Optimal. Leaf size=24 \[ \frac {4}{x}+2 x-\frac {x}{-x+(3+x) \log (4)} \]
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Rubi [B] time = 0.31, antiderivative size = 58, normalized size of antiderivative = 2.42, number of steps used = 5, number of rules used = 5, integrand size = 98, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {1680, 12, 1814, 21, 8} \begin {gather*} 2 x-\frac {12 (1-\log (4)) \log (4)-x \left (4+4 \log ^2(4)-5 \log (4)\right )}{x (1-\log (4)) (x (1-\log (4))-3 \log (4))} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 12
Rule 21
Rule 1680
Rule 1814
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\operatorname {Subst}\left (\int \frac {2 \left (16 x^4 (1-\log (4))^4-9 \log ^2(4) \left (8-10 \log (4)-\log ^2(4)\right )+24 x (1-\log (4)) \log (4) \left (4-11 \log (4)+4 \log ^2(4)\right )-8 x^2 (1-\log (4))^2 \left (4-5 \log (4)+13 \log ^2(4)\right )\right )}{\left (4 x^2 (-1+\log (4))^2-9 \log ^2(4)\right )^2} \, dx,x,x+\frac {-6 \log (4)+6 \log ^2(4)}{4 \left (1-2 \log (4)+\log ^2(4)\right )}\right )\\ &=2 \operatorname {Subst}\left (\int \frac {16 x^4 (1-\log (4))^4-9 \log ^2(4) \left (8-10 \log (4)-\log ^2(4)\right )+24 x (1-\log (4)) \log (4) \left (4-11 \log (4)+4 \log ^2(4)\right )-8 x^2 (1-\log (4))^2 \left (4-5 \log (4)+13 \log ^2(4)\right )}{\left (4 x^2 (-1+\log (4))^2-9 \log ^2(4)\right )^2} \, dx,x,x+\frac {-6 \log (4)+6 \log ^2(4)}{4 \left (1-2 \log (4)+\log ^2(4)\right )}\right )\\ &=-\frac {12 (1-\log (4)) \log (4)-x \left (4-5 \log (4)+4 \log ^2(4)\right )}{x (x (1-\log (4))-3 \log (4)) (1-\log (4))}+\frac {\operatorname {Subst}\left (\int \frac {72 x^2 (1-\log (4))^2 \log ^2(4)-162 \log ^4(4)}{4 x^2 (-1+\log (4))^2-9 \log ^2(4)} \, dx,x,x+\frac {-6 \log (4)+6 \log ^2(4)}{4 \left (1-2 \log (4)+\log ^2(4)\right )}\right )}{9 \log ^2(4)}\\ &=-\frac {12 (1-\log (4)) \log (4)-x \left (4-5 \log (4)+4 \log ^2(4)\right )}{x (x (1-\log (4))-3 \log (4)) (1-\log (4))}+2 \operatorname {Subst}\left (\int 1 \, dx,x,x+\frac {-6 \log (4)+6 \log ^2(4)}{4 \left (1-2 \log (4)+\log ^2(4)\right )}\right )\\ &=2 x-\frac {12 (1-\log (4)) \log (4)-x \left (4-5 \log (4)+4 \log ^2(4)\right )}{x (x (1-\log (4))-3 \log (4)) (1-\log (4))}\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 0.09, size = 97, normalized size = 4.04 \begin {gather*} 2 x+\frac {36 \log ^2(4)}{x \log ^2(64)}+\frac {36 \log ^4(4)-24 \log ^3(4) (3+\log (64))-\log (4) \log (64) (24+5 \log (64))+\log ^2(4) \left (36+48 \log (64)-14 \log ^2(64)\right )+2 \log ^2(64) \left (2+\log ^2(64)\right )}{(-1+\log (4)) \log ^2(64) (x (-1+\log (4))+\log (64))} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.87, size = 79, normalized size = 3.29 \begin {gather*} \frac {2 \, {\left (x^{3} + 4 \, {\left (x^{3} + 3 \, x^{2} + 2 \, x + 6\right )} \log \relax (2)^{2} - {\left (4 \, x^{3} + 6 \, x^{2} + 5 \, x + 12\right )} \log \relax (2) + 2 \, x\right )}}{4 \, {\left (x^{2} + 3 \, x\right )} \log \relax (2)^{2} + x^{2} - 2 \, {\left (2 \, x^{2} + 3 \, x\right )} \log \relax (2)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 87, normalized size = 3.62 \begin {gather*} \frac {2 \, {\left (4 \, x \log \relax (2)^{2} - 4 \, x \log \relax (2) + x\right )}}{4 \, \log \relax (2)^{2} - 4 \, \log \relax (2) + 1} + \frac {2 \, {\left (8 \, x \log \relax (2)^{2} - 5 \, x \log \relax (2) + 24 \, \log \relax (2)^{2} + 2 \, x - 12 \, \log \relax (2)\right )}}{{\left (2 \, x^{2} \log \relax (2) - x^{2} + 6 \, x \log \relax (2)\right )} {\left (2 \, \log \relax (2) - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 37, normalized size = 1.54
