Optimal. Leaf size=34 \[ \frac {-2+x}{x^2 \left (5+\frac {e^2 \left (-1+\frac {2-x}{3}\right )}{x}+x-\log (5)\right )^2} \]
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Rubi [B] time = 0.77, antiderivative size = 273, normalized size of antiderivative = 8.03, number of steps used = 10, number of rules used = 6, integrand size = 213, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {6, 2074, 614, 618, 206, 638} \begin {gather*} \frac {27 \left (e^4+9 \left (25+\log ^2(5)-14 \log (5)+\log (625)\right )-6 e^2 (3-\log (5))\right ) \left (6 x-e^2+15-3 \log (5)\right )}{\left (e^4-6 e^2 (3-\log (5))+9 (5-\log (5))^2\right )^2 \left (-3 x^2-x \left (15-e^2-3 \log (5)\right )+e^2\right )}-\frac {9 \left (-\left (x \left (e^4+9 \left (25+\log ^2(5)-14 \log (5)+\log (625)\right )-6 e^2 (3-\log (5))\right )\right )+2 e^4-3 e^2 (12-\log (625))+9 (5-\log (5)) (10-\log (25))\right )}{\left (12 e^2+\left (e^2-3 (5-\log (5))\right )^2\right ) \left (-3 x^2-x \left (15-e^2-3 \log (5)\right )+e^2\right )^2}-\frac {27 \left (6 x-e^2+15-3 \log (5)\right )}{\left (12 e^2+\left (e^2-3 (5-\log (5))\right )^2\right ) \left (-3 x^2-x \left (15-e^2-3 \log (5)\right )+e^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 206
Rule 614
Rule 618
Rule 638
Rule 2074
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-540+e^2 (45-9 x)-81 x+81 x^2+(108-27 x) \log (5)}{-2025 x^4-405 x^5-27 x^6+e^6 \left (1+3 x+3 x^2+x^3\right )+e^4 \left (-45 x-99 x^2-63 x^3-9 x^4\right )+e^2 \left (675 x^2+945 x^3+297 x^4+27 x^5\right )+\left (2025 x^3+810 x^4+81 x^5+e^4 \left (9 x+18 x^2+9 x^3\right )+e^2 \left (-270 x^2-324 x^3-54 x^4\right )\right ) \log (5)+\left (-405 x^3-81 x^4+e^2 \left (27 x^2+27 x^3\right )\right ) \log ^2(5)+x^3 \left (-3375+27 \log ^3(5)\right )} \, dx\\ &=\int \left (-\frac {27}{\left (e^2-3 x^2-x \left (15-e^2-3 \log (5)\right )\right )^2}+\frac {18 \left (-30+4 e^2-x \left (27-e^2-3 \log (5)\right )+3 \log (25)\right )}{\left (e^2-3 x^2-x \left (15-e^2-3 \log (5)\right )\right )^3}\right ) \, dx\\ &=18 \int \frac {-30+4 e^2-x \left (27-e^2-3 \log (5)\right )+3 \log (25)}{\left (e^2-3 x^2+x \left (-15+e^2+3 \log (5)\right )\right )^3} \, dx-27 \int \frac {1}{\left (e^2-3 x^2-x \left (15-e^2-3 \log (5)\right )\right )^2} \, dx\\ &=-\frac {27 \left (15-e^2+6 x-3 \log (5)\right )}{\left (12 e^2+\left (e^2-3 (5-\log (5))\right )^2\right ) \left (e^2-3 x^2-x \left (15-e^2-3 \log (5)\right )\right )}-\frac {9 \left (2 e^4+9 (5-\log (5)) (10-\log (25))-3 e^2 (12-\log (625))-x \left (e^4-6 e^2 (3-\log (5))+9 \left (25-14 \log (5)+\log ^2(5)+\log (625)\right )\right )\right )}{\left (12 e^2+\left (e^2-3 (5-\log (5))\right )^2\right ) \left (e^2-3 x^2-x \left (15-e^2-3 \log (5)\right )\right )^2}+27 \int \frac {1}{\left (e^2-3 x^2+x \left (-15+e^2+3 \log (5)\right )\right )^2} \, dx-\frac {162 \int \frac {1}{e^2-3 x^2+x \left (-15+e^2+3 \log (5)\right )} \, dx}{12 e^2+\left (e^2-3 (5-\log (5))\right )^2}\\ &=-\frac {9 \left (2 e^4+9 (5-\log (5)) (10-\log (25))-3 e^2 (12-\log (625))-x \left (e^4-6 e^2 (3-\log (5))+9 \left (25-14 \log (5)+\log ^2(5)+\log (625)\right )\right )\right )}{\left (12 e^2+\left (e^2-3 (5-\log (5))\right )^2\right ) \left (e^2-3 x^2-x \left (15-e^2-3 \log (5)\right )\right )^2}+\frac {162 \int \frac {1}{e^2-3 x^2+x \left (-15+e^2+3 \log (5)\right )} \, dx}{12 e^2+\left (e^2-3 (5-\log (5))\right )^2}+\frac {324 \operatorname {Subst}\left (\int \frac {1}{12 e^2-x^2+\left (e^2-3 (5-\log (5))\right )^2} \, dx,x,-15+e^2-6 x+3 \log (5)\right )}{12 e^2+\left (e^2-3 (5-\log (5))\right )^2}\\ &=\frac {324 \tanh ^{-1}\left (\frac {e^2-3 (5+2 x-\log (5))}{\sqrt {e^4-6 e^2 (3-\log (5))+9 (5-\log (5))^2}}\right )}{\left (e^4-6 e^2 (3-\log (5))+9 (5-\log (5))^2\right )^{3/2}}-\frac {9 \left (2 e^4+9 (5-\log (5)) (10-\log (25))-3 e^2 (12-\log (625))-x \left (e^4-6 e^2 (3-\log (5))+9 \left (25-14 \log (5)+\log ^2(5)+\log (625)\right )\right )\right )}{\left (12 e^2+\left (e^2-3 (5-\log (5))\right )^2\right ) \left (e^2-3 x^2-x \left (15-e^2-3 \log (5)\right )\right )^2}-\frac {324 \operatorname {Subst}\left (\int \frac {1}{12 e^2-x^2+\left (e^2-3 (5-\log (5))\right )^2} \, dx,x,-15+e^2-6 x+3 \log (5)\right )}{12 e^2+\left (e^2-3 (5-\log (5))\right )^2}\\ &=-\frac {9 \left (2 e^4+9 (5-\log (5)) (10-\log (25))-3 e^2 (12-\log (625))-x \left (e^4-6 e^2 (3-\log (5))+9 \left (25-14 \log (5)+\log ^2(5)+\log (625)\right )\right )\right )}{\left (12 e^2+\left (e^2-3 (5-\log (5))\right )^2\right ) \left (e^2-3 x^2-x \left (15-e^2-3 \log (5)\right )\right )^2}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.04, size = 27, normalized size = 0.79 \begin {gather*} -\frac {9 (2-x)}{\left (e^2 (1+x)-3 x (5+x-\log (5))\right )^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.88, size = 83, normalized size = 2.44 \begin {gather*} \frac {9 \, {\left (x - 2\right )}}{9 \, x^{4} + 9 \, x^{2} \log \relax (5)^{2} + 90 \, x^{3} + 225 \, x^{2} + {\left (x^{2} + 2 \, x + 1\right )} e^{4} - 6 \, {\left (x^{3} + 6 \, x^{2} + 5 \, x\right )} e^{2} - 6 \, {\left (3 \, x^{3} + 15 \, x^{2} - {\left (x^{2} + x\right )} e^{2}\right )} \log \relax (5)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.79, size = 29, normalized size = 0.85
