Optimal. Leaf size=21 \[ x \left (5+x-5 \log ^2(x)+\log \left (\left (-2+\frac {x}{4}\right ) x\right )\right ) \]
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Rubi [B] time = 0.19, antiderivative size = 44, normalized size of antiderivative = 2.10, number of steps used = 12, number of rules used = 7, integrand size = 50, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.140, Rules used = {6742, 698, 2295, 2296, 2487, 29, 8} \begin {gather*} x^2+5 x-5 x \log ^2(x)+8 \log (8-x)+8 \log (x)-(8-x) \log \left (-\frac {1}{4} (8-x) x\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 29
Rule 698
Rule 2295
Rule 2296
Rule 2487
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {-48-9 x+2 x^2+80 \log (x)-10 x \log (x)+40 \log ^2(x)-5 x \log ^2(x)}{-8+x}+\log \left (\frac {1}{4} (-8+x) x\right )\right ) \, dx\\ &=\int \frac {-48-9 x+2 x^2+80 \log (x)-10 x \log (x)+40 \log ^2(x)-5 x \log ^2(x)}{-8+x} \, dx+\int \log \left (\frac {1}{4} (-8+x) x\right ) \, dx\\ &=-\left ((8-x) \log \left (-\frac {1}{4} (8-x) x\right )\right )-2 \int 1 \, dx+8 \int \frac {1}{x} \, dx+\int \left (\frac {-48-9 x+2 x^2}{-8+x}-10 \log (x)-5 \log ^2(x)\right ) \, dx\\ &=-2 x+8 \log (x)-(8-x) \log \left (-\frac {1}{4} (8-x) x\right )-5 \int \log ^2(x) \, dx-10 \int \log (x) \, dx+\int \frac {-48-9 x+2 x^2}{-8+x} \, dx\\ &=8 x+8 \log (x)-10 x \log (x)-5 x \log ^2(x)-(8-x) \log \left (-\frac {1}{4} (8-x) x\right )+10 \int \log (x) \, dx+\int \left (7+\frac {8}{-8+x}+2 x\right ) \, dx\\ &=5 x+x^2+8 \log (8-x)+8 \log (x)-5 x \log ^2(x)-(8-x) \log \left (-\frac {1}{4} (8-x) x\right )\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.05, size = 25, normalized size = 1.19 \begin {gather*} 5 x+x^2-5 x \log ^2(x)+x \log \left (\frac {1}{4} (-8+x) x\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 26, normalized size = 1.24 \begin {gather*} -5 \, x \log \relax (x)^{2} + x^{2} + x \log \left (\frac {1}{4} \, x^{2} - 2 \, x\right ) + 5 \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 30, normalized size = 1.43 \begin {gather*} -5 \, x \log \relax (x)^{2} + x^{2} - x {\left (2 \, \log \relax (2) - 5\right )} + x \log \left (x - 8\right ) + x \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.55, size = 30, normalized size = 1.43
method | result | size |
default | \(x^{2}+5 x -5 x \ln \relax (x )^{2}-2 x \ln \relax (2)+x \ln \left (x^{2}-8 x \right )\) | \(30\) |
risch | \(x \ln \left (-8+x \right )-5 x \ln \relax (x )^{2}+x \ln \relax (x )-\frac {i x \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (-8+x \right )\right ) \mathrm {csgn}\left (i x \left (-8+x \right )\right )}{2}+\frac {i x \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \left (-8+x \right )\right )^{2}}{2}+\frac {i x \pi \,\mathrm {csgn}\left (i \left (-8+x \right )\right ) \mathrm {csgn}\left (i x \left (-8+x \right )\right )^{2}}{2}-\frac {i x \pi \mathrm {csgn}\left (i x \left (-8+x \right )\right )^{3}}{2}-2 x \ln \relax (2)+x^{2}+5 x\) | \(112\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 39, normalized size = 1.86 \begin {gather*} -5 \, x \log \relax (x)^{2} + x^{2} - 2 \, x {\left (\log \relax (2) + 1\right )} + {\left (x - 8\right )} \log \left (x - 8\right ) + x \log \relax (x) + 7 \, x + 8 \, \log \left (x - 8\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.00, size = 26, normalized size = 1.24 \begin {gather*} 5\,x-5\,x\,{\ln \relax (x)}^2+x\,\ln \left (\frac {x^2}{4}-2\,x\right )+x^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.56, size = 41, normalized size = 1.95 \begin {gather*} x^{2} - 5 x \log {\relax (x )}^{2} + 5 x + \left (x - \frac {4}{3}\right ) \log {\left (\frac {x^{2}}{4} - 2 x \right )} + \frac {4 \log {\left (x^{2} - 8 x \right )}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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