Optimal. Leaf size=45 \[ \frac {9 \tan ^{-1}\left (\frac {x^2+1}{\sqrt {2}}\right )}{8 \sqrt {2}}+\frac {5-7 x^2}{8 \left (x^4+2 x^2+3\right )} \]
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Rubi [A] time = 0.07, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1594, 1663, 1660, 12, 618, 204} \begin {gather*} \frac {5-7 x^2}{8 \left (x^4+2 x^2+3\right )}+\frac {9 \tan ^{-1}\left (\frac {x^2+1}{\sqrt {2}}\right )}{8 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 204
Rule 618
Rule 1594
Rule 1660
Rule 1663
Rubi steps
\begin {align*} \int \frac {-x+2 x^3+4 x^5}{\left (3+2 x^2+x^4\right )^2} \, dx &=\int \frac {x \left (-1+2 x^2+4 x^4\right )}{\left (3+2 x^2+x^4\right )^2} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {-1+2 x+4 x^2}{\left (3+2 x+x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac {5-7 x^2}{8 \left (3+2 x^2+x^4\right )}+\frac {1}{16} \operatorname {Subst}\left (\int \frac {18}{3+2 x+x^2} \, dx,x,x^2\right )\\ &=\frac {5-7 x^2}{8 \left (3+2 x^2+x^4\right )}+\frac {9}{8} \operatorname {Subst}\left (\int \frac {1}{3+2 x+x^2} \, dx,x,x^2\right )\\ &=\frac {5-7 x^2}{8 \left (3+2 x^2+x^4\right )}-\frac {9}{4} \operatorname {Subst}\left (\int \frac {1}{-8-x^2} \, dx,x,2 \left (1+x^2\right )\right )\\ &=\frac {5-7 x^2}{8 \left (3+2 x^2+x^4\right )}+\frac {9 \tan ^{-1}\left (\frac {1+x^2}{\sqrt {2}}\right )}{8 \sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 45, normalized size = 1.00 \begin {gather*} \frac {9 \tan ^{-1}\left (\frac {x^2+1}{\sqrt {2}}\right )}{8 \sqrt {2}}+\frac {5-7 x^2}{8 \left (x^4+2 x^2+3\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-x+2 x^3+4 x^5}{\left (3+2 x^2+x^4\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.12, size = 47, normalized size = 1.04 \begin {gather*} \frac {9 \, \sqrt {2} {\left (x^{4} + 2 \, x^{2} + 3\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (x^{2} + 1\right )}\right ) - 14 \, x^{2} + 10}{16 \, {\left (x^{4} + 2 \, x^{2} + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.11, size = 38, normalized size = 0.84 \begin {gather*} \frac {9}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (x^{2} + 1\right )}\right ) - \frac {7 \, x^{2} - 5}{8 \, {\left (x^{4} + 2 \, x^{2} + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 41, normalized size = 0.91 \begin {gather*} \frac {9 \sqrt {2}\, \arctan \left (\frac {\left (2 x^{2}+2\right ) \sqrt {2}}{4}\right )}{16}+\frac {-\frac {7 x^{2}}{4}+\frac {5}{4}}{2 x^{4}+4 x^{2}+6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\frac {7 \, x^{2} - 5}{8 \, {\left (x^{4} + 2 \, x^{2} + 3\right )}} + \frac {9}{4} \, \int \frac {x}{x^{4} + 2 \, x^{2} + 3}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.19, size = 42, normalized size = 0.93 \begin {gather*} \frac {9\,\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x^2}{2}+\frac {\sqrt {2}}{2}\right )}{16}-\frac {\frac {7\,x^2}{8}-\frac {5}{8}}{x^4+2\,x^2+3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 44, normalized size = 0.98 \begin {gather*} \frac {5 - 7 x^{2}}{8 x^{4} + 16 x^{2} + 24} + \frac {9 \sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} x^{2}}{2} + \frac {\sqrt {2}}{2} \right )}}{16} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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