Optimal. Leaf size=27 \[ \log \left (e^x \left (x+\frac {1}{3} \left (\left (16+e^{2 x}\right )^2+x-x^2\right )\right )\right ) \]
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Rubi [F] time = 0.54, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {260+96 e^{2 x}+5 e^{4 x}+2 x-x^2}{256+32 e^{2 x}+e^{4 x}+4 x-x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (5-\frac {2 \left (510+32 e^{2 x}+9 x-2 x^2\right )}{256+32 e^{2 x}+e^{4 x}+4 x-x^2}\right ) \, dx\\ &=5 x-2 \int \frac {510+32 e^{2 x}+9 x-2 x^2}{256+32 e^{2 x}+e^{4 x}+4 x-x^2} \, dx\\ &=5 x-2 \int \left (\frac {510}{256+32 e^{2 x}+e^{4 x}+4 x-x^2}+\frac {32 e^{2 x}}{256+32 e^{2 x}+e^{4 x}+4 x-x^2}-\frac {9 x}{-256-32 e^{2 x}-e^{4 x}-4 x+x^2}+\frac {2 x^2}{-256-32 e^{2 x}-e^{4 x}-4 x+x^2}\right ) \, dx\\ &=5 x-4 \int \frac {x^2}{-256-32 e^{2 x}-e^{4 x}-4 x+x^2} \, dx+18 \int \frac {x}{-256-32 e^{2 x}-e^{4 x}-4 x+x^2} \, dx-64 \int \frac {e^{2 x}}{256+32 e^{2 x}+e^{4 x}+4 x-x^2} \, dx-1020 \int \frac {1}{256+32 e^{2 x}+e^{4 x}+4 x-x^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.19, size = 25, normalized size = 0.93 \begin {gather*} x+\log \left (256+32 e^{2 x}+e^{4 x}+4 x-x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 23, normalized size = 0.85 \begin {gather*} x + \log \left (-x^{2} + 4 \, x + e^{\left (4 \, x\right )} + 32 \, e^{\left (2 \, x\right )} + 256\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 23, normalized size = 0.85 \begin {gather*} x + \log \left (-x^{2} + 4 \, x + e^{\left (4 \, x\right )} + 32 \, e^{\left (2 \, x\right )} + 256\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 24, normalized size = 0.89
method | result | size |
risch | \(x +\ln \left ({\mathrm e}^{4 x}+32 \,{\mathrm e}^{2 x}-x^{2}+4 x +256\right )\) | \(24\) |
norman | \(x +\ln \left (x^{2}-{\mathrm e}^{4 x}-4 x -32 \,{\mathrm e}^{2 x}-256\right )\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 23, normalized size = 0.85 \begin {gather*} x + \log \left (-x^{2} + 4 \, x + e^{\left (4 \, x\right )} + 32 \, e^{\left (2 \, x\right )} + 256\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.22, size = 23, normalized size = 0.85 \begin {gather*} x+\ln \left (x^2-32\,{\mathrm {e}}^{2\,x}-{\mathrm {e}}^{4\,x}-4\,x-256\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 22, normalized size = 0.81 \begin {gather*} x + \log {\left (- x^{2} + 4 x + e^{4 x} + 32 e^{2 x} + 256 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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