3.69.21 \(\int \frac {-4 x-6 x^2-2 x^3+e^x (2 x+2 x^2)+e^{\frac {x^2+\log ^2(-2 x+e^x x-x^2)}{x}} (-2 x-3 x^2-x^3+e^x (x+x^2)+(-4-4 x+e^x (2+2 x)) \log (-2 x+e^x x-x^2)+(2-e^x+x) \log ^2(-2 x+e^x x-x^2))}{-2 x+e^x x-x^2} \, dx\)

Optimal. Leaf size=28 \[ -1+x \left (2+e^{x+\frac {\log ^2\left (\left (-2+e^x-x\right ) x\right )}{x}}+x\right ) \]

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Rubi [F]  time = 180.00, 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*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

$Aborted

Rubi steps

Aborted

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Mathematica [A]  time = 0.15, size = 27, normalized size = 0.96 \begin {gather*} x \left (2+e^{x+\frac {\log ^2\left (-x \left (2-e^x+x\right )\right )}{x}}+x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.57, size = 34, normalized size = 1.21 \begin {gather*} x^{2} + x e^{\left (\frac {x^{2} + \log \left (-x^{2} + x e^{x} - 2 \, x\right )^{2}}{x}\right )} + 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x-exp(x)+2)*log(exp(x)*x-x^2-2*x)^2+((2*x+2)*exp(x)-4*x-4)*log(exp(x)*x-x^2-2*x)+(x^2+x)*exp(x)-x
^3-3*x^2-2*x)*exp((log(exp(x)*x-x^2-2*x)^2+x^2)/x)+(2*x^2+2*x)*exp(x)-2*x^3-6*x^2-4*x)/(exp(x)*x-x^2-2*x),x, a
lgorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x-exp(x)+2)*log(exp(x)*x-x^2-2*x)^2+((2*x+2)*exp(x)-4*x-4)*log(exp(x)*x-x^2-2*x)+(x^2+x)*exp(x)-x
^3-3*x^2-2*x)*exp((log(exp(x)*x-x^2-2*x)^2+x^2)/x)+(2*x^2+2*x)*exp(x)-2*x^3-6*x^2-4*x)/(exp(x)*x-x^2-2*x),x, a
lgorithm="giac")

[Out]

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

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maple [C]  time = 0.34, size = 812, normalized size = 29.00

