Optimal. Leaf size=42 \[ 4 \log \left (x^2+1\right )-\frac {3}{2} \log \left (x^2+2\right )+\frac {25 x^2+24}{2 \left (x^4+3 x^2+2\right )} \]
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Rubi [A] time = 0.05, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {1663, 1660, 632, 31} \begin {gather*} \frac {25 x^2+24}{2 \left (x^4+3 x^2+2\right )}+4 \log \left (x^2+1\right )-\frac {3}{2} \log \left (x^2+2\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 632
Rule 1660
Rule 1663
Rubi steps
\begin {align*} \int \frac {x \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {4+x+3 x^2+5 x^3}{\left (2+3 x+x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac {24+25 x^2}{2 \left (2+3 x^2+x^4\right )}-\frac {1}{2} \operatorname {Subst}\left (\int \frac {-13-5 x}{2+3 x+x^2} \, dx,x,x^2\right )\\ &=\frac {24+25 x^2}{2 \left (2+3 x^2+x^4\right )}-\frac {3}{2} \operatorname {Subst}\left (\int \frac {1}{2+x} \, dx,x,x^2\right )+4 \operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,x^2\right )\\ &=\frac {24+25 x^2}{2 \left (2+3 x^2+x^4\right )}+4 \log \left (1+x^2\right )-\frac {3}{2} \log \left (2+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 42, normalized size = 1.00 \begin {gather*} 4 \log \left (x^2+1\right )-\frac {3}{2} \log \left (x^2+2\right )+\frac {25 x^2+24}{2 \left (x^4+3 x^2+2\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 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.23, size = 57, normalized size = 1.36 \begin {gather*} \frac {25 \, x^{2} - 3 \, {\left (x^{4} + 3 \, x^{2} + 2\right )} \log \left (x^{2} + 2\right ) + 8 \, {\left (x^{4} + 3 \, x^{2} + 2\right )} \log \left (x^{2} + 1\right ) + 24}{2 \, {\left (x^{4} + 3 \, x^{2} + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 40, normalized size = 0.95 \begin {gather*} \frac {25 \, x^{2} + 24}{2 \, {\left (x^{2} + 2\right )} {\left (x^{2} + 1\right )}} - \frac {3}{2} \, \log \left (x^{2} + 2\right ) + 4 \, \log \left (x^{2} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 36, normalized size = 0.86 \begin {gather*} 4 \ln \left (x^{2}+1\right )-\frac {3 \ln \left (x^{2}+2\right )}{2}-\frac {1}{2 \left (x^{2}+1\right )}+\frac {13}{x^{2}+2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 38, normalized size = 0.90 \begin {gather*} \frac {25 \, x^{2} + 24}{2 \, {\left (x^{4} + 3 \, x^{2} + 2\right )}} - \frac {3}{2} \, \log \left (x^{2} + 2\right ) + 4 \, \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 = 0.05, size = 37, normalized size = 0.88 \begin {gather*} 4\,\ln \left (x^2+1\right )-\frac {3\,\ln \left (x^2+2\right )}{2}+\frac {\frac {25\,x^2}{2}+12}{x^4+3\,x^2+2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 36, normalized size = 0.86 \begin {gather*} \frac {25 x^{2} + 24}{2 x^{4} + 6 x^{2} + 4} + 4 \log {\left (x^{2} + 1 \right )} - \frac {3 \log {\left (x^{2} + 2 \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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