3.80.99 \(\int \frac {-5+11 x+3 x^2+(1-3 x) \log (\frac {e^x x^2}{-2+6 x})+(3-9 x+(-1+3 x) \log (\frac {e^x x^2}{-2+6 x})) \log (-3+\log (\frac {e^x x^2}{-2+6 x}))}{3-9 x+(-1+3 x) \log (\frac {e^x x^2}{-2+6 x})} \, dx\)

Optimal. Leaf size=29 \[ -x+x \log \left (-3+\log \left (\frac {e^x x}{5-\frac {2-x}{x}}\right )\right ) \]

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Rubi [F]  time = 0.72, 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 {-5+11 x+3 x^2+(1-3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )+\left (3-9 x+(-1+3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )\right ) \log \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right )}{3-9 x+(-1+3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-5 + 11*x + 3*x^2 + (1 - 3*x)*Log[(E^x*x^2)/(-2 + 6*x)] + (3 - 9*x + (-1 + 3*x)*Log[(E^x*x^2)/(-2 + 6*x)]
)*Log[-3 + Log[(E^x*x^2)/(-2 + 6*x)]])/(3 - 9*x + (-1 + 3*x)*Log[(E^x*x^2)/(-2 + 6*x)]),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-5+11 x+3 x^2+(1-3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )+\left (3-9 x+(-1+3 x) \log \left (\frac {e^x x^2}{-2+6 x}\right )\right ) \log \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right )}{(1-3 x) \left (3-\log \left (\frac {e^x x^2}{-2+6 x}\right )\right )} \, dx\\ &=\int \left (\frac {-5+11 x+3 x^2+\log \left (\frac {e^x x^2}{-2+6 x}\right )-3 x \log \left (\frac {e^x x^2}{-2+6 x}\right )}{(-1+3 x) \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right )}+\log \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right )\right ) \, dx\\ &=\int \frac {-5+11 x+3 x^2+\log \left (\frac {e^x x^2}{-2+6 x}\right )-3 x \log \left (\frac {e^x x^2}{-2+6 x}\right )}{(-1+3 x) \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right )} \, dx+\int \log \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right ) \, dx\\ &=\int \left (-1+\frac {-2+2 x+3 x^2}{(-1+3 x) \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right )}\right ) \, dx+\int \log \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right ) \, dx\\ &=-x+\int \frac {-2+2 x+3 x^2}{(-1+3 x) \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right )} \, dx+\int \log \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right ) \, dx\\ &=-x+\int \left (\frac {1}{-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )}+\frac {x}{-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )}-\frac {1}{(-1+3 x) \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right )}\right ) \, dx+\int \log \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right ) \, dx\\ &=-x+\int \frac {1}{-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )} \, dx+\int \frac {x}{-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )} \, dx-\int \frac {1}{(-1+3 x) \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right )} \, dx+\int \log \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right ) \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 24, normalized size = 0.83 \begin {gather*} -x+x \log \left (-3+\log \left (\frac {e^x x^2}{-2+6 x}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-5 + 11*x + 3*x^2 + (1 - 3*x)*Log[(E^x*x^2)/(-2 + 6*x)] + (3 - 9*x + (-1 + 3*x)*Log[(E^x*x^2)/(-2 +
6*x)])*Log[-3 + Log[(E^x*x^2)/(-2 + 6*x)]])/(3 - 9*x + (-1 + 3*x)*Log[(E^x*x^2)/(-2 + 6*x)]),x]

[Out]

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

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fricas [A]  time = 0.74, size = 24, normalized size = 0.83 \begin {gather*} x \log \left (\log \left (\frac {x^{2} e^{x}}{2 \, {\left (3 \, x - 1\right )}}\right ) - 3\right ) - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*x-1)*log(x^2*exp(x)/(6*x-2))-9*x+3)*log(log(x^2*exp(x)/(6*x-2))-3)+(-3*x+1)*log(x^2*exp(x)/(6*x
-2))+3*x^2+11*x-5)/((3*x-1)*log(x^2*exp(x)/(6*x-2))-9*x+3),x, algorithm="fricas")

[Out]

x*log(log(1/2*x^2*e^x/(3*x - 1)) - 3) - x

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giac [A]  time = 0.58, size = 23, normalized size = 0.79 \begin {gather*} x \log \left (x + \log \left (\frac {x^{2}}{2 \, {\left (3 \, x - 1\right )}}\right ) - 3\right ) - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*x-1)*log(x^2*exp(x)/(6*x-2))-9*x+3)*log(log(x^2*exp(x)/(6*x-2))-3)+(-3*x+1)*log(x^2*exp(x)/(6*x
-2))+3*x^2+11*x-5)/((3*x-1)*log(x^2*exp(x)/(6*x-2))-9*x+3),x, algorithm="giac")

