Optimal. Leaf size=59 \[ \frac {13 x}{x^4+2 x^2+3}-\frac {2 \left (13 x^2+18\right ) x}{\left (x^4+2 x^2+3\right )^2}+\frac {2 \left (1-2 x^2\right )}{\left (x^4+2 x^2+3\right )^2} \]
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Rubi [A] time = 0.09, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.140, Rules used = {1673, 1678, 1588, 1663, 1660, 8} \[ \frac {13 x}{x^4+2 x^2+3}-\frac {2 \left (13 x^2+18\right ) x}{\left (x^4+2 x^2+3\right )^2}+\frac {2 \left (1-2 x^2\right )}{\left (x^4+2 x^2+3\right )^2} \]
Antiderivative was successfully verified.
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Rule 8
Rule 1588
Rule 1660
Rule 1663
Rule 1673
Rule 1678
Rubi steps
\begin {align*} \int \frac {9-40 x-18 x^2+174 x^4+24 x^5+26 x^6-39 x^8}{\left (3+2 x^2+x^4\right )^3} \, dx &=\int \frac {x \left (-40+24 x^4\right )}{\left (3+2 x^2+x^4\right )^3} \, dx+\int \frac {9-18 x^2+174 x^4+26 x^6-39 x^8}{\left (3+2 x^2+x^4\right )^3} \, dx\\ &=-\frac {2 x \left (18+13 x^2\right )}{\left (3+2 x^2+x^4\right )^2}+\frac {1}{96} \int \frac {3744-2496 x^2-3744 x^4}{\left (3+2 x^2+x^4\right )^2} \, dx+\frac {1}{2} \operatorname {Subst}\left (\int \frac {-40+24 x^2}{\left (3+2 x+x^2\right )^3} \, dx,x,x^2\right )\\ &=\frac {2 \left (1-2 x^2\right )}{\left (3+2 x^2+x^4\right )^2}-\frac {2 x \left (18+13 x^2\right )}{\left (3+2 x^2+x^4\right )^2}+\frac {13 x}{3+2 x^2+x^4}+\frac {1}{32} \operatorname {Subst}\left (\int 0 \, dx,x,x^2\right )\\ &=\frac {2 \left (1-2 x^2\right )}{\left (3+2 x^2+x^4\right )^2}-\frac {2 x \left (18+13 x^2\right )}{\left (3+2 x^2+x^4\right )^2}+\frac {13 x}{3+2 x^2+x^4}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 28, normalized size = 0.47 \[ \frac {13 x^5-4 x^2+3 x+2}{\left (x^4+2 x^2+3\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 38, normalized size = 0.64 \[ \frac {13 \, x^{5} - 4 \, x^{2} + 3 \, x + 2}{x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.44, size = 28, normalized size = 0.47 \[ \frac {13 \, x^{5} - 4 \, x^{2} + 3 \, x + 2}{{\left (x^{4} + 2 \, x^{2} + 3\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 30, normalized size = 0.51 \[ -\frac {-13 x^{5}+4 x^{2}-3 x -2}{\left (x^{4}+2 x^{2}+3\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 38, normalized size = 0.64 \[ \frac {13 \, x^{5} - 4 \, x^{2} + 3 \, x + 2}{x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 28, normalized size = 0.47 \[ \frac {13\,x^5-4\,x^2+3\,x+2}{{\left (x^4+2\,x^2+3\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 36, normalized size = 0.61 \[ - \frac {- 13 x^{5} + 4 x^{2} - 3 x - 2}{x^{8} + 4 x^{6} + 10 x^{4} + 12 x^{2} + 9} \]
Verification of antiderivative is not currently implemented for this CAS.
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