Optimal. Leaf size=59 \[ \frac {2 \left (1-2 x^2\right )}{\left (3+2 x^2+x^4\right )^2}-\frac {2 x \left (18+13 x^2\right )}{\left (3+2 x^2+x^4\right )^2}+\frac {13 x}{3+2 x^2+x^4} \]
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Rubi [A]
time = 0.06, 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 = {1687, 1692,
1602, 1677, 1674, 8} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 1602
Rule 1674
Rule 1677
Rule 1687
Rule 1692
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} \text {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} \text {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 \begin {gather*} \frac {2+3 x-4 x^2+13 x^5}{\left (3+2 x^2+x^4\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 30, normalized size = 0.51
method | result | size |
gosper | \(\frac {13 x^{5}-4 x^{2}+3 x +2}{\left (x^{4}+2 x^{2}+3\right )^{2}}\) | \(29\) |
norman | \(\frac {13 x^{5}-4 x^{2}+3 x +2}{\left (x^{4}+2 x^{2}+3\right )^{2}}\) | \(29\) |
risch | \(\frac {13 x^{5}-4 x^{2}+3 x +2}{\left (x^{4}+2 x^{2}+3\right )^{2}}\) | \(29\) |
default | \(-\frac {-13 x^{5}+4 x^{2}-3 x -2}{\left (x^{4}+2 x^{2}+3\right )^{2}}\) | \(30\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 38, normalized size = 0.64 \begin {gather*} \frac {13 \, x^{5} - 4 \, x^{2} + 3 \, x + 2}{x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 38, normalized size = 0.64 \begin {gather*} \frac {13 \, x^{5} - 4 \, x^{2} + 3 \, x + 2}{x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 36, normalized size = 0.61 \begin {gather*} - \frac {- 13 x^{5} + 4 x^{2} - 3 x - 2}{x^{8} + 4 x^{6} + 10 x^{4} + 12 x^{2} + 9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.66, size = 28, normalized size = 0.47 \begin {gather*} \frac {13 \, x^{5} - 4 \, x^{2} + 3 \, x + 2}{{\left (x^{4} + 2 \, x^{2} + 3\right )}^{2}} \end {gather*}
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 \begin {gather*} \frac {13\,x^5-4\,x^2+3\,x+2}{{\left (x^4+2\,x^2+3\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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