Integrand size = 150, antiderivative size = 35 \[ \int \frac {-1968+1760 x-560 x^2+1280 x^4-512 x^5+\left (-656+1024 x-1136 x^2+1024 x^3-256 x^4\right ) \log (\log (4))}{75645 x^2-67650 x^3+34805 x^4+10880 x^5-7520 x^6+2560 x^7+1280 x^8+\left (50430 x^2-61910 x^3+43840 x^4-7360 x^5-2560 x^6+2560 x^7\right ) \log (\log (4))+\left (8405 x^2-13120 x^3+11680 x^4-5120 x^5+1280 x^6\right ) \log ^2(\log (4))} \, dx=\frac {4}{5 x \left (4+\frac {25 x^2}{4 \left (x-x^2\right )^2}\right ) (3+x+\log (\log (4)))} \]
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Leaf count is larger than twice the leaf count of optimal. \(104\) vs. \(2(35)=70\).
Time = 0.21 (sec) , antiderivative size = 104, normalized size of antiderivative = 2.97, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.027, Rules used = {2099, 632, 210, 652} \[ \int \frac {-1968+1760 x-560 x^2+1280 x^4-512 x^5+\left (-656+1024 x-1136 x^2+1024 x^3-256 x^4\right ) \log (\log (4))}{75645 x^2-67650 x^3+34805 x^4+10880 x^5-7520 x^6+2560 x^7+1280 x^8+\left (50430 x^2-61910 x^3+43840 x^4-7360 x^5-2560 x^6+2560 x^7\right ) \log (\log (4))+\left (8405 x^2-13120 x^3+11680 x^4-5120 x^5+1280 x^6\right ) \log ^2(\log (4))} \, dx=-\frac {80 (-16 x (5+\log (\log (4)))+119+32 \log (\log (4)))}{41 \left (16 x^2-32 x+41\right ) \left (281+16 \log ^2(\log (4))+128 \log (\log (4))\right )}-\frac {16 (4+\log (\log (4)))^2}{5 (3+\log (\log (4))) \left (281+16 \log ^2(\log (4))+128 \log (\log (4))\right ) (x+3+\log (\log (4)))}+\frac {16}{205 x (3+\log (\log (4)))} \]
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Rule 210
Rule 632
Rule 652
Rule 2099
Rubi steps \begin{align*} \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)))) \text {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)))) \text {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{align*}
Time = 0.13 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.91 \[ \int \frac {-1968+1760 x-560 x^2+1280 x^4-512 x^5+\left (-656+1024 x-1136 x^2+1024 x^3-256 x^4\right ) \log (\log (4))}{75645 x^2-67650 x^3+34805 x^4+10880 x^5-7520 x^6+2560 x^7+1280 x^8+\left (50430 x^2-61910 x^3+43840 x^4-7360 x^5-2560 x^6+2560 x^7\right ) \log (\log (4))+\left (8405 x^2-13120 x^3+11680 x^4-5120 x^5+1280 x^6\right ) \log ^2(\log (4))} \, dx=\frac {16 (-1+x)^2}{5 x \left (41-32 x+16 x^2\right ) (3+x+\log (\log (4)))} \]
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Time = 1.40 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.06
method | result | size |
norman | \(\frac {\frac {16}{5}-\frac {32}{5} x +\frac {16}{5} x^{2}}{x \left (16 x^{2}-32 x +41\right ) \left (3+x +\ln \left (2 \ln \left (2\right )\right )\right )}\) | \(37\) |
gosper | \(\frac {16 \left (-1+x \right )^{2}}{5 x \left (16 x^{2} \ln \left (2 \ln \left (2\right )\right )+16 x^{3}-32 x \ln \left (2 \ln \left (2\right )\right )+16 x^{2}+41 \ln \left (2 \ln \left (2\right )\right )-55 x +123\right )}\) | \(53\) |
parallelrisch | \(\frac {256 x^{2}-512 x +256}{80 x \left (16 x^{2} \ln \left (2 \ln \left (2\right )\right )+16 x^{3}-32 x \ln \left (2 \ln \left (2\right )\right )+16 x^{2}+41 \ln \left (2 \ln \left (2\right )\right )-55 x +123\right )}\) | \(58\) |
