Integrand size = 146, antiderivative size = 20 \[ \int \frac {-414720+552960 \log (2)-276480 \log ^2(2)+61440 \log ^3(2)-5120 \log ^4(2)+\left (-414720+552960 \log (2)-276480 \log ^2(2)+61440 \log ^3(2)-5120 \log ^4(2)\right ) \log (4)+\left (-155520+207360 \log (2)-103680 \log ^2(2)+23040 \log ^3(2)-1920 \log ^4(2)\right ) \log ^2(4)+\left (-25920+34560 \log (2)-17280 \log ^2(2)+3840 \log ^3(2)-320 \log ^4(2)\right ) \log ^3(4)+\left (-1620+2160 \log (2)-1080 \log ^2(2)+240 \log ^3(2)-20 \log ^4(2)\right ) \log ^4(4)}{x^{21} \log ^4(4)} \, dx=\frac {(-3+\log (2))^4 (4+\log (4))^4}{x^{20} \log ^4(4)} \]
Time = 0.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {-414720+552960 \log (2)-276480 \log ^2(2)+61440 \log ^3(2)-5120 \log ^4(2)+\left (-414720+552960 \log (2)-276480 \log ^2(2)+61440 \log ^3(2)-5120 \log ^4(2)\right ) \log (4)+\left (-155520+207360 \log (2)-103680 \log ^2(2)+23040 \log ^3(2)-1920 \log ^4(2)\right ) \log ^2(4)+\left (-25920+34560 \log (2)-17280 \log ^2(2)+3840 \log ^3(2)-320 \log ^4(2)\right ) \log ^3(4)+\left (-1620+2160 \log (2)-1080 \log ^2(2)+240 \log ^3(2)-20 \log ^4(2)\right ) \log ^4(4)}{x^{21} \log ^4(4)} \, dx=\frac {(-3+\log (2))^4 (4+\log (4))^4}{x^{20} \log ^4(4)} \]
Integrate[(-414720 + 552960*Log[2] - 276480*Log[2]^2 + 61440*Log[2]^3 - 51 20*Log[2]^4 + (-414720 + 552960*Log[2] - 276480*Log[2]^2 + 61440*Log[2]^3 - 5120*Log[2]^4)*Log[4] + (-155520 + 207360*Log[2] - 103680*Log[2]^2 + 230 40*Log[2]^3 - 1920*Log[2]^4)*Log[4]^2 + (-25920 + 34560*Log[2] - 17280*Log [2]^2 + 3840*Log[2]^3 - 320*Log[2]^4)*Log[4]^3 + (-1620 + 2160*Log[2] - 10 80*Log[2]^2 + 240*Log[2]^3 - 20*Log[2]^4)*Log[4]^4)/(x^21*Log[4]^4),x]
Time = 0.15 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.007, Rules used = {15}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {-414720-5120 \log ^4(2)+61440 \log ^3(2)-276480 \log ^2(2)+\left (-414720-5120 \log ^4(2)+61440 \log ^3(2)-276480 \log ^2(2)+552960 \log (2)\right ) \log (4)+\left (-155520-1920 \log ^4(2)+23040 \log ^3(2)-103680 \log ^2(2)+207360 \log (2)\right ) \log ^2(4)+\left (-25920-320 \log ^4(2)+3840 \log ^3(2)-17280 \log ^2(2)+34560 \log (2)\right ) \log ^3(4)+\left (-1620-20 \log ^4(2)+240 \log ^3(2)-1080 \log ^2(2)+2160 \log (2)\right ) \log ^4(4)+552960 \log (2)}{x^{21} \log ^4(4)} \, dx\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {(3-\log (2))^4 (4+\log (4))^4}{x^{20} \log ^4(4)}\) |
Int[(-414720 + 552960*Log[2] - 276480*Log[2]^2 + 61440*Log[2]^3 - 5120*Log [2]^4 + (-414720 + 552960*Log[2] - 276480*Log[2]^2 + 61440*Log[2]^3 - 5120 *Log[2]^4)*Log[4] + (-155520 + 207360*Log[2] - 103680*Log[2]^2 + 23040*Log [2]^3 - 1920*Log[2]^4)*Log[4]^2 + (-25920 + 34560*Log[2] - 17280*Log[2]^2 + 3840*Log[2]^3 - 320*Log[2]^4)*Log[4]^3 + (-1620 + 2160*Log[2] - 1080*Log [2]^2 + 240*Log[2]^3 - 20*Log[2]^4)*Log[4]^4)/(x^21*Log[4]^4),x]
3.29.78.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(54\) vs. \(2(23)=46\).
