3.89.18 \(\int \frac {-2+x+((-12+7 x) \log (3)+(12-7 x) \log (x)) \log (-\log (3)+\log (x))}{(-256 x^7+256 x^8-64 x^9) \log (3)+(256 x^7-256 x^8+64 x^9) \log (x)} \, dx\)

Optimal. Leaf size=20 \[ \frac {\log (-\log (3)+\log (x))}{64 (-2+x) x^6} \]

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Rubi [F]  time = 0.98, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-2+x+((-12+7 x) \log (3)+(12-7 x) \log (x)) \log (-\log (3)+\log (x))}{\left (-256 x^7+256 x^8-64 x^9\right ) \log (3)+\left (256 x^7-256 x^8+64 x^9\right ) \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-2 + x + ((-12 + 7*x)*Log[3] + (12 - 7*x)*Log[x])*Log[-Log[3] + Log[x]])/((-256*x^7 + 256*x^8 - 64*x^9)*L
og[3] + (256*x^7 - 256*x^8 + 64*x^9)*Log[x]),x]

[Out]

ExpIntegralEi[-6*Log[x/3]]/93312 + ExpIntegralEi[-5*Log[x/3]]/62208 + ExpIntegralEi[-4*Log[x/3]]/41472 + ExpIn
tegralEi[-3*Log[x/3]]/27648 + ExpIntegralEi[-2*Log[x/3]]/18432 + ExpIntegralEi[-Log[x/3]]/12288 - Log[Log[x/3]
]/(128*x^6) - Log[Log[x/3]]/(256*x^5) - Log[Log[x/3]]/(512*x^4) - Log[Log[x/3]]/(1024*x^3) - Log[Log[x/3]]/(20
48*x^2) - Log[Log[x/3]]/(4096*x) - Defer[Int][1/((-2 + x)*x^7*(Log[3] - Log[x])), x]/64 - (3*Defer[Subst][Defe
r[Int][Log[Log[x]]/(-2 + 3*x)^2, x], x, x/3])/4096

