3.96.33 \(\int \frac {12+22 x^2-2 x^3+e^{e^{-4+x}} (-x^2+2 x^3+e^{-4+x} (-x^3+x^4))}{-12 x-9 x^2+22 x^3-x^4+e^{e^{-4+x}} (-x^3+x^4)} \, dx\)

Optimal. Leaf size=31 \[ \log (1-x)+\log \left (6+3 \left (5+\frac {4}{x}\right )-x+e^{e^{-4+x}} x\right ) \]

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Rubi [F]  time = 5.23, 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 {12+22 x^2-2 x^3+e^{e^{-4+x}} \left (-x^2+2 x^3+e^{-4+x} \left (-x^3+x^4\right )\right )}{-12 x-9 x^2+22 x^3-x^4+e^{e^{-4+x}} \left (-x^3+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(12 + 22*x^2 - 2*x^3 + E^E^(-4 + x)*(-x^2 + 2*x^3 + E^(-4 + x)*(-x^3 + x^4)))/(-12*x - 9*x^2 + 22*x^3 - x^
4 + E^E^(-4 + x)*(-x^3 + x^4)),x]

[Out]

20*Defer[Int][(12 + 21*x - x^2 + E^E^(-4 + x)*x^2)^(-1), x] + Defer[Int][E^E^(-4 + x)/(12 + 21*x - x^2 + E^E^(
-4 + x)*x^2), x] + 32*Defer[Int][1/((-1 + x)*(12 + 21*x - x^2 + E^E^(-4 + x)*x^2)), x] + Defer[Int][E^E^(-4 +
x)/((-1 + x)*(12 + 21*x - x^2 + E^E^(-4 + x)*x^2)), x] - 12*Defer[Int][1/(x*(12 + 21*x - x^2 + E^E^(-4 + x)*x^
2)), x] - 2*Defer[Int][x/(12 + 21*x - x^2 + E^E^(-4 + x)*x^2), x] + 2*Defer[Int][(E^E^(-4 + x)*x)/(12 + 21*x -
x^2 + E^E^(-4 + x)*x^2), x] + Defer[Int][(E^(-4 + E^(-4 + x) + x)*x^2)/(12 + 21*x - x^2 + E^E^(-4 + x)*x^2),
x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-12-22 x^2+2 x^3-e^{e^{-4+x}} \left (-x^2+2 x^3+e^{-4+x} \left (-x^3+x^4\right )\right )}{(1-x) x \left (12+21 x-x^2+e^{e^{-4+x}} x^2\right )} \, dx\\ &=\int \left (\frac {12}{(-1+x) x \left (12+21 x-x^2+e^{e^{-4+x}} x^2\right )}+\frac {22 x}{(-1+x) \left (12+21 x-x^2+e^{e^{-4+x}} x^2\right )}-\frac {e^{e^{-4+x}} x}{(-1+x) \left (12+21 x-x^2+e^{e^{-4+x}} x^2\right )}+\frac {e^{-4+e^{-4+x}+x} x^2}{12+21 x-x^2+e^{e^{-4+x}} x^2}-\frac {2 x^2}{(-1+x) \left (12+21 x-x^2+e^{e^{-4+x}} x^2\right )}+\frac {2 e^{e^{-4+x}} x^2}{(-1+x) \left (12+21 x-x^2+e^{e^{-4+x}} x^2\right )}\right ) \, dx\\ &=-\left (2 \int \frac {x^2}{(-1+x) \left (12+21 x-x^2+e^{e^{-4+x}} x^2\right )} \, dx\right )+2 \int \frac {e^{e^{-4+x}} x^2}{(-1+x) \left (12+21 x-x^2+e^{e^{-4+x}} x^2\right )} \, dx+12 \int \frac {1}{(-1+x) x \left (12+21 x-x^2+e^{e^{-4+x}} x^2\right )} \, dx+22 \int \frac {x}{(-1+x) \left (12+21 x-x^2+e^{e^{-4+x}} x^2\right )} \, dx-\int \frac {e^{e^{-4+x}} x}{(-1+x) \left (12+21 x-x^2+e^{e^{-4+x}} x^2\right )} \, dx+\int \frac {e^{-4+e^{-4+x}+x} x^2}{12+21 x-x^2+e^{e^{-4+x}} x^2} \, dx\\ &=-\left (2 \int \left (\frac {1}{12+21 x-x^2+e^{e^{-4+x}} x^2}+\frac {1}{(-1+x) \left (12+21 x-x^2+e^{e^{-4+x}} x^2\right )}+\frac {x}{12+21 