3.67.2 \(\int \frac {2 x-3 x^2-20 x^4+e^{2-x} (4+9 x+14 x^3-6 x^4)+(-5 x^2-6 x^4+e^{2-x} (2+4 x-3 x^2+4 x^3-2 x^4)) \log (x)+(-x^2-e^{2-x} x^2) \log ^2(x)}{x^2} \, dx\)

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

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Rubi [F]  time = 2.97, 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 {2 x-3 x^2-20 x^4+e^{2-x} \left (4+9 x+14 x^3-6 x^4\right )+\left (-5 x^2-6 x^4+e^{2-x} \left (2+4 x-3 x^2+4 x^3-2 x^4\right )\right ) \log (x)+\left (-x^2-e^{2-x} x^2\right ) \log ^2(x)}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(2*x - 3*x^2 - 20*x^4 + E^(2 - x)*(4 + 9*x + 14*x^3 - 6*x^4) + (-5*x^2 - 6*x^4 + E^(2 - x)*(2 + 4*x - 3*x^
2 + 4*x^3 - 2*x^4))*Log[x] + (-x^2 - E^(2 - x)*x^2)*Log[x]^2)/x^2,x]

[Out]

(-6*E^(2 - x))/x + 6*E^(2 - x)*x^2 - 6*x^3 + 2*E^2*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, -x] + 2*Log[x] +
3*E^(2 - x)*Log[x] - 2*E^2*EulerGamma*Log[x] - (2*E^(2 - x)*Log[x])/x + 2*x*Log[x] + 2*E^(2 - x)*x^2*Log[x] -
(5*x + 2*x^3)*Log[x] + 2*E^2*ExpIntegralEi[-x]*Log[x] - 2*E^2*(ExpIntegralE[1, x] + ExpIntegralEi[-x])*Log[x]
- E^2*Log[x]^2 - x*Log[x]^2 - Defer[Int][E^(2 - x)*Log[x]^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {2-3 x-20 x^3-5 x \log (x)-6 x^3 \log (x)-x \log ^2(x)}{x}-\frac {e^{2-x} \left (-4-9 x-14 x^3+6 x^4-2 \log (x)-4 x \log (x)+3 x^2 \log (x)-4 x^3 \log (x)+2 x^4 \log (x)+x^2 \log ^2(x)\right )}{x^2}\right ) \, dx\\ &=\int \frac {2-3 x-20 x^3-5 x \log (x)-6 x^3 \log (x)-x \log ^2(x)}{x} \, dx-\int \frac {e^{2-x} \left (-4-9 x-14 x^3+6 x^4-2 \log (x)-4 x \log (x)+3 x^2 \log (x)-4 x^3 \log (x)+2 x^4 \log (x)+x^2 \log ^2(x)\right )}{x^2} \, dx\\ &=\int \left (\frac {2-3 x-20 x^3}{x}-\left (5+6 x^2\right ) \log (x)-\log ^2(x)\right ) \, dx-\int \left (\frac {e^{2-x} \left (-4-9 x-14 x^3+6 x^4\right )}{x^2}+\frac {e^{2-x} \left (-2-4 x+3 x^2-4 x^3+2 x^4\right ) \log (x)}{x^2}+e^{2-x} \log ^2(x)\right ) \, dx\\ &=\int \frac {2-3 x-20 x^3}{x} \, dx-\int \frac {e^{2-x} \left (-4-9 x-14 x^3+6 x^4\right )}{x^2} \, dx-\int \left (5+6 x^2\right ) \log (x) \, dx-\int \frac {e^{2-x} \left (-2-4 x+3 x^2-4 x^3+2 x^4\right ) \log (x)}{x^2} \, dx-\int \log ^2(x) \, dx-\int e^{2-x} \log ^2(x) \, dx\\ &=3 e^{2-x} \log (x)-\frac {2 e^{2-x} \log (x)}{x}+2 e^{2-x} x^2 \log (x)-\left (5 x+2 x^3\right ) \log (x)+2 e^2 \text {Ei}(-x) \log (x)-x \log ^2(x)+2 \int \log (x) \, dx+\int \left (-3+\frac {2}{x}-20 x^2\right ) \, dx+\int \left (5+2 x^2\right ) \, dx-\int \left (-\frac {4 e^{2-x}}{x^2}-\frac {9 e^{2-x}}{x}-14 e^{2-x} x+6 e^{2-x} x^2\right ) \, dx+\int \frac {e^{2-x} \left (2-3 x-2 x^3-2 e^x x \text {Ei}(-x)\right )}{x^2} \, dx-\int e^{2-x} \log ^2(x) \, dx\\ &=-6 x^3+2 \log (x)+3 e^{2-x} \log (x)-\frac {2 e^{2-x} \log (x)}{x}+2 x \log (x)+2 e^{2-x} x^2 \log (x)-\left (5 x+2 x^3\right ) \log (x)+2 e^2 \text {Ei}(-x) \log (x)-x \log ^2(x)+4 \int \frac {e^{2-x}}{x^2} \, dx-6 \int e^{2-x} x^2 \, dx+9 \int \frac {e^{2-x}}{x} \, dx+14 \int e^{2-x} x \, dx+\int \left (\frac {e^{2-x} \left (2-3 x-2 x^3\right )}{x^2}-\frac {2 e^2 \text {Ei}(-x)}{x}\right ) \, dx-\int e^{2-x} \log ^2(x) \, dx\\ &=-\frac {4 e^{2-x}}{x}-14 e^{2-x} x+6 e^{2-x} x^2-6 x^3+9 e^2 \text {Ei}(-x)+2 \log (x)+3 e^{2-x} \log (x)-\frac {2 e^{2-x} \log (x)}{x}+2 x \log (x)+2 e^{2-x} x^2 \log (x)-\left (5 x+2 x^3\right ) \log (x)+2 e^2 \text {Ei}(-x) \log (x)-x \log ^2(x)-4 \int \frac {e^{2-x}}{x} \, dx-12 \int e^{2-x} x \, dx+14 \int e^{2-x} \, dx-\left (2 e^2\right ) \int \frac {\text {Ei}(-x)}{x} \, dx+\int \frac {e^{2-x} \left (2-3 x-2 x^3\right )}{x^2} \, dx-\int e^{2-x} \log ^2(x) \, dx\\ &=-14 e^{2-x}-\frac {4 e^{2-x}}{x}-2 e^{2-x} x+6 e^{2-x} x^2-6 x^3+5 e^2 \text {Ei}(-x)+2 \log (x)+3 e^{2-x} \log (x)-\frac {2 e^{2-x} \log (x)}{x}+2 x \log (x)+2 e^{2-x} x^2 \log (x)-\left (5 x+2 x^3\right ) \log (x)+2 e^2 \text {Ei}(-x) \log (x)-2 e^2 (E_1(x)+\text {Ei}(-x)) \log (x)-x \log ^2(x)-12 \int e^{2-x} \, dx+\left (2 e^2\right ) \int \frac {E_1(x)}{x} \, dx+\int \left (\frac {2 e^{2-x}}{x^2}-\frac {3 e^{2-x}}{x}-2 e^{2-x} x\right ) \, dx-\int e^{2-x} \log ^2(x) \, dx\\ &=-2 e^{2-x}-\frac {4 e^{2-x}}{x}-2 e^{2-x} x+6 e^{2-x} x^2-6 x^3+5 e^2 \text {Ei}(-x)+2 e^2 x \, _3F_3(1,1,1;2,2,2;-x)+2 \log (x)+3 e^{2-x} \log (x)-2 e^2 \gamma \log (x)-\frac {2 e^{2-x} \log (x)}{x}+2 x \log (x)+2 e^{2-x} x^2 \log (x)-\left (5 x+2 x^3\right ) \log (x)+2 e^2 \text {Ei}(-x) \log (x)-2 e^2 (E_1(x)+\text {Ei}(-x)) \log (x)-e^2 \log ^2(x)-x \log ^2(x)+2 \int \frac {e^{2-x}}{x^2} \, dx-2 \int e^{2-x} x \, dx-3 \int \frac {e^{2-x}}{x} \, dx-\int e^{2-x} \log ^2(x) \, dx\\ &=-2 e^{2-x}-\frac {6 e^{2-x}}{x}+6 e^{2-x} x^2-6 x^3+2 e^2 \text {Ei}(-x)+2 e^2 x \, _3F_3(1,1,1;2,2,2;-x)+2 \log (x)+3 e^{2-x} \log (x)-2 e^2 \gamma \log (x)-\frac {2 e^{2-x} \log (x)}{x}+2 x \log (x)+2 e^{2-x} x^2 \log (x)-\left (5 x+2 x^3\right ) \log (x)+2 e^2 \text {Ei}(-x) \log (x)-2 e^2 (E_1(x)+\text {Ei}(-x)) \log (x)-e^2 \log ^2(x)-x \log ^2(x)-2 \int e^{2-x} \, dx-2 \int \frac {e^{2-x}}{x} \, dx-\int e^{2-x} \log ^2(x) \, dx\\ &=-\frac {6 e^{2-x}}{x}+6 e^{2-x} x^2-6 x^3+2 e^2 x \, _3F_3(1,1,1;2,2,2;-x)+2 \log (x)+3 e^{2-x} \log (x)-2 e^2 \gamma \log (x)-\frac {2 e^{2-x} \log (x)}{x}+2 x \log (x)+2 e^{2-x} x^2 \log (x)-\left (5 x+2 x^3\right ) \log (x)+2 e^2 \text {Ei}(-x) \log (x)-2 e^2 (E_1(x)+\text {Ei}(-x)) \log (x)-e^2 \log ^2(x)-x \log ^2(x)-\int e^{2-x} \log ^2(x) \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.09, size = 67, normalized size = 2.31 \begin {gather*} \frac {e^{-x} \left (-6 e^x x^4+6 e^2 \left (-1+x^3\right )+\left (e^2-e^x x\right ) \left (-2+3 x+2 x^3\right ) \log (x)+x \left (e^2-e^x x\right ) \log ^2(x)\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2*x - 3*x^2 - 20*x^4 + E^(2 - x)*(4 + 9*x + 14*x^3 - 6*x^4) + (-5*x^2 - 6*x^4 + E^(2 - x)*(2 + 4*x
- 3*x^2 + 4*x^3 - 2*x^4))*Log[x] + (-x^2 - E^(2 - x)*x^2)*Log[x]^2)/x^2,x]

