3.85.40 \(\int \frac {-3 x+3 x^3+(-x+x^2) \log (3)+e^x (1+x-3 x^2+x^3) \log (3)+(3 x-e^x x \log (3)+(x-2 x^2) \log (3)) \log (x)}{-3 x^3+3 x^4+e^x (x^2-x^3) \log (3)+(-3 x^2+e^x x \log (3)+(x^2-x^3) \log (3)) \log (x)+x \log (3) \log ^2(x)} \, dx\)

Optimal. Leaf size=30 \[ \log \left (\frac {x-x^2+\log (x)}{-e^x+\frac {3 x}{\log (3)}-\log (x)}\right ) \]

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Rubi [F]  time = 3.44, 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 {-3 x+3 x^3+\left (-x+x^2\right ) \log (3)+e^x \left (1+x-3 x^2+x^3\right ) \log (3)+\left (3 x-e^x x \log (3)+\left (x-2 x^2\right ) \log (3)\right ) \log (x)}{-3 x^3+3 x^4+e^x \left (x^2-x^3\right ) \log (3)+\left (-3 x^2+e^x x \log (3)+\left (x^2-x^3\right ) \log (3)\right ) \log (x)+x \log (3) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-3*x + 3*x^3 + (-x + x^2)*Log[3] + E^x*(1 + x - 3*x^2 + x^3)*Log[3] + (3*x - E^x*x*Log[3] + (x - 2*x^2)*L
og[3])*Log[x])/(-3*x^3 + 3*x^4 + E^x*(x^2 - x^3)*Log[3] + (-3*x^2 + E^x*x*Log[3] + (x^2 - x^3)*Log[3])*Log[x]
+ x*Log[3]*Log[x]^2),x]

[Out]

