Optimal. Leaf size=30 \[ -\frac{x^6}{4 \left (x^4+1\right )}+\frac{3 x^2}{4}-\frac{3}{4} \tan ^{-1}\left (x^2\right ) \]
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Rubi [A] time = 0.0123403, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {28, 275, 288, 321, 203} \[ -\frac{x^6}{4 \left (x^4+1\right )}+\frac{3 x^2}{4}-\frac{3}{4} \tan ^{-1}\left (x^2\right ) \]
Antiderivative was successfully verified.
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Rule 28
Rule 275
Rule 288
Rule 321
Rule 203
Rubi steps
\begin{align*} \int \frac{x^9}{1+2 x^4+x^8} \, dx &=\int \frac{x^9}{\left (1+x^4\right )^2} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^4}{\left (1+x^2\right )^2} \, dx,x,x^2\right )\\ &=-\frac{x^6}{4 \left (1+x^4\right )}+\frac{3}{4} \operatorname{Subst}\left (\int \frac{x^2}{1+x^2} \, dx,x,x^2\right )\\ &=\frac{3 x^2}{4}-\frac{x^6}{4 \left (1+x^4\right )}-\frac{3}{4} \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,x^2\right )\\ &=\frac{3 x^2}{4}-\frac{x^6}{4 \left (1+x^4\right )}-\frac{3}{4} \tan ^{-1}\left (x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0143612, size = 24, normalized size = 0.8 \[ \frac{1}{4} \left (x^2 \left (\frac{1}{x^4+1}+2\right )-3 \tan ^{-1}\left (x^2\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 25, normalized size = 0.8 \begin{align*}{\frac{{x}^{2}}{2}}+{\frac{{x}^{2}}{4\,{x}^{4}+4}}-{\frac{3\,\arctan \left ({x}^{2} \right ) }{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.50002, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{2} \, x^{2} + \frac{x^{2}}{4 \,{\left (x^{4} + 1\right )}} - \frac{3}{4} \, \arctan \left (x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.46552, size = 77, normalized size = 2.57 \begin{align*} \frac{2 \, x^{6} + 3 \, x^{2} - 3 \,{\left (x^{4} + 1\right )} \arctan \left (x^{2}\right )}{4 \,{\left (x^{4} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.127484, size = 22, normalized size = 0.73 \begin{align*} \frac{x^{2}}{2} + \frac{x^{2}}{4 x^{4} + 4} - \frac{3 \operatorname{atan}{\left (x^{2} \right )}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11941, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{2} \, x^{2} + \frac{x^{2}}{4 \,{\left (x^{4} + 1\right )}} - \frac{3}{4} \, \arctan \left (x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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