3.79.19 \(\int \frac {1242+3728 x+3730 x^2+1244 x^3+e^{\frac {2 x-x^2}{2+2 x}} (2 x^3-2 x^4-x^5)+(-2 x-4 x^2-2 x^3+e^{\frac {2 x-x^2}{2+2 x}} (4 x^2-4 x^3-2 x^4)) \log (x)+e^{\frac {2 x-x^2}{2+2 x}} (2 x-2 x^2-x^3) \log ^2(x)}{2 x^3+4 x^4+2 x^5+(4 x^2+8 x^3+4 x^4) \log (x)+(2 x+4 x^2+2 x^3) \log ^2(x)} \, dx\) [7819]

Optimal. Leaf size=28 \[ e^{x-\frac {3 x}{2+\frac {2}{x}}}-\frac {621+x}{x+\log (x)} \]

[Out]

exp(x-3/(2/x+2)*x)-(621+x)/(x+ln(x))

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Rubi [F]
time = 9.71, 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 {1242+3728 x+3730 x^2+1244 x^3+e^{\frac {2 x-x^2}{2+2 x}} \left (2 x^3-2 x^4-x^5\right )+\left (-2 x-4 x^2-2 x^3+e^{\frac {2 x-x^2}{2+2 x}} \left (4 x^2-4 x^3-2 x^4\right )\right ) \log (x)+e^{\frac {2 x-x^2}{2+2 x}} \left (2 x-2 x^2-x^3\right ) \log ^2(x)}{2 x^3+4 x^4+2 x^5+\left (4 x^2+8 x^3+4 x^4\right ) \log (x)+\left (2 x+4 x^2+2 x^3\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(1242 + 3728*x + 3730*x^2 + 1244*x^3 + E^((2*x - x^2)/(2 + 2*x))*(2*x^3 - 2*x^4 - x^5) + (-2*x - 4*x^2 - 2
*x^3 + E^((2*x - x^2)/(2 + 2*x))*(4*x^2 - 4*x^3 - 2*x^4))*Log[x] + E^((2*x - x^2)/(2 + 2*x))*(2*x - 2*x^2 - x^
3)*Log[x]^2)/(2*x^3 + 4*x^4 + 2*x^5 + (4*x^2 + 8*x^3 + 4*x^4)*Log[x] + (2*x + 4*x^2 + 2*x^3)*Log[x]^2),x]

[Out]

