3.29.78 \(\int \frac {-414720+552960 \log (2)-276480 \log ^2(2)+61440 \log ^3(2)-5120 \log ^4(2)+(-414720+552960 \log (2)-276480 \log ^2(2)+61440 \log ^3(2)-5120 \log ^4(2)) \log (4)+(-155520+207360 \log (2)-103680 \log ^2(2)+23040 \log ^3(2)-1920 \log ^4(2)) \log ^2(4)+(-25920+34560 \log (2)-17280 \log ^2(2)+3840 \log ^3(2)-320 \log ^4(2)) \log ^3(4)+(-1620+2160 \log (2)-1080 \log ^2(2)+240 \log ^3(2)-20 \log ^4(2)) \log ^4(4)}{x^{21} \log ^4(4)} \, dx\)

Optimal. Leaf size=20 \[ \frac {(-3+\log (2))^4 (4+\log (4))^4}{x^{20} \log ^4(4)} \]

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 22, normalized size of antiderivative = 1.10, number of steps used = 2, number of rules used = 2, integrand size = 146, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.014, Rules used = {12, 30} \begin {gather*} \frac {(3-\log (2))^4 (4+\log (4))^4}{x^{20} \log ^4(4)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-414720 + 552960*Log[2] - 276480*Log[2]^2 + 61440*Log[2]^3 - 5120*Log[2]^4 + (-414720 + 552960*Log[2] - 2
76480*Log[2]^2 + 61440*Log[2]^3 - 5120*Log[2]^4)*Log[4] + (-155520 + 207360*Log[2] - 103680*Log[2]^2 + 23040*L
og[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)*Lo
g[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]

[Out]

((3 - Log[2])^4*(4 + Log[4])^4)/(x^20*Log[4]^4)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\frac {\left (20 (3-\log (2))^4 (4+\log (4))^4\right ) \int \frac {1}{x^{21}} \, dx}{\log ^4(4)}\\ &=\frac {(3-\log (2))^4 (4+\log (4))^4}{x^{20} \log ^4(4)}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 20, normalized size = 1.00 \begin {gather*} \frac {(-3+\log (2))^4 (4+\log (4))^4}{x^{20} \log ^4(4)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-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 + 2
3040*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]

[Out]

((-3 + Log[2])^4*(4 + Log[4])^4)/(x^20*Log[4]^4)

________________________________________________________________________________________

fricas [B]  time = 0.99, size = 54, normalized size = 2.70 \begin {gather*} \frac {\log \relax (2)^{8} - 4 \, \log \relax (2)^{7} - 18 \, \log \relax (2)^{6} + 68 \, \log \relax (2)^{5} + 145 \, \log \relax (2)^{4} - 408 \, \log \relax (2)^{3} - 648 \, \log \relax (2)^{2} + 864 \, \log \relax (2) + 1296}{x^{20} \log \relax (2)^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/16*(16*(-20*log(2)^4+240*log(2)^3-1080*log(2)^2+2160*log(2)-1620)*log(2)^4+8*(-320*log(2)^4+3840*l
og(2)^3-17280*log(2)^2+34560*log(2)-25920)*log(2)^3+4*(-1920*log(2)^4+23040*log(2)^3-103680*log(2)^2+207360*lo
g(2)-155520)*log(2)^2+2*(-5120*log(2)^4+61440*log(2)^3-276480*log(2)^2+552960*log(2)-414720)*log(2)-5120*log(2
)^4+61440*log(2)^3-276480*log(2)^2+552960*log(2)-414720)/x^21/log(2)^4,x, algorithm="fricas")

[Out]

(log(2)^8 - 4*log(2)^7 - 18*log(2)^6 + 68*log(2)^5 + 145*log(2)^4 - 408*log(2)^3 - 648*log(2)^2 + 864*log(2) +
 1296)/(x^20*log(2)^4)

