Optimal. Leaf size=26 \[ 4-\frac {1}{x}+x-\frac {x}{3 (5-x-\log (8+x))} \]
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Rubi [F]
time = 0.66, 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 {600-165 x+554 x^2-168 x^3-6 x^4+3 x^5+\left (-240+18 x-226 x^2+19 x^3+6 x^4\right ) \log (8+x)+\left (24+3 x+24 x^2+3 x^3\right ) \log ^2(8+x)}{600 x^2-165 x^3-6 x^4+3 x^5+\left (-240 x^2+18 x^3+6 x^4\right ) \log (8+x)+\left (24 x^2+3 x^3\right ) \log ^2(8+x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {600-165 x+554 x^2-168 x^3-6 x^4+3 x^5+\left (-240+18 x-226 x^2+19 x^3+6 x^4\right ) \log (8+x)+3 \left (8+x+8 x^2+x^3\right ) \log ^2(8+x)}{3 x^2 (8+x) (5-x-\log (8+x))^2} \, dx\\ &=\frac {1}{3} \int \frac {600-165 x+554 x^2-168 x^3-6 x^4+3 x^5+\left (-240+18 x-226 x^2+19 x^3+6 x^4\right ) \log (8+x)+3 \left (8+x+8 x^2+x^3\right ) \log ^2(8+x)}{x^2 (8+x) (5-x-\log (8+x))^2} \, dx\\ &=\frac {1}{3} \int \left (\frac {3 \left (1+x^2\right )}{x^2}-\frac {x (9+x)}{(8+x) (-5+x+\log (8+x))^2}+\frac {1}{-5+x+\log (8+x)}\right ) \, dx\\ &=-\left (\frac {1}{3} \int \frac {x (9+x)}{(8+x) (-5+x+\log (8+x))^2} \, dx\right )+\frac {1}{3} \int \frac {1}{-5+x+\log (8+x)} \, dx+\int \frac {1+x^2}{x^2} \, dx\\ &=\frac {1}{3} \int \frac {1}{-5+x+\log (8+x)} \, dx-\frac {1}{3} \int \left (\frac {1}{(-5+x+\log (8+x))^2}+\frac {x}{(-5+x+\log (8+x))^2}-\frac {8}{(8+x) (-5+x+\log (8+x))^2}\right ) \, dx+\int \left (1+\frac {1}{x^2}\right ) \, dx\\ &=-\frac {1}{x}+x-\frac {1}{3} \int \frac {1}{(-5+x+\log (8+x))^2} \, dx-\frac {1}{3} \int \frac {x}{(-5+x+\log (8+x))^2} \, dx+\frac {1}{3} \int \frac {1}{-5+x+\log (8+x)} \, dx+\frac {8}{3} \int \frac {1}{(8+x) (-5+x+\log (8+x))^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.05, size = 24, normalized size = 0.92 \begin {gather*} \frac {1}{3} \left (-\frac {3}{x}+3 x+\frac {x}{-5+x+\log (8+x)}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(50\) vs.
\(2(24)=48\).
time = 2.02, size = 51, normalized size = 1.96
method | result | size |
risch | \(\frac {x^{2}-1}{x}+\frac {x}{3 \ln \left (x +8\right )+3 x -15}\) | \(23\) |
norman | \(\frac {5+x^{3}-\frac {73 x}{3}-\ln \left (x +8\right )^{2} x +\frac {29 x \ln \left (x +8\right )}{3}-\ln \left (x +8\right )}{x \left (\ln \left (x +8\right )+x -5\right )}+\ln \left (x +8\right )\) | \(49\) |
derivativedivides | \(\frac {-211 x +911-195 \ln \left (x +8\right )-38 \left (x +8\right )^{2}+3 \left (x +8\right )^{3}+3 \ln \left (x +8\right ) \left (x +8\right )^{2}}{3 x \left (\ln \left (x +8\right )+x -5\right )}\) | \(51\) |
default | \(\frac {-211 x +911-195 \ln \left (x +8\right )-38 \left (x +8\right )^{2}+3 \left (x +8\right )^{3}+3 \ln \left (x +8\right ) \left (x +8\right )^{2}}{3 x \left (\ln \left (x +8\right )+x -5\right )}\) | \(51\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 43 vs.
\(2 (20) = 40\).
time = 0.33, size = 43, normalized size = 1.65 \begin {gather*} \frac {3 \, x^{3} - 14 \, x^{2} + 3 \, {\left (x^{2} - 1\right )} \log \left (x + 8\right ) - 3 \, x + 15}{3 \, {\left (x^{2} + x \log \left (x + 8\right ) - 5 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 43 vs.
\(2 (20) = 40\).
time = 0.38, size = 43, normalized size = 1.65 \begin {gather*} \frac {3 \, x^{3} - 14 \, x^{2} + 3 \, {\left (x^{2} - 1\right )} \log \left (x + 8\right ) - 3 \, x + 15}{3 \, {\left (x^{2} + x \log \left (x + 8\right ) - 5 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 17, normalized size = 0.65 \begin {gather*} x + \frac {x}{3 x + 3 \log {\left (x + 8 \right )} - 15} - \frac {1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 19, normalized size = 0.73 \begin {gather*} x + \frac {x}{3 \, {\left (x + \log \left (x + 8\right ) - 5\right )}} - \frac {1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.86, size = 32, normalized size = 1.23 \begin {gather*} x-\frac {\ln \left (x+8\right )+x\,\left (\frac {\ln \left (x+8\right )}{3}-\frac {2}{3}\right )-5}{x\,\left (x+\ln \left (x+8\right )-5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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