Optimal. Leaf size=42 \[ \frac{25 x^2+24}{2 \left (x^4+3 x^2+2\right )}+4 \log \left (x^2+1\right )-\frac{3}{2} \log \left (x^2+2\right ) \]
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Rubi [A] time = 0.0491766, antiderivative size = 42, normalized size of antiderivative = 1., 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} \[ \frac{25 x^2+24}{2 \left (x^4+3 x^2+2\right )}+4 \log \left (x^2+1\right )-\frac{3}{2} \log \left (x^2+2\right ) \]
Antiderivative was successfully verified.
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Rule 1663
Rule 1660
Rule 632
Rule 31
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*}
Mathematica [A] time = 0.0170239, size = 42, normalized size = 1. \[ \frac{25 x^2+24}{2 \left (x^4+3 x^2+2\right )}+4 \log \left (x^2+1\right )-\frac{3}{2} \log \left (x^2+2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 36, normalized size = 0.9 \begin{align*} -{\frac{3\,\ln \left ({x}^{2}+2 \right ) }{2}}+13\, \left ({x}^{2}+2 \right ) ^{-1}+4\,\ln \left ({x}^{2}+1 \right ) -{\frac{1}{2\,{x}^{2}+2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.965276, size = 51, normalized size = 1.21 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.00301, size = 144, normalized size = 3.43 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.150097, size = 36, normalized size = 0.86 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12106, size = 54, normalized size = 1.29 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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