Time = 0.10 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.75
| method | result | size |
| gosper | \(\frac {\ln \left (2\right )^{8}-4 \ln \left (2\right )^{7}-18 \ln \left (2\right )^{6}+68 \ln \left (2\right )^{5}+145 \ln \left (2\right )^{4}-408 \ln \left (2\right )^{3}-648 \ln \left (2\right )^{2}+864 \ln \left (2\right )+1296}{x^{20} \ln \left (2\right )^{4}}\) | \(55\) |
| norman | \(\frac {\ln \left (2\right )^{8}-4 \ln \left (2\right )^{7}-18 \ln \left (2\right )^{6}+68 \ln \left (2\right )^{5}+145 \ln \left (2\right )^{4}-408 \ln \left (2\right )^{3}-648 \ln \left (2\right )^{2}+864 \ln \left (2\right )+1296}{x^{20} \ln \left (2\right )^{4}}\) | \(55\) |
| risch | \(\frac {\ln \left (2\right )^{4}}{x^{20}}-\frac {4 \ln \left (2\right )^{3}}{x^{20}}-\frac {18 \ln \left (2\right )^{2}}{x^{20}}+\frac {68 \ln \left (2\right )}{x^{20}}+\frac {145}{x^{20}}-\frac {408}{x^{20} \ln \left (2\right )}-\frac {648}{x^{20} \ln \left (2\right )^{2}}+\frac {864}{x^{20} \ln \left (2\right )^{3}}+\frac {1296}{x^{20} \ln \left (2\right )^{4}}\) | \(76\) |
| default | \(-\frac {8 \left (-20 \ln \left (2\right )^{4}+240 \ln \left (2\right )^{3}-1080 \ln \left (2\right )^{2}+2160 \ln \left (2\right )-1620\right ) \ln \left (2\right )^{4}+4 \left (-320 \ln \left (2\right )^{4}+3840 \ln \left (2\right )^{3}-17280 \ln \left (2\right )^{2}+34560 \ln \left (2\right )-25920\right ) \ln \left (2\right )^{3}+2 \left (-1920 \ln \left (2\right )^{4}+23040 \ln \left (2\right )^{3}-103680 \ln \left (2\right )^{2}+207360 \ln \left (2\right )-155520\right ) \ln \left (2\right )^{2}+\left (-5120 \ln \left (2\right )^{4}+61440 \ln \left (2\right )^{3}-276480 \ln \left (2\right )^{2}+552960 \ln \left (2\right )-414720\right ) \ln \left (2\right )-2560 \ln \left (2\right )^{4}+30720 \ln \left (2\right )^{3}-138240 \ln \left (2\right )^{2}+276480 \ln \left (2\right )-207360}{160 \ln \left (2\right )^{4} x^{20}}\) | \(151\) |
| parallelrisch | \(-\frac {16 \left (-20 \ln \left (2\right )^{4}+240 \ln \left (2\right )^{3}-1080 \ln \left (2\right )^{2}+2160 \ln \left (2\right )-1620\right ) \ln \left (2\right )^{4}+8 \left (-320 \ln \left (2\right )^{4}+3840 \ln \left (2\right )^{3}-17280 \ln \left (2\right )^{2}+34560 \ln \left (2\right )-25920\right ) \ln \left (2\right )^{3}+4 \left (-1920 \ln \left (2\right )^{4}+23040 \ln \left (2\right )^{3}-103680 \ln \left (2\right )^{2}+207360 \ln \left (2\right )-155520\right ) \ln \left (2\right )^{2}+2 \left (-5120 \ln \left (2\right )^{4}+61440 \ln \left (2\right )^{3}-276480 \ln \left (2\right )^{2}+552960 \ln \left (2\right )-414720\right ) \ln \left (2\right )-5120 \ln \left (2\right )^{4}+61440 \ln \left (2\right )^{3}-276480 \ln \left (2\right )^{2}+552960 \ln \left (2\right )-414720}{320 x^{20} \ln \left (2\right )^{4}}\) | \(152\) |
