Optimal. Leaf size=32 \[ \log (x)+\frac {\log \left (x^2 \left (-x+x^2 (3+x)\right )\right )}{e^8 \left (1-\frac {x}{3}\right )} \]
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Rubi [A] time = 0.44, antiderivative size = 31, normalized size of antiderivative = 0.97, number of steps used = 14, number of rules used = 6, integrand size = 95, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.063, Rules used = {12, 6688, 6742, 632, 31, 2525} \begin {gather*} \frac {3 \log \left (-x^3 \left (-x^2-3 x+1\right )\right )}{e^8 (3-x)}+\log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 31
Rule 632
Rule 2525
Rule 6688
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {-27+117 x+9 x^2-15 x^3+e^8 \left (-9+33 x-10 x^2-3 x^3+x^4\right )+\left (-3 x+9 x^2+3 x^3\right ) \log \left (-x^3+3 x^4+x^5\right )}{-9 x+33 x^2-10 x^3-3 x^4+x^5} \, dx}{e^8}\\ &=\frac {\int \left (\frac {e^8}{x}+\frac {9-36 x-15 x^2}{x \left (3-10 x+x^3\right )}+\frac {3 \log \left (x^3 \left (-1+3 x+x^2\right )\right )}{(-3+x)^2}\right ) \, dx}{e^8}\\ &=\log (x)+\frac {\int \frac {9-36 x-15 x^2}{x \left (3-10 x+x^3\right )} \, dx}{e^8}+\frac {3 \int \frac {\log \left (x^3 \left (-1+3 x+x^2\right )\right )}{(-3+x)^2} \, dx}{e^8}\\ &=\log (x)+\frac {3 \log \left (-x^3 \left (1-3 x-x^2\right )\right )}{e^8 (3-x)}+\frac {\int \left (-\frac {78}{17 (-3+x)}+\frac {3}{x}+\frac {3 (20+9 x)}{17 \left (-1+3 x+x^2\right )}\right ) \, dx}{e^8}+\frac {3 \int \frac {-3+12 x+5 x^2}{x \left (3-10 x+x^3\right )} \, dx}{e^8}\\ &=-\frac {78 \log (3-x)}{17 e^8}+\log (x)+\frac {3 \log (x)}{e^8}+\frac {3 \log \left (-x^3 \left (1-3 x-x^2\right )\right )}{e^8 (3-x)}+\frac {3 \int \frac {20+9 x}{-1+3 x+x^2} \, dx}{17 e^8}+\frac {3 \int \left (\frac {26}{17 (-3+x)}-\frac {1}{x}+\frac {-20-9 x}{17 \left (-1+3 x+x^2\right )}\right ) \, dx}{e^8}\\ &=\log (x)+\frac {3 \log \left (-x^3 \left (1-3 x-x^2\right )\right )}{e^8 (3-x)}+\frac {3 \int \frac {-20-9 x}{-1+3 x+x^2} \, dx}{17 e^8}+\frac {\left (3 \left (9-\sqrt {13}\right )\right ) \int \frac {1}{\frac {3}{2}+\frac {\sqrt {13}}{2}+x} \, dx}{34 e^8}+\frac {\left (3 \left (9+\sqrt {13}\right )\right ) \int \frac {1}{\frac {3}{2}-\frac {\sqrt {13}}{2}+x} \, dx}{34 e^8}\\ &=\log (x)+\frac {3 \left (9+\sqrt {13}\right ) \log \left (3-\sqrt {13}+2 x\right )}{34 e^8}+\frac {3 \left (9-\sqrt {13}\right ) \log \left (3+\sqrt {13}+2 x\right )}{34 e^8}+\frac {3 \log \left (-x^3 \left (1-3 x-x^2\right )\right )}{e^8 (3-x)}-\frac {\left (3 \left (9-\sqrt {13}\right )\right ) \int \frac {1}{\frac {3}{2}+\frac {\sqrt {13}}{2}+x} \, dx}{34 e^8}-\frac {\left (3 \left (9+\sqrt {13}\right )\right ) \int \frac {1}{\frac {3}{2}-\frac {\sqrt {13}}{2}+x} \, dx}{34 e^8}\\ &=\log (x)+\frac {3 \log \left (-x^3 \left (1-3 x-x^2\right )\right )}{e^8 (3-x)}\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 0.15, size = 78, normalized size = 2.44 \begin {gather*} \frac {3 \left (9+\sqrt {13}\right ) \log \left (-3+\sqrt {13}-2 x\right )}{34 e^8}+\log (x)-\frac {3 \left (9+\sqrt {13}\right ) \log \left (3-\sqrt {13}+2 x\right )}{34 e^8}-\frac {3 \log \left (x^3 \left (-1+3 x+x^2\right )\right )}{e^8 (-3+x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 34, normalized size = 1.06 \begin {gather*} \frac {{\left ({\left (x - 3\right )} e^{8} \log \relax (x) - 3 \, \log \left (x^{5} + 3 \, x^{4} - x^{3}\right )\right )} e^{\left (-8\right )}}{x - 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 38, normalized size = 1.19 \begin {gather*} \frac {{\left (x e^{8} \log \relax (x) - 3 \, e^{8} \log \relax (x) - 3 \, \log \left (x^{5} + 3 \, x^{4} - x^{3}\right )\right )} e^{\left (-8\right )}}{x - 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 28, normalized size = 0.88
method | result | size |
risch | \(-\frac {3 \,{\mathrm e}^{-8} \ln \left (x^{5}+3 x^{4}-x^{3}\right )}{x -3}+\ln \relax (x )\) | \(28\) |
norman | \(-\frac {3 \,{\mathrm e}^{-8} \ln \left (x^{5}+3 x^{4}-x^{3}\right )}{x -3}+\ln \relax (x )\) | \(30\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.71, size = 308, normalized size = 9.62 \begin {gather*} -\frac {1}{7514} \, {\left ({\left (1161 \, \sqrt {13} \log \left (\frac {2 \, x - \sqrt {13} + 3}{2 \, x + \sqrt {13} + 3}\right ) - \frac {1326}{x - 3} + 4329 \, \log \left (x^{2} + 3 \, x - 1\right ) - 1144 \, \log \left (x - 3\right ) - 7514 \, \log \relax (x)\right )} e^{8} + {\left (261 \, \sqrt {13} \log \left (\frac {2 \, x - \sqrt {13} + 3}{2 \, x + \sqrt {13} + 3}\right ) + \frac {11934}{x - 3} - 949 \, \log \left (x^{2} + 3 \, x - 1\right ) - 5616 \, \log \left (x - 3\right )\right )} e^{8} + 3 \, {\left (83 \, \sqrt {13} \log \left (\frac {2 \, x - \sqrt {13} + 3}{2 \, x + \sqrt {13} + 3}\right ) - \frac {3978}{x - 3} - 273 \, \log \left (x^{2} + 3 \, x - 1\right ) + 546 \, \log \left (x - 3\right )\right )} e^{8} - 33 \, {\left (47 \, \sqrt {13} \log \left (\frac {2 \, x - \sqrt {13} + 3}{2 \, x + \sqrt {13} + 3}\right ) - \frac {442}{x - 3} + 117 \, \log \left (x^{2} + 3 \, x - 1\right ) - 234 \, \log \left (x - 3\right )\right )} e^{8} - 20 \, {\left (6 \, \sqrt {13} \log \left (\frac {2 \, x - \sqrt {13} + 3}{2 \, x + \sqrt {13} + 3}\right ) + \frac {663}{x - 3} - 65 \, \log \left (x^{2} + 3 \, x - 1\right ) + 130 \, \log \left (x - 3\right )\right )} e^{8} + \frac {22542 \, {\left (\log \left (x^{2} + 3 \, x - 1\right ) + 3 \, \log \relax (x)\right )}}{x - 3}\right )} e^{\left (-8\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.75, size = 27, normalized size = 0.84 \begin {gather*} \ln \relax (x)-\frac {3\,\ln \left (x^5+3\,x^4-x^3\right )\,{\mathrm {e}}^{-8}}{x-3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.24, size = 27, normalized size = 0.84 \begin {gather*} \log {\relax (x )} - \frac {3 \log {\left (x^{5} + 3 x^{4} - x^{3} \right )}}{x e^{8} - 3 e^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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