Optimal. Leaf size=25 \[ \frac {x^2}{x+\frac {x^2}{(-6+x-\log (4+3 x))^2}} \]
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Rubi [F]
time = 3.20, 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 {5184+432 x-1740 x^2+518 x^3-62 x^4+3 x^5+\left (3456+864 x-1010 x^2+194 x^3-12 x^4\right ) \log (4+3 x)+\left (864+360 x-192 x^2+18 x^3\right ) \log ^2(4+3 x)+\left (96+56 x-12 x^2\right ) \log ^3(4+3 x)+(4+3 x) \log ^4(4+3 x)}{5184+720 x-1604 x^2+491 x^3-62 x^4+3 x^5+\left (3456+960 x-952 x^2+188 x^3-12 x^4\right ) \log (4+3 x)+\left (864+368 x-186 x^2+18 x^3\right ) \log ^2(4+3 x)+\left (96+56 x-12 x^2\right ) \log ^3(4+3 x)+(4+3 x) \log ^4(4+3 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 {5184+432 x-1740 x^2+518 x^3-62 x^4+3 x^5-2 \left (-1728-432 x+505 x^2-97 x^3+6 x^4\right ) \log (4+3 x)+6 (-6+x)^2 (4+3 x) \log ^2(4+3 x)+\left (96+56 x-12 x^2\right ) \log ^3(4+3 x)+(4+3 x) \log ^4(4+3 x)}{(4+3 x) \left (36-11 x+x^2-2 (-6+x) \log (4+3 x)+\log ^2(4+3 x)\right )^2} \, dx\\ &=\int \left (1+\frac {x^2 \left (-8-31 x+6 x^2-2 \log (4+3 x)-6 x \log (4+3 x)\right )}{(4+3 x) \left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )^2}-\frac {2 x}{36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)}\right ) \, dx\\ &=x-2 \int \frac {x}{36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)} \, dx+\int \frac {x^2 \left (-8-31 x+6 x^2-2 \log (4+3 x)-6 x \log (4+3 x)\right )}{(4+3 x) \left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )^2} \, dx\\ &=x-2 \int \frac {x}{36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)} \, dx+\int \left (-\frac {4 \left (-8-31 x+6 x^2-2 \log (4+3 x)-6 x \log (4+3 x)\right )}{9 \left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )^2}+\frac {x \left (-8-31 x+6 x^2-2 \log (4+3 x)-6 x \log (4+3 x)\right )}{3 \left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )^2}+\frac {16 \left (-8-31 x+6 x^2-2 \log (4+3 x)-6 x \log (4+3 x)\right )}{9 (4+3 x) \left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )^2}\right ) \, dx\\ &=x+\frac {1}{3} \int \frac {x \left (-8-31 x+6 x^2-2 \log (4+3 x)-6 x \log (4+3 x)\right )}{\left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )^2} \, dx-\frac {4}{9} \int \frac {-8-31 x+6 x^2-2 \log (4+3 x)-6 x \log (4+3 x)}{\left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )^2} \, dx+\frac {16}{9} \int \frac {-8-31 x+6 x^2-2 \log (4+3 x)-6 x \log (4+3 x)}{(4+3 x) \left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )^2} \, dx-2 \int \frac {x}{36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)} \, dx\\ &=x-\frac {16}{9 \left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )}+\frac {1}{3} \int \left (-\frac {8 x}{\left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )^2}-\frac {31 x^2}{\left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )^2}+\frac {6 x^3}{\left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )^2}-\frac {2 x \log (4+3 x)}{\left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )^2}-\frac {6 x^2 \log (4+3 x)}{\left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )^2}\right ) \, dx-\frac {4}{9} \int \left (-\frac {8}{\left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )^2}-\frac {31 x}{\left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )^2}+\frac {6 x^2}{\left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )^2}-\frac {2 \log (4+3 x)}{\left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )^2}-\frac {6 x \log (4+3 x)}{\left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )^2}\right ) \, dx-2 \int \frac {x}{36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)} \, dx\\ &=x-\frac {16}{9 \left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )}-\frac {2}{3} \int \frac {x \log (4+3 x)}{\left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )^2} \, dx+\frac {8}{9} \int \frac {\log (4+3 x)}{\left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )^2} \, dx+2 \int \frac {x^3}{\left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )^2} \, dx-2 \int \frac {x^2 \log (4+3 x)}{\left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )^2} \, dx-2 \int \frac {x}{36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)} \, dx-\frac {8}{3} \int \frac {x}{\left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )^2} \, dx-\frac {8}{3} \int \frac {x^2}{\left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )^2} \, dx+\frac {8}{3} \int \frac {x \log (4+3 x)}{\left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )^2} \, dx+\frac {32}{9} \int \frac {1}{\left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )^2} \, dx-\frac {31}{3} \int \frac {x^2}{\left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )^2} \, dx+\frac {124}{9} \int \frac {x}{\left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.10, size = 44, normalized size = 1.76 \begin {gather*} \frac {x (6-x+\log (4+3 x))^2}{36-11 x+x^2-2 (-6+x) \log (4+3 x)+\log ^2(4+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. \(116\) vs.
