3.19.22 \(\int \frac {-x+2 x^2-x^3+e^x (-4+5 x-x^2)+e^x (7 x-5 x^2+x^3) \log (\frac {3}{x})}{-x^2+2 x^3-x^4+e^x (4 x-5 x^2+x^3) \log (\frac {3}{x})} \, dx\)

Optimal. Leaf size=28 \[ \log \left (x+\frac {e^x (4-x) x \log \left (\frac {3}{x}\right )}{-x+x^2}\right ) \]

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Rubi [F]  time = 12.28, 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 {-x+2 x^2-x^3+e^x \left (-4+5 x-x^2\right )+e^x \left (7 x-5 x^2+x^3\right ) \log \left (\frac {3}{x}\right )}{-x^2+2 x^3-x^4+e^x \left (4 x-5 x^2+x^3\right ) \log \left (\frac {3}{x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-x + 2*x^2 - x^3 + E^x*(-4 + 5*x - x^2) + E^x*(7*x - 5*x^2 + x^3)*Log[3/x])/(-x^2 + 2*x^3 - x^4 + E^x*(4*
x - 5*x^2 + x^3)*Log[3/x]),x]

[Out]

x - Log[1 - x] + Log[4 - x] + Log[Log[3/x]] - 4*Defer[Int][((-1 + x)*x - E^x*(-4 + x)*Log[3/x])^(-1), x] - 12*
Defer[Int][1/((-4 + x)*((-1 + x)*x - E^x*(-4 + x)*Log[3/x])), x] + 2*Defer[Int][x/((-1 + x)*x - E^x*(-4 + x)*L
og[3/x]), x] - Defer[Int][x^2/((-1 + x)*x - E^x*(-4 + x)*Log[3/x]), x] + Defer[Int][x/(Log[3/x]*((-1 + x)*x -
E^x*(-4 + x)*Log[3/x])), x] + Defer[Int][1/(Log[3/x]*(-((-1 + x)*x) + E^x*(-4 + x)*Log[3/x])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x-2 x^2+x^3-e^x \left (-4+5 x-x^2\right )-e^x \left (7 x-5 x^2+x^3\right ) \log \left (\frac {3}{x}\right )}{(1-x) x \left (x-x^2-4 e^x \log \left (\frac {3}{x}\right )+e^x x \log \left (\frac {3}{x}\right )\right )} \, dx\\ &=\int \left (-\frac {-4+5 x-x^2-4 \log \left (\frac {3}{x}\right )+12 x \log \left (\frac {3}{x}\right )-6 x^2 \log \left (\frac {3}{x}\right )+x^3 \log \left (\frac {3}{x}\right )}{(-4+x) \log \left (\frac {3}{x}\right ) \left (-x+x^2+4 e^x \log \left (\frac {3}{x}\right )-e^x x \log \left (\frac {3}{x}\right )\right )}+\frac {-4+5 x-x^2+7 x \log \left (\frac {3}{x}\right )-5 x^2 \log \left (\frac {3}{x}\right )+x^3 \log \left (\frac {3}{x}\right )}{(-4+x) (-1+x) x \log \left (\frac {3}{x}\right )}\right ) \, dx\\ &=-\int \frac {-4+5 x-x^2-4 \log \left (\frac {3}{x}\right )+12 x \log \left (\frac {3}{x}\right )-6 x^2 \log \left (\frac {3}{x}\right )+x^3 \log \left (\frac {3}{x}\right )}{(-4+x) \log \left (\frac {3}{x}\right ) \left (-x+x^2+4 e^x \log \left (\frac {3}{x}\right )-e^x x \log \left (\frac {3}{x}\right )\right )} \, dx+\int \frac {-4+5 x-x^2+7 x \log \left (\frac {3}{x}\right )-5 x^2 \log \left (\frac {3}{x}\right )+x^3 \log \left (\frac {3}{x}\right )}{(-4+x) (-1+x) x \log \left (\frac {3}{x}\right )} \, dx\\ &=\int \left (\frac {7-5 x+x^2}{4-5 x+x^2}-\frac {1}{x \log \left (\frac {3}{x}\right )}\right ) \, dx-\int \frac {4-5 x+x^2-\left (-4+12 x-6 x^2+x^3\right ) \log \left (\frac {3}{x}\right )}{(4-x) \log \left (\frac {3}{x}\right ) \left ((-1+x) x-e^x (-4+x) \log \left (\frac {3}{x}\right )\right )} \, dx\\ &=\int \frac {7-5 x+x^2}{4-5 x+x^2} \, dx-\int \frac {1}{x \log \left (\frac {3}{x}\right )} \, dx-\int \left (-\frac {4}{(-4+x) \left (-x+x^2+4 e^x \log \left (\frac {3}{x}\right )-e^x x \log \left (\frac {3}{x}\right )\right )}+\frac {12 x}{(-4+x) \left (-x+x^2+4 e^x \log \left (\frac {3}{x}\right )-e^x x \log \left (\frac {3}{x}\right )\right )}-\frac {6 x^2}{(-4+x) \left (-x+x^2+4 e^x \log \left (\frac {3}{x}\right )-e^x x \log \left (\frac {3}{x}\right )\right )}+\frac {x^3}{(-4+x) \left (-x+x^2+4 e^x \log \left (\frac {3}{x}\right )-e^x x \log \left (\frac {3}{x}\right )\right )}-\frac {4}{(-4+x) \log \left (\frac {3}{x}\right ) \left (-x+x^2+4 e^x \log \left (\frac {3}{x}\right )-e^x x \log \left (\frac {3}{x}\right )\right )}+\frac {5 x}{(-4+x) \log \left (\frac {3}{x}\right ) \left (-x+x^2+4 e^x \log \left (\frac {3}{x}\right )-e^x x \log \left (\frac {3}{x}\right )\right )}-\frac {x^2}{(-4+x) \log \left (\frac {3}{x}\right ) \left (-x+x^2+4 e^x \log \left (\frac {3}{x}\right )-e^x x \log \left (\frac {3}{x}\right )\right )}\right ) \, dx\\ &=4 \int \frac {1}{(-4+x) \left (-x+x^2+4 e^x \log \left (\frac {3}{x}\right )-e^x x \log \left (\frac {3}{x}\right )\right )} \, dx+4 \int \frac {1}{(-4+x) \log \left (\frac {3}{x}\right ) \left (-x+x^2+4 e^x \log \left (\frac {3}{x}\right )-e^x x \log \left (\frac {3}{x}\right )\right )} \, dx-5 \int \frac {x}{(-4+x) \log \left (\frac {3}{x}\right ) \left (-x+x^2+4 e^x \log \left (\frac {3}{x}\right )-e^x x \log \left (\frac {3}{x}\right )\right )} \, dx+6 \int \frac {x^2}{(-4+x) \left (-x+x^2+4 e^x \log \left (\frac {3}{x}\right )-e^x x \log \left (\frac {3}{x}\right )\right )} \, dx-12 \int \frac {x}{(-4+x) \left (-x+x^2+4 e^x \log \left (\frac {3}{x}\right )-e^x x \log \left (\frac {3}{x}\right )\right )} \, dx+\int \left (1+\frac {3}{4-5 x+x^2}\right ) \, dx-\int \frac {x^3}{(-4+x) \left (-x+x^2+4 e^x \log \left (\frac {3}{x}\right )-e^x x \log \left (\frac {3}{x}\right )\right )} \, dx+\int \frac {x^2}{(-4+x) \log \left (\frac {3}{x}\right ) \left (-x+x^2+4 e^x \log \left (\frac {3}{x}\right )-e^x x \log \left (\frac {3}{x}\right )\right )} \, dx+\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (\frac {3}{x}\right )\right )\\ &=x+\log \left (\log \left (\frac {3}{x}\right )\right )+3 \int \frac {1}{4-5 x+x^2} \, dx+4 \int \frac {1}{(-4+x) \left ((-1+x) x-e^x (-4+x) \log \left (\frac {3}{x}\right )\right )} \, dx+4 \int \frac {1}{(-4+x) \log \left (\frac {3}{x}\right ) \left ((-1+x) x-e^x (-4+x) \log \left (\frac {3}{x}\right )\right )} \, dx-5 \int \frac {x}{(-4+x) \log \left (\frac {3}{x}\right ) \left ((-1+x) x-e^x (-4+x) \log \left (\frac {3}{x}\right )\right )} \, dx+6 \int \frac {x^2}{(-4+x) \left ((-1+x) x-e^x (-4+x) \log \left (\frac {3}{x}\right )\right )} \, dx-12 \int \frac {x}{(-4+x) \left ((-1+x) x-e^x (-4+x) \log \left (\frac {3}{x}\right )\right )} \, dx-\int \frac {x^3}{(-4+x) \left ((-1+x) x-e^x (-4+x) \log \left (\frac {3}{x}\right )\right )} \, dx+\int \frac {x^2}{(-4+x) \log \left (\frac {3}{x}\right ) \left ((-1+x) x-e^x (-4+x) \log \left (\frac {3}{x}\right )\right )} \, dx\\ &=x+\log \left (\log \left (\frac {3}{x}\right )\right )+4 \int \frac {1}{(-4+x) \left ((-1+x) x-e^x (-4+x) \log \left (\frac {3}{x}\right )\right )} \, dx+4 \int \frac {1}{(-4+x) \log \left (\frac {3}{x}\right ) \left ((-1+x) x-e^x (-4+x) \log \left (\frac {3}{x}\right )\right )} \, dx-5 \int \left (\frac {4}{(-4+x) \log \left (\frac {3}{x}\right ) \left (-x+x^2+4 e^x \log \left (\frac {3}{x}\right )-e^x x \log \left (\frac {3}{x}\right )\right )}-\frac {1}{\log \left (\frac {3}{x}\right ) \left (x-x^2-4 e^x \log \left (\frac {3}{x}\right )+e^x x \log \left (\frac {3}{x}\right )\right )}\right ) \, dx+6 \int \left (\frac {4}{-x+x^2+4 e^x \log \left (\frac {3}{x}\right )-e^x x \log \left (\frac {3}{x}\right )}+\frac {16}{(-4+x) \left (-x+x^2+4 e^x \log \left (\frac {3}{x}\right )-e^x x \log \left (\frac {3}{x}\right )\right )}+\frac {x}{-x+x^2+4 e^x \log \left (\frac {3}{x}\right )-e^x x \log \left (\frac {3}{x}\right )}\right ) \, dx-12 \int \left (\frac {1}{-x+x^2+4 e^x \log \left (\frac {3}{x}\right )-e^x x \log \left (\frac {3}{x}\right )}+\frac {4}{(-4+x) \left (-x+x^2+4 e^x \log \left (\frac {3}{x}\right )-e^x x \log \left (\frac {3}{x}\right )\right )}\right ) \, dx+\int \frac {1}{-4+x} \, dx-\int \frac {1}{-1+x} \, dx-\int \left (\frac {16}{-x+x^2+4 e^x \log \left (\frac {3}{x}\right )-e^x x \log \left (\frac {3}{x}\right )}+\frac {64}{(-4+x) \left (-x+x^2+4 e^x \log \left (\frac {3}{x}\right )-e^x x \log \left (\frac {3}{x}\right )\right )}+\frac {4 x}{-x+x^2+4 e^x \log \left (\frac {3}{x}\right )-e^x x \log \left (\frac {3}{x}\right )}+\frac {x^2}{-x+x^2+4 e^x \log \left (\frac {3}{x}\right )-e^x x \log \left (\frac {3}{x}\right )}\right ) \, dx+\int \left (\frac {16}{(-4+x) \log \left (\frac {3}{x}\right ) \left (-x+x^2+4 e^x \log \left (\frac {3}{x}\right )-e^x x \log \left (\frac {3}{x}\right )\right )}+\frac {x}{\log \left (\frac {3}{x}\right ) \left (-x+x^2+4 e^x \log \left (\frac {3}{x}\right )-e^x x \log \left (\frac {3}{x}\right )\right )}-\frac {4}{\log \left (\frac {3}{x}\right ) \left (x-x^2-4 e^x \log \left (\frac {3}{x}\right )+e^x x \log \left (\frac {3}{x}\right )\right )}\right ) \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.21, size = 40, normalized size = 1.43 \begin {gather*} -\log (1-x)+\log \left (-x+x^2+4 e^x \log \left (\frac {3}{x}\right )-e^x x \log \left (\frac {3}{x}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-x + 2*x^2 - x^3 + E^x*(-4 + 5*x - x^2) + E^x*(7*x - 5*x^2 + x^3)*Log[3/x])/(-x^2 + 2*x^3 - x^4 + E
^x*(4*x - 5*x^2 + x^3)*Log[3/x]),x]

