Optimal. Leaf size=30 \[ \left (5+\log \left (\frac {x}{3 \left (-\frac {1}{5}+x\right )^2}-\log \left (e^3 \left (4+x^2\right )\right )\right )\right )^4 \]
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Rubi [F] time = 25.94, 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 {50000+247000 x+57500 x^2-162500 x^3+375000 x^4+\left (30000+148200 x+34500 x^2-97500 x^3+225000 x^4\right ) \log \left (\frac {25 x+\left (-3+30 x-75 x^2\right ) \log \left (e^3 \left (4+x^2\right )\right )}{3-30 x+75 x^2}\right )+\left (6000+29640 x+6900 x^2-19500 x^3+45000 x^4\right ) \log ^2\left (\frac {25 x+\left (-3+30 x-75 x^2\right ) \log \left (e^3 \left (4+x^2\right )\right )}{3-30 x+75 x^2}\right )+\left (400+1976 x+460 x^2-1300 x^3+3000 x^4\right ) \log ^3\left (\frac {25 x+\left (-3+30 x-75 x^2\right ) \log \left (e^3 \left (4+x^2\right )\right )}{3-30 x+75 x^2}\right )}{100 x-500 x^2+25 x^3-125 x^4+\left (-12+180 x-903 x^2+1545 x^3-225 x^4+375 x^5\right ) \log \left (e^3 \left (4+x^2\right )\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 \left (-100-494 x-115 x^2+325 x^3-750 x^4\right ) \left (5+\log \left (-\frac {9-115 x+225 x^2+3 (1-5 x)^2 \log \left (4+x^2\right )}{3 (1-5 x)^2}\right )\right )^3}{\left (4-20 x+x^2-5 x^3\right ) \left (9-115 x+225 x^2+3 (1-5 x)^2 \log \left (4+x^2\right )\right )} \, dx\\ &=4 \int \frac {\left (-100-494 x-115 x^2+325 x^3-750 x^4\right ) \left (5+\log \left (-\frac {9-115 x+225 x^2+3 (1-5 x)^2 \log \left (4+x^2\right )}{3 (1-5 x)^2}\right )\right )^3}{\left (4-20 x+x^2-5 x^3\right ) \left (9-115 x+225 x^2+3 (1-5 x)^2 \log \left (4+x^2\right )\right )} \, dx\\ &=4 \int \left (\frac {125 \left (100+494 x+115 x^2-325 x^3+750 x^4\right )}{(-1+5 x) \left (4+x^2\right ) \left (9-115 x+225 x^2+3 \log \left (4+x^2\right )-30 x \log \left (4+x^2\right )+75 x^2 \log \left (4+x^2\right )\right )}+\frac {75 \left (100+494 x+115 x^2-325 x^3+750 x^4\right ) \log \left (-\frac {9-115 x+225 x^2+3 (1-5 x)^2 \log \left (4+x^2\right )}{3 (1-5 x)^2}\right )}{(-1+5 x) \left (4+x^2\right ) \left (9-115 x+225 x^2+3 \log \left (4+x^2\right )-30 x \log \left (4+x^2\right )+75 x^2 \log \left (4+x^2\right )\right )}+\frac {15 \left (100+494 x+115 x^2-325 x^3+750 x^4\right ) \log ^2\left (-\frac {9-115 x+225 x^2+3 (1-5 x)^2 \log \left (4+x^2\right )}{3 (1-5 x)^2}\right )}{(-1+5 x) \left (4+x^2\right ) \left (9-115 x+225 x^2+3 \log \left (4+x^2\right )-30 x \log \left (4+x^2\right )+75 x^2 \log \left (4+x^2\right )\right )}+\frac {\left (100+494 x+115 x^2-325 x^3+750 x^4\right ) \log ^3\left (-\frac {9-115 x+225 x^2+3 (1-5 x)^2 \log \left (4+x^2\right )}{3 (1-5 x)^2}\right )}{(-1+5 x) \left (4+x^2\right ) \left (9-115 x+225 x^2+3 \log \left (4+x^2\right )-30 x \log \left (4+x^2\right )+75 x^2 \log \left (4+x^2\right )\right )}\right ) \, dx\\ &=4 \int \frac {\left (100+494 x+115 x^2-325 x^3+750 x^4\right ) \log ^3\left (-\frac {9-115 x+225 x^2+3 (1-5 x)^2 \log \left (4+x^2\right )}{3 (1-5 x)^2}\right )}{(-1+5 x) \left (4+x^2\right ) \left (9-115 x+225 x^2+3 \log \left (4+x^2\right )-30 x \log \left (4+x^2\right )+75 x^2 \log \left (4+x^2\right )\right )} \, dx+60 \int \frac {\left (100+494 x+115 x^2-325 x^3+750 x^4\right ) \log ^2\left (-\frac {9-115 x+225 x^2+3 (1-5 x)^2 \log \left (4+x^2\right )}{3 (1-5 x)^2}\right )}{(-1+5 x) \left (4+x^2\right ) \left (9-115 x+225 x^2+3 \log \left (4+x^2\right )-30 x \log \left (4+x^2\right )+75 x^2 \log \left (4+x^2\right )\right )} \, dx+300 \int \frac {\left (100+494 x+115 x^2-325 x^3+750 x^4\right ) \log \left (-\frac {9-115 x+225 x^2+3 (1-5 x)^2 \log \left (4+x^2\right )}{3 (1-5 x)^2}\right )}{(-1+5 x) \left (4+x^2\right ) \left (9-115 x+225 x^2+3 \log \left (4+x^2\right )-30 x \log \left (4+x^2\right )+75 x^2 \log \left (4+x^2\right )\right )} \, dx+500 \int \frac {100+494 x+115 x^2-325 x^3+750 x^4}{(-1+5 x) \left (4+x^2\right ) \left (9-115 x+225 x^2+3 \log \left (4+x^2\right )-30 x \log \left (4+x^2\right )+75 x^2 \log \left (4+x^2\right )\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [B] time = 0.09, size = 169, normalized size = 5.63 \begin {gather*} 4 \left (125 \log \left (-\frac {9-115 x+225 x^2+3 (1-5 x)^2 \log \left (4+x^2\right )}{3 (1-5 x)^2}\right )+\frac {75}{2} \log ^2\left (-\frac {9-115 x+225 x^2+3 (1-5 x)^2 \log \left (4+x^2\right )}{3 (1-5 x)^2}\right )+5 \log ^3\left (-\frac {9-115 x+225 x^2+3 (1-5 x)^2 \log \left (4+x^2\right )}{3 (1-5 x)^2}\right )+\frac {1}{4} \log ^4\left (-\frac {9-115 x+225 x^2+3 (1-5 x)^2 \log \left (4+x^2\right )}{3 (1-5 x)^2}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.58, size = 173, normalized size = 5.77 \begin {gather*} \log \left (-\frac {3 \, {\left (25 \, x^{2} - 10 \, x + 1\right )} \log \left ({\left (x^{2} + 4\right )} e^{3}\right ) - 25 \, x}{3 \, {\left (25 \, x^{2} - 10 \, x + 1\right )}}\right )^{4} + 20 \, \log \left (-\frac {3 \, {\left (25 \, x^{2} - 10 \, x + 1\right )} \log \left ({\left (x^{2} + 4\right )} e^{3}\right ) - 25 \, x}{3 \, {\left (25 \, x^{2} - 10 \, x + 1\right )}}\right )^{3} + 150 \, \log \left (-\frac {3 \, {\left (25 \, x^{2} - 10 \, x + 1\right )} \log \left ({\left (x^{2} + 4\right )} e^{3}\right ) - 25 \, x}{3 \, {\left (25 \, x^{2} - 10 \, x + 1\right )}}\right )^{2} + 500 \, \log \left (-\frac {3 \, {\left (25 \, x^{2} - 10 \, x + 1\right )} \log \left ({\left (x^{2} + 4\right )} e^{3}\right ) - 25 \, x}{3 \, {\left (25 \, x^{2} - 10 \, x + 1\right )}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {4 \, {\left (93750 \, x^{4} + {\left (750 \, x^{4} - 325 \, x^{3} + 115 \, x^{2} + 494 \, x + 100\right )} \log \left (-\frac {3 \, {\left (25 \, x^{2} - 10 \, x + 1\right )} \log \left ({\left (x^{2} + 4\right )} e^{3}\right ) - 25 \, x}{3 \, {\left (25 \, x^{2} - 10 \, x + 1\right )}}\right )^{3} - 40625 \, x^{3} + 15 \, {\left (750 \, x^{4} - 325 \, x^{3} + 115 \, x^{2} + 494 \, x + 100\right )} \log \left (-\frac {3 \, {\left (25 \, x^{2} - 10 \, x + 1\right )} \log \left ({\left (x^{2} + 4\right )} e^{3}\right ) - 25 \, x}{3 \, {\left (25 \, x^{2} - 10 \, x + 1\right )}}\right )^{2} + 14375 \, x^{2} + 75 \, {\left (750 \, x^{4} - 325 \, x^{3} + 115 \, x^{2} + 494 \, x + 100\right )} \log \left (-\frac {3 \, {\left (25 \, x^{2} - 10 \, x + 1\right )} \log \left ({\left (x^{2} + 4\right )} e^{3}\right ) - 25 \, x}{3 \, {\left (25 \, x^{2} - 10 \, x + 1\right )}}\right ) + 61750 \, x + 12500\right )}}{125 \, x^{4} - 25 \, x^{3} + 500 \, x^{2} - 3 \, {\left (125 \, x^{5} - 75 \, x^{4} + 515 \, x^{3} - 301 \, x^{2} + 60 \, x - 4\right )} \log \left ({\left (x^{2} + 4\right )} e^{3}\right ) - 100 \, x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[\int \frac {\left (3000 x^{4}-1300 x^{3}+460 x^{2}+1976 x +400\right ) \ln \left (\frac {\left (-75 x^{2}+30 x -3\right ) \ln \left (\left (x^{2}+4\right ) {\mathrm e}^{3}\right )+25 x}{75 x^{2}-30 x +3}\right )^{3}+\left (45000 x^{4}-19500 