3.15.85 \(\int \frac {-64+10368 \log (4)-629856 \log ^2(4)[phone] \log ^3(4)-172186884 \log ^4(4)}{x \log ^4(4)-4 x \log ^4(4) \log (x)+4 x \log ^4(4) \log ^2(x)} \, dx\)

Optimal. Leaf size=23 \[ \left (-81+\frac {2}{\log (4)}\right )^4 \left (1-\frac {4}{-2+\frac {1}{\log (x)}}\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 22, normalized size of antiderivative = 0.96, number of steps used = 4, number of rules used = 2, integrand size = 54, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {12, 32} \begin {gather*} -\frac {2 (2-81 \log (4))^4}{\log ^4(4) (1-2 \log (x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-64 + 10368*Log[4] - 629856*Log[4]^2 + 17006112*Log[4]^3 - 172186884*Log[4]^4)/(x*Log[4]^4 - 4*x*Log[4]^4
*Log[x] + 4*x*Log[4]^4*Log[x]^2),x]

[Out]

(-2*(2 - 81*Log[4])^4)/(Log[4]^4*(1 - 2*Log[x]))

Rule 12

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

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\left (\left (4 (2-81 \log (4))^4\right ) \int \frac {1}{x \log ^4(4)-4 x \log ^4(4) \log (x)+4 x \log ^4(4) \log ^2(x)} \, dx\right )\\ &=-\left (\left (4 (2-81 \log (4))^4\right ) \operatorname {Subst}\left (\int \frac {1}{(-1+2 x)^2 \log ^4(4)} \, dx,x,\log (x)\right )\right )\\ &=-\frac {\left (4 (2-81 \log (4))^4\right ) \operatorname {Subst}\left (\int \frac {1}{(-1+2 x)^2} \, dx,x,\log (x)\right )}{\log ^4(4)}\\ &=-\frac {2 (2-81 \log (4))^4}{\log ^4(4) (1-2 \log (x))}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 22, normalized size = 0.96 \begin {gather*} -\frac {4 (2-81 \log (4))^4}{\log ^4(4) (2-4 \log (x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-64 + 10368*Log[4] - 629856*Log[4]^2 + 17006112*Log[4]^3 - 172186884*Log[4]^4)/(x*Log[4]^4 - 4*x*Lo
g[4]^4*Log[x] + 4*x*Log[4]^4*Log[x]^2),x]

[Out]

(-4*(2 - 81*Log[4])^4)/(Log[4]^4*(2 - 4*Log[x]))

________________________________________________________________________________________

fricas [A]  time = 1.09, size = 43, normalized size = 1.87 \begin {gather*} \frac {2 \, {\left (43046721 \, \log \relax (2)^{4} - 2125764 \, \log \relax (2)^{3} + 39366 \, \log \relax (2)^{2} - 324 \, \log \relax (2) + 1\right )}}{2 \, \log \relax (2)^{4} \log \relax (x) - \log \relax (2)^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2754990144*log(2)^4+136048896*log(2)^3-2519424*log(2)^2+20736*log(2)-64)/(64*x*log(2)^4*log(x)^2-6
4*x*log(2)^4*log(x)+16*x*log(2)^4),x, algorithm="fricas")

[Out]

2*(43046721*log(2)^4 - 2125764*log(2)^3 + 39366*log(2)^2 - 324*log(2) + 1)/(2*log(2)^4*log(x) - log(2)^4)

________________________________________________________________________________________

giac [A]  time = 0.19, size = 38, normalized size = 1.65 \begin {gather*} \frac {2 \, {\left (43046721 \, \log \relax (2)^{4} - 2125764 \, \log \relax (2)^{3} + 39366 \, \log \relax (2)^{2} - 324 \, \log \relax (2) + 1\right )}}{{\left (2 \, \log \relax (x) - 1\right )} \log \relax (2)^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2754990144*log(2)^4+136048896*log(2)^3-2519424*log(2)^2+20736*log(2)-64)/(64*x*log(2)^4*log(x)^2-6
4*x*log(2)^4*log(x)+16*x*log(2)^4),x, algorithm="giac")

[Out]

2*(43046721*log(2)^4 - 2125764*log(2)^3 + 39366*log(2)^2 - 324*log(2) + 1)/((2*log(x) - 1)*log(2)^4)

