3.90.87 \(\int \frac {9-6 x^2-3 \log (x)+(-6+8 x-8 x^2+6 x^3+(3-3 x) \log (x)) \log (\frac {6-2 x+6 x^2-3 \log (x)}{x})}{(-6 x+2 x^2-6 x^3+3 x \log (x)) \log (\frac {6-2 x+6 x^2-3 \log (x)}{x})} \, dx\) [8987]

Optimal. Leaf size=32 \[ \log \left (5 e^{-x} x \log \left (1+3 \left (x+\frac {2-x+x^2-\log (x)}{x}\right )\right )\right ) \]

[Out]

ln(5*x/exp(x)*ln(3*(2-ln(x)+x^2-x)/x+3*x+1))

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Rubi [F]
time = 2.29, 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 {9-6 x^2-3 \log (x)+\left (-6+8 x-8 x^2+6 x^3+(3-3 x) \log (x)\right ) \log \left (\frac {6-2 x+6 x^2-3 \log (x)}{x}\right )}{\left (-6 x+2 x^2-6 x^3+3 x \log (x)\right ) \log \left (\frac {6-2 x+6 x^2-3 \log (x)}{x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(9 - 6*x^2 - 3*Log[x] + (-6 + 8*x - 8*x^2 + 6*x^3 + (3 - 3*x)*Log[x])*Log[(6 - 2*x + 6*x^2 - 3*Log[x])/x])
/((-6*x + 2*x^2 - 6*x^3 + 3*x*Log[x])*Log[(6 - 2*x + 6*x^2 - 3*Log[x])/x]),x]

[Out]