int(1/16*(16*(-20*ln(2)^4+240*ln(2)^3-1080*ln(2)^2+2160*ln(2)-1620)*ln(2)^ 4+8*(-320*ln(2)^4+3840*ln(2)^3-17280*ln(2)^2+34560*ln(2)-25920)*ln(2)^3+4* (-1920*ln(2)^4+23040*ln(2)^3-103680*ln(2)^2+207360*ln(2)-155520)*ln(2)^2+2 *(-5120*ln(2)^4+61440*ln(2)^3-276480*ln(2)^2+552960*ln(2)-414720)*ln(2)-51 20*ln(2)^4+61440*ln(2)^3-276480*ln(2)^2+552960*ln(2)-414720)/x^21/ln(2)^4, x,method=_RETURNVERBOSE)
(ln(2)^8-4*ln(2)^7-18*ln(2)^6+68*ln(2)^5+145*ln(2)^4-408*ln(2)^3-648*ln(2) ^2+864*ln(2)+1296)/x^20/ln(2)^4
Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (20) = 40\).
Time = 0.22 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.70 \[ \int \frac {-414720+552960 \log (2)-276480 \log ^2(2)+61440 \log ^3(2)-5120 \log ^4(2)+\left (-414720+552960 \log (2)-276480 \log ^2(2)+61440 \log ^3(2)-5120 \log ^4(2)\right ) \log (4)+\left (-155520+207360 \log (2)-103680 \log ^2(2)+23040 \log ^3(2)-1920 \log ^4(2)\right ) \log ^2(4)+\left (-25920+34560 \log (2)-17280 \log ^2(2)+3840 \log ^3(2)-320 \log ^4(2)\right ) \log ^3(4)+\left (-1620+2160 \log (2)-1080 \log ^2(2)+240 \log ^3(2)-20 \log ^4(2)\right ) \log ^4(4)}{x^{21} \log ^4(4)} \, dx=\frac {\log \left (2\right )^{8} - 4 \, \log \left (2\right )^{7} - 18 \, \log \left (2\right )^{6} + 68 \, \log \left (2\right )^{5} + 145 \, \log \left (2\right )^{4} - 408 \, \log \left (2\right )^{3} - 648 \, \log \left (2\right )^{2} + 864 \, \log \left (2\right ) + 1296}{x^{20} \log \left (2\right )^{4}} \]
integrate(1/16*(16*(-20*log(2)^4+240*log(2)^3-1080*log(2)^2+2160*log(2)-16 20)*log(2)^4+8*(-320*log(2)^4+3840*log(2)^3-17280*log(2)^2+34560*log(2)-25 920)*log(2)^3+4*(-1920*log(2)^4+23040*log(2)^3-103680*log(2)^2+207360*log( 2)-155520)*log(2)^2+2*(-5120*log(2)^4+61440*log(2)^3-276480*log(2)^2+55296 0*log(2)-414720)*log(2)-5120*log(2)^4+61440*log(2)^3-276480*log(2)^2+55296 0*log(2)-414720)/x^21/log(2)^4,x, algorithm=\
(log(2)^8 - 4*log(2)^7 - 18*log(2)^6 + 68*log(2)^5 + 145*log(2)^4 - 408*lo g(2)^3 - 648*log(2)^2 + 864*log(2) + 1296)/(x^20*log(2)^4)
Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (24) = 48\).
