Optimal. Leaf size=61 \[ 49 x^2-\frac {27 x^4}{4}+\frac {5 x^6}{6}-\frac {206+207 x^2}{2 \left (2+3 x^2+x^4\right )}-\frac {5}{2} \log \left (1+x^2\right )-144 \log \left (2+x^2\right ) \]
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Rubi [A]
time = 0.08, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {1677, 1674,
1671, 646, 31} \begin {gather*} \frac {5 x^6}{6}-\frac {27 x^4}{4}+49 x^2-\frac {5}{2} \log \left (x^2+1\right )-144 \log \left (x^2+2\right )-\frac {207 x^2+206}{2 \left (x^4+3 x^2+2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 646
Rule 1671
Rule 1674
Rule 1677
Rubi steps
\begin {align*} \int \frac {x^7 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^2} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^3 \left (4+x+3 x^2+5 x^3\right )}{\left (2+3 x+x^2\right )^2} \, dx,x,x^2\right )\\ &=-\frac {206+207 x^2}{2 \left (2+3 x^2+x^4\right )}-\frac {1}{2} \text {Subst}\left (\int \frac {102+53 x-27 x^2+12 x^3-5 x^4}{2+3 x+x^2} \, dx,x,x^2\right )\\ &=-\frac {206+207 x^2}{2 \left (2+3 x^2+x^4\right )}-\frac {1}{2} \text {Subst}\left (\int \left (-98+27 x-5 x^2+\frac {298+293 x}{2+3 x+x^2}\right ) \, dx,x,x^2\right )\\ &=49 x^2-\frac {27 x^4}{4}+\frac {5 x^6}{6}-\frac {206+207 x^2}{2 \left (2+3 x^2+x^4\right )}-\frac {1}{2} \text {Subst}\left (\int \frac {298+293 x}{2+3 x+x^2} \, dx,x,x^2\right )\\ &=49 x^2-\frac {27 x^4}{4}+\frac {5 x^6}{6}-\frac {206+207 x^2}{2 \left (2+3 x^2+x^4\right )}-\frac {5}{2} \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,x^2\right )-144 \text {Subst}\left (\int \frac {1}{2+x} \, dx,x,x^2\right )\\ &=49 x^2-\frac {27 x^4}{4}+\frac {5 x^6}{6}-\frac {206+207 x^2}{2 \left (2+3 x^2+x^4\right )}-\frac {5}{2} \log \left (1+x^2\right )-144 \log \left (2+x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 61, normalized size = 1.00 \begin {gather*} 49 x^2-\frac {27 x^4}{4}+\frac {5 x^6}{6}+\frac {-206-207 x^2}{2 \left (2+3 x^2+x^4\right )}-\frac {5}{2} \log \left (1+x^2\right )-144 \log \left (2+x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 51, normalized size = 0.84
method | result | size |
default | \(\frac {5 x^{6}}{6}-\frac {27 x^{4}}{4}+49 x^{2}+\frac {1}{2 x^{2}+2}-\frac {5 \ln \left (x^{2}+1\right )}{2}-144 \ln \left (x^{2}+2\right )-\frac {104}{x^{2}+2}\) | \(51\) |
norman | \(\frac {-406 x^{2}+\frac {365}{12} x^{6}-\frac {17}{4} x^{8}+\frac {5}{6} x^{10}-370}{x^{4}+3 x^{2}+2}-\frac {5 \ln \left (x^{2}+1\right )}{2}-144 \ln \left (x^{2}+2\right )\) | \(53\) |
risch | \(\frac {5 x^{6}}{6}-\frac {27 x^{4}}{4}+49 x^{2}+\frac {-\frac {207 x^{2}}{2}-103}{x^{4}+3 x^{2}+2}-\frac {5 \ln \left (x^{2}+1\right )}{2}-144 \ln \left (x^{2}+2\right )\) | \(53\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 53, normalized size = 0.87 \begin {gather*} \frac {5}{6} \, x^{6} - \frac {27}{4} \, x^{4} + 49 \, x^{2} - \frac {207 \, x^{2} + 206}{2 \, {\left (x^{4} + 3 \, x^{2} + 2\right )}} - 144 \, \log \left (x^{2} + 2\right ) - \frac {5}{2} \, \log \left (x^{2} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 77, normalized size = 1.26 \begin {gather*} \frac {10 \, x^{10} - 51 \, x^{8} + 365 \, x^{6} + 1602 \, x^{4} - 66 \, x^{2} - 1728 \, {\left (x^{4} + 3 \, x^{2} + 2\right )} \log \left (x^{2} + 2\right ) - 30 \, {\left (x^{4} + 3 \, x^{2} + 2\right )} \log \left (x^{2} + 1\right ) - 1236}{12 \, {\left (x^{4} + 3 \, x^{2} + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 56, normalized size = 0.92 \begin {gather*} \frac {5 x^{6}}{6} - \frac {27 x^{4}}{4} + 49 x^{2} + \frac {- 207 x^{2} - 206}{2 x^{4} + 6 x^{2} + 4} - \frac {5 \log {\left (x^{2} + 1 \right )}}{2} - 144 \log {\left (x^{2} + 2 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.20, size = 58, normalized size = 0.95 \begin {gather*} \frac {5}{6} \, x^{6} - \frac {27}{4} \, x^{4} + 49 \, x^{2} + \frac {293 \, x^{4} + 465 \, x^{2} + 174}{4 \, {\left (x^{4} + 3 \, x^{2} + 2\right )}} - 144 \, \log \left (x^{2} + 2\right ) - \frac {5}{2} \, \log \left (x^{2} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 53, normalized size = 0.87 \begin {gather*} 49\,x^2-144\,\ln \left (x^2+2\right )-\frac {\frac {207\,x^2}{2}+103}{x^4+3\,x^2+2}-\frac {5\,\ln \left (x^2+1\right )}{2}-\frac {27\,x^4}{4}+\frac {5\,x^6}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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