x-x^2+e^{e^{-4+x}} x^2}\right ) \, dx\right )+2 \int \left (\frac {e^{e^{-4+x}}}{12+21 x-x^2+e^{e^{-4+x}} x^2}+\frac {e^{e^{-4+x}}}{(-1+x) \left (12+21 x-x^2+e^{e^{-4+x}} x^2\right )}+\frac {e^{e^{-4+x}} x}{12+21 x-x^2+e^{e^{-4+x}} x^2}\right ) \, dx+12 \int \left (\frac {1}{(-1+x) \left (12+21 x-x^2+e^{e^{-4+x}} x^2\right )}-\frac {1}{x \left (12+21 x-x^2+e^{e^{-4+x}} x^2\right )}\right ) \, dx+22 \int \left (\frac {1}{12+21 x-x^2+e^{e^{-4+x}} x^2}+\frac {1}{(-1+x) \left (12+21 x-x^2+e^{e^{-4+x}} x^2\right )}\right ) \, dx+\int \frac {e^{-4+e^{-4+x}+x} x^2}{12+21 x-x^2+e^{e^{-4+x}} x^2} \, dx-\int \left (\frac {e^{e^{-4+x}}}{12+21 x-x^2+e^{e^{-4+x}} x^2}+\frac {e^{e^{-4+x}}}{(-1+x) \left (12+21 x-x^2+e^{e^{-4+x}} x^2\right )}\right ) \, dx\\ &=-\left (2 \int \frac {1}{12+21 x-x^2+e^{e^{-4+x}} x^2} \, dx\right )+2 \int \frac {e^{e^{-4+x}}}{12+21 x-x^2+e^{e^{-4+x}} x^2} \, dx-2 \int \frac {1}{(-1+x) \left (12+21 x-x^2+e^{e^{-4+x}} x^2\right )} \, dx+2 \int \frac {e^{e^{-4+x}}}{(-1+x) \left (12+21 x-x^2+e^{e^{-4+x}} x^2\right )} \, dx-2 \int \frac {x}{12+21 x-x^2+e^{e^{-4+x}} x^2} \, dx+2 \int \frac {e^{e^{-4+x}} x}{12+21 x-x^2+e^{e^{-4+x}} x^2} \, dx+12 \int \frac {1}{(-1+x) \left (12+21 x-x^2+e^{e^{-4+x}} x^2\right )} \, dx-12 \int \frac {1}{x \left (12+21 x-x^2+e^{e^{-4+x}} x^2\right )} \, dx+22 \int \frac {1}{12+21 x-x^2+e^{e^{-4+x}} x^2} \, dx+22 \int \frac {1}{(-1+x) \left (12+21 x-x^2+e^{e^{-4+x}} x^2\right )} \, dx-\int \frac {e^{e^{-4+x}}}{12+21 x-x^2+e^{e^{-4+x}} x^2} \, dx-\int \frac {e^{e^{-4+x}}}{(-1+x) \left (12+21 x-x^2+e^{e^{-4+x}} x^2\right )} \, dx+\int \frac {e^{-4+e^{-4+x}+x} x^2}{12+21 x-x^2+e^{e^{-4+x}} x^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.56, size = 48, normalized size = 1.55 \begin {gather*} \frac {e^4 \log (1-x)-e^4 \log (x)+e^4 \log \left (12+21 x-x^2+e^{e^{-4+x}} x^2\right )}{e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(12 + 22*x^2 - 2*x^3 + E^E^(-4 + x)*(-x^2 + 2*x^3 + E^(-4 + x)*(-x^3 + x^4)))/(-12*x - 9*x^2 + 22*x^
3 - x^4 + E^E^(-4 + x)*(-x^3 + x^4)),x]

[Out]

(E^4*Log[1 - x] - E^4*Log[x] + E^4*Log[12 + 21*x - x^2 + E^E^(-4 + x)*x^2])/E^4

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fricas [A]  time = 0.55, size = 33, normalized size = 1.06 \begin {gather*} \log \left (x^{2} - x\right ) + \log \left (\frac {x^{2} e^{\left (e^{\left (x - 4\right )}\right )} - x^{2} + 21 \, x + 12}{x^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^4-x^3)*exp(x-4)+2*x^3-x^2)*exp(exp(x-4))-2*x^3+22*x^2+12)/((x^4-x^3)*exp(exp(x-4))-x^4+22*x^3-9
*x^2-12*x),x, algorithm="fricas")