x^{3}+6900 x^{2}+29640 x +6000\right ) \ln \left (\frac {\left (-75 x^{2}+30 x -3\right ) \ln \left (\left (x^{2}+4\right ) {\mathrm e}^{3}\right )+25 x}{75 x^{2}-30 x +3}\right )^{2}+\left (225000 x^{4}-97500 x^{3}+34500 x^{2}+148200 x +30000\right ) \ln \left (\frac {\left (-75 x^{2}+30 x -3\right ) \ln \left (\left (x^{2}+4\right ) {\mathrm e}^{3}\right )+25 x}{75 x^{2}-30 x +3}\right )+375000 x^{4}-162500 x^{3}+57500 x^{2}+247000 x +50000}{\left (375 x^{5}-225 x^{4}+1545 x^{3}-903 x^{2}+180 x -12\right ) \ln \left (\left (x^{2}+4\right ) {\mathrm e}^{3}\right )-125 x^{4}+25 x^{3}-500 x^{2}+100 x}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -4 \, \int \frac {93750 \, x^{4} + {\left (750 \, x^{4} - 325 \, x^{3} + 115 \, x^{2} + 494 \, x + 100\right )} \log \left (-\frac {3 \, {\left (25 \, x^{2} - 10 \, x + 1\right )} \log \left ({\left (x^{2} + 4\right )} e^{3}\right ) - 25 \, x}{3 \, {\left (25 \, x^{2} - 10 \, x + 1\right )}}\right )^{3} - 40625 \, x^{3} + 15 \, {\left (750 \, x^{4} - 325 \, x^{3} + 115 \, x^{2} + 494 \, x + 100\right )} \log \left (-\frac {3 \, {\left (25 \, x^{2} - 10 \, x + 1\right )} \log \left ({\left (x^{2} + 4\right )} e^{3}\right ) - 25 \, x}{3 \, {\left (25 \, x^{2} - 10 \, x + 1\right )}}\right )^{2} + 14375 \, x^{2} + 75 \, {\left (750 \, x^{4} - 325 \, x^{3} + 115 \, x^{2} + 494 \, x + 100\right )} \log \left (-\frac {3 \, {\left (25 \, x^{2} - 10 \, x + 1\right )} \log \left ({\left (x^{2} + 4\right )} e^{3}\right ) - 25 \, x}{3 \, {\left (25 \, x^{2} - 10 \, x + 1\right )}}\right ) + 61750 \, x + 12500}{125 \, x^{4} - 25 \, x^{3} + 500 \, x^{2} - 3 \, {\left (125 \, x^{5} - 75 \, x^{4} + 515 \, x^{3} - 301 \, x^{2} + 60 \, x - 4\right )} \log \left ({\left (x^{2} + 4\right )} e^{3}\right ) - 100 \, x}\,{d x} \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 {247000\,x+\ln \left (\frac {25\,x-\ln \left ({\mathrm {e}}^3\,\left (x^2+4\right )\right )\,\left (75\,x^2-30\,x+3\right )}{75\,x^2-30\,x+3}\right )\,\left (225000\,x^4-97500\,x^3+34500\,x^2+148200\,x+30000\right )+{\ln \left (\frac {25\,x-\ln \left ({\mathrm {e}}^3\,\left (x^2+4\right )\right )\,\left (75\,x^2-30\,x+3\right )}{75\,x^2-30\,x+3}\right )}^3\,\left (3000\,x^4-1300\,x^3+460\,x^2+1976\,x+400\right )+{\ln \left (\frac {25\,x-\ln \left ({\mathrm {e}}^3\,\left (x^2+4\right )\right )\,\left (75\,x^2-30\,x+3\right )}{75\,x^2-30\,x+3}\right )}^2\,\left (45000\,x^4-19500\,x^3+6900\,x^2+29640\,x+6000\right )+57500\,x^2-162500\,x^3+375000\,x^4+50000}{100\,x-500\,x^2+25\,x^3-125\,x^4+\ln \left ({\mathrm {e}}^3\,\left (x^2+4\right )\right )\,\left (375\,x^5-225\,x^4+1545\,x^3-903\,x^2+180\,x-12\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.49, size = 141, normalized size = 4.70 \begin {gather*} \log {\left (\frac {25 x + \left (- 75 x^{2} + 30 x - 3\right ) \log {\left (\left (x^{2} + 4\right ) e^{3} \right )}}{75 x^{2} - 30 x + 3} \right )}^{4} + 20 \log {\left (\frac {25 x + \left (- 75 x^{2} + 30 x - 3\right ) \log {\left (\left (x^{2} + 4\right ) e^{3} \right )}}{75 x^{2} - 30 x + 3} \right )}^{3} + 150 \log {\left (\frac {25 x + \left (- 75 x^{2} + 30 x - 3\right ) \log {\left (\left (x^{2} + 4\right ) e^{3} \right )}}{75 x^{2} - 30 x + 3} \right )}^{2} + 500 \log {\left (- \frac {25 x}{75 x^{2} - 30 x + 3} + \log {\left (\left (x^{2} + 4\right ) e^{3} \right )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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