Time = 0.06 (sec) , antiderivative size = 65, normalized size of antiderivative = 3.25 \[ \int \frac {-414720+552960 \log (2)-276480 \log ^2(2)+61440 \log ^3(2)-5120 \log ^4(2)+\left (-414720+552960 \log (2)-276480 \log ^2(2)+61440 \log ^3(2)-5120 \log ^4(2)\right ) \log (4)+\left (-155520+207360 \log (2)-103680 \log ^2(2)+23040 \log ^3(2)-1920 \log ^4(2)\right ) \log ^2(4)+\left (-25920+34560 \log (2)-17280 \log ^2(2)+3840 \log ^3(2)-320 \log ^4(2)\right ) \log ^3(4)+\left (-1620+2160 \log (2)-1080 \log ^2(2)+240 \log ^3(2)-20 \log ^4(2)\right ) \log ^4(4)}{x^{21} \log ^4(4)} \, dx=- \frac {-25920 - 17280 \log {\left (2 \right )} - 2900 \log {\left (2 \right )}^{4} - 1360 \log {\left (2 \right )}^{5} - 20 \log {\left (2 \right )}^{8} + 80 \log {\left (2 \right )}^{7} + 360 \log {\left (2 \right )}^{6} + 8160 \log {\left (2 \right )}^{3} + 12960 \log {\left (2 \right )}^{2}}{20 x^{20} \log {\left (2 \right )}^{4}} \]
integrate(1/16*(16*(-20*ln(2)**4+240*ln(2)**3-1080*ln(2)**2+2160*ln(2)-162 0)*ln(2)**4+8*(-320*ln(2)**4+3840*ln(2)**3-17280*ln(2)**2+34560*ln(2)-2592 0)*ln(2)**3+4*(-1920*ln(2)**4+23040*ln(2)**3-103680*ln(2)**2+207360*ln(2)- 155520)*ln(2)**2+2*(-5120*ln(2)**4+61440*ln(2)**3-276480*ln(2)**2+552960*l n(2)-414720)*ln(2)-5120*ln(2)**4+61440*ln(2)**3-276480*ln(2)**2+552960*ln( 2)-414720)/x**21/ln(2)**4,x)
-(-25920 - 17280*log(2) - 2900*log(2)**4 - 1360*log(2)**5 - 20*log(2)**8 + 80*log(2)**7 + 360*log(2)**6 + 8160*log(2)**3 + 12960*log(2)**2)/(20*x**2 0*log(2)**4)
Leaf count of result is larger than twice the leaf count of optimal. 141 vs. \(2 (20) = 40\).