-x + Log[x] - 2*Defer[Int][(6 - 2*x + 6*x^2 - 3*Log[x])^(-1), x] - 2*Defer[Int][(-6 + 2*x - 6*x^2 + 3*Log[x])^
(-1), x] + 6*Defer[Int][x/((6 - 2*x + 6*x^2 - 3*Log[x])*Log[-2 + 6/x + 6*x - (3*Log[x])/x]), x] + 3*Defer[Int]
[Log[x]/(x*(6 - 2*x + 6*x^2 - 3*Log[x])*Log[-2 + 6/x + 6*x - (3*Log[x])/x]), x] + 9*Defer[Int][1/(x*(-6 + 2*x
- 6*x^2 + 3*Log[x])*Log[-2 + 6/x + 6*x - (3*Log[x])/x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-9+6 x^2+3 \log (x)-\left (-6+8 x-8 x^2+6 x^3+(3-3 x) \log (x)\right ) \log \left (\frac {6-2 x+6 x^2-3 \log (x)}{x}\right )}{\left (6 x-2 x^2+6 x^3-3 x \log (x)\right ) \log \left (-2+\frac {6}{x}+6 x-\frac {3 \log (x)}{x}\right )} \, dx\\ &=\int \left (-\frac {8}{6-2 x+6 x^2-3 \log (x)}+\frac {6}{x \left (6-2 x+6 x^2-3 \log (x)\right )}+\frac {8 x}{6-2 x+6 x^2-3 \log (x)}-\frac {6 x^2}{6-2 x+6 x^2-3 \log (x)}+\frac {3 \log (x)}{6-2 x+6 x^2-3 \log (x)}-\frac {3 \log (x)}{x \left (6-2 x+6 x^2-3 \log (x)\right )}+\frac {6 x}{\left (6-2 x+6 x^2-3 \log (x)\right ) \log \left (-2+\frac {6}{x}+6 x-\frac {3 \log (x)}{x}\right )}+\frac {3 \log (x)}{x \left (6-2 x+6 x^2-3 \log (x)\right ) \log \left (-2+\frac {6}{x}+6 x-\frac {3 \log (x)}{x}\right )}+\frac {9}{x \left (-6+2 x-6 x^2+3 \log (x)\right ) \log \left (-2+\frac {6}{x}+6 x-\frac {3 \log (x)}{x}\right )}\right ) \, dx\\ &=3 \int \frac {\log (x)}{6-2 x+6 x^2-3 \log (x)} \, dx-3 \int \frac {\log (x)}{x \left (6-2 x+6 x^2-3 \log (x)\right )} \, dx+3 \int \frac {\log (x)}{x \left (6-2 x+6 x^2-3 \log (x)\right ) \log \left (-2+\frac {6}{x}+6 x-\frac {3 \log (x)}{x}\right )} \, dx+6 \int \frac {1}{x \left (6-2 x+6 x^2-3 \log (x)\right )} \, dx-6 \int \frac {x^2}{6-2 x+6 x^2-3 \log (x)} \, dx+6 \int \frac {x}{\left (6-2 x+6 x^2-3 \log (x)\right ) \log \left (-2+\frac {6}{x}+6 x-\frac {3 \log (x)}{x}\right )} \, dx-8 \int \frac {1}{6-2 x+6 x^2-3 \log (x)} \, dx+8 \int \frac {x}{6-2 x+6 x^2-3 \log (x)} \, dx+9 \int \frac {1}{x \left (-6+2 x-6 x^2+3 \log (x)\right ) \log \left (-2+\frac {6}{x}+6 x-\frac {3 \log (x)}{x}\right )} \, dx\\ &=3 \int \left (-\frac {1}{3}+\frac {2 \left (3-x+3 x^2\right )}{3 \left (6-2 x+6 x^2-3 \log (x)\right )}\right ) \, dx-3 \int \left (-\frac {1}{3 x}+\frac {2 \left (3-x+3 x^2\right )}{3 x \left (6-2 x+6 x^2-3 \log (x)\right )}\right ) \, dx+3 \int \frac {\log (x)}{x \left (6-2 x+6 x^2-3 \log (x)\right ) \log \left (-2+\frac {6}{x}+6 x-\frac {3 \log (x)}{x}\right )} \, dx+6 \int \frac {1}{x \left (6-2 x+6 x^2-3 \log (x)\right )} \, dx-6 \int \frac {x^2}{6-2 x+6 x^2-3 \log (x)} \, dx+6 \int \frac {x}{\left (6-2 x+6 x^2-3 \log (x)\right ) \log \left (-2+\frac {6}{x}+6 x-\frac {3 \log (x)}{x}\right )} \, dx-8 \int \frac {1}{6-2 x+6 x^2-3 \log (x)} \, dx+8 \int \frac {x}{6-2 x+6 x^2-3 \log (x)} \, dx+9 \int \frac {1}{x \left (-6+2 x-6 x^2+3 \log (x)\right ) \log \left (-2+\frac {6}{x}+6 x-\frac {3 \log (x)}{x}\right )} \, dx\\ &=-x+\log (x)+2 \int \frac {3-x+3 x^2}{6-2 x+6 x^2-3 \log (x)} \, dx-2 \int \frac {3-x+3 x^2}{x \left (6-2 x+6 x^2-3 \log (x)\right )} \, dx+3 \int \frac {\log (x)}{x \left (6-2 x+6 x^2-3 \log (x)\right ) \log \left (-2+\frac {6}{x}+6 x-\frac {3 \log (x)}{x}\right )} \, dx+6 \int \frac {1}{x \left (6-2 x+6 x^2-3 \log (x)\right )} \, dx-6 \int \frac {x^2}{6-2 x+6 x^2-3 \log (x)} \, dx+6 \int \frac {x}{\left (6-2 x+6 x^2-3 \log (x)\right ) \log \left (-2+\frac {6}{x}+6 x-\frac {3 \log (x)}{x}\right )} \, dx-8 \int \frac {1}{6-2 x+6 x^2-3 \log (x)} \, dx+8 \int \frac {x}{6-2 x+6 x^2-3 \log (x)} \, dx+9 \int \frac {1}{x \left (-6+2 x-6 x^2+3 \log (x)\right ) \log \left (-2+\frac {6}{x}+6 x-\frac {3 \log (x)}{x}\right )} \, dx\\ &=-x+\log (x)+2 \int \left (\frac {3}{6-2 x+6 x^2-3 \log (x)}-\frac {x}{6-2 x+6 x^2-3 \log (x)}+\frac {3 x^2}{6-2 x+6 x^2-3 \log (x)}\right ) \, dx-2 \int \left (\frac {3}{x \left (6-2 x+6 x^2-3 \log (x)\right )}+\frac {3 x}{6-2 x+6 x^2-3 \log (x)}+\frac {1}{-6+2 x-6 x^2+3 \log (x)}\right ) \, dx+3 \int \frac {\log (x)}{x \left (6-2 x+6 x^2-3 \log (x)\right ) \log \left (-2+\frac {6}{x}+6 x-\frac {3 \log (x)}{x}\right )} \, dx+6 \int \frac {1}{x \left (6-2 x+6 x^2-3 \log (x)\right )} \, dx-6 \int \frac {x^2}{6-2 x+6 x^2-3 \log (x)} \, dx+6 \int \frac {x}{\left (6-2 x+6 x^2-3 \log (x)\right ) \log \left (-2+\frac {6}{x}+6 x-\frac {3 \log (x)}{x}\right )} \, dx-8 \int \frac {1}{6-2 x+6 x^2-3 \log (x)} \, dx+8 \int \frac {x}{6-2 x+6 x^2-3 \log (x)} \, dx+9 \int \frac {1}{x \left (-6+2 x-6 x^2+3 \log (x)\right ) \log \left (-2+\frac {6}{x}+6 x-\frac {3 \log (x)}{x}\right )} \, dx\\ &=-x+\log (x)-2 \int \frac {x}{6-2 x+6 x^2-3 \log (x)} \, dx-2 \int \frac {1}{-6+2 x-6 x^2+3 \log (x)} \, dx+3 \int \frac {\log (x)}{x \left (6-2 x+6 x^2-3 \log (x)\right ) \log \left (-2+\frac {6}{x}+6 x-\frac {3 \log (x)}{x}\right )} \, dx+6 \int \frac {1}{6-2 x+6 x^2-3 \log (x)} \, dx-6 \int \frac {x}{6-2 x+6 x^2-3 \log (x)} \, dx+6 \int \frac {x}{\left (6-2 x+6 x^2-3 \log (x)\right ) \log \left (-2+\frac {6}{x}+6 x-\frac {3 \log (x)}{x}\right )} \, dx-8 \int \frac {1}{6-2 x+6 x^2-3 \log (x)} \, dx+8 \int \frac {x}{6-2 x+6 x^2-3 \log (x)} \, dx+9 \int \frac {1}{x \left (-6+2 x-6 x^2+3 \log (x)\right ) \log \left (-2+\frac {6}{x}+6 x-\frac {3 \log (x)}{x}\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.05, size = 26, normalized size = 0.81 \begin {gather*} -x+\log (x)+\log \left (\log \left (\frac {6-2 x+6 x^2-3 \log (x)}{x}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(9 - 6*x^2 - 3*Log[x] + (-6 + 8*x - 8*x^2 + 6*x^3 + (3 - 3*x)*Log[x])*Log[(6 - 2*x + 6*x^2 - 3*Log[x
])/x])/((-6*x + 2*x^2 - 6*x^3 + 3*x*Log[x])*Log[(6 - 2*x + 6*x^2 - 3*Log[x])/x]),x]

