Optimal. Leaf size=35 \[ \frac {4}{5 x \left (4+\frac {25 x^2}{4 \left (x-x^2\right )^2}\right ) (3+x+\log (\log (4)))} \]
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Rubi [B] time = 0.42, antiderivative size = 104, normalized size of antiderivative = 2.97, number of steps used = 7, number of rules used = 4, integrand size = 150, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.027, Rules used = {2074, 618, 204, 638} \begin {gather*} -\frac {80 (-16 x (5+\log (\log (4)))+119+32 \log (\log (4)))}{41 \left (16 x^2-32 x+41\right ) \left (281+16 \log ^2(\log (4))+128 \log (\log (4))\right )}-\frac {16 (4+\log (\log (4)))^2}{5 (3+\log (\log (4))) \left (281+16 \log ^2(\log (4))+128 \log (\log (4))\right ) (x+3+\log (\log (4)))}+\frac {16}{205 x (3+\log (\log (4)))} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 638
Rule 2074
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {16}{205 x^2 (3+\log (\log (4)))}-\frac {1280 (5+\log (\log (4)))}{41 \left (41-32 x+16 x^2\right ) \left (281+128 \log (\log (4))+16 \log ^2(\log (4))\right )}+\frac {16 (4+\log (\log (4)))^2}{5 (3+\log (\log (4))) (3+x+\log (\log (4)))^2 \left (281+128 \log (\log (4))+16 \log ^2(\log (4))\right )}+\frac {2560 (86+9 \log (\log (4))+x (39+16 \log (\log (4))))}{41 \left (41-32 x+16 x^2\right )^2 \left (281+128 \log (\log (4))+16 \log ^2(\log (4))\right )}\right ) \, dx\\ &=\frac {16}{205 x (3+\log (\log (4)))}-\frac {16 (4+\log (\log (4)))^2}{5 (3+\log (\log (4))) (3+x+\log (\log (4))) \left (281+128 \log (\log (4))+16 \log ^2(\log (4))\right )}+\frac {2560 \int \frac {86+9 \log (\log (4))+x (39+16 \log (\log (4)))}{\left (41-32 x+16 x^2\right )^2} \, dx}{41 \left (281+128 \log (\log (4))+16 \log ^2(\log (4))\right )}-\frac {(1280 (5+\log (\log (4)))) \int \frac {1}{41-32 x+16 x^2} \, dx}{41 \left (281+128 \log (\log (4))+16 \log ^2(\log (4))\right )}\\ &=\frac {16}{205 x (3+\log (\log (4)))}-\frac {16 (4+\log (\log (4)))^2}{5 (3+\log (\log (4))) (3+x+\log (\log (4))) \left (281+128 \log (\log (4))+16 \log ^2(\log (4))\right )}-\frac {80 (119+32 \log (\log (4))-16 x (5+\log (\log (4))))}{41 \left (41-32 x+16 x^2\right ) \left (281+128 \log (\log (4))+16 \log ^2(\log (4))\right )}+\frac {(1280 (5+\log (\log (4)))) \int \frac {1}{41-32 x+16 x^2} \, dx}{41 \left (281+128 \log (\log (4))+16 \log ^2(\log (4))\right )}+\frac {(2560 (5+\log (\log (4)))) \operatorname {Subst}\left (\int \frac {1}{-1600-x^2} \, dx,x,-32+32 x\right )}{41 \left (281+128 \log (\log (4))+16 \log ^2(\log (4))\right )}\\ &=\frac {16}{205 x (3+\log (\log (4)))}+\frac {64 \tan ^{-1}\left (\frac {4 (1-x)}{5}\right ) (5+\log (\log (4)))}{41 \left (281+128 \log (\log (4))+16 \log ^2(\log (4))\right )}-\frac {16 (4+\log (\log (4)))^2}{5 (3+\log (\log (4))) (3+x+\log (\log (4))) \left (281+128 \log (\log (4))+16 \log ^2(\log (4))\right )}-\frac {80 (119+32 \log (\log (4))-16 x (5+\log (\log (4))))}{41 \left (41-32 x+16 x^2\right ) \left (281+128 \log (\log (4))+16 \log ^2(\log (4))\right )}-\frac {(2560 (5+\log (\log (4)))) \operatorname {Subst}\left (\int \frac {1}{-1600-x^2} \, dx,x,-32+32 x\right )}{41 \left (281+128 \log (\log (4))+16 \log ^2(\log (4))\right )}\\ &=\frac {16}{205 x (3+\log (\log (4)))}-\frac {16 (4+\log (\log (4)))^2}{5 (3+\log (\log (4))) (3+x+\log (\log (4))) \left (281+128 \log (\log (4))+16 \log ^2(\log (4))\right )}-\frac {80 (119+32 \log (\log (4))-16 x (5+\log (\log (4))))}{41 \left (41-32 x+16 x^2\right ) \left (281+128 \log (\log (4))+16 \log ^2(\log (4))\right )}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.12, size = 32, normalized size = 0.91 \begin {gather*} \frac {16 (-1+x)^2}{5 x \left (41-32 x+16 x^2\right ) (3+x+\log (\log (4)))} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 51, normalized size = 1.46 \begin {gather*} \frac {16 \, {\left (x^{2} - 2 \, x + 1\right )}}{5 \, {\left (16 \, x^{4} + 16 \, x^{3} - 55 \, x^{2} + {\left (16 \, x^{3} - 32 \, x^{2} + 41 \, x\right )} \log \left (2 \, \log \relax (2)\right ) + 123 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 59, normalized size = 1.69 \begin {gather*} \frac {16 \, {\left (x^{2} - 2 \, x + 1\right )}}{5 \, {\left (16 \, x^{4} + 16 \, x^{3} \log \left (2 \, \log \relax (2)\right ) + 16 \, x^{3} - 32 \, x^{2} \log \left (2 \, \log \relax (2)\right ) - 55 \, x^{2} + 41 \, x \log \left (2 \, \log \relax (2)\right ) + 123 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.43, size = 37, normalized size = 1.06