\(2(25)=50\).
time = 3.32, size = 117, normalized size = 4.68
method | result | size |
risch | \(x -\frac {x^{2}}{\ln \left (4+3 x \right )^{2}-2 \ln \left (4+3 x \right ) x +x^{2}+12 \ln \left (4+3 x \right )-11 x +36}\) | \(43\) |
derivativedivides | \(\frac {\left (4+3 x \right )^{3}+8712-1218 x -22 \left (4+3 x \right )^{2}+198 \ln \left (4+3 x \right )^{2}+2904 \ln \left (4+3 x \right )-6 \ln \left (4+3 x \right ) \left (4+3 x \right )^{2}+9 \ln \left (4+3 x \right )^{2} \left (4+3 x \right )}{3 \left (4+3 x \right )^{2}-18 \ln \left (4+3 x \right ) \left (4+3 x \right )+27 \ln \left (4+3 x \right )^{2}+924-369 x +396 \ln \left (4+3 x \right )}\) | \(117\) |
default | \(\frac {\left (4+3 x \right )^{3}+8712-1218 x -22 \left (4+3 x \right )^{2}+198 \ln \left (4+3 x \right )^{2}+2904 \ln \left (4+3 x \right )-6 \ln \left (4+3 x \right ) \left (4+3 x \right )^{2}+9 \ln \left (4+3 x \right )^{2} \left (4+3 x \right )}{3 \left (4+3 x \right )^{2}-18 \ln \left (4+3 x \right ) \left (4+3 x \right )+27 \ln \left (4+3 x \right )^{2}+924-369 x +396 \ln \left (4+3 x \right )}\) | \(117\) |
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. 67 vs.
\(2 (25) = 50\).
time = 0.33, size = 67, normalized size = 2.68 \begin {gather*} \frac {x^{3} + x \log \left (3 \, x + 4\right )^{2} - 12 \, x^{2} - 2 \, {\left (x^{2} - 6 \, x\right )} \log \left (3 \, x + 4\right ) + 36 \, x}{x^{2} - 2 \, {\left (x - 6\right )} \log \left (3 \, x + 4\right ) + \log \left (3 \, x + 4\right )^{2} - 11 \, x + 36} \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. 67 vs.
\(2 (25) = 50\).
time = 0.42, size = 67, normalized size = 2.68 \begin {gather*} \frac {x^{3} + x \log \left (3 \, x + 4\right )^{2} - 12 \, x^{2} - 2 \, {\left (x^{2} - 6 \, x\right )} \log \left (3 \, x + 4\right ) + 36 \, x}{x^{2} - 2 \, {\left (x - 6\right )} \log \left (3 \, x + 4\right ) + \log \left (3 \, x + 4\right )^{2} - 11 \, x + 36} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.11, size = 32, normalized size = 1.28 \begin {gather*} - \frac {x^{2}}{x^{2} - 11 x + \left (12 - 2 x\right ) \log {\left (3 x + 4 \right )} + \log {\left (3 x + 4 \right )}^{2} + 36} + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.49, size = 42, normalized size = 1.68 \begin {gather*} x - \frac {x^{2}}{x^{2} - 2 \, x \log \left (3 \, x + 4\right ) + \log \left (3 \, x + 4\right )^{2} - 11 \, x + 12 \, \log \left (3 \, x + 4\right ) + 36} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {432\,x+{\ln \left (3\,x+4\right )}^2\,\left (18\,x^3-192\,x^2+360\,x+864\right )+{\ln \left (3\,x+4\right )}^4\,\left (3\,x+4\right )+{\ln \left (3\,x+4\right )}^3\,\left (-12\,x^2+56\,x+96\right )-1740\,x^2+518\,x^3-62\,x^4+3\,x^5+\ln \left (3\,x+4\right )\,\left (-12\,x^4+194\,x^3-1010\,x^2+864\,x+3456\right )+5184}{720\,x+{\ln \left (3\,x+4\right )}^2\,\left (18\,x^3-186\,x^2+368\,x+864\right )+{\ln \left (3\,x+4\right )}^4\,\left (3\,x+4\right )+{\ln \left (3\,x+4\right )}^3\,\left (-12\,x^2+56\,x+96\right )-1604\,x^2+491\,x^3-62\,x^4+3\,x^5+\ln \left (3\,x+4\right )\,\left (-12\,x^4+188\,x^3-952\,x^2+960\,x+3456\right )+5184} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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