[Out]

(-6*E^x*x^4 + 6*E^2*(-1 + x^3) + (E^2 - E^x*x)*(-2 + 3*x + 2*x^3)*Log[x] + x*(E^2 - E^x*x)*Log[x]^2)/(E^x*x)

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fricas [B]  time = 0.62, size = 77, normalized size = 2.66 \begin {gather*} -\frac {6 \, x^{4} + {\left (x^{2} - x e^{\left (-x + 2\right )}\right )} \log \relax (x)^{2} - 6 \, {\left (x^{3} - 1\right )} e^{\left (-x + 2\right )} + {\left (2 \, x^{4} + 3 \, x^{2} - {\left (2 \, x^{3} + 3 \, x - 2\right )} e^{\left (-x + 2\right )} - 2 \, x\right )} \log \relax (x)}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2*exp(2-x)-x^2)*log(x)^2+((-2*x^4+4*x^3-3*x^2+4*x+2)*exp(2-x)-6*x^4-5*x^2)*log(x)+(-6*x^4+14*x^
3+9*x+4)*exp(2-x)-20*x^4-3*x^2+2*x)/x^2,x, algorithm="fricas")

[Out]

-(6*x^4 + (x^2 - x*e^(-x + 2))*log(x)^2 - 6*(x^3 - 1)*e^(-x + 2) + (2*x^4 + 3*x^2 - (2*x^3 + 3*x - 2)*e^(-x +
2) - 2*x)*log(x))/x

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giac [B]  time = 0.15, size = 104, normalized size = 3.59 \begin {gather*} -\frac {2 \, x^{4} \log \relax (x) - 2 \, x^{3} e^{\left (-x + 2\right )} \log \relax (x) + 6 \, x^{4} - 6 \, x^{3} e^{\left (-x + 2\right )} + x^{2} \log \relax (x)^{2} - x e^{\left (-x + 2\right )} \log \relax (x)^{2} + 3 \, x^{2} \log \relax (x) - 3 \, x e^{\left (-x + 2\right )} \log \relax (x) - 2 \, x \log \relax (x) + 2 \, e^{\left (-x + 2\right )} \log \relax (x) + 6 \, e^{\left (-x + 2\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2*exp(2-x)-x^2)*log(x)^2+((-2*x^4+4*x^3-3*x^2+4*x+2)*exp(2-x)-6*x^4-5*x^2)*log(x)+(-6*x^4+14*x^
3+9*x+4)*exp(2-x)-20*x^4-3*x^2+2*x)/x^2,x, algorithm="giac")

[Out]

-(2*x^4*log(x) - 2*x^3*e^(-x + 2)*log(x) + 6*x^4 - 6*x^3*e^(-x + 2) + x^2*log(x)^2 - x*e^(-x + 2)*log(x)^2 + 3
*x^2*log(x) - 3*x*e^(-x + 2)*log(x) - 2*x*log(x) + 2*e^(-x + 2)*log(x) + 6*e^(-x + 2))/x

