3.36.40 \(\int \frac {-2+15 x-8 x^2+(1-6 x+3 x^2) \log (x)}{(-3 x+9 x^2-3 x^3+(x-3 x^2+x^3) \log (x)) \log (-54 x+162 x^2-54 x^3+(18 x-54 x^2+18 x^3) \log (4)+(18 x-54 x^2+18 x^3+(-6 x+18 x^2-6 x^3) \log (4)) \log (x))} \, dx\)

Optimal. Leaf size=22 \[ \log \left (\log \left (6 x \left (-(-1+x)^2+x\right ) (-3+\log (4)) (-3+\log (x))\right )\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 21.45, 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+15 x-8 x^2+\left (1-6 x+3 x^2\right ) \log (x)}{\left (-3 x+9 x^2-3 x^3+\left (x-3 x^2+x^3\right ) \log (x)\right ) \log \left (-54 x+162 x^2-54 x^3+\left (18 x-54 x^2+18 x^3\right ) \log (4)+\left (18 x-54 x^2+18 x^3+\left (-6 x+18 x^2-6 x^3\right ) \log (4)\right ) \log (x)\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-2 + 15*x - 8*x^2 + (1 - 6*x + 3*x^2)*Log[x])/((-3*x + 9*x^2 - 3*x^3 + (x - 3*x^2 + x^3)*Log[x])*Log[-54*
x + 162*x^2 - 54*x^3 + (18*x - 54*x^2 + 18*x^3)*Log[4] + (18*x - 54*x^2 + 18*x^3 + (-6*x + 18*x^2 - 6*x^3)*Log
[4])*Log[x]]),x]

[Out]

(-18*Defer[Int][1/((3 + Sqrt[5] - 2*x)*(-3 + Log[x])*Log[-6*x*(1 - 3*x + x^2)*(-3 + Log[4])*(-3 + Log[x])]), x
])/Sqrt[5] - 2*Defer[Int][1/(x*(-3 + Log[x])*Log[-6*x*(1 - 3*x + x^2)*(-3 + Log[4])*(-3 + Log[x])]), x] - (6*(
5 + 3*Sqrt[5])*Defer[Int][1/((-3 - Sqrt[5] + 2*x)*(-3 + Log[x])*Log[-6*x*(1 - 3*x + x^2)*(-3 + Log[4])*(-3 + L
og[x])]), x])/5 - (18*Defer[Int][1/((-3 + Sqrt[5] + 2*x)*(-3 + Log[x])*Log[-6*x*(1 - 3*x + x^2)*(-3 + Log[4])*
(-3 + Log[x])]), x])/Sqrt[5] - (6*(5 - 3*Sqrt[5])*Defer[Int][1/((-3 + Sqrt[5] + 2*x)*(-3 + Log[x])*Log[-6*x*(1
- 3*x + x^2)*(-3 + Log[4])*(-3 + Log[x])]), x])/5 - (24*Defer[Int][Log[x]/((3 + Sqrt[5] - 2*x)*(-3 + Log[x])*
Log[-6*x*(1 - 3*x + x^2)*(-3 + Log[4])*(-3 + Log[x])]), x])/Sqrt[5] + 6*Sqrt[5]*Defer[Int][Log[x]/((3 + Sqrt[5
] - 2*x)*(-3 + Log[x])*Log[-6*x*(1 - 3*x + x^2)*(-3 + Log[4])*(-3 + Log[x])]), x] + Defer[Int][Log[x]/(x*(-3 +
Log[x])*Log[-6*x*(1 - 3*x + x^2)*(-3 + Log[4])*(-3 + Log[x])]), x] + (2*(5 + 3*Sqrt[5])*Defer[Int][Log[x]/((-
3 - Sqrt[5] + 2*x)*(-3 + Log[x])*Log[-6*x*(1 - 3*x + x^2)*(-3 + Log[4])*(-3 + Log[x])]), x])/5 - (24*Defer[Int
][Log[x]/((-3 + Sqrt[5] + 2*x)*(-3 + Log[x])*Log[-6*x*(1 - 3*x + x^2)*(-3 + Log[4])*(-3 + Log[x])]), x])/Sqrt[
5] + 6*Sqrt[5]*Defer[Int][Log[x]/((-3 + Sqrt[5] + 2*x)*(-3 + Log[x])*Log[-6*x*(1 - 3*x + x^2)*(-3 + Log[4])*(-
3 + Log[x])]), x] + (2*(5 - 3*Sqrt[5])*Defer[Int][Log[x]/((-3 + Sqrt[5] + 2*x)*(-3 + Log[x])*Log[-6*x*(1 - 3*x
+ x^2)*(-3 + Log[4])*(-3 + Log[x])]), x])/5