risch | \(\frac {\frac {16}{5}-\frac {32}{5} x +\frac {16}{5} x^{2}}{\left (16 x^{2} \ln \left (2\right )+16 x^{2} \ln \left (\ln \left (2\right )\right )+16 x^{3}-32 x \ln \left (2\right )-32 x \ln \left (\ln \left (2\right )\right )+16 x^{2}+41 \ln \left (2\right )+41 \ln \left (\ln \left (2\right )\right )-55 x +123\right ) x}\) | \(68\) |
default | \(-\frac {1280 \left (\left (-\frac {5}{16}-\frac {\ln \left (2 \ln \left (2\right )\right )}{16}\right ) x +\frac {119}{256}+\frac {\ln \left (2 \ln \left (2\right )\right )}{8}\right )}{41 \left (16 \ln \left (2 \ln \left (2\right )\right )^{2}+128 \ln \left (2 \ln \left (2\right )\right )+281\right ) \left (x^{2}-2 x +\frac {41}{16}\right )}+\frac {16}{5 \left (41 \ln \left (2 \ln \left (2\right )\right )+123\right ) x}-\frac {16 \left (\ln \left (2 \ln \left (2\right )\right )^{2}+8 \ln \left (2 \ln \left (2\right )\right )+16\right )}{5 \left (16 \ln \left (2 \ln \left (2\right )\right )^{2}+128 \ln \left (2 \ln \left (2\right )\right )+281\right ) \left (\ln \left (2 \ln \left (2\right )\right )+3\right ) \left (3+x +\ln \left (2 \ln \left (2\right )\right )\right )}\) | \(127\) |
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Time = 0.25 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.46 \[ \int \frac {-1968+1760 x-560 x^2+1280 x^4-512 x^5+\left (-656+1024 x-1136 x^2+1024 x^3-256 x^4\right ) \log (\log (4))}{75645 x^2-67650 x^3+34805 x^4+10880 x^5-7520 x^6+2560 x^7+1280 x^8+\left (50430 x^2-61910 x^3+43840 x^4-7360 x^5-2560 x^6+2560 x^7\right ) \log (\log (4))+\left (8405 x^2-13120 x^3+11680 x^4-5120 x^5+1280 x^6\right ) \log ^2(\log (4))} \, dx=\frac {16 \, {\left (x^{2} - 2 \, x + 1\right )}}{5 \, {\left (16 \, x^{4} + 16 \, x^{3} - 55 \, x^{2} + {\left (16 \, x^{3} - 32 \, x^{2} + 41 \, x\right )} \log \left (2 \, \log \left (2\right )\right ) + 123 \, x\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (29) = 58\).
Time = 31.36 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.89 \[ \int \frac {-1968+1760 x-560 x^2+1280 x^4-512 x^5+\left (-656+1024 x-1136 x^2+1024 x^3-256 x^4\right ) \log (\log (4))}{75645 x^2-67650 x^3+34805 x^4+10880 x^5-7520 x^6+2560 x^7+1280 x^8+\left (50430 x^2-61910 x^3+43840 x^4-7360 x^5-2560 x^6+2560 x^7\right ) \log (\log (4))+\left (8405 x^2-13120 x^3+11680 x^4-5120 x^5+1280 x^6\right ) \log ^2(\log (4))} \, dx=- \frac {- 16 x^{2} + 32 x - 16}{80 x^{4} + x^{3} \cdot \left (80 \log {\left (\log {\left (2 \right )} \right )} + 80 \log {\left (2 \right )} + 80\right ) + x^{2} \left (-275 - 160 \log {\left (2 \right )} - 160 \log {\left (\log {\left (2 \right )} \right )}\right ) + x \left (205 \log {\left (\log {\left (2 \right )} \right )} + 205 \log {\left (2 \right )} + 615\right )} \]
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Time = 0.17 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.54 \[ \int \frac {-1968+1760 x-560 x^2+1280 x^4-512 x^5+\left (-656+1024 x-1136 x^2+1024 x^3-256 x^4\right ) \log (\log (4))}{75645 x^2-67650 x^3+34805 x^4+10880 x^5-7520 x^6+2560 x^7+1280 x^8+\left (50430 x^2-61910 x^3+43840 x^4-7360 x^5-2560 x^6+2560 x^7\right ) \log (\log (4))+\left (8405 x^2-13120 x^3+11680 x^4-5120 x^5+1280 x^6\right ) \log ^2(\log (4))} \, dx=\frac {16 \, {\left (x^{2} - 2 \, x + 1\right )}}{5 \, {\left (16 \, x^{4} + 16 \, x^{3} {\left (\log \left (2 \, \log \left (2\right )\right ) + 1\right )} - x^{2} {\left (32 \, \log \left (2 \, \log \left (2\right )\right ) + 55\right )} + 41 \, x {\left (\log \left (2 \, \log \left (2\right )\right ) + 3\right )}\right )}} \]