[Out]

-Log[1 - x] + Log[-x + x^2 + 4*E^x*Log[3/x] - E^x*x*Log[3/x]]

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fricas [A]  time = 0.75, size = 42, normalized size = 1.50 \begin {gather*} x - \log \left (x - 1\right ) + \log \left (x - 4\right ) + \log \left (\frac {{\left ({\left (x - 4\right )} e^{x} \log \left (\frac {3}{x}\right ) - x^{2} + x\right )} e^{\left (-x\right )}}{x - 4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^3-5*x^2+7*x)*exp(x)*log(3/x)+(-x^2+5*x-4)*exp(x)-x^3+2*x^2-x)/((x^3-5*x^2+4*x)*exp(x)*log(3/x)-x
^4+2*x^3-x^2),x, algorithm="fricas")

[Out]

x - log(x - 1) + log(x - 4) + log(((x - 4)*e^x*log(3/x) - x^2 + x)*e^(-x)/(x - 4))

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giac [A]  time = 0.42, size = 35, normalized size = 1.25 \begin {gather*} \log \left (x e^{x} \log \left (\frac {3}{x}\right ) - x^{2} - 4 \, e^{x} \log \left (\frac {3}{x}\right ) + x\right ) - \log \left (x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^3-5*x^2+7*x)*exp(x)*log(3/x)+(-x^2+5*x-4)*exp(x)-x^3+2*x^2-x)/((x^3-5*x^2+4*x)*exp(x)*log(3/x)-x
^4+2*x^3-x^2),x, algorithm="giac")

