3.51.15 \(\int \frac {12+2 x+(732-1248 x+582 x^2-27 x^3-27 x^4) \log (x)+(-444+466 x-72 x^2-27 x^3) \log (x) \log (\frac {6+x}{x \log (x)})+(90-39 x-9 x^2) \log (x) \log ^2(\frac {6+x}{x \log (x)})+(-6-x) \log (x) \log ^3(\frac {6+x}{x \log (x)})}{(-750+1225 x-585 x^2+27 x^3+27 x^4) \log (x)+(450-465 x+72 x^2+27 x^3) \log (x) \log (\frac {6+x}{x \log (x)})+(-90+39 x+9 x^2) \log (x) \log ^2(\frac {6+x}{x \log (x)})+(6+x) \log (x) \log ^3(\frac {6+x}{x \log (x)})} \, dx\)

Optimal. Leaf size=26 \[ -x+\frac {x}{\left (-5+3 x+\log \left (\frac {1+\frac {6}{x}}{\log (x)}\right )\right )^2} \]

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Rubi [F]  time = 1.70, 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 {12+2 x+\left (732-1248 x+582 x^2-27 x^3-27 x^4\right ) \log (x)+\left (-444+466 x-72 x^2-27 x^3\right ) \log (x) \log \left (\frac {6+x}{x \log (x)}\right )+\left (90-39 x-9 x^2\right ) \log (x) \log ^2\left (\frac {6+x}{x \log (x)}\right )+(-6-x) \log (x) \log ^3\left (\frac {6+x}{x \log (x)}\right )}{\left (-750+1225 x-585 x^2+27 x^3+27 x^4\right ) \log (x)+\left (450-465 x+72 x^2+27 x^3\right ) \log (x) \log \left (\frac {6+x}{x \log (x)}\right )+\left (-90+39 x+9 x^2\right ) \log (x) \log ^2\left (\frac {6+x}{x \log (x)}\right )+(6+x) \log (x) \log ^3\left (\frac {6+x}{x \log (x)}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(12 + 2*x + (732 - 1248*x + 582*x^2 - 27*x^3 - 27*x^4)*Log[x] + (-444 + 466*x - 72*x^2 - 27*x^3)*Log[x]*Lo
g[(6 + x)/(x*Log[x])] + (90 - 39*x - 9*x^2)*Log[x]*Log[(6 + x)/(x*Log[x])]^2 + (-6 - x)*Log[x]*Log[(6 + x)/(x*
Log[x])]^3)/((-750 + 1225*x - 585*x^2 + 27*x^3 + 27*x^4)*Log[x] + (450 - 465*x + 72*x^2 + 27*x^3)*Log[x]*Log[(
6 + x)/(x*Log[x])] + (-90 + 39*x + 9*x^2)*Log[x]*Log[(6 + x)/(x*Log[x])]^2 + (6 + x)*Log[x]*Log[(6 + x)/(x*Log
[x])]^3),x]

[Out]

