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

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 = {} $$\begin{array}{c}\int \frac{-2+x+((-12+7x)\log (3)+(12-7x)\log (x))\log (-\log (3)+\log (x))}{(-256x^7+256x^8-64x^9)\log (3)+(256x^7-256x^8+64x^9)\log (x)}dx\end{array}$$

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

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

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 $$\begin{array}{c}\frac{\log (-\log (3)+\log (x))}{64(x^7-2x^6)}\end{array}$$

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 $$\begin{array}{c}\frac{1}{4096}(\frac{1}{x-2}-\frac{x^5+2x^4+4x^3+8x^2+16x+32}{x^6})\log (-\log (3)+\log (x))\end{array}$$

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	$\frac{\ln (\ln (x)-\ln (3))}{64x^6(x-2)}$	$19$

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 $$\begin{array}{c}\frac{\log (-\log (3)+\log (x))}{64(x^7-2x^6)}\end{array}$$

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 $$\begin{array}{c}-\frac{\ln (\ln \left(\frac{x}{3}\right))}{128x^6-64x^7}\end{array}$$

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 $$\begin{array}{c}\frac{\log (\log (x)-\log (3))}{64x^7-128x^6}\end{array}$$

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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