3.22.93 \(\int \frac {-4 x^2-x^3-2 x^4+e^{-x-x^2 \log (x)+x^2 \log (\frac {4+x+2 x^2}{x^2})} (-4-9 x-16 x^2-6 x^3-2 x^4+(-8 x^2-2 x^3-4 x^4) \log (x)+(8 x^2+2 x^3+4 x^4) \log (\frac {4+x+2 x^2}{x^2}))}{4 x^2+x^3+2 x^4+e^{-x-x^2 \log (x)+x^2 \log (\frac {4+x+2 x^2}{x^2})} (4 x+x^2+2 x^3)} \, dx\)

Optimal. Leaf size=36 \[ -x+\log \left (\frac {e^{-x+x^2 \left (-\log (x)+\log \left (2+\frac {4+x}{x^2}\right )\right )}+x}{x}\right ) \]

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Rubi [F]  time = 47.43, 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 {-4 x^2-x^3-2 x^4+e^{-x-x^2 \log (x)+x^2 \log \left (\frac {4+x+2 x^2}{x^2}\right )} \left (-4-9 x-16 x^2-6 x^3-2 x^4+\left (-8 x^2-2 x^3-4 x^4\right ) \log (x)+\left (8 x^2+2 x^3+4 x^4\right ) \log \left (\frac {4+x+2 x^2}{x^2}\right )\right )}{4 x^2+x^3+2 x^4+e^{-x-x^2 \log (x)+x^2 \log \left (\frac {4+x+2 x^2}{x^2}\right )} \left (4 x+x^2+2 x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-4*x^2 - x^3 - 2*x^4 + E^(-x - x^2*Log[x] + x^2*Log[(4 + x + 2*x^2)/x^2])*(-4 - 9*x - 16*x^2 - 6*x^3 - 2*
x^4 + (-8*x^2 - 2*x^3 - 4*x^4)*Log[x] + (8*x^2 + 2*x^3 + 4*x^4)*Log[(4 + x + 2*x^2)/x^2]))/(4*x^2 + x^3 + 2*x^
4 + E^(-x - x^2*Log[x] + x^2*Log[(4 + x + 2*x^2)/x^2])*(4*x + x^2 + 2*x^3)),x]

[Out]