________________________________________________________________________________________

giac [B]  time = 0.24, size = 141, normalized size = 7.05 \begin {gather*} \frac {{\left (\log \relax (2)^{4} - 12 \, \log \relax (2)^{3} + 54 \, \log \relax (2)^{2} - 108 \, \log \relax (2) + 81\right )} \log \relax (2)^{4} + 8 \, {\left (\log \relax (2)^{4} - 12 \, \log \relax (2)^{3} + 54 \, \log \relax (2)^{2} - 108 \, \log \relax (2) + 81\right )} \log \relax (2)^{3} + 16 \, \log \relax (2)^{4} + 24 \, {\left (\log \relax (2)^{4} - 12 \, \log \relax (2)^{3} + 54 \, \log \relax (2)^{2} - 108 \, \log \relax (2) + 81\right )} \log \relax (2)^{2} - 192 \, \log \relax (2)^{3} + 32 \, {\left (\log \relax (2)^{4} - 12 \, \log \relax (2)^{3} + 54 \, \log \relax (2)^{2} - 108 \, \log \relax (2) + 81\right )} \log \relax (2) + 864 \, \log \relax (2)^{2} - 1728 \, \log \relax (2) + 1296}{x^{20} \log \relax (2)^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/16*(16*(-20*log(2)^4+240*log(2)^3-1080*log(2)^2+2160*log(2)-1620)*log(2)^4+8*(-320*log(2)^4+3840*l
og(2)^3-17280*log(2)^2+34560*log(2)-25920)*log(2)^3+4*(-1920*log(2)^4+23040*log(2)^3-103680*log(2)^2+207360*lo
g(2)-155520)*log(2)^2+2*(-5120*log(2)^4+61440*log(2)^3-276480*log(2)^2+552960*log(2)-414720)*log(2)-5120*log(2
)^4+61440*log(2)^3-276480*log(2)^2+552960*log(2)-414720)/x^21/log(2)^4,x, algorithm="giac")

[Out]

((log(2)^4 - 12*log(2)^3 + 54*log(2)^2 - 108*log(2) + 81)*log(2)^4 + 8*(log(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) + 81)*log(2) + 864*log(2)^2 - 1728*lo
g(2) + 1296)/(x^20*log(2)^4)

________________________________________________________________________________________

maple [B]  time = 0.06, size = 55, normalized size = 2.75




method result size



gosper \(\frac {\ln \relax (2)^{8}-4 \ln \relax (2)^{7}-18 \ln \relax (2)^{6}+68 \ln \relax (2)^{5}+145 \ln \relax (2)^{4}-408 \ln \relax (2)^{3}-648 \ln \relax (2)^{2}+864 \ln \relax (2)+1296}{x^{20} \ln \relax (2)^{4}}\) \(55\)
norman \(\frac {\ln \relax (2)^{8}-4 \ln \relax (2)^{7}-18 \ln \relax (2)^{6}+68 \ln \relax (2)^{5}+145 \ln \relax (2)^{4}-408 \ln \relax (2)^{3}-648 \ln \relax (2)^{2}+864 \ln \relax (2)+1296}{x^{20} \ln \relax (2)^{4}}\) \(55\)
risch \(\frac {\ln \relax (2)^{4}}{x^{20}}-\frac {4 \ln \relax (2)^{3}}{x^{20}}-\frac {18 \ln \relax (2)^{2}}{x^{20}}+\frac {68 \ln \relax (2)}{x^{20}}+\frac {145}{x^{20}}-\frac {408}{x^{20} \ln \relax (2)}-\frac {648}{x^{20} \ln \relax (2)^{2}}+\frac {864}{x^{20} \ln \relax (2)^{3}}+\frac {1296}{x^{20} \ln \relax (2)^{4}}\) \(76\)
default \(-\frac {8 \left (-20 \ln \relax (2)^{4}+240 \ln \relax (2)^{3}-1080 \ln \relax (2)^{2}+2160 \ln \relax (2)-1620\right ) \ln \relax (2)^{4}+4 \left (-320 \ln \relax (2)^{4}+3840 \ln \relax (2)^{3}-17280 \ln \relax (2)^{2}+34560 \ln \relax (2)-25920\right ) \ln \relax (2)^{3}+2 \left (-1920 \ln \relax (2)^{4}+23040 \ln \relax (2)^{3}-103680 \ln \relax (2)^{2}+207360 \ln \relax (2)-155520\right ) \ln \relax (2)^{2}+\left (-5120 \ln \relax (2)^{4}+61440 \ln \relax (2)^{3}-276480 \ln \relax (2)^{2}+552960 \ln \relax (2)-414720\right ) \ln \relax (2)-2560 \ln \relax (2)^{4}+30720 \ln \relax (2)^{3}-138240 \ln \relax (2)^{2}+276480 \ln \relax (2)-207360}{160 \ln \relax (2)^{4} x^{20}}\) \(151\)



Verification of antiderivative is not currently implemented for this CAS.