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2-15 x+8 x^2-\left (1-6 x+3 x^2\right ) \log (x)}{x \left (1-3 x+x^2\right ) (3-\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )} \, dx\\ &=\int \left (\frac {-2+15 x-8 x^2+\log (x)-6 x \log (x)+3 x^2 \log (x)}{x (-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )}-\frac {(-3+x) \left (-2+15 x-8 x^2+\log (x)-6 x \log (x)+3 x^2 \log (x)\right )}{\left (1-3 x+x^2\right ) (-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )}\right ) \, dx\\ &=\int \frac {-2+15 x-8 x^2+\log (x)-6 x \log (x)+3 x^2 \log (x)}{x (-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )} \, dx-\int \frac {(-3+x) \left (-2+15 x-8 x^2+\log (x)-6 x \log (x)+3 x^2 \log (x)\right )}{\left (1-3 x+x^2\right ) (-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )} \, dx\\ &=\int \left (\frac {15}{(-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )}-\frac {2}{x (-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )}-\frac {8 x}{(-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )}-\frac {6 \log (x)}{(-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )}+\frac {\log (x)}{x (-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )}+\frac {3 x \log (x)}{(-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )}\right ) \, dx-\int \left (\frac {6}{\left (1-3 x+x^2\right ) (-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )}-\frac {47 x}{\left (1-3 x+x^2\right ) (-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )}+\frac {39 x^2}{\left (1-3 x+x^2\right ) (-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )}-\frac {8 x^3}{\left (1-3 x+x^2\right ) (-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )}-\frac {3 \log (x)}{\left (1-3 x+x^2\right ) (-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )}+\frac {19 x \log (x)}{\left (1-3 x+x^2\right ) (-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )}-\frac {15 x^2 \log (x)}{\left (1-3 x+x^2\right ) (-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )}+\frac {3 x^3 \log (x)}{\left (1-3 x+x^2\right ) (-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )}\right ) \, dx\\ &=-\left (2 \int \frac {1}{x (-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )} \, dx\right )+3 \int \frac {x \log (x)}{(-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )} \, dx+3 \int \frac {\log (x)}{\left (1-3 x+x^2\right ) (-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )} \, dx-3 \int \frac {x^3 \log (x)}{\left (1-3 x+x^2\right ) (-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )} \, dx-6 \int \frac {1}{\left (1-3 x+x^2\right ) (-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )} \, dx-6 \int \frac {\log (x)}{(-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )} \, dx-8 \int \frac {x}{(-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )} \, dx+8 \int \frac {x^3}{\left (1-3 x+x^2\right ) (-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )} \, dx+15 \int \frac {1}{(-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )} \, dx+15 \int \frac {x^2 \log (x)}{\left (1-3 x+x^2\right ) (-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )} \, dx-19 \int \frac {x \log (x)}{\left (1-3 x+x^2\right ) (-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )} \, dx-39 \int \frac {x^2}{\left (1-3 x+x^2\right ) (-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )} \, dx+47 \int \frac {x}{\left (1-3 x+x^2\right ) (-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )} \, dx+\int \frac {\log (x)}{x (-3+\log (x)) \log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.32, size = 21, normalized size = 0.95 \begin {gather*} \log \left (\log \left (-6 x \left (1-3 x+x^2\right ) (-3+\log (4)) (-3+\log (x))\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2 + 15*x - 8*x^2 + (1 - 6*x + 3*x^2)*Log[x])/((-3*x + 9*x^2 - 3*x^3 + (x - 3*x^2 + x^3)*Log[x])*Lo
g[-54*x + 162*x^2 - 54*x^3 + (18*x - 54*x^2 + 18*x^3)*Log[4] + (18*x - 54*x^2 + 18*x^3 + (-6*x + 18*x^2 - 6*x^
3)*Log[4])*Log[x]]),x]

[Out]

Log[Log[-6*x*(1 - 3*x + x^2)*(-3 + Log[4])*(-3 + Log[x])]]

________________________________________________________________________________________

fricas [B]  time = 0.65, size = 62, normalized size = 2.82 \begin {gather*} \log \left (\log \left (-54 \, x^{3} + 162 \, x^{2} + 36 \, {\left (x^{3} - 3 \, x^{2} + x\right )} \log \relax (2) + 6 \, {\left (3 \, x^{3} - 9 \, x^{2} - 2 \, {\left (x^{3} - 3 \, x^{2} + x\right )} \log \relax (2) + 3 \, x\right )} \log \relax (x) - 54 \, x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^2-6*x+1)*log(x)-8*x^2+15*x-2)/((x^3-3*x^2+x)*log(x)-3*x^3+9*x^2-3*x)/log((2*(-6*x^3+18*x^2-6*x
)*log(2)+18*x^3-54*x^2+18*x)*log(x)+2*(18*x^3-54*x^2+18*x)*log(2)-54*x^3+162*x^2-54*x),x, algorithm="fricas")