method	result	size
risch	\(x^{2}+x \,x^{\frac {2 i \pi }{x}} \left (x -{\mathrm e}^{x}+2\right )^{\frac {2 i \pi }{x}} x^{\frac {i \pi \,\mathrm {csgn}\left (i x \right )}{x}} \left (x -{\mathrm e}^{x}+2\right )^{\frac {i \pi \,\mathrm {csgn}\left (i x \right )}{x}} x^{-\frac {i \pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{x}-2-x \right )\right )}{x}} \left (x -{\mathrm e}^{x}+2\right )^{-\frac {i \pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{x}-2-x \right )\right )}{x}} \left (x -{\mathrm e}^{x}+2\right )^{\frac {2 \ln \relax (x )}{x}} x^{-\frac {i \pi \,\mathrm {csgn}\left (i x \left ({\mathrm e}^{x}-2-x \right )\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-2-x \right )\right )}{x}} \left (x -{\mathrm e}^{x}+2\right )^{-\frac {i \pi \,\mathrm {csgn}\left (i x \left ({\mathrm e}^{x}-2-x \right )\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-2-x \right )\right )}{x}} x^{-\frac {i \mathrm {csgn}\left (i x \left ({\mathrm e}^{x}-2-x \right )\right ) \pi }{x}} \left (x -{\mathrm e}^{x}+2\right )^{-\frac {i \mathrm {csgn}\left (i x \left ({\mathrm e}^{x}-2-x \right )\right ) \pi }{x}} x^{-\frac {2 i \pi }{x}} \left (x -{\mathrm e}^{x}+2\right )^{-\frac {2 i \pi }{x}} {\mathrm e}^{\frac {-\pi ^{2} \mathrm {csgn}\left (i x \left ({\mathrm e}^{x}-2-x \right )\right )^{4} \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-2-x \right )\right )^{2}-2 \pi ^{2} \mathrm {csgn}\left (i x \left ({\mathrm e}^{x}-2-x \right )\right )^{3} \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-2-x \right )\right )^{2} \mathrm {csgn}\left (i x \right )-\pi ^{2} \mathrm {csgn}\left (i x \left ({\mathrm e}^{x}-2-x \right )\right )^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-2-x \right )\right )^{2}-2 \pi ^{2} \mathrm {csgn}\left (i x \left ({\mathrm e}^{x}-2-x \right )\right )^{5} \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-2-x \right )\right )+2 \pi ^{2} \mathrm {csgn}\left (i x \left ({\mathrm e}^{x}-2-x \right )\right )^{3} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-2-x \right )\right )-\pi ^{2} \mathrm {csgn}\left (i x \left ({\mathrm e}^{x}-2-x \right )\right )^{6}+2 \pi ^{2} \mathrm {csgn}\left (i x \left ({\mathrm e}^{x}-2-x \right )\right )^{5} \mathrm {csgn}\left (i x \right )-\pi ^{2} \mathrm {csgn}\left (i x \left ({\mathrm e}^{x}-2-x \right )\right )^{4} \mathrm {csgn}\left (i x \right )^{2}-4 \pi ^{2} \mathrm {csgn}\left (i x \left ({\mathrm e}^{x}-2-x \right )\right )^{4} \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-2-x \right )\right )-4 \pi ^{2} \mathrm {csgn}\left (i x \left ({\mathrm e}^{x}-2-x \right )\right )^{3} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-2-x \right )\right )-4 \pi ^{2} \mathrm {csgn}\left (i x \left ({\mathrm e}^{x}-2-x \right )\right )^{5}+4 \pi ^{2} \mathrm {csgn}\left (i x \left ({\mathrm e}^{x}-2-x \right )\right )^{4} \mathrm {csgn}\left (i x \right )-4 \pi ^{2} \mathrm {csgn}\left (i x \left ({\mathrm e}^{x}-2-x \right )\right )^{4}+4 \pi ^{2} \mathrm {csgn}\left (i x \left ({\mathrm e}^{x}-2-x \right )\right )^{2} \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-2-x \right )\right )+4 \pi ^{2} \mathrm {csgn}\left (i x \left ({\mathrm e}^{x}-2-x \right )\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x}-2-x \right )\right )+4 \pi ^{2} \mathrm {csgn}\left (i x \left ({\mathrm e}^{x}-2-x \right )\right )^{3}-4 \pi ^{2} \mathrm {csgn}\left (i x \left ({\mathrm e}^{x}-2-x \right )\right )^{2} \mathrm {csgn}\left (i x \right )+8 \pi ^{2} \mathrm {csgn}\left (i x \left ({\mathrm e}^{x}-2-x \right )\right )^{2}-4 \pi ^{2}+4 \ln \relax (x )^{2}+4 \ln \left (x -{\mathrm e}^{x}+2\right )^{2}+4 x^{2}}{4 x}}+2 x\)	\(812\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((x-exp(x)+2)*ln(exp(x)*x-x^2-2*x)^2+((2*x+2)*exp(x)-4*x-4)*ln(exp(x)*x-x^2-2*x)+(x^2+x)*exp(x)-x^3-3*x^2
-2*x)*exp((ln(exp(x)*x-x^2-2*x)^2+x^2)/x)+(2*x^2+2*x)*exp(x)-2*x^3-6*x^2-4*x)/(exp(x)*x-x^2-2*x),x,method=_RET
URNVERBOSE)

[Out]