E^(((2 - x)*x)/(2*(1 + x))) + 622*Defer[Int][(x + Log[x])^(-2), x] + 621*Defer[Int][1/(x*(x + Log[x])^2), x] +
Defer[Int][x/(x + Log[x])^2, x] - Defer[Int][(x + Log[x])^(-1), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-\frac {x^2}{2+2 x}} \left (2 e^{\frac {x^2}{2+2 x}} (1+x)^2 (621+622 x)-e^{\frac {x}{1+x}} x^3 \left (-2+2 x+x^2\right )-2 x \left (e^{\frac {x^2}{2+2 x}} (1+x)^2+e^{\frac {x}{1+x}} x \left (-2+2 x+x^2\right )\right ) \log (x)-e^{\frac {x}{1+x}} x \left (-2+2 x+x^2\right ) \log ^2(x)\right )}{2 x (1+x)^2 (x+\log (x))^2} \, dx\\ &=\frac {1}{2} \int \frac {e^{-\frac {x^2}{2+2 x}} \left (2 e^{\frac {x^2}{2+2 x}} (1+x)^2 (621+622 x)-e^{\frac {x}{1+x}} x^3 \left (-2+2 x+x^2\right )-2 x \left (e^{\frac {x^2}{2+2 x}} (1+x)^2+e^{\frac {x}{1+x}} x \left (-2+2 x+x^2\right )\right ) \log (x)-e^{\frac {x}{1+x}} x \left (-2+2 x+x^2\right ) \log ^2(x)\right )}{x (1+x)^2 (x+\log (x))^2} \, dx\\ &=\frac {1}{2} \int \left (-\frac {e^{\frac {x}{1+x}-\frac {x^2}{2+2 x}} x^2 \left (-2+2 x+x^2\right )}{(1+x)^2 (x+\log (x))^2}-\frac {2 e^{\frac {x}{1+x}-\frac {x^2}{2+2 x}} x \left (-2+2 x+x^2\right ) \log (x)}{(1+x)^2 (x+\log (x))^2}-\frac {e^{\frac {x}{1+x}-\frac {x^2}{2+2 x}} \left (-2+2 x+x^2\right ) \log ^2(x)}{(1+x)^2 (x+\log (x))^2}-\frac {2 (-621-622 x+x \log (x))}{x (x+\log (x))^2}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {e^{\frac {x}{1+x}-\frac {x^2}{2+2 x}} x^2 \left (-2+2 x+x^2\right )}{(1+x)^2 (x+\log (x))^2} \, dx\right )-\frac {1}{2} \int \frac {e^{\frac {x}{1+x}-\frac {x^2}{2+2 x}} \left (-2+2 x+x^2\right ) \log ^2(x)}{(1+x)^2 (x+\log (x))^2} \, dx-\int \frac {e^{\frac {x}{1+x}-\frac {x^2}{2+2 x}} x \left (-2+2 x+x^2\right ) \log (x)}{(1+x)^2 (x+\log (x))^2} \, dx-\int \frac {-621-622 x+x \log (x)}{x (x+\log (x))^2} \, dx\\ &=-\left (\frac {1}{2} \int \frac {e^{-\frac {(-2+x) x}{2 (1+x)}} x^2 \left (-2+2 x+x^2\right )}{(1+x)^2 (x+\log (x))^2} \, dx\right )-\frac {1}{2} \int \frac {e^{-\frac {(-2+x) x}{2 (1+x)}} \left (-2+2 x+x^2\right ) \log ^2(x)}{(1+x)^2 (x+\log (x))^2} \, dx-\int \frac {e^{-\frac {(-2+x) x}{2 (1+x)}} x \left (-2+2 x+x^2\right ) \log (x)}{(1+x)^2 (x+\log (x))^2} \, dx-\int \left (\frac {-621-622 x-x^2}{x (x+\log (x))^2}+\frac {1}{x+\log (x)}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \left (-\frac {3 e^{-\frac {(-2+x) x}{2 (1+x)}}}{(x+\log (x))^2}+\frac {e^{-\frac {(-2+x) x}{2 (1+x)}} x^2}{(x+\log (x))^2}-\frac {3 e^{-\frac {(-2+x) x}{2 (1+x)}}}{(1+x)^2 (x+\log (x))^2}+\frac {6 e^{-\frac {(-2+x) x}{2 (1+x)}}}{(1+x) (x+\log (x))^2}\right ) \, dx\right )-\frac {1}{2} \int \left (\frac {e^{-\frac {(-2+x) x}{2 (1+x)}} \left (-2+2 x+x^2\right )}{(1+x)^2}+\frac {e^{-\frac {(-2+x) x}{2 (1+x)}} x^2 \left (-2+2 x+x^2\right )}{(1+x)^2 (x+\log (x))^2}-\frac {2 e^{-\frac {(-2+x) x}{2 (1+x)}} x \left (-2+2 x+x^2\right )}{(1+x)^2 (x+\log (x))}\right ) \, dx-\int \frac {-621-622 x-x^2}{x (x+\log (x))^2} \, dx-\int \frac {1}{x+\log (x)} \, dx-\int \left (-\frac {e^{-\frac {(-2+x) x}{2 (1+x)}} x^2 \left (-2+2 x+x^2\right )}{(1+x)^2 (x+\log (x))^2}+\frac {e^{-\frac {(-2+x) x}{2 (1+x)}} x \left (-2+2 x+x^2\right )}{(1+x)^2 (x+\log (x))}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {e^{-\frac {(-2+x) x}{2 (1+x)}} \left (-2+2 x+x^2\right )}{(1+x)^2} \, dx\right )-\frac {1}{2} \int \frac {e^{-\frac {(-2+x) x}{2 (1+x)}} x^2}{(x+\log (x))^2} \, dx-\frac {1}{2} \int \frac {e^{-\frac {(-2+x) x}{2 (1+x)}} x^2 \left (-2+2 x+x^2\right )}{(1+x)^2 (x+\log (x))^2} \, dx+\frac {3}{2} \int \frac {e^{-\frac {(-2+x) x}{2 (1+x)}}}{(x+\log (x))^2} \, dx+\frac {3}{2} \int \frac {e^{-\frac {(-2+x) x}{2 (1+x)}}}{(1+x)^2 (x+\log (x))^2} \, dx-3 \int \frac {e^{-\frac {(-2+x) x}{2 (1+x)}}}{(1+x) (x+\log (x))^2} \, dx+\int \frac {e^{-\frac {(-2+x) x}{2 (1+x)}} x^2 \left (-2+2 x+x^2\right )}{(1+x)^2 (x+\log (x))^2} \, dx-\int \frac {1}{x+\log (x)} \, dx-\int \left (-\frac {622}{(x+\log (x))^2}-\frac {621}{x (x+\log (x))^2}-\frac {x}{(x+\log (x))^2}\right ) \, dx\\ &=e^{\frac {(2-x) x}{2 (1+x)}}-\frac {1}{2} \int \frac {e^{-\frac {(-2+x) x}{2 (1+x)}} x^2}{(x+\log (x))^2} \, dx-\frac {1}{2} \int \left (-\frac {3 e^{-\frac {(-2+x) x}{2 (1+x)}}}{(x+\log (x))^2}+\frac {e^{-\frac {(-2+x) x}{2 (1+x)}} x^2}{(x+\log (x))^2}-\frac {3 e^{-\frac {(-2+x) x}{2 (1+x)}}}{(1+x)^2 (x+\log (x))^2}+\frac {6 e^{-\frac {(-2+x) x}{2 (1+x)}}}{(1+x) (x+\log (x))^2}\right ) \, dx+\frac {3}{2} \int \frac {e^{-\frac {(-2+x) x}{2 (1+x)}}}{(x+\log (x))^2} \, dx+\frac {3}{2} \int \frac {e^{-\frac {(-2+x) x}{2 (1+x)}}}{(1+x)^2 (x+\log (x))^2} \, dx-3 \int \frac {e^{-\frac {(-2+x) x}{2 (1+x)}}}{(1+x) (x+\log (x))^2} \, dx+621 \int \frac {1}{x (x+\log (x))^2} \, dx+622 \int \frac {1}{(x+\log (x))^2} \, dx+\int \frac {x}{(x+\log (x))^2} \, dx-\int \frac {1}{x+\log (x)} \, dx+\int \left (-\frac {3 e^{-\frac {(-2+x) x}{2 (1+x)}}}{(x+\log (x))^2}+\frac {e^{-\frac {(-2+x) x}{2 (1+x)}} x^2}{(x+\log (x))^2}-\frac {3 e^{-\frac {(-2+x) x}{2 (1+x)}}}{(1+x)^2 (x+\log (x))^2}+\frac {6 e^{-\frac {(-2+x) x}{2 (1+x)}}}{(1+x) (x+\log (x))^2}\right ) \, dx\\ &=e^{\frac {(2-x) x}{2 (1+x)}}-2 \left (\frac {1}{2} \int \frac {e^{-\frac {(-2+x) x}{2 (1+x)}} x^2}{(x+\log (x))^2} \, dx\right )+2 \left (\frac {3}{2} \int \frac {e^{-\frac {(-2+x) x}{2 (1+x)}}}{(x+\log (x))^2} \, dx\right )+2 \left (\frac {3}{2} \int \frac {e^{-\frac {(-2+x) x}{2 (1+x)}}}{(1+x)^2 (x+\log (x))^2} \, dx\right )-3 \int \frac {e^{-\frac {(-2+x) x}{2 (1+x)}}}{(x+\log (x))^2} \, dx-3 \int \frac {e^{-\frac {(-2+x) x}{2 (1+x)}}}{(1+x)^2 (x+\log (x))^2} \, dx-2 \left (3 \int \frac {e^{-\frac {(-2+x) x}{2 (1+x)}}}{(1+x) (x+\log (x))^2} \, dx\right )+6 \int \frac {e^{-\frac {(-2+x) x}{2 (1+x)}}}{(1+x) (x+\log (x))^2} \, dx+621 \int \frac {1}{x (x+\log (x))^2} \, dx+622 \int \frac {1}{(x+\log (x))^2} \, dx+\int \frac {x}{(x+\log (x))^2} \, dx+\int \frac {e^{-\frac {(-2+x) x}{2 (1+x)}} x^2}{(x+\log (x))^2} \, dx-\int \frac {1}{x+\log (x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.20, size = 33, normalized size = 1.18 \begin {gather*} \frac {1}{2} \left (2 e^{-\frac {(-2+x) x}{2 (1+x)}}-\frac {2 (621+x)}{x+\log (x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1242 + 3728*x + 3730*x^2 + 1244*x^3 + E^((2*x - x^2)/(2 + 2*x))*(2*x^3 - 2*x^4 - x^5) + (-2*x - 4*x
^2 - 2*x^3 + E^((2*x - x^2)/(2 + 2*x))*(4*x^2 - 4*x^3 - 2*x^4))*Log[x] + E^((2*x - x^2)/(2 + 2*x))*(2*x - 2*x^
2 - x^3)*Log[x]^2)/(2*x^3 + 4*x^4 + 2*x^5 + (4*x^2 + 8*x^3 + 4*x^4)*Log[x] + (2*x + 4*x^2 + 2*x^3)*Log[x]^2),x
]