[Out]

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

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maple [C]  time = 0.47, size = 187, normalized size = 6.45

method	result	size
risch	\(\ln \left (-\ln \relax (2)-\ln \relax (3)+2 \ln \relax (x )+\ln \left ({\mathrm e}^{x}\right )-\ln \left (x -\frac {1}{3}\right )-\frac {i \pi \,\mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x -\frac {1}{3}}\right ) \left (-\mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x -\frac {1}{3}}\right )+\mathrm {csgn}\left (i {\mathrm e}^{x}\right )\right ) \left (-\mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x -\frac {1}{3}}\right )+\mathrm {csgn}\left (\frac {i}{x -\frac {1}{3}}\right )\right )}{2}-\frac {i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \left (-\mathrm {csgn}\left (i x^{2}\right )+\mathrm {csgn}\left (i x \right )\right )^{2}}{2}-\frac {i \pi \,\mathrm {csgn}\left (\frac {i x^{2} {\mathrm e}^{x}}{x -\frac {1}{3}}\right ) \left (-\mathrm {csgn}\left (\frac {i x^{2} {\mathrm e}^{x}}{x -\frac {1}{3}}\right )+\mathrm {csgn}\left (i x^{2}\right )\right ) \left (-\mathrm {csgn}\left (\frac {i x^{2} {\mathrm e}^{x}}{x -\frac {1}{3}}\right )+\mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x -\frac {1}{3}}\right )\right )}{2}-3\right ) x -x\)	\(187\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((3*x-1)*ln(x^2*exp(x)/(6*x-2))-9*x+3)*ln(ln(x^2*exp(x)/(6*x-2))-3)+(-3*x+1)*ln(x^2*exp(x)/(6*x-2))+3*x^2
+11*x-5)/((3*x-1)*ln(x^2*exp(x)/(6*x-2))-9*x+3),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.49, size = 26, normalized size = 0.90 \begin {gather*} x \log \left (x - \log \relax (2) - \log \left (3 \, x - 1\right ) + 2 \, \log \relax (x) - 3\right ) - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*x-1)*log(x^2*exp(x)/(6*x-2))-9*x+3)*log(log(x^2*exp(x)/(6*x-2))-3)+(-3*x+1)*log(x^2*exp(x)/(6*x
-2))+3*x^2+11*x-5)/((3*x-1)*log(x^2*exp(x)/(6*x-2))-9*x+3),x, algorithm="maxima")

[Out]

x*log(x - log(2) - log(3*x - 1) + 2*log(x) - 3) - x

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mupad [B]  time = 5.42, size = 21, normalized size = 0.72 \begin {gather*} x\,\left (\ln \left (\ln \left (\frac {x^2\,{\mathrm {e}}^x}{6\,x-2}\right )-3\right )-1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((11*x + log(log((x^2*exp(x))/(6*x - 2)) - 3)*(log((x^2*exp(x))/(6*x - 2))*(3*x - 1) - 9*x + 3) - log((x^2*
exp(x))/(6*x - 2))*(3*x - 1) + 3*x^2 - 5)/(log((x^2*exp(x))/(6*x - 2))*(3*x - 1) - 9*x + 3),x)

[Out]

x*(log(log((x^2*exp(x))/(6*x - 2)) - 3) - 1)

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sympy [B]  time = 1.45, size = 41, normalized size = 1.41 \begin {gather*} - x + \left (x - \frac {1}{18}\right ) \log {\left (\log {\left (\frac {x^{2} e^{x}}{6 x - 2} \right )} - 3 \right )} + \frac {\log {\left (\log {\left (\frac {x^{2} e^{x}}{6 x - 2} \right )} - 3 \right )}}{18} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*x-1)*ln(x**2*exp(x)/(6*x-2))-9*x+3)*ln(ln(x**2*exp(x)/(6*x-2))-3)+(-3*x+1)*ln(x**2*exp(x)/(6*x-
2))+3*x**2+11*x-5)/((3*x-1)*ln(x**2*exp(x)/(6*x-2))-9*x+3),x)

[Out]

-x + (x - 1/18)*log(log(x**2*exp(x)/(6*x - 2)) - 3) + log(log(x**2*exp(x)/(6*x - 2)) - 3)/18

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