[Out]

log(log(-54*x^3 + 162*x^2 + 36*(x^3 - 3*x^2 + x)*log(2) + 6*(3*x^3 - 9*x^2 - 2*(x^3 - 3*x^2 + x)*log(2) + 3*x)
*log(x) - 54*x))

________________________________________________________________________________________

giac [B]  time = 0.24, size = 71, normalized size = 3.23 \begin {gather*} \log \left (\log \relax (2) + \log \left (-6 \, x^{2} \log \relax (2) \log \relax (x) + 18 \, x^{2} \log \relax (2) + 9 \, x^{2} \log \relax (x) + 18 \, x \log \relax (2) \log \relax (x) - 27 \, x^{2} - 54 \, x \log \relax (2) - 27 \, x \log \relax (x) - 6 \, \log \relax (2) \log \relax (x) + 81 \, x + 18 \, \log \relax (2) + 9 \, \log \relax (x) - 27\right ) + \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^2-6*x+1)*log(x)-8*x^2+15*x-2)/((x^3-3*x^2+x)*log(x)-3*x^3+9*x^2-3*x)/log((2*(-6*x^3+18*x^2-6*x
)*log(2)+18*x^3-54*x^2+18*x)*log(x)+2*(18*x^3-54*x^2+18*x)*log(2)-54*x^3+162*x^2-54*x),x, algorithm="giac")

[Out]

log(log(2) + log(-6*x^2*log(2)*log(x) + 18*x^2*log(2) + 9*x^2*log(x) + 18*x*log(2)*log(x) - 27*x^2 - 54*x*log(
2) - 27*x*log(x) - 6*log(2)*log(x) + 81*x + 18*log(2) + 9*log(x) - 27) + log(x))

________________________________________________________________________________________

maple [C]  time = 0.19, size = 309, normalized size = 14.05

method	result	size
risch	\(\ln \left (\ln \left (x^{2}-3 x +1\right )+\frac {i \left (-2 \pi \mathrm {csgn}\left (i x \left (\ln \relax (x )-3\right ) \left (x^{2}-3 x +1\right )\right )^{2}-\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (\ln \relax (x )-3\right ) \left (x^{2}-3 x +1\right )\right ) \mathrm {csgn}\left (i x \left (\ln \relax (x )-3\right ) \left (x^{2}-3 x +1\right )\right )+\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \left (\ln \relax (x )-3\right ) \left (x^{2}-3 x +1\right )\right )^{2}-\pi \,\mathrm {csgn}\left (i \left (\ln \relax (x )-3\right )\right ) \mathrm {csgn}\left (i \left (x^{2}-3 x +1\right )\right ) \mathrm {csgn}\left (i \left (\ln \relax (x )-3\right ) \left (x^{2}-3 x +1\right )\right )+\pi \,\mathrm {csgn}\left (i \left (\ln \relax (x )-3\right )\right ) \mathrm {csgn}\left (i \left (\ln \relax (x )-3\right ) \left (x^{2}-3 x +1\right )\right )^{2}+\pi \,\mathrm {csgn}\left (i \left (x^{2}-3 x +1\right )\right ) \mathrm {csgn}\left (i \left (\ln \relax (x )-3\right ) \left (x^{2}-3 x +1\right )\right )^{2}-\pi \mathrm {csgn}\left (i \left (\ln \relax (x )-3\right ) \left (x^{2}-3 x +1\right )\right )^{3}+\pi \,\mathrm {csgn}\left (i \left (\ln \relax (x )-3\right ) \left (x^{2}-3 x +1\right )\right ) \mathrm {csgn}\left (i x \left (\ln \relax (x )-3\right ) \left (x^{2}-3 x +1\right )\right )^{2}+\pi \mathrm {csgn}\left (i x \left (\ln \relax (x )-3\right ) \left (x^{2}-3 x +1\right )\right )^{3}-2 i \ln \left (\ln \relax (x )-3\right )-2 i \ln \relax (3)-2 i \ln \relax (x )-2 i \ln \relax (2)+2 \pi \right )}{2}\right )\)	\(309\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*x^2-6*x+1)*ln(x)-8*x^2+15*x-2)/((x^3-3*x^2+x)*ln(x)-3*x^3+9*x^2-3*x)/ln((2*(-6*x^3+18*x^2-6*x)*ln(2)+1
8*x^3-54*x^2+18*x)*ln(x)+2*(18*x^3-54*x^2+18*x)*ln(2)-54*x^3+162*x^2-54*x),x,method=_RETURNVERBOSE)