[Out]

(2/E^(((-2 + x)*x)/(2*(1 + x))) - (2*(621 + x))/(x + Log[x]))/2

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Maple [A]
time = 20.65, size = 25, normalized size = 0.89

method	result	size
risch	\({\mathrm e}^{-\frac {\left (x -2\right ) x}{2 \left (x +1\right )}}-\frac {621+x}{x +\ln \left (x \right )}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^3-2*x^2+2*x)*exp((-x^2+2*x)/(2*x+2))*ln(x)^2+((-2*x^4-4*x^3+4*x^2)*exp((-x^2+2*x)/(2*x+2))-2*x^3-4*x^
2-2*x)*ln(x)+(-x^5-2*x^4+2*x^3)*exp((-x^2+2*x)/(2*x+2))+1244*x^3+3730*x^2+3728*x+1242)/((2*x^3+4*x^2+2*x)*ln(x
)^2+(4*x^4+8*x^3+4*x^2)*ln(x)+2*x^5+4*x^4+2*x^3),x,method=_RETURNVERBOSE)

[Out]

exp(-1/2*(x-2)*x/(x+1))-(621+x)/(x+ln(x))

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Maxima [A]
time = 0.34, size = 41, normalized size = 1.46 \begin {gather*} -\frac {{\left ({\left (x + 621\right )} e^{\left (\frac {1}{2} \, x\right )} - {\left (x e^{\frac {3}{2}} + e^{\frac {3}{2}} \log \left (x\right )\right )} e^{\left (-\frac {3}{2 \, {\left (x + 1\right )}}\right )}\right )} e^{\left (-\frac {1}{2} \, x\right )}}{x + \log \left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^3-2*x^2+2*x)*exp((-x^2+2*x)/(2+2*x))*log(x)^2+((-2*x^4-4*x^3+4*x^2)*exp((-x^2+2*x)/(2+2*x))-2*x
^3-4*x^2-2*x)*log(x)+(-x^5-2*x^4+2*x^3)*exp((-x^2+2*x)/(2+2*x))+1244*x^3+3730*x^2+3728*x+1242)/((2*x^3+4*x^2+2
*x)*log(x)^2+(4*x^4+8*x^3+4*x^2)*log(x)+2*x^5+4*x^4+2*x^3),x, algorithm="maxima")

[Out]

-((x + 621)*e^(1/2*x) - (x*e^(3/2) + e^(3/2)*log(x))*e^(-3/2/(x + 1)))*e^(-1/2*x)/(x + log(x))

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Fricas [A]
time = 0.35, size = 47, normalized size = 1.68 \begin {gather*} \frac {x e^{\left (-\frac {x^{2} - 2 \, x}{2 \, {\left (x + 1\right )}}\right )} + e^{\left (-\frac {x^{2} - 2 \, x}{2 \, {\left (x + 1\right )}}\right )} \log \left (x\right ) - x - 621}{x + \log \left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^3-2*x^2+2*x)*exp((-x^2+2*x)/(2+2*x))*log(x)^2+((-2*x^4-4*x^3+4*x^2)*exp((-x^2+2*x)/(2+2*x))-2*x
^3-4*x^2-2*x)*log(x)+(-x^5-2*x^4+2*x^3)*exp((-x^2+2*x)/(2+2*x))+1244*x^3+3730*x^2+3728*x+1242)/((2*x^3+4*x^2+2
*x)*log(x)^2+(4*x^4+8*x^3+4*x^2)*log(x)+2*x^5+4*x^4+2*x^3),x, algorithm="fricas")