[Out]

log(x*e^x*log(3/x) - x^2 - 4*e^x*log(3/x) + x) - log(x - 1)

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maple [A]  time = 0.36, size = 37, normalized size = 1.32

method	result	size
norman	\(-\ln \left (x -1\right )+\ln \left (-x \,{\mathrm e}^{x} \ln \left (\frac {3}{x}\right )+x^{2}+4 \,{\mathrm e}^{x} \ln \left (\frac {3}{x}\right )-x \right )\)	\(37\)
default	\(-\ln \left (x -1\right )+\ln \left (x \,{\mathrm e}^{x} \ln \relax (x )-{\mathrm e}^{x} x \left (\ln \left (\frac {3}{x}\right )+\ln \relax (x )\right )+x^{2}-4 \,{\mathrm e}^{x} \ln \relax (x )+4 \,{\mathrm e}^{x} \left (\ln \left (\frac {3}{x}\right )+\ln \relax (x )\right )-x \right )\)	\(55\)
risch	\(x +\ln \left (x -4\right )-\ln \left (x -1\right )+\ln \left (\ln \relax (x )+\frac {i \left (2 i {\mathrm e}^{x} \ln \relax (3) x -8 i {\mathrm e}^{x} \ln \relax (3)-2 i x^{2}+2 i x \right ) {\mathrm e}^{-x}}{2 x -8}\right )\)	\(55\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^3-5*x^2+7*x)*exp(x)*ln(3/x)+(-x^2+5*x-4)*exp(x)-x^3+2*x^2-x)/((x^3-5*x^2+4*x)*exp(x)*ln(3/x)-x^4+2*x^3
-x^2),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [B]  time = 0.99, size = 67, normalized size = 2.39 \begin {gather*} -\log \left (x - 1\right ) + \log \left (x - 4\right ) + \log \left (-\frac {x^{2} - {\left (x \log \relax (3) - {\left (x - 4\right )} \log \relax (x) - 4 \, \log \relax (3)\right )} e^{x} - x}{x \log \relax (3) - {\left (x - 4\right )} \log \relax (x) - 4 \, \log \relax (3)}\right ) + \log \left (-\log \relax (3) + \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^3-5*x^2+7*x)*exp(x)*log(3/x)+(-x^2+5*x-4)*exp(x)-x^3+2*x^2-x)/((x^3-5*x^2+4*x)*exp(x)*log(3/x)-x
^4+2*x^3-x^2),x, algorithm="maxima")

[Out]

-log(x - 1) + log(x - 4) + log(-(x^2 - (x*log(3) - (x - 4)*log(x) - 4*log(3))*e^x - x)/(x*log(3) - (x - 4)*log
(x) - 4*log(3))) + log(-log(3) + log(x))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {x+{\mathrm {e}}^x\,\left (x^2-5\,x+4\right )-2\,x^2+x^3-{\mathrm {e}}^x\,\ln \left (\frac {3}{x}\right )\,\left (x^3-5\,x^2+7\,x\right )}{x^2-2\,x^3+x^4-{\mathrm {e}}^x\,\ln \left (\frac {3}{x}\right )\,\left (x^3-5\,x^2+4\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + exp(x)*(x^2 - 5*x + 4) - 2*x^2 + x^3 - exp(x)*log(3/x)*(7*x - 5*x^2 + x^3))/(x^2 - 2*x^3 + x^4 - exp(
x)*log(3/x)*(4*x - 5*x^2 + x^3)),x)

[Out]

int((x + exp(x)*(x^2 - 5*x + 4) - 2*x^2 + x^3 - exp(x)*log(3/x)*(7*x - 5*x^2 + x^3))/(x^2 - 2*x^3 + x^4 - exp(
x)*log(3/x)*(4*x - 5*x^2 + x^3)), x)

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sympy [A]  time = 0.97, size = 39, normalized size = 1.39 \begin {gather*} \log {\left (x - 4 \right )} - \log {\left (x - 1 \right )} + \log {\left (\frac {- x^{2} + x}{x \log {\left (\frac {3}{x} \right )} - 4 \log {\left (\frac {3}{x} \right )}} + e^{x} \right )} + \log {\left (\log {\left (\frac {3}{x} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**3-5*x**2+7*x)*exp(x)*ln(3/x)+(-x**2+5*x-4)*exp(x)-x**3+2*x**2-x)/((x**3-5*x**2+4*x)*exp(x)*ln(3
/x)-x**4+2*x**3-x**2),x)

[Out]

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

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