Time = 0.22 (sec) , antiderivative size = 141, normalized size of antiderivative = 7.05 \[ \int \frac {-414720+552960 \log (2)-276480 \log ^2(2)+61440 \log ^3(2)-5120 \log ^4(2)+\left (-414720+552960 \log (2)-276480 \log ^2(2)+61440 \log ^3(2)-5120 \log ^4(2)\right ) \log (4)+\left (-155520+207360 \log (2)-103680 \log ^2(2)+23040 \log ^3(2)-1920 \log ^4(2)\right ) \log ^2(4)+\left (-25920+34560 \log (2)-17280 \log ^2(2)+3840 \log ^3(2)-320 \log ^4(2)\right ) \log ^3(4)+\left (-1620+2160 \log (2)-1080 \log ^2(2)+240 \log ^3(2)-20 \log ^4(2)\right ) \log ^4(4)}{x^{21} \log ^4(4)} \, dx=\frac {{\left (\log \left (2\right )^{4} - 12 \, \log \left (2\right )^{3} + 54 \, \log \left (2\right )^{2} - 108 \, \log \left (2\right ) + 81\right )} \log \left (2\right )^{4} + 8 \, {\left (\log \left (2\right )^{4} - 12 \, \log \left (2\right )^{3} + 54 \, \log \left (2\right )^{2} - 108 \, \log \left (2\right ) + 81\right )} \log \left (2\right )^{3} + 16 \, \log \left (2\right )^{4} + 24 \, {\left (\log \left (2\right )^{4} - 12 \, \log \left (2\right )^{3} + 54 \, \log \left (2\right )^{2} - 108 \, \log \left (2\right ) + 81\right )} \log \left (2\right )^{2} - 192 \, \log \left (2\right )^{3} + 32 \, {\left (\log \left (2\right )^{4} - 12 \, \log \left (2\right )^{3} + 54 \, \log \left (2\right )^{2} - 108 \, \log \left (2\right ) + 81\right )} \log \left (2\right ) + 864 \, \log \left (2\right )^{2} - 1728 \, \log \left (2\right ) + 1296}{x^{20} \log \left (2\right )^{4}} \]
integrate(1/16*(16*(-20*log(2)^4+240*log(2)^3-1080*log(2)^2+2160*log(2)-16 20)*log(2)^4+8*(-320*log(2)^4+3840*log(2)^3-17280*log(2)^2+34560*log(2)-25 920)*log(2)^3+4*(-1920*log(2)^4+23040*log(2)^3-103680*log(2)^2+207360*log( 2)-155520)*log(2)^2+2*(-5120*log(2)^4+61440*log(2)^3-276480*log(2)^2+55296 0*log(2)-414720)*log(2)-5120*log(2)^4+61440*log(2)^3-276480*log(2)^2+55296 0*log(2)-414720)/x^21/log(2)^4,x, algorithm=\
((log(2)^4 - 12*log(2)^3 + 54*log(2)^2 - 108*log(2) + 81)*log(2)^4 + 8*(lo g(2)^4 - 12*log(2)^3 + 54*log(2)^2 - 108*log(2) + 81)*log(2)^3 + 16*log(2) ^4 + 24*(log(2)^4 - 12*log(2)^3 + 54*log(2)^2 - 108*log(2) + 81)*log(2)^2 - 192*log(2)^3 + 32*(log(2)^4 - 12*log(2)^3 + 54*log(2)^2 - 108*log(2) + 8 1)*log(2) + 864*log(2)^2 - 1728*log(2) + 1296)/(x^20*log(2)^4)
Leaf count of result is larger than twice the leaf count of optimal. 141 vs. \(2 (20) = 40\).
Time = 0.26 (sec) , antiderivative size = 141, normalized size of antiderivative = 7.05 \[ \int \frac {-414720+552960 \log (2)-276480 \log ^2(2)+61440 \log ^3(2)-5120 \log ^4(2)+\left (-414720+552960 \log (2)-276480 \log ^2(2)+61440 \log ^3(2)-5120 \log ^4(2)\right ) \log (4)+\left (-155520+207360 \log (2)-103680 \log ^2(2)+23040 \log ^3(2)-1920 \log ^4(2)\right ) \log ^2(4)+\left (-25920+34560 \log (2)-17280 \log ^2(2)+3840 \log ^3(2)-320 \log ^4(2)\right ) \log ^3(4)+\left (-1620+2160 \log (2)-1080 \log ^2(2)+240 \log ^3(2)-20 \log ^4(2)\right ) \log ^4(4)}{x^{21} \log ^4(4)} \, dx=\frac {{\left (\log \left (2\right )^{4} - 12 \, \log \left (2\right )^{3} + 54 \, \log \left (2\right )^{2} - 108 \, \log \left (2\right ) + 81\right )} \log \left (2\right )^{4} + 8 \, {\left (\log \left (2\right )^{4} - 12 \, \log \left (2\right )^{3} + 54 \, \log \left (2\right )^{2} - 108 \, \log \left (2\right ) + 81\right )} \log \left (2\right )^{3} + 16 \, \log \left (2\right )^{4} + 24 \, {\left (\log \left (2\right )^{4} - 12 \, \log \left (2\right )^{3} + 54 \, \log \left (2\right )^{2} - 108 \, \log \left (2\right ) + 81\right )} \log \left (2\right )^{2} - 192 \, \log \left (2\right )^{3} + 32 \, {\left (\log \left (2\right )^{4} - 12 \, \log \left (2\right )^{3} + 54 \, \log \left (2\right )^{2} - 108 \, \log \left (2\right ) + 81\right )} \log \left (2\right ) + 864 \, \log \left (2\right )^{2} - 1728 \, \log \left (2\right ) + 1296}{x^{20} \log \left (2\right )^{4}} \]
integrate(1/16*(16*(-20*log(2)^4+240*log(2)^3-1080*log(2)^2+2160*log(2)-16 20)*log(2)^4+8*(-320*log(2)^4+3840*log(2)^3-17280*log(2)^2+34560*log(2)-25 920)*log(2)^3+4*(-1920*log(2)^4+23040*log(2)^3-103680*log(2)^2+207360*log( 2)-155520)*log(2)^2+2*(-5120*log(2)^4+61440*log(2)^3-276480*log(2)^2+55296 0*log(2)-414720)*log(2)-5120*log(2)^4+61440*log(2)^3-276480*log(2)^2+55296 0*log(2)-414720)/x^21/log(2)^4,x, algorithm=\
((log(2)^4 - 12*log(2)^3 + 54*log(2)^2 - 108*log(2) + 81)*log(2)^4 + 8*(lo g(2)^4 - 12*log(2)^3 + 54*log(2)^2 - 108*log(2) + 81)*log(2)^3 + 16*log(2) ^4 + 24*(log(2)^4 - 12*log(2)^3 + 54*log(2)^2 - 108*log(2) + 81)*log(2)^2 - 192*log(2)^3 + 32*(log(2)^4 - 12*log(2)^3 + 54*log(2)^2 - 108*log(2) + 8 1)*log(2) + 864*log(2)^2 - 1728*log(2) + 1296)/(x^20*log(2)^4)
Time = 9.00 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {-414720+552960 \log (2)-276480 \log ^2(2)+61440 \log ^3(2)-5120 \log ^4(2)+\left (-414720+552960 \log (2)-276480 \log ^2(2)+61440 \log ^3(2)-5120 \log ^4(2)\right ) \log (4)+\left (-155520+207360 \log (2)-103680 \log ^2(2)+23040 \log ^3(2)-1920 \log ^4(2)\right ) \log ^2(4)+\left (-25920+34560 \log (2)-17280 \log ^2(2)+3840 \log ^3(2)-320 \log ^4(2)\right ) \log ^3(4)+\left (-1620+2160 \log (2)-1080 \log ^2(2)+240 \log ^3(2)-20 \log ^4(2)\right ) \log ^4(4)}{x^{21} \log ^4(4)} \, dx=\frac {{\left (\ln \left (2\right )-{\ln \left (2\right )}^2+6\right )}^4}{x^{20}\,{\ln \left (2\right )}^4} \]
int(-((log(2)*(276480*log(2)^2 - 552960*log(2) - 61440*log(2)^3 + 5120*log (2)^4 + 414720))/8 - 34560*log(2) + 17280*log(2)^2 - 3840*log(2)^3 + 320*l og(2)^4 + log(2)^4*(1080*log(2)^2 - 2160*log(2) - 240*log(2)^3 + 20*log(2) ^4 + 1620) + (log(2)^3*(17280*log(2)^2 - 34560*log(2) - 3840*log(2)^3 + 32 0*log(2)^4 + 25920))/2 + (log(2)^2*(103680*log(2)^2 - 207360*log(2) - 2304 0*log(2)^3 + 1920*log(2)^4 + 155520))/4 + 25920)/(x^21*log(2)^4),x)