} + 2 \, x^{2} + 2\right ) + 1}{4 \,{\left (x^{4} + 2 \, x^{2} + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.128267, size = 26, normalized size = 0.81 \begin{align*} \frac{\log{\left (x^{4} + 2 x^{2} + 2 \right )}}{4} + \frac{1}{4 x^{4} + 8 x^{2} + 8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05774, size = 38, normalized size = 1.19 \begin{align*} \frac{1}{4 \,{\left (x^{4} + 2 \, x^{2} + 2\right )}} + \frac{1}{4} \, \log \left (x^{4} + 2 \, x^{2} + 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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