Optimal. Leaf size=62 \[ -\frac{x \left (9 x^2+5\right )}{8 \left (x^4+3 x^2+2\right )}-\frac{1}{3 x^3}+\frac{11}{4 x}+\frac{21}{2} \tan ^{-1}(x)-\frac{71 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{8 \sqrt{2}} \]
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Rubi [A] time = 0.0842406, antiderivative size = 62, 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} \[ -\frac{x \left (9 x^2+5\right )}{8 \left (x^4+3 x^2+2\right )}-\frac{1}{3 x^3}+\frac{11}{4 x}+\frac{21}{2} \tan ^{-1}(x)-\frac{71 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{8 \sqrt{2}} \]
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^4 \left (2+3 x^2+x^4\right )^2} \, dx &=-\frac{x \left (5+9 x^2\right )}{8 \left (2+3 x^2+x^4\right )}-\frac{1}{4} \int \frac{-8+10 x^2-\frac{39 x^4}{2}+\frac{9 x^6}{2}}{x^4 \left (2+3 x^2+x^4\right )} \, dx\\ &=-\frac{x \left (5+9 x^2\right )}{8 \left (2+3 x^2+x^4\right )}-\frac{1}{4} \int \left (-\frac{4}{x^4}+\frac{11}{x^2}-\frac{42}{1+x^2}+\frac{71}{2 \left (2+x^2\right )}\right ) \, dx\\ &=-\frac{1}{3 x^3}+\frac{11}{4 x}-\frac{x \left (5+9 x^2\right )}{8 \left (2+3 x^2+x^4\right )}-\frac{71}{8} \int \frac{1}{2+x^2} \, dx+\frac{21}{2} \int \frac{1}{1+x^2} \, dx\\ &=-\frac{1}{3 x^3}+\frac{11}{4 x}-\frac{x \left (5+9 x^2\right )}{8 \left (2+3 x^2+x^4\right )}+\frac{21}{2} \tan ^{-1}(x)-\frac{71 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{8 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.0533273, size = 56, normalized size = 0.9 \[ \frac{1}{48} \left (-\frac{6 x \left (9 x^2+5\right )}{x^4+3 x^2+2}-\frac{16}{x^3}+\frac{132}{x}+504 \tan ^{-1}(x)-213 \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 48, normalized size = 0.8 \begin{align*} -{\frac{13\,x}{8\,{x}^{2}+16}}-{\frac{71\,\sqrt{2}}{16}\arctan \left ({\frac{x\sqrt{2}}{2}} \right ) }+{\frac{x}{2\,{x}^{2}+2}}+{\frac{21\,\arctan \left ( x \right ) }{2}}-{\frac{1}{3\,{x}^{3}}}+{\frac{11}{4\,x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.47063, size = 70, normalized size = 1.13 \begin{align*} -\frac{71}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + \frac{39 \, x^{6} + 175 \, x^{4} + 108 \, x^{2} - 16}{24 \,{\left (x^{7} + 3 \, x^{5} + 2 \, x^{3}\right )}} + \frac{21}{2} \, \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58719, size = 213, normalized size = 3.44 \begin{align*} \frac{78 \, x^{6} + 350 \, x^{4} - 213 \, \sqrt{2}{\left (x^{7} + 3 \, x^{5} + 2 \, x^{3}\right )} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + 216 \, x^{2} + 504 \,{\left (x^{7} + 3 \, x^{5} + 2 \, x^{3}\right )} \arctan \left (x\right ) - 32}{48 \,{\left (x^{7} + 3 \, x^{5} + 2 \, x^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.225061, size = 56, normalized size = 0.9 \begin{align*} \frac{21 \operatorname{atan}{\left (x \right )}}{2} - \frac{71 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}}{16} + \frac{39 x^{6} + 175 x^{4} + 108 x^{2} - 16}{24 x^{7} + 72 x^{5} + 48 x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11528, size = 70, normalized size = 1.13 \begin{align*} -\frac{71}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - \frac{9 \, x^{3} + 5 \, x}{8 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}} + \frac{33 \, x^{2} - 4}{12 \, x^{3}} + \frac{21}{2} \, \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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