[Out]

log(x^2 - x) + log((x^2*e^(e^(x - 4)) - x^2 + 21*x + 12)/x^2)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, x^{3} - 22 \, x^{2} - {\left (2 \, x^{3} - x^{2} + {\left (x^{4} - x^{3}\right )} e^{\left (x - 4\right )}\right )} e^{\left (e^{\left (x - 4\right )}\right )} - 12}{x^{4} - 22 \, x^{3} + 9 \, x^{2} - {\left (x^{4} - x^{3}\right )} e^{\left (e^{\left (x - 4\right )}\right )} + 12 \, x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^4-x^3)*exp(x-4)+2*x^3-x^2)*exp(exp(x-4))-2*x^3+22*x^2+12)/((x^4-x^3)*exp(exp(x-4))-x^4+22*x^3-9
*x^2-12*x),x, algorithm="giac")

[Out]

integrate((2*x^3 - 22*x^2 - (2*x^3 - x^2 + (x^4 - x^3)*e^(x - 4))*e^(e^(x - 4)) - 12)/(x^4 - 22*x^3 + 9*x^2 -
(x^4 - x^3)*e^(e^(x - 4)) + 12*x), x)

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maple [A]  time = 0.24, size = 30, normalized size = 0.97

method	result	size
norman	\(-\ln \relax (x )+\ln \left (x -1\right )+\ln \left ({\mathrm e}^{{\mathrm e}^{x -4}} x^{2}-x^{2}+21 x +12\right )\)	\(30\)
risch	\(\ln \left (x^{2}-x \right )+\ln \left ({\mathrm e}^{{\mathrm e}^{x -4}}-\frac {x^{2}-21 x -12}{x^{2}}\right )\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((x^4-x^3)*exp(x-4)+2*x^3-x^2)*exp(exp(x-4))-2*x^3+22*x^2+12)/((x^4-x^3)*exp(exp(x-4))-x^4+22*x^3-9*x^2-1
2*x),x,method=_RETURNVERBOSE)

[Out]

-ln(x)+ln(x-1)+ln(exp(exp(x-4))*x^2-x^2+21*x+12)

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maxima [A]  time = 0.39, size = 31, normalized size = 1.00 \begin {gather*} \log \left (x - 1\right ) + \log \relax (x) + \log \left (\frac {x^{2} e^{\left (e^{\left (x - 4\right )}\right )} - x^{2} + 21 \, x + 12}{x^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^4-x^3)*exp(x-4)+2*x^3-x^2)*exp(exp(x-4))-2*x^3+22*x^2+12)/((x^4-x^3)*exp(exp(x-4))-x^4+22*x^3-9
*x^2-12*x),x, algorithm="maxima")

[Out]

log(x - 1) + log(x) + log((x^2*e^(e^(x - 4)) - x^2 + 21*x + 12)/x^2)

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mupad [B]  time = 0.59, size = 31, normalized size = 1.00 \begin {gather*} \ln \left (x\,\left (x-1\right )\right )+\ln \left (\frac {21\,x-x^2+x^2\,{\mathrm {e}}^{{\mathrm {e}}^{x-4}}+12}{x^2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(exp(x - 4))*(exp(x - 4)*(x^3 - x^4) + x^2 - 2*x^3) - 22*x^2 + 2*x^3 - 12)/(12*x + exp(exp(x - 4))*(x^
3 - x^4) + 9*x^2 - 22*x^3 + x^4),x)

[Out]

log(x*(x - 1)) + log((21*x - x^2 + x^2*exp(exp(x - 4)) + 12)/x^2)

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sympy [A]  time = 0.31, size = 26, normalized size = 0.84 \begin {gather*} \log {\left (x^{2} - x \right )} + \log {\left (e^{e^{x - 4}} + \frac {- x^{2} + 21 x + 12}{x^{2}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x**4-x**3)*exp(x-4)+2*x**3-x**2)*exp(exp(x-4))-2*x**3+22*x**2+12)/((x**4-x**3)*exp(exp(x-4))-x**4
+22*x**3-9*x**2-12*x),x)

[Out]

log(x**2 - x) + log(exp(exp(x - 4)) + (-x**2 + 21*x + 12)/x**2)

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