(-5*Defer[Int][(2 + 4/x^2 + x^(-1))^x^2/((2 + 4/x^2 + x^(-1))^x^2 + E^x*x^(1 + x^2)), x])/2 + ((8*I)*Defer[Int
][(2 + 4/x^2 + x^(-1))^x^2/((-1 + I*Sqrt[31] - 4*x)*((2 + 4/x^2 + x^(-1))^x^2 + E^x*x^(1 + x^2))), x])/Sqrt[31
] - Defer[Int][((2 + 4/x^2 + x^(-1))^x^2*x)/((2 + 4/x^2 + x^(-1))^x^2 + E^x*x^(1 + x^2)), x] + 2*Log[2 + 4/x^2
+ x^(-1)]*Defer[Int][((2 + 4/x^2 + x^(-1))^x^2*x)/((2 + 4/x^2 + x^(-1))^x^2 + E^x*x^(1 + x^2)), x] - 2*Log[x]
*Defer[Int][((2 + 4/x^2 + x^(-1))^x^2*x)/((2 + 4/x^2 + x^(-1))^x^2 + E^x*x^(1 + x^2)), x] - ((16*I)*Defer[Int]
[(E^x*x^(1 + x^2))/((-1 + I*Sqrt[31] - 4*x)*((2 + 4/x^2 + x^(-1))^x^2 + E^x*x^(1 + x^2))), x])/Sqrt[31] - ((4*
I)*Defer[Int][(E^x*x^(2 + x^2))/((-1 + I*Sqrt[31] - 4*x)*((2 + 4/x^2 + x^(-1))^x^2 + E^x*x^(1 + x^2))), x])/Sq
rt[31] - ((8*I)*Defer[Int][(E^x*x^(3 + x^2))/((-1 + I*Sqrt[31] - 4*x)*((2 + 4/x^2 + x^(-1))^x^2 + E^x*x^(1 + x
^2))), x])/Sqrt[31] - (15*(31 + I*Sqrt[31])*Defer[Int][(2 + 4/x^2 + x^(-1))^x^2/((1 - I*Sqrt[31] + 4*x)*((2 +
4/x^2 + x^(-1))^x^2 + E^x*x^(1 + x^2))), x])/62 + ((8*I)*Defer[Int][(2 + 4/x^2 + x^(-1))^x^2/((1 + I*Sqrt[31]
+ 4*x)*((2 + 4/x^2 + x^(-1))^x^2 + E^x*x^(1 + x^2))), x])/Sqrt[31] - (15*(31 - I*Sqrt[31])*Defer[Int][(2 + 4/x
^2 + x^(-1))^x^2/((1 + I*Sqrt[31] + 4*x)*((2 + 4/x^2 + x^(-1))^x^2 + E^x*x^(1 + x^2))), x])/62 - ((16*I)*Defer
[Int][(E^x*x^(1 + x^2))/((1 + I*Sqrt[31] + 4*x)*((2 + 4/x^2 + x^(-1))^x^2 + E^x*x^(1 + x^2))), x])/Sqrt[31] -
((4*I)*Defer[Int][(E^x*x^(2 + x^2))/((1 + I*Sqrt[31] + 4*x)*((2 + 4/x^2 + x^(-1))^x^2 + E^x*x^(1 + x^2))), x])
/Sqrt[31] - ((8*I)*Defer[Int][(E^x*x^(3 + x^2))/((1 + I*Sqrt[31] + 4*x)*((2 + 4/x^2 + x^(-1))^x^2 + E^x*x^(1 +
x^2))), x])/Sqrt[31] - Defer[Int][(2 + 4/x^2 + x^(-1))^x^2/(x*(E^x*x^(1 + x^2) + ((4 + x + 2*x^2)/x^2)^x^2)),
x] - ((8*I)*Defer[Int][Defer[Int][((2 + 4/x^2 + x^(-1))^x^2*x)/((2 + 4/x^2 + x^(-1))^x^2 + E^x*x^(1 + x^2)),
x]/(-1 + I*Sqrt[31] - 4*x), x])/Sqrt[31] + 6*Defer[Int][Defer[Int][((2 + 4/x^2 + x^(-1))^x^2*x)/((2 + 4/x^2 +
x^(-1))^x^2 + E^x*x^(1 + x^2)), x]/x, x] - (8*(31 + I*Sqrt[31])*Defer[Int][Defer[Int][((2 + 4/x^2 + x^(-1))^x^
2*x)/((2 + 4/x^2 + x^(-1))^x^2 + E^x*x^(1 + x^2)), x]/(1 - I*Sqrt[31] + 4*x), x])/31 - ((8*I)*Defer[Int][Defer
[Int][((2 + 4/x^2 + x^(-1))^x^2*x)/((2 + 4/x^2 + x^(-1))^x^2 + E^x*x^(1 + x^2)), x]/(1 + I*Sqrt[31] + 4*x), x]