-x - 6*Defer[Int][x/(-5 + 3*x + Log[(6 + x)/(x*Log[x])])^3, x] + 12*Defer[Int][1/((6 + x)*(-5 + 3*x + Log[(6 +
x)/(x*Log[x])])^3), x] + 2*Defer[Int][1/(Log[x]*(-5 + 3*x + Log[(6 + x)/(x*Log[x])])^3), x] + Defer[Int][(-5
+ 3*x + Log[(6 + x)/(x*Log[x])])^(-2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-2 (6+x)+\log (x) \left (3 \left (-244+416 x-194 x^2+9 x^3+9 x^4\right )+\left (444-466 x+72 x^2+27 x^3\right ) \log \left (\frac {6+x}{x \log (x)}\right )+\left (-90+39 x+9 x^2\right ) \log ^2\left (\frac {6+x}{x \log (x)}\right )+(6+x) \log ^3\left (\frac {6+x}{x \log (x)}\right )\right )}{(6+x) \log (x) \left (5-3 x-\log \left (\frac {6+x}{x \log (x)}\right )\right )^3} \, dx\\ &=\int \left (-1-\frac {2 \left (-6-x-6 \log (x)+18 x \log (x)+3 x^2 \log (x)\right )}{(6+x) \log (x) \left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^3}+\frac {1}{\left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^2}\right ) \, dx\\ &=-x-2 \int \frac {-6-x-6 \log (x)+18 x \log (x)+3 x^2 \log (x)}{(6+x) \log (x) \left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^3} \, dx+\int \frac {1}{\left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^2} \, dx\\ &=-x-2 \int \left (-\frac {6}{(6+x) \left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^3}+\frac {18 x}{(6+x) \left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^3}+\frac {3 x^2}{(6+x) \left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^3}-\frac {6}{(6+x) \log (x) \left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^3}-\frac {x}{(6+x) \log (x) \left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^3}\right ) \, dx+\int \frac {1}{\left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^2} \, dx\\ &=-x+2 \int \frac {x}{(6+x) \log (x) \left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^3} \, dx-6 \int \frac {x^2}{(6+x) \left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^3} \, dx+12 \int \frac {1}{(6+x) \left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^3} \, dx+12 \int \frac {1}{(6+x) \log (x) \left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^3} \, dx-36 \int \frac {x}{(6+x) \left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^3} \, dx+\int \frac {1}{\left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^2} \, dx\\ &=-x+2 \int \left (\frac {1}{\log (x) \left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^3}-\frac {6}{(6+x) \log (x) \left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^3}\right ) \, dx-6 \int \left (-\frac {6}{\left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^3}+\frac {x}{\left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^3}+\frac {36}{(6+x) \left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^3}\right ) \, dx+12 \int \frac {1}{(6+x) \left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^3} \, dx+12 \int \frac {1}{(6+x) \log (x) \left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^3} \, dx-36 \int \left (\frac {1}{\left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^3}-\frac {6}{(6+x) \left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^3}\right ) \, dx+\int \frac {1}{\left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^2} \, dx\\ &=-x+2 \int \frac {1}{\log (x) \left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^3} \, dx-6 \int \frac {x}{\left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^3} \, dx+12 \int \frac {1}{(6+x) \left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^3} \, dx+\int \frac {1}{\left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.13, size = 25, normalized size = 0.96 \begin {gather*} -x+\frac {x}{\left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(12 + 2*x + (732 - 1248*x + 582*x^2 - 27*x^3 - 27*x^4)*Log[x] + (-444 + 466*x - 72*x^2 - 27*x^3)*Log
[x]*Log[(6 + x)/(x*Log[x])] + (90 - 39*x - 9*x^2)*Log[x]*Log[(6 + x)/(x*Log[x])]^2 + (-6 - x)*Log[x]*Log[(6 +
x)/(x*Log[x])]^3)/((-750 + 1225*x - 585*x^2 + 27*x^3 + 27*x^4)*Log[x] + (450 - 465*x + 72*x^2 + 27*x^3)*Log[x]
*Log[(6 + x)/(x*Log[x])] + (-90 + 39*x + 9*x^2)*Log[x]*Log[(6 + x)/(x*Log[x])]^2 + (6 + x)*Log[x]*Log[(6 + x)/
(x*Log[x])]^3),x]

[Out]

-x + x/(-5 + 3*x + Log[(6 + x)/(x*Log[x])])^2

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fricas [B]  time = 0.70, size = 100, normalized size = 3.85 \begin {gather*} -\frac {9 \, x^{3} + x \log \left (\frac {x + 6}{x \log \relax (x)}\right )^{2} - 30 \, x^{2} + 2 \, {\left (3 \, x^{2} - 5 \, x\right )} \log \left (\frac {x + 6}{x \log \relax (x)}\right ) + 24 \, x}{9 \, x^{2} + 2 \, {\left (3 \, x - 5\right )} \log \left (\frac {x + 6}{x \log \relax (x)}\right ) + \log \left (\frac {x + 6}{x \log \relax (x)}\right )^{2} - 30 \, x + 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x-6)*log(x)*log((x+6)/x/log(x))^3+(-9*x^2-39*x+90)*log(x)*log((x+6)/x/log(x))^2+(-27*x^3-72*x^2+4
66*x-444)*log(x)*log((x+6)/x/log(x))+(-27*x^4-27*x^3+582*x^2-1248*x+732)*log(x)+2*x+12)/((x+6)*log(x)*log((x+6
)/x/log(x))^3+(9*x^2+39*x-90)*log(x)*log((x+6)/x/log(x))^2+(27*x^3+72*x^2-465*x+450)*log(x)*log((x+6)/x/log(x)
)+(27*x^4+27*x^3-585*x^2+1225*x-750)*log(x)),x, algorithm="fricas")