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maple [B]  time = 0.08, size = 97, normalized size = 3.34

method	result	size
risch	\(\left ({\mathrm e}^{2-x}-x \right ) \ln \relax (x )^{2}-\frac {\left (2 x^{4}-2 x^{3} {\mathrm e}^{2-x}+3 x^{2}-3 x \,{\mathrm e}^{2-x}+2 \,{\mathrm e}^{2-x}\right ) \ln \relax (x )}{x}+\frac {-6 x^{4}+6 x^{3} {\mathrm e}^{2-x}+2 x \ln \relax (x )-6 \,{\mathrm e}^{2-x}}{x}\)	\(97\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^2*exp(2-x)-x^2)*ln(x)^2+((-2*x^4+4*x^3-3*x^2+4*x+2)*exp(2-x)-6*x^4-5*x^2)*ln(x)+(-6*x^4+14*x^3+9*x+4)
*exp(2-x)-20*x^4-3*x^2+2*x)/x^2,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -2 \, x^{3} \log \relax (x) - 6 \, x^{3} + 6 \, {\rm Ei}\left (-x\right ) e^{2} + 6 \, {\left (x^{2} e^{2} + 2 \, x e^{2} + 2 \, e^{2}\right )} e^{\left (-x\right )} - 14 \, {\left (x e^{2} + e^{2}\right )} e^{\left (-x\right )} - 4 \, e^{2} \Gamma \left (-1, x\right ) - 5 \, x \log \relax (x) + 3 \, e^{\left (-x + 2\right )} \log \relax (x) + 2 \, x - \frac {x^{2} \log \relax (x)^{2} - 2 \, x^{2} \log \relax (x) + 2 \, x^{2} - {\left (x e^{2} \log \relax (x)^{2} + 2 \, {\left (x^{3} e^{2} - e^{2}\right )} \log \relax (x)\right )} e^{\left (-x\right )}}{x} - \int \frac {2 \, {\left (x^{3} e^{2} - e^{2}\right )} e^{\left (-x\right )}}{x^{2}}\,{d x} + 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2*exp(2-x)-x^2)*log(x)^2+((-2*x^4+4*x^3-3*x^2+4*x+2)*exp(2-x)-6*x^4-5*x^2)*log(x)+(-6*x^4+14*x^
3+9*x+4)*exp(2-x)-20*x^4-3*x^2+2*x)/x^2,x, algorithm="maxima")

[Out]

-2*x^3*log(x) - 6*x^3 + 6*Ei(-x)*e^2 + 6*(x^2*e^2 + 2*x*e^2 + 2*e^2)*e^(-x) - 14*(x*e^2 + e^2)*e^(-x) - 4*e^2*
gamma(-1, x) - 5*x*log(x) + 3*e^(-x + 2)*log(x) + 2*x - (x^2*log(x)^2 - 2*x^2*log(x) + 2*x^2 - (x*e^2*log(x)^2
+ 2*(x^3*e^2 - e^2)*log(x))*e^(-x))/x - integrate(2*(x^3*e^2 - e^2)*e^(-x)/x^2, x) + 2*log(x)

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mupad [B]  time = 4.34, size = 83, normalized size = 2.86 \begin {gather*} 2\,\ln \relax (x)-{\ln \relax (x)}^2\,\left (x-{\mathrm {e}}^{2-x}\right )-6\,x^3-\ln \relax (x)\,\left (3\,x-{\mathrm {e}}^{2-x}\,\left (\frac {2\,x^3+3\,x}{x}-\frac {2}{x}\right )+2\,x^3\right )+\frac {{\mathrm {e}}^{2-x}\,\left (6\,x^3-6\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x)^2*(x^2*exp(2 - x) + x^2) - exp(2 - x)*(9*x + 14*x^3 - 6*x^4 + 4) - 2*x + 3*x^2 + 20*x^4 + log(x)*
(5*x^2 - exp(2 - x)*(4*x - 3*x^2 + 4*x^3 - 2*x^4 + 2) + 6*x^4))/x^2,x)

[Out]

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

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sympy [B]  time = 0.40, size = 70, normalized size = 2.41 \begin {gather*} - 6 x^{3} - x \log {\relax (x )}^{2} + \left (- 2 x^{3} - 3 x\right ) \log {\relax (x )} + 2 \log {\relax (x )} + \frac {\left (2 x^{3} \log {\relax (x )} + 6 x^{3} + x \log {\relax (x )}^{2} + 3 x \log {\relax (x )} - 2 \log {\relax (x )} - 6\right ) e^{2 - x}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**2*exp(2-x)-x**2)*ln(x)**2+((-2*x**4+4*x**3-3*x**2+4*x+2)*exp(2-x)-6*x**4-5*x**2)*ln(x)+(-6*x**
4+14*x**3+9*x+4)*exp(2-x)-20*x**4-3*x**2+2*x)/x**2,x)

[Out]

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

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