[Out]

ln(ln(x^2-3*x+1)+1/2*I*(-2*Pi*csgn(I*x*(ln(x)-3)*(x^2-3*x+1))^2-Pi*csgn(I*x)*csgn(I*(ln(x)-3)*(x^2-3*x+1))*csg
n(I*x*(ln(x)-3)*(x^2-3*x+1))+Pi*csgn(I*x)*csgn(I*x*(ln(x)-3)*(x^2-3*x+1))^2-Pi*csgn(I*(ln(x)-3))*csgn(I*(x^2-3
*x+1))*csgn(I*(ln(x)-3)*(x^2-3*x+1))+Pi*csgn(I*(ln(x)-3))*csgn(I*(ln(x)-3)*(x^2-3*x+1))^2+Pi*csgn(I*(x^2-3*x+1
))*csgn(I*(ln(x)-3)*(x^2-3*x+1))^2-Pi*csgn(I*(ln(x)-3)*(x^2-3*x+1))^3+Pi*csgn(I*(ln(x)-3)*(x^2-3*x+1))*csgn(I*
x*(ln(x)-3)*(x^2-3*x+1))^2+Pi*csgn(I*x*(ln(x)-3)*(x^2-3*x+1))^3-2*I*ln(ln(x)-3)-2*I*ln(3)-2*I*ln(x)-2*I*ln(2)+
2*Pi))

________________________________________________________________________________________

maxima [C]  time = 0.79, size = 32, normalized size = 1.45 \begin {gather*} \log \left (i \, \pi + \log \relax (3) + \log \relax (2) + \log \left (x^{2} - 3 \, x + 1\right ) + \log \relax (x) + \log \left (2 \, \log \relax (2) - 3\right ) + \log \left (\log \relax (x) - 3\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^2-6*x+1)*log(x)-8*x^2+15*x-2)/((x^3-3*x^2+x)*log(x)-3*x^3+9*x^2-3*x)/log((2*(-6*x^3+18*x^2-6*x
)*log(2)+18*x^3-54*x^2+18*x)*log(x)+2*(18*x^3-54*x^2+18*x)*log(2)-54*x^3+162*x^2-54*x),x, algorithm="maxima")

[Out]

log(I*pi + log(3) + log(2) + log(x^2 - 3*x + 1) + log(x) + log(2*log(2) - 3) + log(log(x) - 3))

________________________________________________________________________________________

mupad [B]  time = 3.03, size = 69, normalized size = 3.14 \begin {gather*} \ln \left (\ln \left (\ln \relax (x)\,\left (18\,x-2\,\ln \relax (2)\,\left (6\,x^3-18\,x^2+6\,x\right )-54\,x^2+18\,x^3\right )-54\,x+2\,\ln \relax (2)\,\left (18\,x^3-54\,x^2+18\,x\right )+162\,x^2-54\,x^3\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(15*x + log(x)*(3*x^2 - 6*x + 1) - 8*x^2 - 2)/(log(log(x)*(18*x - 2*log(2)*(6*x - 18*x^2 + 6*x^3) - 54*x^
2 + 18*x^3) - 54*x + 2*log(2)*(18*x - 54*x^2 + 18*x^3) + 162*x^2 - 54*x^3)*(3*x - log(x)*(x - 3*x^2 + x^3) - 9
*x^2 + 3*x^3)),x)

[Out]

log(log(log(x)*(18*x - 2*log(2)*(6*x - 18*x^2 + 6*x^3) - 54*x^2 + 18*x^3) - 54*x + 2*log(2)*(18*x - 54*x^2 + 1
8*x^3) + 162*x^2 - 54*x^3))

________________________________________________________________________________________

sympy [B]  time = 1.04, size = 66, normalized size = 3.00 \begin {gather*} \log {\left (\log {\left (- 54 x^{3} + 162 x^{2} - 54 x + \left (36 x^{3} - 108 x^{2} + 36 x\right ) \log {\relax (2 )} + \left (18 x^{3} - 54 x^{2} + 18 x + \left (- 12 x^{3} + 36 x^{2} - 12 x\right ) \log {\relax (2 )}\right ) \log {\relax (x )} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x**2-6*x+1)*ln(x)-8*x**2+15*x-2)/((x**3-3*x**2+x)*ln(x)-3*x**3+9*x**2-3*x)/ln((2*(-6*x**3+18*x**
2-6*x)*ln(2)+18*x**3-54*x**2+18*x)*ln(x)+2*(18*x**3-54*x**2+18*x)*ln(2)-54*x**3+162*x**2-54*x),x)

[Out]

log(log(-54*x**3 + 162*x**2 - 54*x + (36*x**3 - 108*x**2 + 36*x)*log(2) + (18*x**3 - 54*x**2 + 18*x + (-12*x**
3 + 36*x**2 - 12*x)*log(2))*log(x)))

________________________________________________________________________________________