[Out]

-(9*x^3 + x*log((x + 6)/(x*log(x)))^2 - 30*x^2 + 2*(3*x^2 - 5*x)*log((x + 6)/(x*log(x))) + 24*x)/(9*x^2 + 2*(3
*x - 5)*log((x + 6)/(x*log(x))) + log((x + 6)/(x*log(x)))^2 - 30*x + 25)

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giac [B]  time = 3.04, size = 466, normalized size = 17.92 \begin {gather*} -x + \frac {3 \, x^{3} \log \relax (x) + 18 \, x^{2} \log \relax (x) - x^{2} - 6 \, x \log \relax (x) - 6 \, x}{27 \, x^{4} \log \relax (x) + 18 \, x^{3} \log \left (x + 6\right ) \log \relax (x) + 3 \, x^{2} \log \left (x + 6\right )^{2} \log \relax (x) - 18 \, x^{3} \log \relax (x)^{2} - 6 \, x^{2} \log \left (x + 6\right ) \log \relax (x)^{2} + 3 \, x^{2} \log \relax (x)^{3} - 18 \, x^{3} \log \relax (x) \log \left (\log \relax (x)\right ) - 6 \, x^{2} \log \left (x + 6\right ) \log \relax (x) \log \left (\log \relax (x)\right ) + 6 \, x^{2} \log \relax (x)^{2} \log \left (\log \relax (x)\right ) + 3 \, x^{2} \log \relax (x) \log \left (\log \relax (x)\right )^{2} + 72 \, x^{3} \log \relax (x) + 78 \, x^{2} \log \left (x + 6\right ) \log \relax (x) + 18 \, x \log \left (x + 6\right )^{2} \log \relax (x) - 78 \, x^{2} \log \relax (x)^{2} - 36 \, x \log \left (x + 6\right ) \log \relax (x)^{2} + 18 \, x \log \relax (x)^{3} - 78 \, x^{2} \log \relax (x) \log \left (\log \relax (x)\right ) - 36 \, x \log \left (x + 6\right ) \log \relax (x) \log \left (\log \relax (x)\right ) + 36 \, x \log \relax (x)^{2} \log \left (\log \relax (x)\right ) + 18 \, x \log \relax (x) \log \left (\log \relax (x)\right )^{2} - 9 \, x^{3} - 6 \, x^{2} \log \left (x + 6\right ) - x \log \left (x + 6\right )^{2} - 513 \, x^{2} \log \relax (x) - 214 \, x \log \left (x + 6\right ) \log \relax (x) - 6 \, \log \left (x + 6\right )^{2} \log \relax (x) + 215 \, x \log \relax (x)^{2} + 12 \, \log \left (x + 6\right ) \log \relax (x)^{2} - 6 \, \log \relax (x)^{3} + 6 \, x^{2} \log \left (\log \relax (x)\right ) + 2 \, x \log \left (x + 6\right ) \log \left (\log \relax (x)\right ) + 214 \, x \log \relax (x) \log \left (\log \relax (x)\right ) + 12 \, \log \left (x + 6\right ) \log \relax (x) \log \left (\log \relax (x)\right ) - 12 \, \log \relax (x)^{2} \log \left (\log \relax (x)\right ) - x \log \left (\log \relax (x)\right )^{2} - 6 \, \log \relax (x) \log \left (\log \relax (x)\right )^{2} - 24 \, x^{2} - 26 \, x \log \left (x + 6\right ) - 6 \, \log \left (x + 6\right )^{2} + 656 \, x \log \relax (x) + 72 \, \log \left (x + 6\right ) \log \relax (x) - 66 \, \log \relax (x)^{2} + 26 \, x \log \left (\log \relax (x)\right ) + 12 \, \log \left (x + 6\right ) \log \left (\log \relax (x)\right ) - 72 \, \log \relax (x) \log \left (\log \relax (x)\right ) - 6 \, \log \left (\log \relax (x)\right )^{2} + 155 \, x + 60 \, \log \left (x + 6\right ) - 210 \, \log \relax (x) - 60 \, \log \left (\log \relax (x)\right ) - 150} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x-6)*log(x)*log((x+6)/x/log(x))^3+(-9*x^2-39*x+90)*log(x)*log((x+6)/x/log(x))^2+(-27*x^3-72*x^2+4
66*x-444)*log(x)*log((x+6)/x/log(x))+(-27*x^4-27*x^3+582*x^2-1248*x+732)*log(x)+2*x+12)/((x+6)*log(x)*log((x+6
)/x/log(x))^3+(9*x^2+39*x-90)*log(x)*log((x+6)/x/log(x))^2+(27*x^3+72*x^2-465*x+450)*log(x)*log((x+6)/x/log(x)
)+(27*x^4+27*x^3-585*x^2+1225*x-750)*log(x)),x, algorithm="giac")