x^2+x*x^(2*I/x*Pi)*(x-exp(x)+2)^(2*I/x*Pi)*x^(I/x*Pi*csgn(I*x))*(x-exp(x)+2)^(I/x*Pi*csgn(I*x))*x^(-I/x*Pi*csg
n(I*(exp(x)-2-x)))*(x-exp(x)+2)^(-I/x*Pi*csgn(I*(exp(x)-2-x)))*(x-exp(x)+2)^(2*ln(x)/x)*x^(-I/x*Pi*csgn(I*x*(e
xp(x)-2-x))*csgn(I*x)*csgn(I*(exp(x)-2-x)))*(x-exp(x)+2)^(-I/x*Pi*csgn(I*x*(exp(x)-2-x))*csgn(I*x)*csgn(I*(exp
(x)-2-x)))*x^(-I*csgn(I*x*(exp(x)-2-x))*Pi/x)*(x-exp(x)+2)^(-I*csgn(I*x*(exp(x)-2-x))*Pi/x)*x^(-2*I/x*Pi)*(x-e
xp(x)+2)^(-2*I/x*Pi)*exp(1/4*(-Pi^2*csgn(I*x*(exp(x)-2-x))^4*csgn(I*(exp(x)-2-x))^2-2*Pi^2*csgn(I*x*(exp(x)-2-
x))^3*csgn(I*(exp(x)-2-x))^2*csgn(I*x)-Pi^2*csgn(I*x*(exp(x)-2-x))^2*csgn(I*x)^2*csgn(I*(exp(x)-2-x))^2-2*Pi^2
*csgn(I*x*(exp(x)-2-x))^5*csgn(I*(exp(x)-2-x))+2*Pi^2*csgn(I*x*(exp(x)-2-x))^3*csgn(I*x)^2*csgn(I*(exp(x)-2-x)
)-Pi^2*csgn(I*x*(exp(x)-2-x))^6+2*Pi^2*csgn(I*x*(exp(x)-2-x))^5*csgn(I*x)-Pi^2*csgn(I*x*(exp(x)-2-x))^4*csgn(I
*x)^2-4*Pi^2*csgn(I*x*(exp(x)-2-x))^4*csgn(I*(exp(x)-2-x))-4*Pi^2*csgn(I*x*(exp(x)-2-x))^3*csgn(I*x)*csgn(I*(e
xp(x)-2-x))-4*Pi^2*csgn(I*x*(exp(x)-2-x))^5+4*Pi^2*csgn(I*x*(exp(x)-2-x))^4*csgn(I*x)-4*Pi^2*csgn(I*x*(exp(x)-
2-x))^4+4*Pi^2*csgn(I*x*(exp(x)-2-x))^2*csgn(I*(exp(x)-2-x))+4*Pi^2*csgn(I*x*(exp(x)-2-x))*csgn(I*x)*csgn(I*(e
xp(x)-2-x))+4*Pi^2*csgn(I*x*(exp(x)-2-x))^3-4*Pi^2*csgn(I*x*(exp(x)-2-x))^2*csgn(I*x)+8*Pi^2*csgn(I*x*(exp(x)-
2-x))^2-4*Pi^2+4*ln(x)^2+4*ln(x-exp(x)+2)^2+4*x^2)/x)+2*x

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maxima [A]  time = 0.44, size = 49, normalized size = 1.75 \begin {gather*} x^{2} + x e^{\left (x + \frac {\log \relax (x)^{2}}{x} + \frac {2 \, \log \relax (x) \log \left (-x + e^{x} - 2\right )}{x} + \frac {\log \left (-x + e^{x} - 2\right )^{2}}{x}\right )} + 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x-exp(x)+2)*log(exp(x)*x-x^2-2*x)^2+((2*x+2)*exp(x)-4*x-4)*log(exp(x)*x-x^2-2*x)+(x^2+x)*exp(x)-x
^3-3*x^2-2*x)*exp((log(exp(x)*x-x^2-2*x)^2+x^2)/x)+(2*x^2+2*x)*exp(x)-2*x^3-6*x^2-4*x)/(exp(x)*x-x^2-2*x),x, a
lgorithm="maxima")

[Out]

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

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mupad [B]  time = 4.30, size = 32, normalized size = 1.14 \begin {gather*} 2\,x+x^2+x\,{\mathrm {e}}^{\frac {{\ln \left (x\,{\mathrm {e}}^x-2\,x-x^2\right )}^2}{x}}\,{\mathrm {e}}^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x + exp((x^2 + log(x*exp(x) - 2*x - x^2)^2)/x)*(2*x - log(x*exp(x) - 2*x - x^2)^2*(x - exp(x) + 2) + lo
g(x*exp(x) - 2*x - x^2)*(4*x - exp(x)*(2*x + 2) + 4) + 3*x^2 + x^3 - exp(x)*(x + x^2)) - exp(x)*(2*x + 2*x^2)
+ 6*x^2 + 2*x^3)/(2*x - x*exp(x) + x^2),x)

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x-exp(x)+2)*ln(exp(x)*x-x**2-2*x)**2+((2*x+2)*exp(x)-4*x-4)*ln(exp(x)*x-x**2-2*x)+(x**2+x)*exp(x)
-x**3-3*x**2-2*x)*exp((ln(exp(x)*x-x**2-2*x)**2+x**2)/x)+(2*x**2+2*x)*exp(x)-2*x**3-6*x**2-4*x)/(exp(x)*x-x**2
-2*x),x)

[Out]

Timed out

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