method | result | size |
norman | \(\frac {\frac {16}{5}-\frac {32}{5} x +\frac {16}{5} x^{2}}{x \left (16 x^{2}-32 x +41\right ) \left (3+x +\ln \left (2 \ln \relax (2)\right )\right )}\) | \(37\) |
gosper | \(\frac {16 \left (x -1\right )^{2}}{5 x \left (16 x^{2} \ln \left (2 \ln \relax (2)\right )+16 x^{3}-32 x \ln \left (2 \ln \relax (2)\right )+16 x^{2}+41 \ln \left (2 \ln \relax (2)\right )-55 x +123\right )}\) | \(53\) |
risch | \(\frac {\frac {16}{5}-\frac {32}{5} x +\frac {16}{5} x^{2}}{\left (16 x^{2} \ln \relax (2)+16 x^{2} \ln \left (\ln \relax (2)\right )+16 x^{3}-32 x \ln \relax (2)-32 x \ln \left (\ln \relax (2)\right )+16 x^{2}+41 \ln \relax (2)+41 \ln \left (\ln \relax (2)\right )-55 x +123\right ) x}\) | \(68\) |
default | \(-\frac {1280 \left (\left (-\frac {5}{16}-\frac {\ln \left (2 \ln \relax (2)\right )}{16}\right ) x +\frac {119}{256}+\frac {\ln \left (2 \ln \relax (2)\right )}{8}\right )}{41 \left (16 \ln \left (2 \ln \relax (2)\right )^{2}+128 \ln \left (2 \ln \relax (2)\right )+281\right ) \left (x^{2}-2 x +\frac {41}{16}\right )}+\frac {16}{5 \left (41 \ln \left (2 \ln \relax (2)\right )+123\right ) x}-\frac {16 \left (\ln \left (2 \ln \relax (2)\right )^{2}+8 \ln \left (2 \ln \relax (2)\right )+16\right )}{5 \left (16 \ln \left (2 \ln \relax (2)\right )^{2}+128 \ln \left (2 \ln \relax (2)\right )+281\right ) \left (\ln \left (2 \ln \relax (2)\right )+3\right ) \left (3+x +\ln \left (2 \ln \relax (2)\right )\right )}\) | \(127\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 54, normalized size = 1.54 \begin {gather*} \frac {16 \, {\left (x^{2} - 2 \, x + 1\right )}}{5 \, {\left (16 \, x^{4} + 16 \, x^{3} {\left (\log \left (2 \, \log \relax (2)\right ) + 1\right )} - x^{2} {\left (32 \, \log \left (2 \, \log \relax (2)\right ) + 55\right )} + 41 \, x {\left (\log \left (2 \, \log \relax (2)\right ) + 3\right )}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.34, size = 205, normalized size = 5.86 \begin {gather*} \frac {\frac {\left (\ln \left ({\ln \relax (4)}^{1121190}\right )+738961\,{\ln \left (\ln \relax (4)\right )}^2+261056\,{\ln \left (\ln \relax (4)\right )}^3+52256\,{\ln \left (\ln \relax (4)\right )}^4+5632\,{\ln \left (\ln \relax (4)\right )}^5+256\,{\ln \left (\ln \relax (4)\right )}^6+710649\right )\,x^2}{5\,{\left (\ln \left (\ln \relax (4)\right )+3\right )}^2\,\left (\ln \left ({\ln \relax (4)}^{71936}\right )+25376\,{\ln \left (\ln \relax (4)\right )}^2+4096\,{\ln \left (\ln \relax (4)\right )}^3+256\,{\ln \left (\ln \relax (4)\right )}^4+78961\right )}-\frac {\left (\ln \left ({\ln \relax (4)}^{2242380}\right )+1477922\,{\ln \left (\ln \relax (4)\right )}^2+522112\,{\ln \left (\ln \relax (4)\right )}^3+104512\,{\ln \left (\ln \relax (4)\right )}^4+11264\,{\ln \left (\ln \relax (4)\right )}^5+512\,{\ln \left (\ln \relax (4)\right )}^6+1421298\right )\,x}{5\,{\left (\ln \left (\ln \relax (4)\right )+3\right )}^2\,\left (\ln \left ({\ln \relax (4)}^{71936}\right )+25376\,{\ln \left (\ln \relax (4)\right )}^2+4096\,{\ln \left (\ln \relax (4)\right )}^3+256\,{\ln \left (\ln \relax (4)\right )}^4+78961\right )}+\frac {1}{5}}{x^4+\left (\ln \left (\ln \relax (4)\right )+1\right )\,x^3+\left (-\ln \left ({\ln \relax (4)}^2\right )-\frac {55}{16}\right )\,x^2+\left (\ln \left ({\ln \relax (4)}^{41/16}\right )+\frac {123}{16}\right )\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 76.22, size = 66, normalized size = 1.89 \begin {gather*} - \frac {- 16 x^{2} + 32 x - 16}{80 x^{4} + x^{3} \left (80 \log {\left (\log {\relax (2 )} \right )} + 80 \log {\relax (2 )} + 80\right ) + x^{2} \left (-275 - 160 \log {\relax (2 )} - 160 \log {\left (\log {\relax (2 )} \right )}\right ) + x \left (205 \log {\left (\log {\relax (2 )} \right )} + 205 \log {\relax (2 )} + 615\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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