Optimal. Leaf size=32 \[ x+\frac {e^8}{x (3+x)}-\frac {\log \left (x+x^2\right )}{x-x (4+x)} \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(90\) vs. \(2(32)=64\).
time = 0.64, antiderivative size = 90, normalized size of antiderivative = 2.81, number
of steps used = 33, number of rules used = 16, integrand size = 76, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules
used = {6873, 6874, 46, 90, 84, 78, 75, 2593, 2404, 2341, 2351, 31, 2465, 2442, 36, 29}
\begin {gather*} x+\frac {e^8}{(x+3) x}+\frac {x \log (x)}{9 (x+3)}-\frac {\log (x)}{9}-\frac {\log (x+1)}{3 (x+3)}+\frac {\log (x)}{3 x}+\frac {\log (x+1)}{3 x}-\frac {\log (x)+\log (x+1)-\log (x (x+1))}{(x+3) x} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 46
Rule 75
Rule 78
Rule 84
Rule 90
Rule 2341
Rule 2351
Rule 2404
Rule 2442
Rule 2465
Rule 2593
Rule 6873
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3+7 x+11 x^2+15 x^3+7 x^4+x^5+e^8 \left (-3-5 x-2 x^2\right )+\left (-3-5 x-2 x^2\right ) \log \left (x+x^2\right )}{x^2 (1+x) (3+x)^2} \, dx\\ &=\int \left (\frac {11}{(1+x) (3+x)^2}+\frac {3}{x^2 (1+x) (3+x)^2}+\frac {7}{x (1+x) (3+x)^2}+\frac {15 x}{(1+x) (3+x)^2}+\frac {7 x^2}{(1+x) (3+x)^2}+\frac {x^3}{(1+x) (3+x)^2}-\frac {e^8 (3+2 x)}{x^2 (3+x)^2}-\frac {(3+2 x) \log (x (1+x))}{x^2 (3+x)^2}\right ) \, dx\\ &=3 \int \frac {1}{x^2 (1+x) (3+x)^2} \, dx+7 \int \frac {1}{x (1+x) (3+x)^2} \, dx+7 \int \frac {x^2}{(1+x) (3+x)^2} \, dx+11 \int \frac {1}{(1+x) (3+x)^2} \, dx+15 \int \frac {x}{(1+x) (3+x)^2} \, dx-e^8 \int \frac {3+2 x}{x^2 (3+x)^2} \, dx+\int \frac {x^3}{(1+x) (3+x)^2} \, dx-\int \frac {(3+2 x) \log (x (1+x))}{x^2 (3+x)^2} \, dx\\ &=\frac {e^8}{x (3+x)}+3 \int \left (\frac {1}{9 x^2}-\frac {5}{27 x}+\frac {1}{4 (1+x)}-\frac {1}{18 (3+x)^2}-\frac {7}{108 (3+x)}\right ) \, dx+7 \int \left (\frac {1}{9 x}-\frac {1}{4 (1+x)}+\frac {1}{6 (3+x)^2}+\frac {5}{36 (3+x)}\right ) \, dx+7 \int \left (\frac {1}{4 (1+x)}-\frac {9}{2 (3+x)^2}+\frac {3}{4 (3+x)}\right ) \, dx+11 \int \left (\frac {1}{4 (1+x)}-\frac {1}{2 (3+x)^2}-\frac {1}{4 (3+x)}\right ) \, dx+15 \int \left (-\frac {1}{4 (1+x)}+\frac {3}{2 (3+x)^2}+\frac {1}{4 (3+x)}\right ) \, dx+(\log (x)+\log (1+x)-\log (x (1+x))) \int \frac {3+2 x}{x^2 (3+x)^2} \, dx+\int \left (1-\frac {1}{4 (1+x)}+\frac {27}{2 (3+x)^2}-\frac {27}{4 (3+x)}\right ) \, dx-\int \frac {(3+2 x) \log (x)}{x^2 (3+x)^2} \, dx-\int \frac {(3+2 x) \log (1+x)}{x^2 (3+x)^2} \, dx\\ &=-\frac {1}{3 x}+x+\frac {e^8}{x (3+x)}+\frac {2 \log (x)}{9}-\frac {1}{2} \log (1+x)-\frac {\log (x)+\log (1+x)-\log (x (1+x))}{x (3+x)}+\frac {5}{18} \log (3+x)-\int \left (\frac {\log (x)}{3 x^2}-\frac {\log (x)}{3 (3+x)^2}\right ) \, dx-\int \left (\frac {\log (1+x)}{3 x^2}-\frac {\log (1+x)}{3 (3+x)^2}\right ) \, dx\\ &=-\frac {1}{3 x}+x+\frac {e^8}{x (3+x)}+\frac {2 \log (x)}{9}-\frac {1}{2} \log (1+x)-\frac {\log (x)+\log (1+x)-\log (x (1+x))}{x (3+x)}+\frac {5}{18} \log (3+x)-\frac {1}{3} \int \frac {\log (x)}{x^2} \, dx+\frac {1}{3} \int \frac {\log (x)}{(3+x)^2} \, dx-\frac {1}{3} \int \frac {\log (1+x)}{x^2} \, dx+\frac {1}{3} \int \frac {\log (1+x)}{(3+x)^2} \, dx\\ &=x+\frac {e^8}{x (3+x)}+\frac {2 \log (x)}{9}+\frac {\log (x)}{3 x}+\frac {x \log (x)}{9 (3+x)}-\frac {1}{2} \log (1+x)+\frac {\log (1+x)}{3 x}-\frac {\log (1+x)}{3 (3+x)}-\frac {\log (x)+\log (1+x)-\log (x (1+x))}{x (3+x)}+\frac {5}{18} \log (3+x)-\frac {1}{9} \int \frac {1}{3+x} \, dx-\frac {1}{3} \int \frac {1}{x (1+x)} \, dx+\frac {1}{3} \int \frac {1}{(1+x) (3+x)} \, dx\\ &=x+\frac {e^8}{x (3+x)}+\frac {2 \log (x)}{9}+\frac {\log (x)}{3 x}+\frac {x \log (x)}{9 (3+x)}-\frac {1}{2} \log (1+x)+\frac {\log (1+x)}{3 x}-\frac {\log (1+x)}{3 (3+x)}-\frac {\log (x)+\log (1+x)-\log (x (1+x))}{x (3+x)}+\frac {1}{6} \log (3+x)+\frac {1}{6} \int \frac {1}{1+x} \, dx-\frac {1}{6} \int \frac {1}{3+x} \, dx-\frac {1}{3} \int \frac {1}{x} \, dx+\frac {1}{3} \int \frac {1}{1+x} \, dx\\ &=x+\frac {e^8}{x (3+x)}-\frac {\log (x)}{9}+\frac {\log (x)}{3 x}+\frac {x \log (x)}{9 (3+x)}+\frac {\log (1+x)}{3 x}-\frac {\log (1+x)}{3 (3+x)}-\frac {\log (x)+\log (1+x)-\log (x (1+x))}{x (3+x)}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.13, size = 26, normalized size = 0.81 \begin {gather*} \frac {e^8+x^2 (3+x)+\log (x (1+x))}{x (3+x)} \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. \(73\) vs.
\(2(33)=66\).
time = 0.31, size = 74, normalized size = 2.31
method | result | size |
norman | \(\frac {x^{3}+{\mathrm e}^{8}-9 x +\ln \left (x^{2}+x \right )}{\left (3+x \right ) x}\) | \(27\) |
risch | \(\frac {\ln \left (x^{2}+x \right )}{\left (3+x \right ) x}+\frac {x^{3}+{\mathrm e}^{8}+3 x^{2}}{\left (3+x \right ) x}\) | \(37\) |
default | \(x -\frac {\ln \left (x \right )}{2}+\frac {{\mathrm e}^{8}}{3 x}-\frac {1}{3 x}-\frac {{\mathrm e}^{8}}{3 \left (3+x \right )}-\frac {\ln \left (x +1\right )}{2}+\frac {1+\frac {x}{3}+\frac {3 \ln \left (x^{2}+x \right ) x}{2}+\frac {x^{2} \ln \left (x^{2}+x \right )}{2}+\ln \left (x^{2}+x \right )}{x \left (3+x \right )}\) | \(74\) |
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. 162 vs.
\(2 (31) = 62\).
time = 0.31, size = 162, normalized size = 5.06 \begin {gather*} \frac {1}{36} \, {\left (\frac {6 \, {\left (x + 6\right )}}{x^{2} + 3 \, x} + 7 \, \log \left (x + 3\right ) - 27 \, \log \left (x + 1\right ) + 20 \, \log \left (x\right )\right )} e^{8} + \frac {5}{36} \, {\left (\frac {6}{x + 3} - 5 \, \log \left (x + 3\right ) + 9 \, \log \left (x + 1\right ) - 4 \, \log \left (x\right )\right )} e^{8} - \frac {1}{2} \, {\left (\frac {2}{x + 3} - \log \left (x + 3\right ) + \log \left (x + 1\right )\right )} e^{8} + x + \frac {9 \, {\left (x^{2} + 3 \, x + 2\right )} \log \left (x + 1\right ) - 2 \, {\left (2 \, x^{2} + 6 \, x - 9\right )} \log \left (x\right ) + 6 \, x + 18}{18 \, {\left (x^{2} + 3 \, x\right )}} - \frac {x + 6}{6 \, {\left (x^{2} + 3 \, x\right )}} - \frac {1}{6 \, {\left (x + 3\right )}} - \frac {1}{2} \, \log \left (x + 1\right ) + \frac {2}{9} \, \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 27, normalized size = 0.84 \begin {gather*} \frac {x^{3} + 3 \, x^{2} + e^{8} + \log \left (x^{2} + x\right )}{x^{2} + 3 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.12, size = 24, normalized size = 0.75 \begin {gather*} x + \frac {\log {\left (x^{2} + x \right )}}{x^{2} + 3 x} + \frac {e^{8}}{x^{2} + 3 x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 27, normalized size = 0.84 \begin {gather*} \frac {x^{3} + 3 \, x^{2} + e^{8} + \log \left (x^{2} + x\right )}{x^{2} + 3 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.34, size = 20, normalized size = 0.62 \begin {gather*} x+\frac {\ln \left (x^2+x\right )+{\mathrm {e}}^8}{x\,\left (x+3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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