________________________________________________________________________________________

maple [A]  time = 0.06, size = 39, normalized size = 1.70

method	result	size
default	\(-\frac {-172186884 \ln \relax (2)^{4}+8503056 \ln \relax (2)^{3}-157464 \ln \relax (2)^{2}+1296 \ln \relax (2)-4}{2 \ln \relax (2)^{4} \left (-1+2 \ln \relax (x )\right )}\)	\(39\)
risch	\(\frac {86093442}{-1+2 \ln \relax (x )}-\frac {4251528}{\ln \relax (2) \left (-1+2 \ln \relax (x )\right )}+\frac {78732}{\ln \relax (2)^{2} \left (-1+2 \ln \relax (x )\right )}-\frac {648}{\ln \relax (2)^{3} \left (-1+2 \ln \relax (x )\right )}+\frac {2}{\ln \relax (2)^{4} \left (-1+2 \ln \relax (x )\right )}\)	\(68\)
norman	\(\frac {8 \left (43046721 \ln \relax (2)^{4}-2125764 \ln \relax (2)^{3}+39366 \ln \relax (2)^{2}-324 \ln \relax (2)+1\right ) \ln \relax (x )^{2}}{\ln \relax (2)^{4} \left (-1+2 \ln \relax (x )\right )}-\frac {4 \left (43046721 \ln \relax (2)^{4}-2125764 \ln \relax (2)^{3}+39366 \ln \relax (2)^{2}-324 \ln \relax (2)+1\right ) \ln \relax (x )}{\ln \relax (2)^{4}}\)	\(76\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2754990144*ln(2)^4+136048896*ln(2)^3-2519424*ln(2)^2+20736*ln(2)-64)/(64*x*ln(2)^4*ln(x)^2-64*x*ln(2)^4*
ln(x)+16*x*ln(2)^4),x,method=_RETURNVERBOSE)

[Out]

-1/2*(-172186884*ln(2)^4+8503056*ln(2)^3-157464*ln(2)^2+1296*ln(2)-4)/ln(2)^4/(-1+2*ln(x))

________________________________________________________________________________________

maxima [A]  time = 0.79, size = 43, normalized size = 1.87 \begin {gather*} \frac {2 \, {\left (43046721 \, \log \relax (2)^{4} - 2125764 \, \log \relax (2)^{3} + 39366 \, \log \relax (2)^{2} - 324 \, \log \relax (2) + 1\right )}}{2 \, \log \relax (2)^{4} \log \relax (x) - \log \relax (2)^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2754990144*log(2)^4+136048896*log(2)^3-2519424*log(2)^2+20736*log(2)-64)/(64*x*log(2)^4*log(x)^2-6
4*x*log(2)^4*log(x)+16*x*log(2)^4),x, algorithm="maxima")

[Out]

2*(43046721*log(2)^4 - 2125764*log(2)^3 + 39366*log(2)^2 - 324*log(2) + 1)/(2*log(2)^4*log(x) - log(2)^4)

________________________________________________________________________________________

mupad [B]  time = 1.16, size = 24, normalized size = 1.04 \begin {gather*} \frac {4\,\ln \relax (x)\,{\left (81\,\ln \relax (2)-1\right )}^4}{{\ln \relax (2)}^4\,\left (2\,\ln \relax (x)-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2519424*log(2)^2 - 20736*log(2) - 136048896*log(2)^3 + 2754990144*log(2)^4 + 64)/(16*x*log(2)^4 - 64*x*l
og(2)^4*log(x) + 64*x*log(2)^4*log(x)^2),x)

[Out]

(4*log(x)*(81*log(2) - 1)^4)/(log(2)^4*(2*log(x) - 1))

________________________________________________________________________________________

sympy [B]  time = 0.12, size = 41, normalized size = 1.78 \begin {gather*} \frac {- 4251528 \log {\relax (2 )}^{3} - 648 \log {\relax (2 )} + 2 + 78732 \log {\relax (2 )}^{2} + 86093442 \log {\relax (2 )}^{4}}{2 \log {\relax (2 )}^{4} \log {\relax (x )} - \log {\relax (2 )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2754990144*ln(2)**4+136048896*ln(2)**3-2519424*ln(2)**2+20736*ln(2)-64)/(64*x*ln(2)**4*ln(x)**2-64
*x*ln(2)**4*ln(x)+16*x*ln(2)**4),x)

[Out]

(-4251528*log(2)**3 - 648*log(2) + 2 + 78732*log(2)**2 + 86093442*log(2)**4)/(2*log(2)**4*log(x) - log(2)**4)

________________________________________________________________________________________