Optimal. Leaf size=34 \[ \log \left (-5+\frac {\frac {3}{5+x \left (4-x^2\right )}+\log (x)}{4 (1+x) \log (x)}\right ) \]
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Rubi [F] time = 4.99, 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 {15+27 x+12 x^2-3 x^3-3 x^4+\left (27 x+24 x^2-9 x^3-12 x^4\right ) \log (x)+\left (25 x+40 x^2+16 x^3-10 x^4-8 x^5+x^7\right ) \log ^2(x)}{\left (-15 x-27 x^2-12 x^3+3 x^4+3 x^5\right ) \log (x)+\left (475 x+1735 x^2+2364 x^3+1234 x^4-222 x^5-512 x^6-141 x^7+39 x^8+20 x^9\right ) \log ^2(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 {3 \left (-5-9 x-4 x^2+x^3+x^4\right )+3 x \left (-9-8 x+3 x^2+4 x^3\right ) \log (x)-x \left (-5-4 x+x^3\right )^2 \log ^2(x)}{x \left (5+9 x+4 x^2-x^3-x^4\right ) \log (x) \left (3+\left (-95-176 x-80 x^2+19 x^3+20 x^4\right ) \log (x)\right )} \, dx\\ &=\int \left (\frac {1}{19+39 x+20 x^2}-\frac {1}{x \log (x)}-\frac {3 \left (-176-160 x+57 x^2+80 x^3\right )}{(19+20 x) \left (-5-4 x+x^3\right ) \left (3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)\right )}+\frac {-95-176 x-80 x^2+19 x^3+20 x^4}{x \left (3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)\right )}\right ) \, dx\\ &=-\left (3 \int \frac {-176-160 x+57 x^2+80 x^3}{(19+20 x) \left (-5-4 x+x^3\right ) \left (3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)\right )} \, dx\right )+\int \frac {1}{19+39 x+20 x^2} \, dx-\int \frac {1}{x \log (x)} \, dx+\int \frac {-95-176 x-80 x^2+19 x^3+20 x^4}{x \left (3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)\right )} \, dx\\ &=-\left (3 \int \left (\frac {20}{(19+20 x) \left (3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)\right )}+\frac {-4+3 x^2}{\left (-5-4 x+x^3\right ) \left (3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)\right )}\right ) \, dx\right )+20 \int \frac {1}{19+20 x} \, dx-20 \int \frac {1}{20+20 x} \, dx+\int \left (-\frac {176}{3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)}-\frac {95}{x \left (3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)\right )}-\frac {80 x}{3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)}+\frac {19 x^2}{3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)}+\frac {20 x^3}{3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)}\right ) \, dx-\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log (x)\right )\\ &=-\log (1+x)+\log (19+20 x)-\log (\log (x))-3 \int \frac {-4+3 x^2}{\left (-5-4 x+x^3\right ) \left (3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)\right )} \, dx+19 \int \frac {x^2}{3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)} \, dx+20 \int \frac {x^3}{3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)} \, dx-60 \int \frac {1}{(19+20 x) \left (3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)\right )} \, dx-80 \int \frac {x}{3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)} \, dx-95 \int \frac {1}{x \left (3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)\right )} \, dx-176 \int \frac {1}{3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)} \, dx\\ &=-\log (1+x)+\log (19+20 x)-\log (\log (x))-3 \int \left (-\frac {4}{\left (-5-4 x+x^3\right ) \left (3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)\right )}+\frac {3 x^2}{\left (-5-4 x+x^3\right ) \left (3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)\right )}\right ) \, dx+19 \int \frac {x^2}{3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)} \, dx+20 \int \frac {x^3}{3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)} \, dx-60 \int \frac {1}{(19+20 x) \left (3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)\right )} \, dx-80 \int \frac {x}{3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)} \, dx-95 \int \frac {1}{x \left (3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)\right )} \, dx-176 \int \frac {1}{3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)} \, dx\\ &=-\log (1+x)+\log (19+20 x)-\log (\log (x))-9 \int \frac {x^2}{\left (-5-4 x+x^3\right ) \left (3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)\right )} \, dx+12 \int \frac {1}{\left (-5-4 x+x^3\right ) \left (3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)\right )} \, dx+19 \int \frac {x^2}{3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)} \, dx+20 \int \frac {x^3}{3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)} \, dx-60 \int \frac {1}{(19+20 x) \left (3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)\right )} \, dx-80 \int \frac {x}{3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)} \, dx-95 \int \frac {1}{x \left (3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)\right )} \, dx-176 \int \frac {1}{3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 11.71, size = 70, normalized size = 2.06 \begin {gather*} -\log (1+x)+\log (19+20 x)-\log \left ((19+20 x) \left (5+4 x-x^3\right )\right )-\log (\log (x))+\log \left (3-95 \log (x)-176 x \log (x)-80 x^2 \log (x)+19 x^3 \log (x)+20 x^4 \log (x)\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.52, size = 67, normalized size = 1.97 \begin {gather*} \log \left (20 \, x + 19\right ) - \log \left (x + 1\right ) + \log \left (\frac {{\left (20 \, x^{4} + 19 \, x^{3} - 80 \, x^{2} - 176 \, x - 95\right )} \log \relax (x) + 3}{20 \, x^{4} + 19 \, x^{3} - 80 \, x^{2} - 176 \, x - 95}\right ) - \log \left (\log \relax (x)\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 58, normalized size = 1.71 \begin {gather*} \log \left (20 \, x^{4} \log \relax (x) + 19 \, x^{3} \log \relax (x) - 80 \, x^{2} \log \relax (x) - 176 \, x \log \relax (x) - 95 \, \log \relax (x) + 3\right ) - \log \left (x^{4} + x^{3} - 4 \, x^{2} - 9 \, x - 5\right ) - \log \left (\log \relax (x)\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 47, normalized size = 1.38
method | result | size |
risch | \(-\ln \left (x +1\right )+\ln \left (20 x +19\right )-\ln \left (\ln \relax (x )\right )+\ln \left (\ln \relax (x )+\frac {3}{20 x^{4}+19 x^{3}-80 x^{2}-176 x -95}\right )\) | \(47\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.40, size = 67, normalized size = 1.97 \begin {gather*} \log \left (20 \, x + 19\right ) - \log \left (x + 1\right ) + \log \left (\frac {{\left (20 \, x^{4} + 19 \, x^{3} - 80 \, x^{2} - 176 \, x - 95\right )} \log \relax (x) + 3}{20 \, x^{4} + 19 \, x^{3} - 80 \, x^{2} - 176 \, x - 95}\right ) - \log \left (\log \relax (x)\right ) \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 = -0.03 \begin {gather*} \int -\frac {27\,x+\ln \relax (x)\,\left (-12\,x^4-9\,x^3+24\,x^2+27\,x\right )+{\ln \relax (x)}^2\,\left (x^7-8\,x^5-10\,x^4+16\,x^3+40\,x^2+25\,x\right )+12\,x^2-3\,x^3-3\,x^4+15}{\ln \relax (x)\,\left (-3\,x^5-3\,x^4+12\,x^3+27\,x^2+15\,x\right )-{\ln \relax (x)}^2\,\left (20\,x^9+39\,x^8-141\,x^7-512\,x^6-222\,x^5+1234\,x^4+2364\,x^3+1735\,x^2+475\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: PolynomialError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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