Integrand size = 56, antiderivative size = 18 \[ \int \frac {-4 x^5+48 x^5 \log ^2(4)+12 x^5 \log (x)}{320 \log ^6(4)+240 \log ^4(4) \log (x)+60 \log ^2(4) \log ^2(x)+5 \log ^3(x)} \, dx=\frac {2 x^6}{5 \left (4 \log ^2(4)+\log (x)\right )^2} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.43 (sec) , antiderivative size = 227, normalized size of antiderivative = 12.61, number of steps used = 16, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {6, 2641, 6820, 12, 2343, 2346, 2209, 2413, 6617} \[ \int \frac {-4 x^5+48 x^5 \log ^2(4)+12 x^5 \log (x)}{320 \log ^6(4)+240 \log ^4(4) \log (x)+60 \log ^2(4) \log ^2(x)+5 \log ^3(x)} \, dx=\frac {36}{5} e^{-24 \log ^2(4)} \operatorname {ExpIntegralEi}\left (6 \left (\log (x)+4 \log ^2(4)\right )\right )-\frac {72}{5} e^{-24 \log ^2(4)} \left (-3 \log (x)+1-12 \log ^2(4)\right ) \operatorname {ExpIntegralEi}\left (6 \left (\log (x)+4 \log ^2(4)\right )\right )-\frac {432}{5} e^{-24 \log ^2(4)} \left (\log (x)+4 \log ^2(4)\right ) \operatorname {ExpIntegralEi}\left (6 \left (\log (x)+4 \log ^2(4)\right )\right )+\frac {36}{5} e^{-24 \log ^2(4)} \left (6 \log (x)+1+24 \log ^2(4)\right ) \operatorname {ExpIntegralEi}\left (6 \left (\log (x)+4 \log ^2(4)\right )\right )+\frac {72 x^6}{5}-\frac {6 x^6 \left (6 \log (x)+1+24 \log ^2(4)\right )}{5 \left (\log (x)+4 \log ^2(4)\right )}+\frac {12 x^6 \left (-3 \log (x)+1-12 \log ^2(4)\right )}{5 \left (\log (x)+4 \log ^2(4)\right )}+\frac {2 x^6 \left (-3 \log (x)+1-12 \log ^2(4)\right )}{5 \left (\log (x)+4 \log ^2(4)\right )^2} \]
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Rule 6
Rule 12
Rule 2209
Rule 2343
Rule 2346
Rule 2413
Rule 2641
Rule 6617
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^5 \left (-4+48 \log ^2(4)\right )+12 x^5 \log (x)}{320 \log ^6(4)+240 \log ^4(4) \log (x)+60 \log ^2(4) \log ^2(x)+5 \log ^3(x)} \, dx \\ & = \int \frac {x^5 \left (-4+48 \log ^2(4)+12 \log (x)\right )}{320 \log ^6(4)+240 \log ^4(4) \log (x)+60 \log ^2(4) \log ^2(x)+5 \log ^3(x)} \, dx \\ & = \int \frac {4 x^5 \left (-1+12 \log ^2(4)+3 \log (x)\right )}{5 \left (4 \log ^2(4)+\log (x)\right )^3} \, dx \\ & = \frac {4}{5} \int \frac {x^5 \left (-1+12 \log ^2(4)+3 \log (x)\right )}{\left (4 \log ^2(4)+\log (x)\right )^3} \, dx \\ & = -\frac {72}{5} e^{-24 \log ^2(4)} \operatorname {ExpIntegralEi}\left (6 \left (4 \log ^2(4)+\log (x)\right )\right ) \left (1-12 \log ^2(4)-3 \log (x)\right )+\frac {2 x^6 \left (1-12 \log ^2(4)-3 \log (x)\right )}{5 \left (4 \log ^2(4)+\log (x)\right )^2}+\frac {12 x^6 \left (1-12 \log ^2(4)-3 \log (x)\right )}{5 \left (4 \log ^2(4)+\log (x)\right )}-\frac {12}{5} \int \left (\frac {18 e^{-24 \log ^2(4)} \operatorname {ExpIntegralEi}\left (6 \left (4 \log ^2(4)+\log (x)\right )\right )}{x}-\frac {x^5 \left (1+24 \log ^2(4)+6 \log (x)\right )}{2 \left (4 \log ^2(4)+\log (x)\right )^2}\right ) \, dx \\ & = -\frac {72}{5} e^{-24 \log ^2(4)} \operatorname {ExpIntegralEi}\left (6 \left (4 \log ^2(4)+\log (x)\right )\right ) \left (1-12 \log ^2(4)-3 \log (x)\right )+\frac {2 x^6 \left (1-12 \log ^2(4)-3 \log (x)\right )}{5 \left (4 \log ^2(4)+\log (x)\right )^2}+\frac {12 x^6 \left (1-12 \log ^2(4)-3 \log (x)\right )}{5 \left (4 \log ^2(4)+\log (x)\right )}+\frac {6}{5} \int \frac {x^5 \left (1+24 \log ^2(4)+6 \log (x)\right )}{\left (4 \log ^2(4)+\log (x)\right )^2} \, dx-\frac {1}{5} \left (216 e^{-24 \log ^2(4)}\right ) \int \frac {\operatorname {ExpIntegralEi}\left (6 \left (4 \log ^2(4)+\log (x)\right )\right )}{x} \, dx \\ & = -\frac {72}{5} e^{-24 \log ^2(4)} \operatorname {ExpIntegralEi}\left (6 \left (4 \log ^2(4)+\log (x)\right )\right ) \left (1-12 \log ^2(4)-3 \log (x)\right )+\frac {2 x^6 \left (1-12 \log ^2(4)-3 \log (x)\right )}{5 \left (4 \log ^2(4)+\log (x)\right )^2}+\frac {12 x^6 \left (1-12 \log ^2(4)-3 \log (x)\right )}{5 \left (4 \log ^2(4)+\log (x)\right )}+\frac {36}{5} e^{-24 \log ^2(4)} \operatorname {ExpIntegralEi}\left (6 \left (4 \log ^2(4)+\log (x)\right )\right ) \left (1+24 \log ^2(4)+6 \log (x)\right )-\frac {6 x^6 \left (1+24 \log ^2(4)+6 \log (x)\right )}{5 \left (4 \log ^2(4)+\log (x)\right )}-\frac {36}{5} \int \left (\frac {6 e^{-24 \log ^2(4)} \operatorname {ExpIntegralEi}\left (6 \left (4 \log ^2(4)+\log (x)\right )\right )}{x}-\frac {x^5}{4 \log ^2(4)+\log (x)}\right ) \, dx-\frac {1}{5} \left (216 e^{-24 \log ^2(4)}\right ) \text {Subst}\left (\int \operatorname {ExpIntegralEi}\left (6 \left (x+4 \log ^2(4)\right )\right ) \, dx,x,\log (x)\right ) \\ & = -\frac {72}{5} e^{-24 \log ^2(4)} \operatorname {ExpIntegralEi}\left (6 \left (4 \log ^2(4)+\log (x)\right )\right ) \left (1-12 \log ^2(4)-3 \log (x)\right )+\frac {2 x^6 \left (1-12 \log ^2(4)-3 \log (x)\right )}{5 \left (4 \log ^2(4)+\log (x)\right )^2}+\frac {12 x^6 \left (1-12 \log ^2(4)-3 \log (x)\right )}{5 \left (4 \log ^2(4)+\log (x)\right )}+\frac {36}{5} e^{-24 \log ^2(4)} \operatorname {ExpIntegralEi}\left (6 \left (4 \log ^2(4)+\log (x)\right )\right ) \left (1+24 \log ^2(4)+6 \log (x)\right )-\frac {6 x^6 \left (1+24 \log ^2(4)+6 \log (x)\right )}{5 \left (4 \log ^2(4)+\log (x)\right )}+\frac {36}{5} \int \frac {x^5}{4 \log ^2(4)+\log (x)} \, dx-\frac {1}{5} \left (36 e^{-24 \log ^2(4)}\right ) \text {Subst}\left (\int \operatorname {ExpIntegralEi}(x) \, dx,x,24 \log ^2(4)+6 \log (x)\right )-\frac {1}{5} \left (216 e^{-24 \log ^2(4)}\right ) \int \frac {\operatorname {ExpIntegralEi}\left (6 \left (4 \log ^2(4)+\log (x)\right )\right )}{x} \, dx \\ & = \frac {36 x^6}{5}-\frac {72}{5} e^{-24 \log ^2(4)} \operatorname {ExpIntegralEi}\left (6 \left (4 \log ^2(4)+\log (x)\right )\right ) \left (1-12 \log ^2(4)-3 \log (x)\right )+\frac {2 x^6 \left (1-12 \log ^2(4)-3 \log (x)\right )}{5 \left (4 \log ^2(4)+\log (x)\right )^2}+\frac {12 x^6 \left (1-12 \log ^2(4)-3 \log (x)\right )}{5 \left (4 \log ^2(4)+\log (x)\right )}-\frac {216}{5} e^{-24 \log ^2(4)} \operatorname {ExpIntegralEi}\left (24 \log ^2(4)+6 \log (x)\right ) \left (4 \log ^2(4)+\log (x)\right )+\frac {36}{5} e^{-24 \log ^2(4)} \operatorname {ExpIntegralEi}\left (6 \left (4 \log ^2(4)+\log (x)\right )\right ) \left (1+24 \log ^2(4)+6 \log (x)\right )-\frac {6 x^6 \left (1+24 \log ^2(4)+6 \log (x)\right )}{5 \left (4 \log ^2(4)+\log (x)\right )}+\frac {36}{5} \text {Subst}\left (\int \frac {e^{6 x}}{x+4 \log ^2(4)} \, dx,x,\log (x)\right )-\frac {1}{5} \left (216 e^{-24 \log ^2(4)}\right ) \text {Subst}\left (\int \operatorname {ExpIntegralEi}\left (6 \left (x+4 \log ^2(4)\right )\right ) \, dx,x,\log (x)\right ) \\ & = \frac {36 x^6}{5}+\frac {36}{5} e^{-24 \log ^2(4)} \operatorname {ExpIntegralEi}\left (6 \left (4 \log ^2(4)+\log (x)\right )\right )-\frac {72}{5} e^{-24 \log ^2(4)} \operatorname {ExpIntegralEi}\left (6 \left (4 \log ^2(4)+\log (x)\right )\right ) \left (1-12 \log ^2(4)-3 \log (x)\right )+\frac {2 x^6 \left (1-12 \log ^2(4)-3 \log (x)\right )}{5 \left (4 \log ^2(4)+\log (x)\right )^2}+\frac {12 x^6 \left (1-12 \log ^2(4)-3 \log (x)\right )}{5 \left (4 \log ^2(4)+\log (x)\right )}-\frac {216}{5} e^{-24 \log ^2(4)} \operatorname {ExpIntegralEi}\left (24 \log ^2(4)+6 \log (x)\right ) \left (4 \log ^2(4)+\log (x)\right )+\frac {36}{5} e^{-24 \log ^2(4)} \operatorname {ExpIntegralEi}\left (6 \left (4 \log ^2(4)+\log (x)\right )\right ) \left (1+24 \log ^2(4)+6 \log (x)\right )-\frac {6 x^6 \left (1+24 \log ^2(4)+6 \log (x)\right )}{5 \left (4 \log ^2(4)+\log (x)\right )}-\frac {1}{5} \left (36 e^{-24 \log ^2(4)}\right ) \text {Subst}\left (\int \operatorname {ExpIntegralEi}(x) \, dx,x,24 \log ^2(4)+6 \log (x)\right ) \\ & = \frac {72 x^6}{5}+\frac {36}{5} e^{-24 \log ^2(4)} \operatorname {ExpIntegralEi}\left (6 \left (4 \log ^2(4)+\log (x)\right )\right )-\frac {72}{5} e^{-24 \log ^2(4)} \operatorname {ExpIntegralEi}\left (6 \left (4 \log ^2(4)+\log (x)\right )\right ) \left (1-12 \log ^2(4)-3 \log (x)\right )+\frac {2 x^6 \left (1-12 \log ^2(4)-3 \log (x)\right )}{5 \left (4 \log ^2(4)+\log (x)\right )^2}+\frac {12 x^6 \left (1-12 \log ^2(4)-3 \log (x)\right )}{5 \left (4 \log ^2(4)+\log (x)\right )}-\frac {432}{5} e^{-24 \log ^2(4)} \operatorname {ExpIntegralEi}\left (24 \log ^2(4)+6 \log (x)\right ) \left (4 \log ^2(4)+\log (x)\right )+\frac {36}{5} e^{-24 \log ^2(4)} \operatorname {ExpIntegralEi}\left (6 \left (4 \log ^2(4)+\log (x)\right )\right ) \left (1+24 \log ^2(4)+6 \log (x)\right )-\frac {6 x^6 \left (1+24 \log ^2(4)+6 \log (x)\right )}{5 \left (4 \log ^2(4)+\log (x)\right )} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {-4 x^5+48 x^5 \log ^2(4)+12 x^5 \log (x)}{320 \log ^6(4)+240 \log ^4(4) \log (x)+60 \log ^2(4) \log ^2(x)+5 \log ^3(x)} \, dx=\frac {2 x^6}{5 \left (4 \log ^2(4)+\log (x)\right )^2} \]