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2-x+(-12+7 x) \log \left (\frac {x}{3}\right ) \log \left (\log \left (\frac {x}{3}\right )\right )}{64 (2-x)^2 x^7 (\log (3)-\log (x))} \, dx\\ &=\frac {1}{64} \int \frac {2-x+(-12+7 x) \log \left (\frac {x}{3}\right ) \log \left (\log \left (\frac {x}{3}\right )\right )}{(2-x)^2 x^7 (\log (3)-\log (x))} \, dx\\ &=\frac {1}{64} \int \left (-\frac {1}{(-2+x) x^7 (\log (3)-\log (x))}-\frac {(-12+7 x) \log \left (\log \left (\frac {x}{3}\right )\right )}{(-2+x)^2 x^7}\right ) \, dx\\ &=-\left (\frac {1}{64} \int \frac {1}{(-2+x) x^7 (\log (3)-\log (x))} \, dx\right )-\frac {1}{64} \int \frac {(-12+7 x) \log \left (\log \left (\frac {x}{3}\right )\right )}{(-2+x)^2 x^7} \, dx\\ &=-\left (\frac {1}{64} \int \frac {1}{(-2+x) x^7 (\log (3)-\log (x))} \, dx\right )-\frac {1}{64} \int \left (\frac {\log \left (\log \left (\frac {x}{3}\right )\right )}{64 (-2+x)^2}-\frac {3 \log \left (\log \left (\frac {x}{3}\right )\right )}{x^7}-\frac {5 \log \left (\log \left (\frac {x}{3}\right )\right )}{4 x^6}-\frac {\log \left (\log \left (\frac {x}{3}\right )\right )}{2 x^5}-\frac {3 \log \left (\log \left (\frac {x}{3}\right )\right )}{16 x^4}-\frac {\log \left (\log \left (\frac {x}{3}\right )\right )}{16 x^3}-\frac {\log \left (\log \left (\frac {x}{3}\right )\right )}{64 x^2}\right ) \, dx\\ &=-\frac {\int \frac {\log \left (\log \left (\frac {x}{3}\right )\right )}{(-2+x)^2} \, dx}{4096}+\frac {\int \frac {\log \left (\log \left (\frac {x}{3}\right )\right )}{x^2} \, dx}{4096}+\frac {\int \frac {\log \left (\log \left (\frac {x}{3}\right )\right )}{x^3} \, dx}{1024}+\frac {3 \int \frac {\log \left (\log \left (\frac {x}{3}\right )\right )}{x^4} \, dx}{1024}+\frac {1}{128} \int \frac {\log \left (\log \left (\frac {x}{3}\right )\right )}{x^5} \, dx-\frac {1}{64} \int \frac {1}{(-2+x) x^7 (\log (3)-\log (x))} \, dx+\frac {5}{256} \int \frac {\log \left (\log \left (\frac {x}{3}\right )\right )}{x^6} \, dx+\frac {3}{64} \int \frac {\log \left (\log \left (\frac {x}{3}\right )\right )}{x^7} \, dx\\ &=-\frac {\log \left (\log \left (\frac {x}{3}\right )\right )}{128 x^6}-\frac {\log \left (\log \left (\frac {x}{3}\right )\right )}{256 x^5}-\frac {\log \left (\log \left (\frac {x}{3}\right )\right )}{512 x^4}-\frac {\log \left (\log \left (\frac {x}{3}\right )\right )}{1024 x^3}-\frac {\log \left (\log \left (\frac {x}{3}\right )\right )}{2048 x^2}-\frac {\log \left (\log \left (\frac {x}{3}\right )\right )}{4096 x}+\frac {\int \frac {1}{x^2 \log \left (\frac {x}{3}\right )} \, dx}{4096}+\frac {\int \frac {1}{x^3 \log \left (\frac {x}{3}\right )} \, dx}{2048}-\frac {3 \operatorname {Subst}\left (\int \frac {\log (\log (x))}{(-2+3 x)^2} \, dx,x,\frac {x}{3}\right )}{4096}+\frac {\int \frac {1}{x^4 \log \left (\frac {x}{3}\right )} \, dx}{1024}+\frac {1}{512} \int \frac {1}{x^5 \log \left (\frac {x}{3}\right )} \, dx+\frac {1}{256} \int \frac {1}{x^6 \log \left (\frac {x}{3}\right )} \, dx+\frac {1}{128} \int \frac {1}{x^7 \log \left (\frac {x}{3}\right )} \, dx-\frac {1}{64} \int \frac {1}{(-2+x) x^7 (\log (3)-\log (x))} \, dx\\ &=-\frac {\log \left (\log \left (\frac {x}{3}\right )\right )}{128 x^6}-\frac {\log \left (\log \left (\frac {x}{3}\right )\right )}{256 x^5}-\frac {\log \left (\log \left (\frac {x}{3}\right )\right )}{512 x^4}-\frac {\log \left (\log \left (\frac {x}{3}\right )\right )}{1024 x^3}-\frac {\log \left (\log \left (\frac {x}{3}\right )\right )}{2048 x^2}-\frac {\log \left (\log \left (\frac {x}{3}\right )\right )}{4096 x}+\frac {\operatorname {Subst}\left (\int \frac {e^{-6 x}}{x} \, dx,x,\log \left (\frac {x}{3}\right )\right )}{93312}+\frac {\operatorname {Subst}\left (\int \frac {e^{-5 x}}{x} \, dx,x,\log \left (\frac {x}{3}\right )\right )}{62208}+\frac {\operatorname {Subst}\left (\int \frac {e^{-4 x}}{x} \, dx,x,\log \left (\frac {x}{3}\right )\right )}{41472}+\frac {\operatorname {Subst}\left (\int \frac {e^{-3 x}}{x} \, dx,x,\log \left (\frac {x}{3}\right )\right )}{27648}+\frac {\operatorname {Subst}\left (\int \frac {e^{-2 x}}{x} \, dx,x,\log \left (\frac {x}{3}\right )\right )}{18432}+\frac {\operatorname {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log \left (\frac {x}{3}\right )\right )}{12288}-\frac {3 \operatorname {Subst}\left (\int \frac {\log (\log (x))}{(-2+3 x)^2} \, dx,x,\frac {x}{3}\right )}{4096}-\frac {1}{64} \int \frac {1}{(-2+x) x^7 (\log (3)-\log (x))} \, dx\\ &=\frac {\text {Ei}\left (-6 \log \left (\frac {x}{3}\right )\right )}{93312}+\frac {\text {Ei}\left (-5 \log \left (\frac {x}{3}\right )\right )}{62208}+\frac {\text {Ei}\left (-4 \log \left (\frac {x}{3}\right )\right )}{41472}+\frac {\text {Ei}\left (-3 \log \left (\frac {x}{3}\right )\right )}{27648}+\frac {\text {Ei}\left (-2 \log \left (\frac {x}{3}\right )\right )}{18432}+\frac {\text {Ei}\left (-\log \left (\frac {x}{3}\right )\right )}{12288}-\frac {\log \left (\log \left (\frac {x}{3}\right )\right )}{128 x^6}-\frac {\log \left (\log \left (\frac {x}{3}\right )\right )}{256 x^5}-\frac {\log \left (\log \left (\frac {x}{3}\right )\right )}{512 x^4}-\frac {\log \left (\log \left (\frac {x}{3}\right )\right )}{1024 x^3}-\frac {\log \left (\log \left (\frac {x}{3}\right )\right )}{2048 x^2}-\frac {\log \left (\log \left (\frac {x}{3}\right )\right )}{4096 x}-\frac {3 \operatorname {Subst}\left (\int \frac {\log (\log (x))}{(-2+3 x)^2} \, dx,x,\frac {x}{3}\right )}{4096}-\frac {1}{64} \int \frac {1}{(-2+x) x^7 (\log (3)-\log (x))} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.31, size = 19, normalized size = 0.95 \begin {gather*} \frac {\log \left (\log \left (\frac {x}{3}\right )\right )}{64 (-2+x) x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2 + x + ((-12 + 7*x)*Log[3] + (12 - 7*x)*Log[x])*Log[-Log[3] + Log[x]])/((-256*x^7 + 256*x^8 - 64*
x^9)*Log[3] + (256*x^7 - 256*x^8 + 64*x^9)*Log[x]),x]