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Time = 0.26 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.69 \[ \int \frac {-1968+1760 x-560 x^2+1280 x^4-512 x^5+\left (-656+1024 x-1136 x^2+1024 x^3-256 x^4\right ) \log (\log (4))}{75645 x^2-67650 x^3+34805 x^4+10880 x^5-7520 x^6+2560 x^7+1280 x^8+\left (50430 x^2-61910 x^3+43840 x^4-7360 x^5-2560 x^6+2560 x^7\right ) \log (\log (4))+\left (8405 x^2-13120 x^3+11680 x^4-5120 x^5+1280 x^6\right ) \log ^2(\log (4))} \, dx=\frac {16 \, {\left (x^{2} - 2 \, x + 1\right )}}{5 \, {\left (16 \, x^{4} + 16 \, x^{3} \log \left (2 \, \log \left (2\right )\right ) + 16 \, x^{3} - 32 \, x^{2} \log \left (2 \, \log \left (2\right )\right ) - 55 \, x^{2} + 41 \, x \log \left (2 \, \log \left (2\right )\right ) + 123 \, x\right )}} \]
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Time = 14.60 (sec) , antiderivative size = 205, normalized size of antiderivative = 5.86 \[ \int \frac {-1968+1760 x-560 x^2+1280 x^4-512 x^5+\left (-656+1024 x-1136 x^2+1024 x^3-256 x^4\right ) \log (\log (4))}{75645 x^2-67650 x^3+34805 x^4+10880 x^5-7520 x^6+2560 x^7+1280 x^8+\left (50430 x^2-61910 x^3+43840 x^4-7360 x^5-2560 x^6+2560 x^7\right ) \log (\log (4))+\left (8405 x^2-13120 x^3+11680 x^4-5120 x^5+1280 x^6\right ) \log ^2(\log (4))} \, dx=\frac {\frac {\left (\ln \left ({\ln \left (4\right )}^{1121190}\right )+738961\,{\ln \left (\ln \left (4\right )\right )}^2+261056\,{\ln \left (\ln \left (4\right )\right )}^3+52256\,{\ln \left (\ln \left (4\right )\right )}^4+5632\,{\ln \left (\ln \left (4\right )\right )}^5+256\,{\ln \left (\ln \left (4\right )\right )}^6+710649\right )\,x^2}{5\,{\left (\ln \left (\ln \left (4\right )\right )+3\right )}^2\,\left (\ln \left ({\ln \left (4\right )}^{71936}\right )+25376\,{\ln \left (\ln \left (4\right )\right )}^2+4096\,{\ln \left (\ln \left (4\right )\right )}^3+256\,{\ln \left (\ln \left (4\right )\right )}^4+78961\right )}-\frac {\left (\ln \left ({\ln \left (4\right )}^{2242380}\right )+1477922\,{\ln \left (\ln \left (4\right )\right )}^2+522112\,{\ln \left (\ln \left (4\right )\right )}^3+104512\,{\ln \left (\ln \left (4\right )\right )}^4+11264\,{\ln \left (\ln \left (4\right )\right )}^5+512\,{\ln \left (\ln \left (4\right )\right )}^6+1421298\right )\,x}{5\,{\left (\ln \left (\ln \left (4\right )\right )+3\right )}^2\,\left (\ln \left ({\ln \left (4\right )}^{71936}\right )+25376\,{\ln \left (\ln \left (4\right )\right )}^2+4096\,{\ln \left (\ln \left (4\right )\right )}^3+256\,{\ln \left (\ln \left (4\right )\right )}^4+78961\right )}+\frac {1}{5}}{x^4+\left (\ln \left (\ln \left (4\right )\right )+1\right )\,x^3+\left (-\ln \left ({\ln \left (4\right )}^2\right )-\frac {55}{16}\right )\,x^2+\left (\ln \left ({\ln \left (4\right )}^{41/16}\right )+\frac {123}{16}\right )\,x} \]
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