[Out]

(x*e^(-1/2*(x^2 - 2*x)/(x + 1)) + e^(-1/2*(x^2 - 2*x)/(x + 1))*log(x) - x - 621)/(x + log(x))

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Sympy [A]
time = 0.26, size = 22, normalized size = 0.79 \begin {gather*} \frac {- x - 621}{x + \log {\left (x \right )}} + e^{\frac {- x^{2} + 2 x}{2 x + 2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**3-2*x**2+2*x)*exp((-x**2+2*x)/(2+2*x))*ln(x)**2+((-2*x**4-4*x**3+4*x**2)*exp((-x**2+2*x)/(2+2*
x))-2*x**3-4*x**2-2*x)*ln(x)+(-x**5-2*x**4+2*x**3)*exp((-x**2+2*x)/(2+2*x))+1244*x**3+3730*x**2+3728*x+1242)/(
(2*x**3+4*x**2+2*x)*ln(x)**2+(4*x**4+8*x**3+4*x**2)*ln(x)+2*x**5+4*x**4+2*x**3),x)

[Out]

(-x - 621)/(x + log(x)) + exp((-x**2 + 2*x)/(2*x + 2))

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Giac [A]
time = 0.52, size = 47, normalized size = 1.68 \begin {gather*} \frac {x e^{\left (-\frac {x^{2} - 2 \, x}{2 \, {\left (x + 1\right )}}\right )} + e^{\left (-\frac {x^{2} - 2 \, x}{2 \, {\left (x + 1\right )}}\right )} \log \left (x\right ) - x - 621}{x + \log \left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^3-2*x^2+2*x)*exp((-x^2+2*x)/(2+2*x))*log(x)^2+((-2*x^4-4*x^3+4*x^2)*exp((-x^2+2*x)/(2+2*x))-2*x
^3-4*x^2-2*x)*log(x)+(-x^5-2*x^4+2*x^3)*exp((-x^2+2*x)/(2+2*x))+1244*x^3+3730*x^2+3728*x+1242)/((2*x^3+4*x^2+2
*x)*log(x)^2+(4*x^4+8*x^3+4*x^2)*log(x)+2*x^5+4*x^4+2*x^3),x, algorithm="giac")

[Out]

(x*e^(-1/2*(x^2 - 2*x)/(x + 1)) + e^(-1/2*(x^2 - 2*x)/(x + 1))*log(x) - x - 621)/(x + log(x))

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Mupad [B]
time = 5.55, size = 60, normalized size = 2.14 \begin {gather*} {\mathrm {e}}^{\frac {2\,x}{2\,x+2}-\frac {x^2}{2\,x+2}}+\frac {1}{x+1}-\frac {\frac {622\,x+621}{x+1}-\frac {x\,\ln \left (x\right )}{x+1}}{x+\ln \left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3728*x - log(x)*(2*x + exp((2*x - x^2)/(2*x + 2))*(4*x^3 - 4*x^2 + 2*x^4) + 4*x^2 + 2*x^3) - exp((2*x - x
^2)/(2*x + 2))*(2*x^4 - 2*x^3 + x^5) + 3730*x^2 + 1244*x^3 - exp((2*x - x^2)/(2*x + 2))*log(x)^2*(2*x^2 - 2*x
+ x^3) + 1242)/(log(x)^2*(2*x + 4*x^2 + 2*x^3) + log(x)*(4*x^2 + 8*x^3 + 4*x^4) + 2*x^3 + 4*x^4 + 2*x^5),x)

[Out]

exp((2*x)/(2*x + 2) - x^2/(2*x + 2)) + 1/(x + 1) - ((622*x + 621)/(x + 1) - (x*log(x))/(x + 1))/(x + log(x))

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