-x - Defer[Int][(-x + x^2 - Log[x])^(-1), x] - Defer[Int][1/(x*(-x + x^2 - Log[x])), x] + (Log[9]*Defer[Int][x
/(-x + x^2 - Log[x]), x])/Log[3] - (Log[27]*Defer[Int][(3*x - E^x*Log[3] - Log[3]*Log[x])^(-1), x])/Log[3] + L
og[3]*Defer[Int][1/(x*(3*x - E^x*Log[3] - Log[3]*Log[x])), x] + (Log[27]*Defer[Int][x/(3*x - E^x*Log[3] - Log[
3]*Log[x]), x])/Log[3] + Log[3]*Defer[Int][Log[x]/(-3*x + E^x*Log[3] + Log[3]*Log[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {(-1+x) \left (-e^x \log (3)+x^2 \left (3+e^x \log (3)\right )+x \left (3+\log (3)-e^x \log (9)\right )\right )-x \left (-3-\log (3)+e^x \log (3)+x \log (9)\right ) \log (x)}{x ((-1+x) x-\log (x)) \left (3 x-e^x \log (3)-\log (3) \log (x)\right )} \, dx\\ &=\int \left (\frac {\log (3)+x^3 \log (3)-x \log (3) \left (1-\frac {\log (9)}{\log (3)}\right )-x^2 \log (3) \left (1+\frac {\log (9)}{\log (3)}\right )-x \log (3) \log (x)}{x \log (3) \left (x-x^2+\log (x)\right )}+\frac {\log ^2(3)-x \log (27)+x^2 \log (27)-x \log ^2(3) \log (x)}{x \log (3) \left (3 x-e^x \log (3)-\log (3) \log (x)\right )}\right ) \, dx\\ &=\frac {\int \frac {\log (3)+x^3 \log (3)-x \log (3) \left (1-\frac {\log (9)}{\log (3)}\right )-x^2 \log (3) \left (1+\frac {\log (9)}{\log (3)}\right )-x \log (3) \log (x)}{x \left (x-x^2+\log (x)\right )} \, dx}{\log (3)}+\frac {\int \frac {\log ^2(3)-x \log (27)+x^2 \log (27)-x \log ^2(3) \log (x)}{x \left (3 x-e^x \log (3)-\log (3) \log (x)\right )} \, dx}{\log (3)}\\ &=\frac {\int \frac {-\left ((-1+x) \left (-\log (3)+x^2 \log (3)-x \log (9)\right )\right )+x \log (3) \log (x)}{x ((-1+x) x-\log (x))} \, dx}{\log (3)}+\frac {\int \left (\frac {\log ^2(3)}{x \left (3 x-e^x \log (3)-\log (3) \log (x)\right )}-\frac {\log (27)}{3 x-e^x \log (3)-\log (3) \log (x)}+\frac {x \log (27)}{3 x-e^x \log (3)-\log (3) \log (x)}+\frac {\log ^2(3) \log (x)}{-3 x+e^x \log (3)+\log (3) \log (x)}\right ) \, dx}{\log (3)}\\ &=\frac {\int \left (-\log (3)+\frac {(-1+x) (\log (3)+x \log (9))}{x \left (-x+x^2-\log (x)\right )}\right ) \, dx}{\log (3)}+\log (3) \int \frac {1}{x \left (3 x-e^x \log (3)-\log (3) \log (x)\right )} \, dx+\log (3) \int \frac {\log (x)}{-3 x+e^x \log (3)+\log (3) \log (x)} \, dx-\frac {\log (27) \int \frac {1}{3 x-e^x \log (3)-\log (3) \log (x)} \, dx}{\log (3)}+\frac {\log (27) \int \frac {x}{3 x-e^x \log (3)-\log (3) \log (x)} \, dx}{\log (3)}\\ &=-x+\frac {\int \frac {(-1+x) (\log (3)+x \log (9))}{x \left (-x+x^2-\log (x)\right )} \, dx}{\log (3)}+\log (3) \int \frac {1}{x \left (3 x-e^x \log (3)-\log (3) \log (x)\right )} \, dx+\log (3) \int \frac {\log (x)}{-3 x+e^x \log (3)+\log (3) \log (x)} \, dx-\frac {\log (27) \int \frac {1}{3 x-e^x \log (3)-\log (3) \log (x)} \, dx}{\log (3)}+\frac {\log (27) \int \frac {x}{3 x-e^x \log (3)-\log (3) \log (x)} \, dx}{\log (3)}\\ &=-x+\frac {\int \left (-\frac {\log (3)}{x \left (-x+x^2-\log (x)\right )}+\frac {x \log (9)}{-x+x^2-\log (x)}+\frac {\log (3) \left (1-\frac {\log (9)}{\log (3)}\right )}{-x+x^2-\log (x)}\right ) \, dx}{\log (3)}+\log (3) \int \frac {1}{x \left (3 x-e^x \log (3)-\log (3) \log (x)\right )} \, dx+\log (3) \int \frac {\log (x)}{-3 x+e^x \log (3)+\log (3) \log (x)} \, dx-\frac {\log (27) \int \frac {1}{3 x-e^x \log (3)-\log (3) \log (x)} \, dx}{\log (3)}+\frac {\log (27) \int \frac {x}{3 x-e^x \log (3)-\log (3) \log (x)} \, dx}{\log (3)}\\ &=-x+\log (3) \int \frac {1}{x \left (3 x-e^x \log (3)-\log (3) \log (x)\right )} \, dx+\log (3) \int \frac {\log (x)}{-3 x+e^x \log (3)+\log (3) \log (x)} \, dx+\frac {\log (9) \int \frac {x}{-x+x^2-\log (x)} \, dx}{\log (3)}-\frac {\log (27) \int \frac {1}{3 x-e^x \log (3)-\log (3) \log (x)} \, dx}{\log (3)}+\frac {\log (27) \int \frac {x}{3 x-e^x \log (3)-\log (3) \log (x)} \, dx}{\log (3)}-\int \frac {1}{-x+x^2-\log (x)} \, dx-\int \frac {1}{x \left (-x+x^2-\log (x)\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 0.49, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-3 x+3 x^3+\left (-x+x^2\right ) \log (3)+e^x \left (1+x-3 x^2+x^3\right ) \log (3)+\left (3 x-e^x x \log (3)+\left (x-2 x^2\right ) \log (3)\right ) \log (x)}{-3 x^3+3 x^4+e^x \left (x^2-x^3\right ) \log (3)+\left (-3 x^2+e^x x \log (3)+\left (x^2-x^3\right ) \log (3)\right ) \log (x)+x \log (3) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(-3*x + 3*x^3 + (-x + x^2)*Log[3] + E^x*(1 + x - 3*x^2 + x^3)*Log[3] + (3*x - E^x*x*Log[3] + (x - 2*
x^2)*Log[3])*Log[x])/(-3*x^3 + 3*x^4 + E^x*(x^2 - x^3)*Log[3] + (-3*x^2 + E^x*x*Log[3] + (x^2 - x^3)*Log[3])*L
og[x] + x*Log[3]*Log[x]^2),x]

[Out]

Integrate[(-3*x + 3*x^3 + (-x + x^2)*Log[3] + E^x*(1 + x - 3*x^2 + x^3)*Log[3] + (3*x - E^x*x*Log[3] + (x - 2*
x^2)*Log[3])*Log[x])/(-3*x^3 + 3*x^4 + E^x*(x^2 - x^3)*Log[3] + (-3*x^2 + E^x*x*Log[3] + (x^2 - x^3)*Log[3])*L
og[x] + x*Log[3]*Log[x]^2), x]