)/Sqrt[31] - (8*(31 - I*Sqrt[31])*Defer[Int][Defer[Int][((2 + 4/x^2 + x^(-1))^x^2*x)/((2 + 4/x^2 + x^(-1))^x^2
+ E^x*x^(1 + x^2)), x]/(1 + I*Sqrt[31] + 4*x), x])/31 - (4*(15 - I*Sqrt[31])*Defer[Int][Defer[Int][(2 + 4/x^2
+ x^(-1))^x^2/((1 + I*Sqrt[31] + 4*x)*((2 + 4/x^2 + x^(-1))^x^2 + E^x*x^(1 + x^2))), x]/(-1 + I*Sqrt[31] - 4*
x), x])/31 - (32*(1 + I*Sqrt[31])*Defer[Int][Defer[Int][(2 + 4/x^2 + x^(-1))^x^2/((1 + I*Sqrt[31] + 4*x)*((2 +
4/x^2 + x^(-1))^x^2 + E^x*x^(1 + x^2))), x]/(-1 + I*Sqrt[31] - 4*x), x])/31 + (4*(23 + (7*I)*Sqrt[31])*Defer[
Int][Defer[Int][(2 + 4/x^2 + x^(-1))^x^2/((1 + I*Sqrt[31] + 4*x)*((2 + 4/x^2 + x^(-1))^x^2 + E^x*x^(1 + x^2)))
, x]/(-1 + I*Sqrt[31] - 4*x), x])/31 + (24*(31 - I*Sqrt[31])*Defer[Int][Defer[Int][(2 + 4/x^2 + x^(-1))^x^2/((
1 + I*Sqrt[31] + 4*x)*((2 + 4/x^2 + x^(-1))^x^2 + E^x*x^(1 + x^2))), x]/x, x])/31 - (3*(31 + (15*I)*Sqrt[31])*
Defer[Int][Defer[Int][(2 + 4/x^2 + x^(-1))^x^2/((1 + I*Sqrt[31] + 4*x)*((2 + 4/x^2 + x^(-1))^x^2 + E^x*x^(1 +
x^2))), x]/x, x])/31 - (3*(217 - (23*I)*Sqrt[31])*Defer[Int][Defer[Int][(2 + 4/x^2 + x^(-1))^x^2/((1 + I*Sqrt[
31] + 4*x)*((2 + 4/x^2 + x^(-1))^x^2 + E^x*x^(1 + x^2))), x]/x, x])/31 - (1024*Defer[Int][Defer[Int][(2 + 4/x^
2 + x^(-1))^x^2/((1 + I*Sqrt[31] + 4*x)*((2 + 4/x^2 + x^(-1))^x^2 + E^x*x^(1 + x^2))), x]/(1 - I*Sqrt[31] + 4*
x), x])/31 + (64*(15 - I*Sqrt[31])*Defer[Int][Defer[Int][(2 + 4/x^2 + x^(-1))^x^2/((1 + I*Sqrt[31] + 4*x)*((2
+ 4/x^2 + x^(-1))^x^2 + E^x*x^(1 + x^2))), x]/(1 - I*Sqrt[31] + 4*x), x])/31 + (64*(1 + I*Sqrt[31])*Defer[Int]
[Defer[Int][(2 + 4/x^2 + x^(-1))^x^2/((1 + I*Sqrt[31] + 4*x)*((2 + 4/x^2 + x^(-1))^x^2 + E^x*x^(1 + x^2))), x]
/(1 - I*Sqrt[31] + 4*x), x])/31 - (68*(15 - I*Sqrt[31])*Defer[Int][Defer[Int][(2 + 4/x^2 + x^(-1))^x^2/((1 + I
*Sqrt[31] + 4*x)*((2 + 4/x^2 + x^(-1))^x^2 + E^x*x^(1 + x^2))), x]/(1 + I*Sqrt[31] + 4*x), x])/31 - (32*(1 + I
*Sqrt[31])*Defer[Int][Defer[Int][(2 + 4/x^2 + x^(-1))^x^2/((1 + I*Sqrt[31] + 4*x)*((2 + 4/x^2 + x^(-1))^x^2 +
E^x*x^(1 + x^2))), x]/(1 + I*Sqrt[31] + 4*x), x])/31 + (12*(23 + (7*I)*Sqrt[31])*Defer[Int][Defer[Int][(2 + 4/
x^2 + x^(-1))^x^2/((1 + I*Sqrt[31] + 4*x)*((2 + 4/x^2 + x^(-1))^x^2 + E^x*x^(1 + x^2))), x]/(1 + I*Sqrt[31] +
4*x), x])/31 + (8*(97 - (15*I)*Sqrt[31])*Defer[Int][Defer[Int][(2 + 4/x^2 + x^(-1))^x^2/((1 + I*Sqrt[31] + 4*x
)*((2 + 4/x^2 + x^(-1))^x^2 + E^x*x^(1 + x^2))), x]/(1 + I*Sqrt[31] + 4*x), x])/31

