Optimal. Leaf size=34 \[ 1-x^2+\frac {\left (-e^5+x-\log \left (\frac {1}{x^2}\right )\right ) \left (x+\log \left (\frac {3+x}{2}\right )\right )}{x} \]
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Rubi [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 0.68, antiderivative size = 172, normalized size of antiderivative = 5.06, number of steps
used = 19, number of rules used = 13, integrand size = 73, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.178, Rules used = {6, 1607,
6874, 1634, 2379, 2438, 2442, 36, 31, 29, 2423, 2338, 2439} \begin {gather*} \frac {2}{3} \text {PolyLog}\left (2,-\frac {3}{x}\right )+\frac {2}{3} \text {PolyLog}\left (2,-\frac {x}{3}\right )-x^2+\frac {1}{3} \log \left (\frac {3}{x}+1\right ) \log \left (\frac {1}{x^2}\right )+\frac {1}{3} \left (\log \left (\frac {1}{x^2}\right )+e^5+2\right ) \log (x)-\frac {1}{3} \left (\log \left (\frac {1}{x^2}\right )+e^5+2\right ) \log (x+3)-\frac {\log \left (\frac {x}{2}+\frac {3}{2}\right ) \left (\log \left (\frac {1}{x^2}\right )+e^5+2\right )}{x}+x+\frac {\log ^2(x)}{3}-\frac {2}{3} \log (3) \log (x)+\frac {1}{3} \left (6-e^5\right ) \log (x)-\frac {2 \log (x)}{3}+\frac {1}{3} \left (3+e^5\right ) \log (x+3)+\frac {2}{3} \log (x+3)+\frac {2 \log \left (\frac {x}{2}+\frac {3}{2}\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 29
Rule 31
Rule 36
Rule 1607
Rule 1634
Rule 2338
Rule 2379
Rule 2423
Rule 2438
Rule 2439
Rule 2442
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\left (6-e^5\right ) x+6 x^2-5 x^3-2 x^4-x \log \left (\frac {1}{x^2}\right )+\left (6+2 x+e^5 (3+x)+(3+x) \log \left (\frac {1}{x^2}\right )\right ) \log \left (\frac {3+x}{2}\right )}{3 x^2+x^3} \, dx\\ &=\int \frac {\left (6-e^5\right ) x+6 x^2-5 x^3-2 x^4-x \log \left (\frac {1}{x^2}\right )+\left (6+2 x+e^5 (3+x)+(3+x) \log \left (\frac {1}{x^2}\right )\right ) \log \left (\frac {3+x}{2}\right )}{x^2 (3+x)} \, dx\\ &=\int \left (\frac {6 \left (1-\frac {e^5}{6}\right )+6 x-5 x^2-2 x^3-\log \left (\frac {1}{x^2}\right )}{x (3+x)}+\frac {\log \left (\frac {3}{2}+\frac {x}{2}\right ) \left (2 \left (1+\frac {e^5}{2}\right )+\log \left (\frac {1}{x^2}\right )\right )}{x^2}\right ) \, dx\\ &=\int \frac {6 \left (1-\frac {e^5}{6}\right )+6 x-5 x^2-2 x^3-\log \left (\frac {1}{x^2}\right )}{x (3+x)} \, dx+\int \frac {\log \left (\frac {3}{2}+\frac {x}{2}\right ) \left (2 \left (1+\frac {e^5}{2}\right )+\log \left (\frac {1}{x^2}\right )\right )}{x^2} \, dx\\ &=-\frac {\log \left (\frac {3}{2}+\frac {x}{2}\right ) \left (2+e^5+\log \left (\frac {1}{x^2}\right )\right )}{x}+\frac {1}{3} \left (2+e^5+\log \left (\frac {1}{x^2}\right )\right ) \log (x)-\frac {1}{3} \left (2+e^5+\log \left (\frac {1}{x^2}\right )\right ) \log (3+x)+2 \int \left (-\frac {\log \left (\frac {3}{2}+\frac {x}{2}\right )}{x^2}+\frac {\log (x)}{3 x}-\frac {\log (3+x)}{3 x}\right ) \, dx+\int \left (\frac {6-e^5+6 x-5 x^2-2 x^3}{x (3+x)}-\frac {\log \left (\frac {1}{x^2}\right )}{x (3+x)}\right ) \, dx\\ &=-\frac {\log \left (\frac {3}{2}+\frac {x}{2}\right ) \left (2+e^5+\log \left (\frac {1}{x^2}\right )\right )}{x}+\frac {1}{3} \left (2+e^5+\log \left (\frac {1}{x^2}\right )\right ) \log (x)-\frac {1}{3} \left (2+e^5+\log \left (\frac {1}{x^2}\right )\right ) \log (3+x)+\frac {2}{3} \int \frac {\log (x)}{x} \, dx-\frac {2}{3} \int \frac {\log (3+x)}{x} \, dx-2 \int \frac {\log \left (\frac {3}{2}+\frac {x}{2}\right )}{x^2} \, dx+\int \frac {6-e^5+6 x-5 x^2-2 x^3}{x (3+x)} \, dx-\int \frac {\log \left (\frac {1}{x^2}\right )}{x (3+x)} \, dx\\ &=\frac {2 \log \left (\frac {3}{2}+\frac {x}{2}\right )}{x}-\frac {\log \left (\frac {3}{2}+\frac {x}{2}\right ) \left (2+e^5+\log \left (\frac {1}{x^2}\right )\right )}{x}-\frac {2}{3} \log (3) \log (x)+\frac {1}{3} \left (2+e^5+\log \left (\frac {1}{x^2}\right )\right ) \log (x)+\frac {\log ^2(x)}{3}-\frac {1}{3} \left (2+e^5+\log \left (\frac {1}{x^2}\right )\right ) \log (3+x)-\frac {1}{3} \int \frac {\log \left (\frac {1}{x^2}\right )}{x} \, dx+\frac {1}{3} \int \frac {\log \left (\frac {1}{x^2}\right )}{3+x} \, dx-\frac {2}{3} \int \frac {\log \left (1+\frac {x}{3}\right )}{x} \, dx-\int \frac {1}{\left (\frac {3}{2}+\frac {x}{2}\right ) x} \, dx+\int \left (1+\frac {6-e^5}{3 x}-2 x+\frac {3+e^5}{3 (3+x)}\right ) \, dx\\ &=x-x^2+\frac {2 \log \left (\frac {3}{2}+\frac {x}{2}\right )}{x}+\frac {1}{3} \log \left (1+\frac {x}{3}\right ) \log \left (\frac {1}{x^2}\right )+\frac {1}{12} \log ^2\left (\frac {1}{x^2}\right )-\frac {\log \left (\frac {3}{2}+\frac {x}{2}\right ) \left (2+e^5+\log \left (\frac {1}{x^2}\right )\right )}{x}+\frac {1}{3} \left (6-e^5\right ) \log (x)-\frac {2}{3} \log (3) \log (x)+\frac {1}{3} \left (2+e^5+\log \left (\frac {1}{x^2}\right )\right ) \log (x)+\frac {\log ^2(x)}{3}+\frac {1}{3} \left (3+e^5\right ) \log (3+x)-\frac {1}{3} \left (2+e^5+\log \left (\frac {1}{x^2}\right )\right ) \log (3+x)+\frac {2 \text {Li}_2\left (-\frac {x}{3}\right )}{3}+\frac {1}{3} \int \frac {1}{\frac {3}{2}+\frac {x}{2}} \, dx-\frac {2}{3} \int \frac {1}{x} \, dx+\frac {2}{3} \int \frac {\log \left (1+\frac {x}{3}\right )}{x} \, dx\\ &=x-x^2+\frac {2 \log \left (\frac {3}{2}+\frac {x}{2}\right )}{x}+\frac {1}{3} \log \left (1+\frac {x}{3}\right ) \log \left (\frac {1}{x^2}\right )+\frac {1}{12} \log ^2\left (\frac {1}{x^2}\right )-\frac {\log \left (\frac {3}{2}+\frac {x}{2}\right ) \left (2+e^5+\log \left (\frac {1}{x^2}\right )\right )}{x}-\frac {2 \log (x)}{3}+\frac {1}{3} \left (6-e^5\right ) \log (x)-\frac {2}{3} \log (3) \log (x)+\frac {1}{3} \left (2+e^5+\log \left (\frac {1}{x^2}\right )\right ) \log (x)+\frac {\log ^2(x)}{3}+\frac {2}{3} \log (3+x)+\frac {1}{3} \left (3+e^5\right ) \log (3+x)-\frac {1}{3} \left (2+e^5+\log \left (\frac {1}{x^2}\right )\right ) \log (3+x)\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.15, size = 36, normalized size = 1.06 \begin {gather*} x-x^2+2 \log (x)-\frac {\left (e^5+\log \left (\frac {1}{x^2}\right )\right ) \log \left (\frac {3+x}{2}\right )}{x}+\log (3+x) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.32, size = 85, normalized size = 2.50
method | result | size |
risch | \(-\frac {\left (i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x \right )^{2}-2 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+i \pi \mathrm {csgn}\left (i x^{2}\right )^{3}+2 \,{\mathrm e}^{5}-4 \ln \left (x \right )\right ) \ln \left (\frac {3}{2}+\frac {x}{2}\right )}{2 x}+2 \ln \left (x \right )+\ln \left (3+x \right )-x^{2}+x\) | \(85\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 46, normalized size = 1.35 \begin {gather*} -\frac {x^{3} - x^{2} - e^{5} \log \left (2\right ) - {\left (x - e^{5} + 2 \, \log \left (x\right )\right )} \log \left (x + 3\right ) - 2 \, {\left (x - \log \left (2\right )\right )} \log \left (x\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 40, normalized size = 1.18 \begin {gather*} -\frac {x^{3} - x^{2} - {\left (x - e^{5} - \log \left (\frac {1}{x^{2}}\right )\right )} \log \left (\frac {1}{2} \, x + \frac {3}{2}\right ) + x \log \left (\frac {1}{x^{2}}\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.35, size = 36, normalized size = 1.06 \begin {gather*} - x^{2} + x + 2 \log {\left (x \right )} + \log {\left (x + 3 \right )} + \frac {\left (- \log {\left (\frac {1}{x^{2}} \right )} - e^{5}\right ) \log {\left (\frac {x}{2} + \frac {3}{2} \right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 52, normalized size = 1.53 \begin {gather*} -\frac {x^{3} - x^{2} + \log \left (2\right ) \log \left (x^{2}\right ) - x \log \left (x + 3\right ) - \log \left (x^{2}\right ) \log \left (x + 3\right ) - 2 \, x \log \left (x\right ) + e^{5} \log \left (\frac {1}{2} \, x + \frac {3}{2}\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.32, size = 43, normalized size = 1.26 \begin {gather*} x+\ln \left (x+3\right )+2\,\ln \left (x\right )-x^2-\frac {\ln \left (\frac {1}{x^2}\right )\,\ln \left (\frac {x}{2}+\frac {3}{2}\right )}{x}-\frac {{\mathrm {e}}^5\,\ln \left (\frac {x}{2}+\frac {3}{2}\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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