Optimal. Leaf size=31 \[ 2-e^{\frac {9}{\frac {2}{x}+x}}+4 x \left (-1-x^2+2 \log (x)\right ) \]
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Rubi [A] time = 0.29, antiderivative size = 28, normalized size of antiderivative = 0.90, number of steps used = 5, number of rules used = 4, integrand size = 65, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {28, 6688, 6706, 2295} \begin {gather*} -4 x^3-e^{\frac {9 x}{x^2+2}}-4 x+8 x \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 2295
Rule 6688
Rule 6706
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {16-32 x^2-44 x^4-12 x^6+e^{\frac {9 x}{2+x^2}} \left (-18+9 x^2\right )+\left (32+32 x^2+8 x^4\right ) \log (x)}{\left (2+x^2\right )^2} \, dx\\ &=\int \left (4-12 x^2+\frac {9 e^{\frac {9 x}{2+x^2}} \left (-2+x^2\right )}{\left (2+x^2\right )^2}+8 \log (x)\right ) \, dx\\ &=4 x-4 x^3+8 \int \log (x) \, dx+9 \int \frac {e^{\frac {9 x}{2+x^2}} \left (-2+x^2\right )}{\left (2+x^2\right )^2} \, dx\\ &=-e^{\frac {9 x}{2+x^2}}-4 x-4 x^3+8 x \log (x)\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.24, size = 28, normalized size = 0.90 \begin {gather*} -e^{\frac {9 x}{2+x^2}}-4 x-4 x^3+8 x \log (x) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 27, normalized size = 0.87 \begin {gather*} -4 \, x^{3} + 8 \, x \log \relax (x) - 4 \, x - e^{\left (\frac {9 \, x}{x^{2} + 2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 27, normalized size = 0.87 \begin {gather*} -4 \, x^{3} + 8 \, x \log \relax (x) - 4 \, x - e^{\left (\frac {9 \, x}{x^{2} + 2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 28, normalized size = 0.90
method | result | size |
risch | \(-4 x^{3}+8 x \ln \relax (x )-4 x -{\mathrm e}^{\frac {9 x}{x^{2}+2}}\) | \(28\) |
default | \(-4 x^{3}-4 x +\frac {-x^{2} {\mathrm e}^{\frac {9 x}{x^{2}+2}}-2 \,{\mathrm e}^{\frac {9 x}{x^{2}+2}}}{x^{2}+2}+8 x \ln \relax (x )\) | \(53\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 27, normalized size = 0.87 \begin {gather*} -4 \, x^{3} + 8 \, x \log \relax (x) - 4 \, x - e^{\left (\frac {9 \, x}{x^{2} + 2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.54, size = 29, normalized size = 0.94 \begin {gather*} 4\,x-{\mathrm {e}}^{\frac {9\,x}{x^2+2}}+8\,x\,\left (\ln \relax (x)-1\right )-4\,x^3 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.38, size = 24, normalized size = 0.77 \begin {gather*} - 4 x^{3} + 8 x \log {\relax (x )} - 4 x - e^{\frac {9 x}{x^{2} + 2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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