method | result | size |
norman | \(\frac {9 x -18}{\left ({\mathrm e}^{2} x +3 x \ln \relax (5)-3 x^{2}+{\mathrm e}^{2}-15 x \right )^{2}}\) | \(29\) |
risch | \(\frac {9 x -18}{x^{2} {\mathrm e}^{4}+6 \,{\mathrm e}^{2} \ln \relax (5) x^{2}-6 x^{3} {\mathrm e}^{2}+9 x^{2} \ln \relax (5)^{2}-18 x^{3} \ln \relax (5)+9 x^{4}+2 x \,{\mathrm e}^{4}+6 x \,{\mathrm e}^{2} \ln \relax (5)-36 x^{2} {\mathrm e}^{2}-90 x^{2} \ln \relax (5)+90 x^{3}+{\mathrm e}^{4}-30 \,{\mathrm e}^{2} x +225 x^{2}}\) | \(96\) |
gosper | \(\frac {9 x -18}{x^{2} {\mathrm e}^{4}+6 \,{\mathrm e}^{2} \ln \relax (5) x^{2}-6 x^{3} {\mathrm e}^{2}+9 x^{2} \ln \relax (5)^{2}-18 x^{3} \ln \relax (5)+9 x^{4}+2 x \,{\mathrm e}^{4}+6 x \,{\mathrm e}^{2} \ln \relax (5)-36 x^{2} {\mathrm e}^{2}-90 x^{2} \ln \relax (5)+90 x^{3}+{\mathrm e}^{4}-30 \,{\mathrm e}^{2} x +225 x^{2}}\) | \(101\) |
default | \(-3 \left (\munderset {\textit {\_R} =\RootOf \left (27 \textit {\_Z}^{6}+\left (-27 \,{\mathrm e}^{2}-81 \ln \relax (5)+405\right ) \textit {\_Z}^{5}+\left (54 \,{\mathrm e}^{2} \ln \relax (5)+81 \ln \relax (5)^{2}-297 \,{\mathrm e}^{2}-810 \ln \relax (5)+9 \,{\mathrm e}^{4}+2025\right ) \textit {\_Z}^{4}+\left (-27 \,{\mathrm e}^{2} \ln \relax (5)^{2}-27 \ln \relax (5)^{3}+324 \,{\mathrm e}^{2} \ln \relax (5)+405 \ln \relax (5)^{2}-9 \,{\mathrm e}^{4} \ln \relax (5)-945 \,{\mathrm e}^{2}-2025 \ln \relax (5)+63 \,{\mathrm e}^{4}-{\mathrm e}^{6}+3375\right ) \textit {\_Z}^{3}+\left (-27 \,{\mathrm e}^{2} \ln \relax (5)^{2}+270 \,{\mathrm e}^{2} \ln \relax (5)-18 \,{\mathrm e}^{4} \ln \relax (5)-675 \,{\mathrm e}^{2}+99 \,{\mathrm e}^{4}-3 \,{\mathrm e}^{6}\right ) \textit {\_Z}^{2}+\left (-9 \,{\mathrm e}^{4} \ln \relax (5)+45 \,{\mathrm e}^{4}-3 \,{\mathrm e}^{6}\right ) \textit {\_Z} -{\mathrm e}^{6}\right )}{\sum }\frac {\left (-9 \textit {\_R}^{2}+\left ({\mathrm e}^{2}+3 \ln \relax (5)+9\right ) \textit {\_R} -5 \,{\mathrm e}^{2}-12 \ln \relax (5)+60\right ) \ln \left (x -\textit {\_R} \right )}{9 \,{\mathrm e}^{4} \ln \relax (5) \textit {\_R}^{2}-324 \,{\mathrm e}^{2} \ln \relax (5) \textit {\_R}^{2}+27 \,{\mathrm e}^{2} \ln \relax (5)^{2} \textit {\_R}^{2}-72 \,{\mathrm e}^{2} \ln \relax (5) \textit {\_R}^{3}+18 \,{\mathrm e}^{2} \ln \relax (5)^{2} \textit {\_R} -180 \textit {\_R} \,{\mathrm e}^{2} \ln \relax (5)+12 \textit {\_R} \,{\mathrm e}^{4} \ln \relax (5)+{\mathrm e}^{6}-15 \,{\mathrm e}^{4}+3 \,{\mathrm e}^{4} \ln \relax (5)-63 \textit {\_R}^{2} {\mathrm e}^{4}+2 \textit {\_R} \,{\mathrm e}^{6}-66 \textit {\_R} \,{\mathrm e}^{4}+450 \,{\mathrm e}^{2} \textit {\_R} -12 \textit {\_R}^{3} {\mathrm e}^{4}+945 \textit {\_R}^{2} {\mathrm e}^{2}-675 \textit {\_R}^{4}-2700 \textit {\_R}^{3}-3375 \textit {\_R}^{2}-108 \textit {\_R}^{3} \ln \relax (5)^{2}+\textit {\_R}^{2} {\mathrm e}^{6}+2025 \textit {\_R}^{2} \ln \relax (5)-405 \textit {\_R}^{2} \ln \relax (5)^{2}+396 \textit {\_R}^{3} {\mathrm e}^{2}+45 \textit {\_R}^{4} {\mathrm e}^{2}+1080 \textit {\_R}^{3} \ln \relax (5)+135 \textit {\_R}^{4} \ln \relax (5)+27 \textit {\_R}^{2} \ln \relax (5)^{3}-54 \textit {\_R}^{5}}\right )\) | \(401\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.49, size = 70, normalized size = 2.06 \begin {gather*} \frac {9 \, {\left (x - 2\right )}}{9 \, x^{4} - 6 \, x^{3} {\left (e^{2} + 3 \, \log \relax (5) - 15\right )} + {\left (6 \, {\left (e^{2} - 15\right )} \log \relax (5) + 9 \, \log \relax (5)^{2} + e^{4} - 36 \, e^{2} + 225\right )} x^{2} + 2 \, {\left (3 \, e^{2} \log \relax (5) + e^{4} - 15 \, e^{2}\right )} x + e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.92, size = 525, normalized size = 15.44 \begin {gather*} \frac {\frac {\left (2187\,\ln \left (125\right )\,\ln \left (625\right )-26244\,{\ln \relax (5)}^2\right )\,x^2}{774\,{\mathrm {e}}^4-8100\,{\mathrm {e}}^2-36\,{\mathrm {e}}^6+{\mathrm {e}}^8-40500\,\ln \relax (5)+\ln \left (5^{12\,{\mathrm {e}}^6-396\,{\mathrm {e}}^4}\right )+5940\,{\mathrm {e}}^2\,\ln \relax (5)-1404\,{\mathrm {e}}^2\,{\ln \relax (5)}^2+108\,{\mathrm {e}}^2\,{\ln \relax (5)}^3+54\,{\mathrm {e}}^4\,{\ln \relax (5)}^2+12150\,{\ln \relax (5)}^2-1620\,{\ln \relax (5)}^3+81\,{\ln \relax (5)}^4+50625}+\frac {9\,\left (774\,{\mathrm {e}}^4-8100\,{\mathrm {e}}^2-36\,{\mathrm {e}}^6+{\mathrm {e}}^8-24300\,\ln \relax (5)-4050\,\ln \left (625\right )+\ln \left (5^{3240\,{\mathrm {e}}^2-396\,{\mathrm {e}}^4+12\,{\mathrm {e}}^6-648\,{\ln \relax (5)}^2}\right )+2700\,{\mathrm {e}}^2\,\ln \relax (5)+1620\,\ln \relax (5)\,\ln \left (625\right )+\ln \left (625\right )\,\ln \left (\frac {1}{5^{108\,{\mathrm {e}}^2}}\right )-972\,{\mathrm {e}}^2\,{\ln \relax (5)}^2+108\,{\mathrm {e}}^2\,{\ln \relax (5)}^3+54\,{\mathrm {e}}^4\,{\ln \relax (5)}^2+5670\,{\ln \relax (5)}^2-972\,{\ln \relax (5)}^3+81\,{\ln \relax (5)}^4+50625\right )\,x}{774\,{\mathrm {e}}^4-8100\,{\mathrm {e}}^2-36\,{\mathrm {e}}^6+{\mathrm {e}}^8-40500\,\ln \relax (5)+\ln \left (5^{12\,{\mathrm {e}}^6-396\,{\mathrm {e}}^4}\right )+5940\,{\mathrm {e}}^2\,\ln \relax (5)-1404\,{\mathrm {e}}^2\,{\ln \relax (5)}^2+108\,{\mathrm {e}}^2\,{\ln \relax (5)}^3+54\,{\mathrm {e}}^4\,{\ln \relax (5)}^2+12150\,{\ln \relax (5)}^2-1620\,{\ln \relax (5)}^3+81\,{\ln \relax (5)}^4+50625}-\frac {\frac {9\,\ln \left (5^{2700\,{\mathrm {e}}^2-180\,{\mathrm {e}}^4+48\,{\mathrm {e}}^6+324\,{\mathrm {e}}^2\,{\ln \relax (5)}^2+324\,{\ln \relax (5)}^3}\right )}{2}-145800\,{\mathrm {e}}^2+13932\,{\mathrm {e}}^4-648\,{\mathrm {e}}^6+18\,{\mathrm {e}}^8-546750\,\ln \relax (5)-\frac {91125\,\ln \left (625\right )}{2}+94770\,{\mathrm {e}}^2\,\ln \relax (5)-6318\,{\mathrm {e}}^4\,\ln \relax (5)+\frac {54675\,\ln \relax (5)\,\ln \left (625\right )}{2}+\frac {9\,\ln \left (625\right )\,\ln \left (5^{27\,{\mathrm {e}}^4-540\,{\mathrm {e}}^2}\right )}{2}-15552\,{\mathrm {e}}^2\,{\ln \relax (5)}^2+486\,{\mathrm {e}}^2\,{\ln \relax (5)}^3+486\,{\mathrm {e}}^4\,{\ln \relax (5)}^2-\frac {10935\,{\ln \relax (5)}^2\,\ln \left (625\right )}{2}+109350\,{\ln \relax (5)}^2-7290\,{\ln \relax (5)}^3+911250}{{\left (\ln \left (\frac {{15625}^{{\mathrm {e}}^2}}{807793566946316088741610050849573099185363389551639556884765625}\right )-18\,{\mathrm {e}}^2+{\mathrm {e}}^4+9\,{\ln \relax (5)}^2+225\right )}^2}}{9\,x^4+\left (90-\ln \left (3814697265625\right )-6\,{\mathrm {e}}^2\right )\,x^3+\left (\ln \left (\frac {{15625}^{{\mathrm {e}}^2}}{807793566946316088741610050849573099185363389551639556884765625}\right )-36\,{\mathrm {e}}^2+{\mathrm {e}}^4+9\,{\ln \relax (5)}^2+225\right )\,x^2+\left (2\,{\mathrm {e}}^4-30\,{\mathrm {e}}^2+\ln \left (5^{6\,{\mathrm {e}}^2}\right )\right )\,x+{\mathrm {e}}^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 4.83, size = 83, normalized size = 2.44 \begin {gather*} - \frac {18 - 9 x}{9 x^{4} + x^{3} \left (- 6 e^{2} - 18 \log {\relax (5 )} + 90\right ) + x^{2} \left (- 36 e^{2} - 90 \log {\relax (5 )} + 9 \log {\relax (5 )}^{2} + e^{4} + 6 e^{2} \log {\relax (5 )} + 225\right ) + x \left (- 30 e^{2} + 6 e^{2} \log {\relax (5 )} + 2 e^{4}\right ) + e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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