[In]

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)-5120*ln(2)^4+61440*ln(2)^3-276480*ln
(2)^2+552960*ln(2)-414720)/x^21/ln(2)^4,x,method=_RETURNVERBOSE)

[Out]

(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

________________________________________________________________________________________

maxima [B]  time = 0.36, size = 141, normalized size = 7.05 \begin {gather*} \frac {{\left (\log \relax (2)^{4} - 12 \, \log \relax (2)^{3} + 54 \, \log \relax (2)^{2} - 108 \, \log \relax (2) + 81\right )} \log \relax (2)^{4} + 8 \, {\left (\log \relax (2)^{4} - 12 \, \log \relax (2)^{3} + 54 \, \log \relax (2)^{2} - 108 \, \log \relax (2) + 81\right )} \log \relax (2)^{3} + 16 \, \log \relax (2)^{4} + 24 \, {\left (\log \relax (2)^{4} - 12 \, \log \relax (2)^{3} + 54 \, \log \relax (2)^{2} - 108 \, \log \relax (2) + 81\right )} \log \relax (2)^{2} - 192 \, \log \relax (2)^{3} + 32 \, {\left (\log \relax (2)^{4} - 12 \, \log \relax (2)^{3} + 54 \, \log \relax (2)^{2} - 108 \, \log \relax (2) + 81\right )} \log \relax (2) + 864 \, \log \relax (2)^{2} - 1728 \, \log \relax (2) + 1296}{x^{20} \log \relax (2)^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/16*(16*(-20*log(2)^4+240*log(2)^3-1080*log(2)^2+2160*log(2)-1620)*log(2)^4+8*(-320*log(2)^4+3840*l
og(2)^3-17280*log(2)^2+34560*log(2)-25920)*log(2)^3+4*(-1920*log(2)^4+23040*log(2)^3-103680*log(2)^2+207360*lo
g(2)-155520)*log(2)^2+2*(-5120*log(2)^4+61440*log(2)^3-276480*log(2)^2+552960*log(2)-414720)*log(2)-5120*log(2
)^4+61440*log(2)^3-276480*log(2)^2+552960*log(2)-414720)/x^21/log(2)^4,x, algorithm="maxima")

[Out]

((log(2)^4 - 12*log(2)^3 + 54*log(2)^2 - 108*log(2) + 81)*log(2)^4 + 8*(log(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) + 81)*log(2) + 864*log(2)^2 - 1728*lo
g(2) + 1296)/(x^20*log(2)^4)

________________________________________________________________________________________

mupad [B]  time = 1.70, size = 20, normalized size = 1.00 \begin {gather*} \frac {{\left (\ln \relax (2)-{\ln \relax (2)}^2+6\right )}^4}{x^{20}\,{\ln \relax (2)}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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*log(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 + 320*log(2)^4 + 25920))/2 + (log(2)^2
*(103680*log(2)^2 - 207360*log(2) - 23040*log(2)^3 + 1920*log(2)^4 + 155520))/4 + 25920)/(x^21*log(2)^4),x)

[Out]

(log(2) - log(2)^2 + 6)^4/(x^20*log(2)^4)

________________________________________________________________________________________

sympy [B]  time = 0.11, size = 65, normalized size = 3.25 \begin {gather*} - \frac {-25920 - 17280 \log {\relax (2 )} - 2900 \log {\relax (2 )}^{4} - 1360 \log {\relax (2 )}^{5} - 20 \log {\relax (2 )}^{8} + 80 \log {\relax (2 )}^{7} + 360 \log {\relax (2 )}^{6} + 8160 \log {\relax (2 )}^{3} + 12960 \log {\relax (2 )}^{2}}{20 x^{20} \log {\relax (2 )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(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)-5120*ln(2)**4+6
1440*ln(2)**3-276480*ln(2)**2+552960*ln(2)-414720)/x**21/ln(2)**4,x)

[Out]

-(-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**20*log(2)**4)

________________________________________________________________________________________