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Time = 0.28 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94
method | result | size |
norman | \(\frac {2 x^{6}}{5 \left (\ln \left (x \right )+16 \ln \left (2\right )^{2}\right )^{2}}\) | \(17\) |
risch | \(\frac {2 x^{6}}{5 \left (\ln \left (x \right )+16 \ln \left (2\right )^{2}\right )^{2}}\) | \(17\) |
default | \(\frac {2 x^{6}}{5 \left (256 \ln \left (2\right )^{4}+32 \ln \left (2\right )^{2} \ln \left (x \right )+\ln \left (x \right )^{2}\right )}\) | \(27\) |
parallelrisch | \(\frac {2 x^{6}}{5 \left (256 \ln \left (2\right )^{4}+32 \ln \left (2\right )^{2} \ln \left (x \right )+\ln \left (x \right )^{2}\right )}\) | \(27\) |
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Time = 0.31 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.44 \[ \int \frac {-4 x^5+48 x^5 \log ^2(4)+12 x^5 \log (x)}{320 \log ^6(4)+240 \log ^4(4) \log (x)+60 \log ^2(4) \log ^2(x)+5 \log ^3(x)} \, dx=\frac {2 \, x^{6}}{5 \, {\left (256 \, \log \left (2\right )^{4} + 32 \, \log \left (2\right )^{2} \log \left (x\right ) + \log \left (x\right )^{2}\right )}} \]
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Time = 0.06 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.50 \[ \int \frac {-4 x^5+48 x^5 \log ^2(4)+12 x^5 \log (x)}{320 \log ^6(4)+240 \log ^4(4) \log (x)+60 \log ^2(4) \log ^2(x)+5 \log ^3(x)} \, dx=\frac {2 x^{6}}{5 \log {\left (x \right )}^{2} + 160 \log {\left (2 \right )}^{2} \log {\left (x \right )} + 1280 \log {\left (2 \right )}^{4}} \]
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\[ \int \frac {-4 x^5+48 x^5 \log ^2(4)+12 x^5 \log (x)}{320 \log ^6(4)+240 \log ^4(4) \log (x)+60 \log ^2(4) \log ^2(x)+5 \log ^3(x)} \, dx=\int { \frac {4 \, {\left (48 \, x^{5} \log \left (2\right )^{2} + 3 \, x^{5} \log \left (x\right ) - x^{5}\right )}}{5 \, {\left (4096 \, \log \left (2\right )^{6} + 768 \, \log \left (2\right )^{4} \log \left (x\right ) + 48 \, \log \left (2\right )^{2} \log \left (x\right )^{2} + \log \left (x\right )^{3}\right )}} \,d x } \]
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Time = 0.30 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.44 \[ \int \frac {-4 x^5+48 x^5 \log ^2(4)+12 x^5 \log (x)}{320 \log ^6(4)+240 \log ^4(4) \log (x)+60 \log ^2(4) \log ^2(x)+5 \log ^3(x)} \, dx=\frac {2 \, x^{6}}{5 \, {\left (256 \, \log \left (2\right )^{4} + 32 \, \log \left (2\right )^{2} \log \left (x\right ) + \log \left (x\right )^{2}\right )}} \]
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Time = 12.88 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \frac {-4 x^5+48 x^5 \log ^2(4)+12 x^5 \log (x)}{320 \log ^6(4)+240 \log ^4(4) \log (x)+60 \log ^2(4) \log ^2(x)+5 \log ^3(x)} \, dx=\frac {2\,x^6}{5\,{\left (\ln \left (x\right )+16\,{\ln \left (2\right )}^2\right )}^2} \]
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