Optimal. Leaf size=53 \[ \frac{x \left (11 x^2+9\right )}{4 \left (x^4+3 x^2+2\right )}-\frac{1}{x}-\frac{19}{2} \tan ^{-1}(x)+\frac{45 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{4 \sqrt{2}} \]
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Rubi [A] time = 0.0729781, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {1669, 1664, 203} \[ \frac{x \left (11 x^2+9\right )}{4 \left (x^4+3 x^2+2\right )}-\frac{1}{x}-\frac{19}{2} \tan ^{-1}(x)+\frac{45 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{4 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 1669
Rule 1664
Rule 203
Rubi steps
\begin{align*} \int \frac{4+x^2+3 x^4+5 x^6}{x^2 \left (2+3 x^2+x^4\right )^2} \, dx &=\frac{x \left (9+11 x^2\right )}{4 \left (2+3 x^2+x^4\right )}-\frac{1}{4} \int \frac{-8+19 x^2-11 x^4}{x^2 \left (2+3 x^2+x^4\right )} \, dx\\ &=\frac{x \left (9+11 x^2\right )}{4 \left (2+3 x^2+x^4\right )}-\frac{1}{4} \int \left (-\frac{4}{x^2}+\frac{38}{1+x^2}-\frac{45}{2+x^2}\right ) \, dx\\ &=-\frac{1}{x}+\frac{x \left (9+11 x^2\right )}{4 \left (2+3 x^2+x^4\right )}-\frac{19}{2} \int \frac{1}{1+x^2} \, dx+\frac{45}{4} \int \frac{1}{2+x^2} \, dx\\ &=-\frac{1}{x}+\frac{x \left (9+11 x^2\right )}{4 \left (2+3 x^2+x^4\right )}-\frac{19}{2} \tan ^{-1}(x)+\frac{45 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{4 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.0493476, size = 51, normalized size = 0.96 \[ \frac{1}{8} \left (\frac{2 x \left (11 x^2+9\right )}{x^4+3 x^2+2}-\frac{8}{x}-76 \tan ^{-1}(x)+45 \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 43, normalized size = 0.8 \begin{align*}{\frac{13\,x}{4\,{x}^{2}+8}}+{\frac{45\,\sqrt{2}}{8}\arctan \left ({\frac{x\sqrt{2}}{2}} \right ) }-{\frac{x}{2\,{x}^{2}+2}}-{\frac{19\,\arctan \left ( x \right ) }{2}}-{x}^{-1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.45781, size = 61, normalized size = 1.15 \begin{align*} \frac{45}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + \frac{7 \, x^{4} - 3 \, x^{2} - 8}{4 \,{\left (x^{5} + 3 \, x^{3} + 2 \, x\right )}} - \frac{19}{2} \, \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.17958, size = 185, normalized size = 3.49 \begin{align*} \frac{14 \, x^{4} + 45 \, \sqrt{2}{\left (x^{5} + 3 \, x^{3} + 2 \, x\right )} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - 6 \, x^{2} - 76 \,{\left (x^{5} + 3 \, x^{3} + 2 \, x\right )} \arctan \left (x\right ) - 16}{8 \,{\left (x^{5} + 3 \, x^{3} + 2 \, x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.201047, size = 49, normalized size = 0.92 \begin{align*} \frac{7 x^{4} - 3 x^{2} - 8}{4 x^{5} + 12 x^{3} + 8 x} - \frac{19 \operatorname{atan}{\left (x \right )}}{2} + \frac{45 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}}{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12253, size = 61, normalized size = 1.15 \begin{align*} \frac{45}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + \frac{7 \, x^{4} - 3 \, x^{2} - 8}{4 \,{\left (x^{5} + 3 \, x^{3} + 2 \, x\right )}} - \frac{19}{2} \, \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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