[Out]

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

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Maple [A]
time = 2.77, size = 27, normalized size = 0.84

method	result	size
default	\(\ln \left (x \right )-x +\ln \left (\ln \left (\frac {-3 \ln \left (x \right )+6 x^{2}-2 x +6}{x}\right )\right )\)	\(27\)
norman	\(\ln \left (x \right )-x +\ln \left (\ln \left (\frac {-3 \ln \left (x \right )+6 x^{2}-2 x +6}{x}\right )\right )\)	\(27\)
risch	\(\ln \left (x \right )-x +\ln \left (\ln \left (x^{2}-\frac {x}{3}-\frac {\ln \left (x \right )}{2}+1\right )+\frac {i \left (-\pi \,\mathrm {csgn}\left (i \left (-x^{2}+\frac {x}{3}+\frac {\ln \left (x \right )}{2}-1\right )\right ) \mathrm {csgn}\left (\frac {i \left (-x^{2}+\frac {x}{3}+\frac {\ln \left (x \right )}{2}-1\right )}{x}\right )^{2}-\pi \,\mathrm {csgn}\left (i \left (-x^{2}+\frac {x}{3}+\frac {\ln \left (x \right )}{2}-1\right )\right ) \mathrm {csgn}\left (\frac {i \left (-x^{2}+\frac {x}{3}+\frac {\ln \left (x \right )}{2}-1\right )}{x}\right ) \mathrm {csgn}\left (\frac {i}{x}\right )+\pi \mathrm {csgn}\left (\frac {i \left (-x^{2}+\frac {x}{3}+\frac {\ln \left (x \right )}{2}-1\right )}{x}\right )^{3}+\pi \mathrm {csgn}\left (\frac {i \left (-x^{2}+\frac {x}{3}+\frac {\ln \left (x \right )}{2}-1\right )}{x}\right )^{2} \mathrm {csgn}\left (\frac {i}{x}\right )-2 i \ln \left (2\right )-2 i \ln \left (3\right )+2 i \ln \left (x \right )\right )}{2}\right )\)	\(191\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-3*x+3)*ln(x)+6*x^3-8*x^2+8*x-6)*ln((-3*ln(x)+6*x^2-2*x+6)/x)-3*ln(x)-6*x^2+9)/(3*x*ln(x)-6*x^3+2*x^2-6
*x)/ln((-3*ln(x)+6*x^2-2*x+6)/x),x,method=_RETURNVERBOSE)

[Out]

ln(x)-x+ln(ln((-3*ln(x)+6*x^2-2*x+6)/x))

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Maxima [A]
time = 0.29, size = 27, normalized size = 0.84 \begin {gather*} -x + \log \left (x\right ) + \log \left (\log \left (6 \, x^{2} - 2 \, x - 3 \, \log \left (x\right ) + 6\right ) - \log \left (x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x + log(x) + log(log(6*x^2 - 2*x - 3*log(x) + 6) - log(x))

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Fricas [A]
time = 0.51, size = 26, normalized size = 0.81 \begin {gather*} -x + \log \left (x\right ) + \log \left (\log \left (\frac {6 \, x^{2} - 2 \, x - 3 \, \log \left (x\right ) + 6}{x}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.26, size = 24, normalized size = 0.75 \begin {gather*} - x + \log {\left (x \right )} + \log {\left (\log {\left (\frac {6 x^{2} - 2 x - 3 \log {\left (x \right )} + 6}{x} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-3*x+3)*ln(x)+6*x**3-8*x**2+8*x-6)*ln((-3*ln(x)+6*x**2-2*x+6)/x)-3*ln(x)-6*x**2+9)/(3*x*ln(x)-6*x
**3+2*x**2-6*x)/ln((-3*ln(x)+6*x**2-2*x+6)/x),x)

[Out]

-x + log(x) + log(log((6*x**2 - 2*x - 3*log(x) + 6)/x))

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Giac [A]
time = 0.44, size = 27, normalized size = 0.84 \begin {gather*} -x + \log \left (x\right ) + \log \left (\log \left (6 \, x^{2} - 2 \, x - 3 \, \log \left (x\right ) + 6\right ) - \log \left (x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x + log(x) + log(log(6*x^2 - 2*x - 3*log(x) + 6) - log(x))

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Mupad [B]
time = 9.24, size = 27, normalized size = 0.84 \begin {gather*} \ln \left (\ln \left (-\frac {2\,x+3\,\ln \left (x\right )-6\,x^2-6}{x}\right )\right )-x+\ln \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*log(x) + log(-(2*x + 3*log(x) - 6*x^2 - 6)/x)*(log(x)*(3*x - 3) - 8*x + 8*x^2 - 6*x^3 + 6) + 6*x^2 - 9)
/(log(-(2*x + 3*log(x) - 6*x^2 - 6)/x)*(6*x - 3*x*log(x) - 2*x^2 + 6*x^3)),x)

[Out]

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

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