Rubi steps

Aborted

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Mathematica [A]  time = 0.43, size = 42, normalized size = 1.17 \begin {gather*} -2 x-\log (x)-x^2 \log (x)+\log \left (\left (2+\frac {4}{x^2}+\frac {1}{x}\right )^{x^2}+e^x x^{1+x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4*x^2 - x^3 - 2*x^4 + E^(-x - x^2*Log[x] + x^2*Log[(4 + x + 2*x^2)/x^2])*(-4 - 9*x - 16*x^2 - 6*x^
3 - 2*x^4 + (-8*x^2 - 2*x^3 - 4*x^4)*Log[x] + (8*x^2 + 2*x^3 + 4*x^4)*Log[(4 + x + 2*x^2)/x^2]))/(4*x^2 + x^3
+ 2*x^4 + E^(-x - x^2*Log[x] + x^2*Log[(4 + x + 2*x^2)/x^2])*(4*x + x^2 + 2*x^3)),x]

[Out]

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

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fricas [A]  time = 0.83, size = 40, normalized size = 1.11 \begin {gather*} -x + \log \left (x + e^{\left (-x^{2} \log \relax (x) + x^{2} \log \left (\frac {2 \, x^{2} + x + 4}{x^{2}}\right ) - x\right )}\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*x^4-2*x^3-8*x^2)*log(x)+(4*x^4+2*x^3+8*x^2)*log((2*x^2+x+4)/x^2)-2*x^4-6*x^3-16*x^2-9*x-4)*exp
(-x^2*log(x)+x^2*log((2*x^2+x+4)/x^2)-x)-2*x^4-x^3-4*x^2)/((2*x^3+x^2+4*x)*exp(-x^2*log(x)+x^2*log((2*x^2+x+4)
/x^2)-x)+2*x^4+x^3+4*x^2),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 2.95, size = 36, normalized size = 1.00 \begin {gather*} -x + \log \left (x + e^{\left (x^{2} \log \left (2 \, x^{2} + x + 4\right ) - 3 \, x^{2} \log \relax (x) - x\right )}\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*x^4-2*x^3-8*x^2)*log(x)+(4*x^4+2*x^3+8*x^2)*log((2*x^2+x+4)/x^2)-2*x^4-6*x^3-16*x^2-9*x-4)*exp
(-x^2*log(x)+x^2*log((2*x^2+x+4)/x^2)-x)-2*x^4-x^3-4*x^2)/((2*x^3+x^2+4*x)*exp(-x^2*log(x)+x^2*log((2*x^2+x+4)
/x^2)-x)+2*x^4+x^3+4*x^2),x, algorithm="giac")

[Out]

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

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maple [C]  time = 0.29, size = 565, normalized size = 15.69