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fricas [A]  time = 0.90, size = 28, normalized size = 0.93 \begin {gather*} \log \left (-x^{2} + x + \log \relax (x)\right ) - \log \left (e^{x} \log \relax (3) + \log \relax (3) \log \relax (x) - 3 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x*log(3)*exp(x)+(-2*x^2+x)*log(3)+3*x)*log(x)+(x^3-3*x^2+x+1)*log(3)*exp(x)+(x^2-x)*log(3)+3*x^3-
3*x)/(x*log(3)*log(x)^2+(x*log(3)*exp(x)+(-x^3+x^2)*log(3)-3*x^2)*log(x)+(-x^3+x^2)*log(3)*exp(x)+3*x^4-3*x^3)
,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.87, size = 32, normalized size = 1.07 \begin {gather*} \log \left (x^{2} - x - \log \relax (x)\right ) - \log \left (-e^{x} \log \relax (3) - \log \relax (3) \log \relax (x) + 3 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x*log(3)*exp(x)+(-2*x^2+x)*log(3)+3*x)*log(x)+(x^3-3*x^2+x+1)*log(3)*exp(x)+(x^2-x)*log(3)+3*x^3-
3*x)/(x*log(3)*log(x)^2+(x*log(3)*exp(x)+(-x^3+x^2)*log(3)-3*x^2)*log(x)+(-x^3+x^2)*log(3)*exp(x)+3*x^4-3*x^3)
,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.04, size = 32, normalized size = 1.07

method	result	size
risch	\(\ln \left (\ln \relax (x )+x -x^{2}\right )-\ln \left (\ln \relax (x )+\frac {\ln \relax (3) {\mathrm e}^{x}-3 x}{\ln \relax (3)}\right )\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x*ln(3)*exp(x)+(-2*x^2+x)*ln(3)+3*x)*ln(x)+(x^3-3*x^2+x+1)*ln(3)*exp(x)+(x^2-x)*ln(3)+3*x^3-3*x)/(x*ln(
3)*ln(x)^2+(x*ln(3)*exp(x)+(-x^3+x^2)*ln(3)-3*x^2)*ln(x)+(-x^3+x^2)*ln(3)*exp(x)+3*x^4-3*x^3),x,method=_RETURN
VERBOSE)

[Out]

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

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maxima [A]  time = 0.49, size = 33, normalized size = 1.10 \begin {gather*} \log \left (-x^{2} + x + \log \relax (x)\right ) - \log \left (\frac {e^{x} \log \relax (3) + \log \relax (3) \log \relax (x) - 3 \, x}{\log \relax (3)}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x*log(3)*exp(x)+(-2*x^2+x)*log(3)+3*x)*log(x)+(x^3-3*x^2+x+1)*log(3)*exp(x)+(x^2-x)*log(3)+3*x^3-
3*x)/(x*log(3)*log(x)^2+(x*log(3)*exp(x)+(-x^3+x^2)*log(3)-3*x^2)*log(x)+(-x^3+x^2)*log(3)*exp(x)+3*x^4-3*x^3)
,x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\ln \relax (x)\,\left (3\,x+\ln \relax (3)\,\left (x-2\,x^2\right )-x\,{\mathrm {e}}^x\,\ln \relax (3)\right )-3\,x+3\,x^3-\ln \relax (3)\,\left (x-x^2\right )+{\mathrm {e}}^x\,\ln \relax (3)\,\left (x^3-3\,x^2+x+1\right )}{\ln \relax (x)\,\left (\ln \relax (3)\,\left (x^2-x^3\right )-3\,x^2+x\,{\mathrm {e}}^x\,\ln \relax (3)\right )-3\,x^3+3\,x^4+{\mathrm {e}}^x\,\ln \relax (3)\,\left (x^2-x^3\right )+x\,\ln \relax (3)\,{\ln \relax (x)}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)*(3*x + log(3)*(x - 2*x^2) - x*exp(x)*log(3)) - 3*x + 3*x^3 - log(3)*(x - x^2) + exp(x)*log(3)*(x -
3*x^2 + x^3 + 1))/(log(x)*(log(3)*(x^2 - x^3) - 3*x^2 + x*exp(x)*log(3)) - 3*x^3 + 3*x^4 + exp(x)*log(3)*(x^2
- x^3) + x*log(3)*log(x)^2),x)

[Out]

int((log(x)*(3*x + log(3)*(x - 2*x^2) - x*exp(x)*log(3)) - 3*x + 3*x^3 - log(3)*(x - x^2) + exp(x)*log(3)*(x -
3*x^2 + x^3 + 1))/(log(x)*(log(3)*(x^2 - x^3) - 3*x^2 + x*exp(x)*log(3)) - 3*x^3 + 3*x^4 + exp(x)*log(3)*(x^2
- x^3) + x*log(3)*log(x)^2), x)

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sympy [A]  time = 0.47, size = 27, normalized size = 0.90 \begin {gather*} - \log {\left (\frac {- 3 x + \log {\relax (3 )} \log {\relax (x )}}{\log {\relax (3 )}} + e^{x} \right )} + \log {\left (- x^{2} + x + \log {\relax (x )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x*ln(3)*exp(x)+(-2*x**2+x)*ln(3)+3*x)*ln(x)+(x**3-3*x**2+x+1)*ln(3)*exp(x)+(x**2-x)*ln(3)+3*x**3-
3*x)/(x*ln(3)*ln(x)**2+(x*ln(3)*exp(x)+(-x**3+x**2)*ln(3)-3*x**2)*ln(x)+(-x**3+x**2)*ln(3)*exp(x)+3*x**4-3*x**
3),x)

[Out]

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

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