method | result | size |
default | \(2 x +\frac {6 \ln \relax (2)}{\left (2 \ln \relax (2)-1\right ) \left (2 x \ln \relax (2)+6 \ln \relax (2)-x \right )}+\frac {4}{x}\) | \(37\) |
risch | \(2 x +\frac {\frac {2 \left (8 \ln \relax (2)^{2}-5 \ln \relax (2)+2\right ) x}{2 \ln \relax (2)-1}+24 \ln \relax (2)}{x \left (2 x \ln \relax (2)+6 \ln \relax (2)-x \right )}\) | \(52\) |
norman | \(\frac {\frac {\left (28 \ln \relax (2)^{2}+5 \ln \relax (2)-2\right ) x^{2}}{3 \ln \relax (2)}+\left (4 \ln \relax (2)-2\right ) x^{3}+24 \ln \relax (2)}{x \left (2 x \ln \relax (2)+6 \ln \relax (2)-x \right )}\) | \(56\) |
gosper | \(\frac {12 x^{3} \ln \relax (2)^{2}+28 x^{2} \ln \relax (2)^{2}-6 x^{3} \ln \relax (2)+5 x^{2} \ln \relax (2)+72 \ln \relax (2)^{2}-2 x^{2}}{3 x \left (2 x \ln \relax (2)+6 \ln \relax (2)-x \right ) \ln \relax (2)}\) | \(69\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.38, size = 64, normalized size = 2.67 \begin {gather*} 2 \, x + \frac {2 \, {\left ({\left (8 \, \log \relax (2)^{2} - 5 \, \log \relax (2) + 2\right )} x + 24 \, \log \relax (2)^{2} - 12 \, \log \relax (2)\right )}}{{\left (4 \, \log \relax (2)^{2} - 4 \, \log \relax (2) + 1\right )} x^{2} + 6 \, {\left (2 \, \log \relax (2)^{2} - \log \relax (2)\right )} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.37, size = 303, normalized size = 12.62 \begin {gather*} \frac {16\,{\ln \relax (2)}^2-4\,\ln \left (16\right )+4}{x\,\left (4\,{\ln \relax (2)}^2-\ln \left (16\right )+1\right )}+\frac {x\,\left (8\,{\ln \relax (2)}^2-\ln \left (256\right )+2\right )}{4\,{\ln \relax (2)}^2-\ln \left (16\right )+1}-\frac {\mathrm {atanh}\left (\frac {\left (12\,\ln \relax (2)\,\ln \left (16\right )-12\,\ln \relax (2)+2\,x\,{\left (4\,{\ln \relax (2)}^2-\ln \left (16\right )+1\right )}^2-24\,{\ln \relax (2)}^2\,\ln \left (16\right )+24\,{\ln \relax (2)}^2-48\,{\ln \relax (2)}^3+96\,{\ln \relax (2)}^4\right )\,\left (5\,\ln \relax (2)-2\,\ln \left (16\right )-5\,\ln \relax (2)\,\ln \left (16\right )-44\,{\ln \relax (2)}^2\,\ln \left (16\right )+18\,{\ln \relax (2)}^2\,\ln \left (256\right )+20\,{\ln \relax (2)}^3+2\,{\ln \left (16\right )}^2\right )}{3\,\ln \relax (2)\,\sqrt {\ln \left (16\right )-4\,\ln \relax (2)}\,\left (4\,{\ln \relax (2)}^2-\ln \left (16\right )+1\right )\,\left (20\,\ln \relax (2)-8\,\ln \left (16\right )-20\,\ln \relax (2)\,\ln \left (16\right )-176\,{\ln \relax (2)}^2\,\ln \left (16\right )+72\,{\ln \relax (2)}^2\,\ln \left (256\right )+80\,{\ln \relax (2)}^3+8\,{\ln \left (16\right )}^2\right )}\right )\,\left (5\,\ln \relax (2)-2\,\ln \left (16\right )-5\,\ln \relax (2)\,\ln \left (16\right )-44\,{\ln \relax (2)}^2\,\ln \left (16\right )+18\,{\ln \relax (2)}^2\,\ln \left (256\right )+20\,{\ln \relax (2)}^3+2\,{\ln \left (16\right )}^2\right )}{3\,\ln \relax (2)\,\sqrt {\ln \left (16\right )-4\,\ln \relax (2)}\,\left (4\,{\ln \relax (2)}^2-\ln \left (16\right )+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.22, size = 60, normalized size = 2.50 \begin {gather*} 2 x + \frac {x \left (- 10 \log {\relax (2 )} + 4 + 16 \log {\relax (2 )}^{2}\right ) - 24 \log {\relax (2 )} + 48 \log {\relax (2 )}^{2}}{x^{2} \left (- 4 \log {\relax (2 )} + 1 + 4 \log {\relax (2 )}^{2}\right ) + x \left (- 6 \log {\relax (2 )} + 12 \log {\relax (2 )}^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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