[Out]

Log[Log[x/3]]/(64*(-2 + x)*x^6)

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fricas [A]  time = 0.54, size = 21, normalized size = 1.05 \begin {gather*} \frac {\log \left (-\log \relax (3) + \log \relax (x)\right )}{64 \, {\left (x^{7} - 2 \, x^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-7*x+12)*log(x)+(7*x-12)*log(3))*log(log(x)-log(3))+x-2)/((64*x^9-256*x^8+256*x^7)*log(x)+(-64*x^
9+256*x^8-256*x^7)*log(3)),x, algorithm="fricas")

[Out]

1/64*log(-log(3) + log(x))/(x^7 - 2*x^6)

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giac [B]  time = 0.19, size = 44, normalized size = 2.20 \begin {gather*} \frac {1}{4096} \, {\left (\frac {1}{x - 2} - \frac {x^{5} + 2 \, x^{4} + 4 \, x^{3} + 8 \, x^{2} + 16 \, x + 32}{x^{6}}\right )} \log \left (-\log \relax (3) + \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-7*x+12)*log(x)+(7*x-12)*log(3))*log(log(x)-log(3))+x-2)/((64*x^9-256*x^8+256*x^7)*log(x)+(-64*x^
9+256*x^8-256*x^7)*log(3)),x, algorithm="giac")

[Out]

1/4096*(1/(x - 2) - (x^5 + 2*x^4 + 4*x^3 + 8*x^2 + 16*x + 32)/x^6)*log(-log(3) + log(x))

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maple [A]  time = 0.20, size = 19, normalized size = 0.95

method	result	size
risch	\(\frac {\ln \left (\ln \relax (x )-\ln \relax (3)\right )}{64 x^{6} \left (x -2\right )}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-7*x+12)*ln(x)+(7*x-12)*ln(3))*ln(ln(x)-ln(3))+x-2)/((64*x^9-256*x^8+256*x^7)*ln(x)+(-64*x^9+256*x^8-25
6*x^7)*ln(3)),x,method=_RETURNVERBOSE)

[Out]

1/64*ln(ln(x)-ln(3))/x^6/(x-2)

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maxima [A]  time = 0.52, size = 21, normalized size = 1.05 \begin {gather*} \frac {\log \left (-\log \relax (3) + \log \relax (x)\right )}{64 \, {\left (x^{7} - 2 \, x^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-7*x+12)*log(x)+(7*x-12)*log(3))*log(log(x)-log(3))+x-2)/((64*x^9-256*x^8+256*x^7)*log(x)+(-64*x^
9+256*x^8-256*x^7)*log(3)),x, algorithm="maxima")

[Out]

1/64*log(-log(3) + log(x))/(x^7 - 2*x^6)

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mupad [B]  time = 6.03, size = 20, normalized size = 1.00 \begin {gather*} -\frac {\ln \left (\ln \left (\frac {x}{3}\right )\right )}{128\,x^6-64\,x^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + log(log(x) - log(3))*(log(3)*(7*x - 12) - log(x)*(7*x - 12)) - 2)/(log(x)*(256*x^7 - 256*x^8 + 64*x^9
) - log(3)*(256*x^7 - 256*x^8 + 64*x^9)),x)

[Out]

-log(log(x/3))/(128*x^6 - 64*x^7)

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sympy [A]  time = 0.53, size = 17, normalized size = 0.85 \begin {gather*} \frac {\log {\left (\log {\relax (x )} - \log {\relax (3 )} \right )}}{64 x^{7} - 128 x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-7*x+12)*ln(x)+(7*x-12)*ln(3))*ln(ln(x)-ln(3))+x-2)/((64*x**9-256*x**8+256*x**7)*ln(x)+(-64*x**9+
256*x**8-256*x**7)*ln(3)),x)

[Out]

log(log(x) - log(3))/(64*x**7 - 128*x**6)

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