[Out]

-x + (3*x^3*log(x) + 18*x^2*log(x) - x^2 - 6*x*log(x) - 6*x)/(27*x^4*log(x) + 18*x^3*log(x + 6)*log(x) + 3*x^2
*log(x + 6)^2*log(x) - 18*x^3*log(x)^2 - 6*x^2*log(x + 6)*log(x)^2 + 3*x^2*log(x)^3 - 18*x^3*log(x)*log(log(x)
) - 6*x^2*log(x + 6)*log(x)*log(log(x)) + 6*x^2*log(x)^2*log(log(x)) + 3*x^2*log(x)*log(log(x))^2 + 72*x^3*log
(x) + 78*x^2*log(x + 6)*log(x) + 18*x*log(x + 6)^2*log(x) - 78*x^2*log(x)^2 - 36*x*log(x + 6)*log(x)^2 + 18*x*
log(x)^3 - 78*x^2*log(x)*log(log(x)) - 36*x*log(x + 6)*log(x)*log(log(x)) + 36*x*log(x)^2*log(log(x)) + 18*x*l
og(x)*log(log(x))^2 - 9*x^3 - 6*x^2*log(x + 6) - x*log(x + 6)^2 - 513*x^2*log(x) - 214*x*log(x + 6)*log(x) - 6
*log(x + 6)^2*log(x) + 215*x*log(x)^2 + 12*log(x + 6)*log(x)^2 - 6*log(x)^3 + 6*x^2*log(log(x)) + 2*x*log(x +
6)*log(log(x)) + 214*x*log(x)*log(log(x)) + 12*log(x + 6)*log(x)*log(log(x)) - 12*log(x)^2*log(log(x)) - x*log
(log(x))^2 - 6*log(x)*log(log(x))^2 - 24*x^2 - 26*x*log(x + 6) - 6*log(x + 6)^2 + 656*x*log(x) + 72*log(x + 6)
*log(x) - 66*log(x)^2 + 26*x*log(log(x)) + 12*log(x + 6)*log(log(x)) - 72*log(x)*log(log(x)) - 6*log(log(x))^2
+ 155*x + 60*log(x + 6) - 210*log(x) - 60*log(log(x)) - 150)

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maple [B]  time = 1.62, size = 84, normalized size = 3.23

method	result	size
norman	\(\frac {-24 x +30 x^{2}+10 \ln \left (\frac {x +6}{x \ln \relax (x )}\right ) x -9 x^{3}-6 \ln \left (\frac {x +6}{x \ln \relax (x )}\right ) x^{2}-\ln \left (\frac {x +6}{x \ln \relax (x )}\right )^{2} x}{\left (\ln \left (\frac {x +6}{x \ln \relax (x )}\right )+3 x -5\right )^{2}}\)	\(84\)
risch	\(-x +\frac {4 x}{\left (2 \ln \left (x +6\right )-i \pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (x +6\right )}{\ln \relax (x )}\right ) \mathrm {csgn}\left (\frac {i \left (x +6\right )}{x \ln \relax (x )}\right )-i \pi \,\mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right ) \mathrm {csgn}\left (i \left (x +6\right )\right ) \mathrm {csgn}\left (\frac {i \left (x +6\right )}{\ln \relax (x )}\right )-i \pi \mathrm {csgn}\left (\frac {i \left (x +6\right )}{\ln \relax (x )}\right )^{3}-i \pi \mathrm {csgn}\left (\frac {i \left (x +6\right )}{x \ln \relax (x )}\right )^{3}-10-2 \ln \relax (x )-2 \ln \left (\ln \relax (x )\right )+6 x +i \pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (x +6\right )}{x \ln \relax (x )}\right )^{2}+i \pi \,\mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right ) \mathrm {csgn}\left (\frac {i \left (x +6\right )}{\ln \relax (x )}\right )^{2}+i \pi \,\mathrm {csgn}\left (i \left (x +6\right )\right ) \mathrm {csgn}\left (\frac {i \left (x +6\right )}{\ln \relax (x )}\right )^{2}+i \pi \,\mathrm {csgn}\left (\frac {i \left (x +6\right )}{\ln \relax (x )}\right ) \mathrm {csgn}\left (\frac {i \left (x +6\right )}{x \ln \relax (x )}\right )^{2}\right )^{2}}\)	\(240\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x-6)*ln(x)*ln((x+6)/x/ln(x))^3+(-9*x^2-39*x+90)*ln(x)*ln((x+6)/x/ln(x))^2+(-27*x^3-72*x^2+466*x-444)*ln
(x)*ln((x+6)/x/ln(x))+(-27*x^4-27*x^3+582*x^2-1248*x+732)*ln(x)+2*x+12)/((x+6)*ln(x)*ln((x+6)/x/ln(x))^3+(9*x^
2+39*x-90)*ln(x)*ln((x+6)/x/ln(x))^2+(27*x^3+72*x^2-465*x+450)*ln(x)*ln((x+6)/x/ln(x))+(27*x^4+27*x^3-585*x^2+
1225*x-750)*ln(x)),x,method=_RETURNVERBOSE)

