Optimal. Leaf size=20 \[ x+(-4+e+x) \log \left (4 \left (-16-x+4 x^2\right )\right ) \]
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Rubi [B] time = 0.21, antiderivative size = 117, normalized size of antiderivative = 5.85, number of steps used = 12, number of rules used = 6, integrand size = 52, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {6728, 1657, 632, 31, 2523, 773} \begin {gather*} x \log \left (16 x^2-4 x-64\right )+x-\frac {1}{8} \left (31+\sqrt {257}-8 e\right ) \log \left (-8 x-\sqrt {257}+1\right )-\frac {1}{8} \left (1-\sqrt {257}\right ) \log \left (-8 x-\sqrt {257}+1\right )-\frac {1}{8} \left (31-\sqrt {257}-8 e\right ) \log \left (-8 x+\sqrt {257}+1\right )-\frac {1}{8} \left (1+\sqrt {257}\right ) \log \left (-8 x+\sqrt {257}+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 632
Rule 773
Rule 1657
Rule 2523
Rule 6728
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {12+e+2 (17-4 e) x-12 x^2}{16+x-4 x^2}+\log \left (-64-4 x+16 x^2\right )\right ) \, dx\\ &=\int \frac {12+e+2 (17-4 e) x-12 x^2}{16+x-4 x^2} \, dx+\int \log \left (-64-4 x+16 x^2\right ) \, dx\\ &=x \log \left (-64-4 x+16 x^2\right )-\int \frac {(1-8 x) x}{16+x-4 x^2} \, dx+\int \left (3-\frac {36-e-(31-8 e) x}{16+x-4 x^2}\right ) \, dx\\ &=x+x \log \left (-64-4 x+16 x^2\right )+\frac {1}{4} \int \frac {128+4 x}{16+x-4 x^2} \, dx-\int \frac {36-e-(31-8 e) x}{16+x-4 x^2} \, dx\\ &=x+x \log \left (-64-4 x+16 x^2\right )+\frac {1}{2} \left (1-\sqrt {257}\right ) \int \frac {1}{\frac {1}{2}-\frac {\sqrt {257}}{2}-4 x} \, dx+\frac {1}{2} \left (1+\sqrt {257}\right ) \int \frac {1}{\frac {1}{2}+\frac {\sqrt {257}}{2}-4 x} \, dx+\frac {1}{2} \left (31-\sqrt {257}-8 e\right ) \int \frac {1}{\frac {1}{2}+\frac {\sqrt {257}}{2}-4 x} \, dx-\frac {1}{2} \left (-31-\sqrt {257}+8 e\right ) \int \frac {1}{\frac {1}{2}-\frac {\sqrt {257}}{2}-4 x} \, dx\\ &=x-\frac {1}{8} \left (1-\sqrt {257}\right ) \log \left (1-\sqrt {257}-8 x\right )-\frac {1}{8} \left (31+\sqrt {257}-8 e\right ) \log \left (1-\sqrt {257}-8 x\right )-\frac {1}{8} \left (1+\sqrt {257}\right ) \log \left (1+\sqrt {257}-8 x\right )-\frac {1}{8} \left (31-\sqrt {257}-8 e\right ) \log \left (1+\sqrt {257}-8 x\right )+x \log \left (-64-4 x+16 x^2\right )\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 0.08, size = 71, normalized size = 3.55 \begin {gather*} x-4 \log \left (1-\sqrt {257}-8 x\right )+e \log \left (1-\sqrt {257}-8 x\right )-4 \log \left (1+\sqrt {257}-8 x\right )+e \log \left (1+\sqrt {257}-8 x\right )+x \log \left (-64-4 x+16 x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 19, normalized size = 0.95 \begin {gather*} {\left (x + e - 4\right )} \log \left (16 \, x^{2} - 4 \, x - 64\right ) + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.21, size = 42, normalized size = 2.10 \begin {gather*} x \log \left (16 \, x^{2} - 4 \, x - 64\right ) + e \log \left (4 \, x^{2} - x - 16\right ) + x - 4 \, \log \left (4 \, x^{2} - x - 16\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.20, size = 32, normalized size = 1.60
method | result | size |
norman | \(x +\left ({\mathrm e}-4\right ) \ln \left (16 x^{2}-4 x -64\right )+\ln \left (16 x^{2}-4 x -64\right ) x\) | \(32\) |
risch | \(\ln \left (16 x^{2}-4 x -64\right ) x +\ln \left (4 x^{2}-x -16\right ) {\mathrm e}-4 \ln \left (4 x^{2}-x -16\right )+x\) | \(43\) |
default | \(x -4 \ln \left (4 x^{2}-x -16\right )+\ln \left (4 x^{2}-x -16\right ) {\mathrm e}+2 x \ln \relax (2)+x \ln \left (4 x^{2}-x -16\right )\) | \(48\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.51, size = 115, normalized size = 5.75 \begin {gather*} -\frac {1}{257} \, \sqrt {257} e \log \left (\frac {8 \, x - \sqrt {257} - 1}{8 \, x + \sqrt {257} - 1}\right ) + 2 \, x {\left (\log \relax (2) - 1\right )} + \frac {1}{257} \, {\left (\sqrt {257} \log \left (\frac {8 \, x - \sqrt {257} - 1}{8 \, x + \sqrt {257} - 1}\right ) + 257 \, \log \left (4 \, x^{2} - x - 16\right )\right )} e + \frac {1}{8} \, {\left (8 \, x - 1\right )} \log \left (4 \, x^{2} - x - 16\right ) + 3 \, x - \frac {31}{8} \, \log \left (4 \, x^{2} - x - 16\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.20, size = 31, normalized size = 1.55 \begin {gather*} x+x\,\ln \left (16\,x^2-4\,x-64\right )+\ln \left (4\,x^2-x-16\right )\,\left (\mathrm {e}-4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.27, size = 29, normalized size = 1.45 \begin {gather*} x \log {\left (16 x^{2} - 4 x - 64 \right )} + x + \left (-4 + e\right ) \log {\left (4 x^{2} - x - 16 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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