[Out]

(-24*x+30*x^2+10*ln((x+6)/x/ln(x))*x-9*x^3-6*ln((x+6)/x/ln(x))*x^2-ln((x+6)/x/ln(x))^2*x)/(ln((x+6)/x/ln(x))+3
*x-5)^2

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maxima [B]  time = 0.45, size = 165, normalized size = 6.35 \begin {gather*} -\frac {9 \, x^{3} + x \log \left (x + 6\right )^{2} + x \log \relax (x)^{2} + x \log \left (\log \relax (x)\right )^{2} - 30 \, x^{2} + 2 \, {\left (3 \, x^{2} - x \log \relax (x) - x \log \left (\log \relax (x)\right ) - 5 \, x\right )} \log \left (x + 6\right ) - 2 \, {\left (3 \, x^{2} - 5 \, x\right )} \log \relax (x) - 2 \, {\left (3 \, x^{2} - x \log \relax (x) - 5 \, x\right )} \log \left (\log \relax (x)\right ) + 24 \, x}{9 \, x^{2} + 2 \, {\left (3 \, x - \log \relax (x) - \log \left (\log \relax (x)\right ) - 5\right )} \log \left (x + 6\right ) + \log \left (x + 6\right )^{2} - 2 \, {\left (3 \, x - 5\right )} \log \relax (x) + \log \relax (x)^{2} - 2 \, {\left (3 \, x - \log \relax (x) - 5\right )} \log \left (\log \relax (x)\right ) + \log \left (\log \relax (x)\right )^{2} - 30 \, x + 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x-6)*log(x)*log((x+6)/x/log(x))^3+(-9*x^2-39*x+90)*log(x)*log((x+6)/x/log(x))^2+(-27*x^3-72*x^2+4
66*x-444)*log(x)*log((x+6)/x/log(x))+(-27*x^4-27*x^3+582*x^2-1248*x+732)*log(x)+2*x+12)/((x+6)*log(x)*log((x+6
)/x/log(x))^3+(9*x^2+39*x-90)*log(x)*log((x+6)/x/log(x))^2+(27*x^3+72*x^2-465*x+450)*log(x)*log((x+6)/x/log(x)
)+(27*x^4+27*x^3-585*x^2+1225*x-750)*log(x)),x, algorithm="maxima")

[Out]

-(9*x^3 + x*log(x + 6)^2 + x*log(x)^2 + x*log(log(x))^2 - 30*x^2 + 2*(3*x^2 - x*log(x) - x*log(log(x)) - 5*x)*
log(x + 6) - 2*(3*x^2 - 5*x)*log(x) - 2*(3*x^2 - x*log(x) - 5*x)*log(log(x)) + 24*x)/(9*x^2 + 2*(3*x - log(x)
- log(log(x)) - 5)*log(x + 6) + log(x + 6)^2 - 2*(3*x - 5)*log(x) + log(x)^2 - 2*(3*x - log(x) - 5)*log(log(x)
) + log(log(x))^2 - 30*x + 25)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {\ln \relax (x)\,\left (x+6\right )\,{\ln \left (\frac {x+6}{x\,\ln \relax (x)}\right )}^3+\ln \relax (x)\,\left (9\,x^2+39\,x-90\right )\,{\ln \left (\frac {x+6}{x\,\ln \relax (x)}\right )}^2+\ln \relax (x)\,\left (27\,x^3+72\,x^2-466\,x+444\right )\,\ln \left (\frac {x+6}{x\,\ln \relax (x)}\right )-2\,x+\ln \relax (x)\,\left (27\,x^4+27\,x^3-582\,x^2+1248\,x-732\right )-12}{\ln \relax (x)\,\left (x+6\right )\,{\ln \left (\frac {x+6}{x\,\ln \relax (x)}\right )}^3+\ln \relax (x)\,\left (9\,x^2+39\,x-90\right )\,{\ln \left (\frac {x+6}{x\,\ln \relax (x)}\right )}^2+\ln \relax (x)\,\left (27\,x^3+72\,x^2-465\,x+450\right )\,\ln \left (\frac {x+6}{x\,\ln \relax (x)}\right )+\ln \relax (x)\,\left (27\,x^4+27\,x^3-585\,x^2+1225\,x-750\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x)*(1248*x - 582*x^2 + 27*x^3 + 27*x^4 - 732) - 2*x + log((x + 6)/(x*log(x)))^3*log(x)*(x + 6) + log
((x + 6)/(x*log(x)))*log(x)*(72*x^2 - 466*x + 27*x^3 + 444) + log((x + 6)/(x*log(x)))^2*log(x)*(39*x + 9*x^2 -
90) - 12)/(log(x)*(1225*x - 585*x^2 + 27*x^3 + 27*x^4 - 750) + log((x + 6)/(x*log(x)))^3*log(x)*(x + 6) + log
((x + 6)/(x*log(x)))*log(x)*(72*x^2 - 465*x + 27*x^3 + 450) + log((x + 6)/(x*log(x)))^2*log(x)*(39*x + 9*x^2 -
90)),x)

[Out]

int(-(log(x)*(1248*x - 582*x^2 + 27*x^3 + 27*x^4 - 732) - 2*x + log((x + 6)/(x*log(x)))^3*log(x)*(x + 6) + log
((x + 6)/(x*log(x)))*log(x)*(72*x^2 - 466*x + 27*x^3 + 444) + log((x + 6)/(x*log(x)))^2*log(x)*(39*x + 9*x^2 -
90) - 12)/(log(x)*(1225*x - 585*x^2 + 27*x^3 + 27*x^4 - 750) + log((x + 6)/(x*log(x)))^3*log(x)*(x + 6) + log
((x + 6)/(x*log(x)))*log(x)*(72*x^2 - 465*x + 27*x^3 + 450) + log((x + 6)/(x*log(x)))^2*log(x)*(39*x + 9*x^2 -
90)), x)

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sympy [B]  time = 0.45, size = 39, normalized size = 1.50 \begin {gather*} - x + \frac {x}{9 x^{2} - 30 x + \left (6 x - 10\right ) \log {\left (\frac {x + 6}{x \log {\relax (x )}} \right )} + \log {\left (\frac {x + 6}{x \log {\relax (x )}} \right )}^{2} + 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x-6)*ln(x)*ln((x+6)/x/ln(x))**3+(-9*x**2-39*x+90)*ln(x)*ln((x+6)/x/ln(x))**2+(-27*x**3-72*x**2+46
6*x-444)*ln(x)*ln((x+6)/x/ln(x))+(-27*x**4-27*x**3+582*x**2-1248*x+732)*ln(x)+2*x+12)/((x+6)*ln(x)*ln((x+6)/x/
ln(x))**3+(9*x**2+39*x-90)*ln(x)*ln((x+6)/x/ln(x))**2+(27*x**3+72*x**2-465*x+450)*ln(x)*ln((x+6)/x/ln(x))+(27*
x**4+27*x**3-585*x**2+1225*x-750)*ln(x)),x)

[Out]

-x + x/(9*x**2 - 30*x + (6*x - 10)*log((x + 6)/(x*log(x))) + log((x + 6)/(x*log(x)))**2 + 25)

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