method	result	size
risch	\(x^{2} \ln \left (x^{2}+\frac {1}{2} x +2\right )-2 x^{2} \ln \relax (x )+\frac {i \pi \,x^{2} \mathrm {csgn}\left (\frac {i}{x^{2}}\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}+\frac {1}{2} x +2\right )}{x^{2}}\right )^{2}}{2}+\frac {i \pi \,x^{2} \mathrm {csgn}\left (i \left (x^{2}+\frac {1}{2} x +2\right )\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}+\frac {1}{2} x +2\right )}{x^{2}}\right )^{2}}{2}-\frac {i \pi \,x^{2} \mathrm {csgn}\left (\frac {i \left (x^{2}+\frac {1}{2} x +2\right )}{x^{2}}\right )^{3}}{2}-\frac {i \pi \,x^{2} \mathrm {csgn}\left (\frac {i}{x^{2}}\right ) \mathrm {csgn}\left (i \left (x^{2}+\frac {1}{2} x +2\right )\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}+\frac {1}{2} x +2\right )}{x^{2}}\right )}{2}+\frac {i \pi \,x^{2} \mathrm {csgn}\left (i x^{2}\right )^{3}}{2}+\frac {i \pi \,x^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )}{2}-i \pi \,x^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+x^{2} \ln \relax (2)-x -\ln \relax (x )-x^{2} \left (\ln \relax (2)-2 \ln \relax (x )+\ln \left (x^{2}+\frac {1}{2} x +2\right )+\frac {i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \left (-\mathrm {csgn}\left (i x^{2}\right )+\mathrm {csgn}\left (i x \right )\right )^{2}}{2}-\frac {i \pi \,\mathrm {csgn}\left (\frac {i \left (x^{2}+\frac {1}{2} x +2\right )}{x^{2}}\right ) \left (-\mathrm {csgn}\left (\frac {i \left (x^{2}+\frac {1}{2} x +2\right )}{x^{2}}\right )+\mathrm {csgn}\left (\frac {i}{x^{2}}\right )\right ) \left (-\mathrm {csgn}\left (\frac {i \left (x^{2}+\frac {1}{2} x +2\right )}{x^{2}}\right )+\mathrm {csgn}\left (i \left (x^{2}+\frac {1}{2} x +2\right )\right )\right )}{2}\right )+\ln \left (x +x^{-x^{2}} x^{-2 x^{2}} 2^{x^{2}} \left (x^{2}+\frac {1}{2} x +2\right )^{x^{2}} {\mathrm e}^{\frac {x \left (i \pi x \mathrm {csgn}\left (i x^{2}\right )^{3}-2 i \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+i \pi x \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-i x \pi \mathrm {csgn}\left (\frac {i \left (x^{2}+\frac {1}{2} x +2\right )}{x^{2}}\right )^{3}+i x \pi \mathrm {csgn}\left (\frac {i \left (x^{2}+\frac {1}{2} x +2\right )}{x^{2}}\right )^{2} \mathrm {csgn}\left (\frac {i}{x^{2}}\right )+i x \pi \mathrm {csgn}\left (\frac {i \left (x^{2}+\frac {1}{2} x +2\right )}{x^{2}}\right )^{2} \mathrm {csgn}\left (i \left (x^{2}+\frac {1}{2} x +2\right )\right )-i x \pi \,\mathrm {csgn}\left (\frac {i \left (x^{2}+\frac {1}{2} x +2\right )}{x^{2}}\right ) \mathrm {csgn}\left (\frac {i}{x^{2}}\right ) \mathrm {csgn}\left (i \left (x^{2}+\frac {1}{2} x +2\right )\right )-2\right )}{2}}\right )\)	\(565\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-4*x^4-2*x^3-8*x^2)*ln(x)+(4*x^4+2*x^3+8*x^2)*ln((2*x^2+x+4)/x^2)-2*x^4-6*x^3-16*x^2-9*x-4)*exp(-x^2*ln
(x)+x^2*ln((2*x^2+x+4)/x^2)-x)-2*x^4-x^3-4*x^2)/((2*x^3+x^2+4*x)*exp(-x^2*ln(x)+x^2*ln((2*x^2+x+4)/x^2)-x)+2*x
^4+x^3+4*x^2),x,method=_RETURNVERBOSE)

[Out]

x^2*ln(x^2+1/2*x+2)-2*x^2*ln(x)+1/2*I*Pi*x^2*csgn(I/x^2)*csgn(I/x^2*(x^2+1/2*x+2))^2+1/2*I*Pi*x^2*csgn(I*(x^2+
1/2*x+2))*csgn(I/x^2*(x^2+1/2*x+2))^2-1/2*I*Pi*x^2*csgn(I/x^2*(x^2+1/2*x+2))^3-1/2*I*Pi*x^2*csgn(I/x^2)*csgn(I
*(x^2+1/2*x+2))*csgn(I/x^2*(x^2+1/2*x+2))+1/2*I*Pi*x^2*csgn(I*x^2)^3+1/2*I*Pi*x^2*csgn(I*x)^2*csgn(I*x^2)-I*Pi
*x^2*csgn(I*x)*csgn(I*x^2)^2+x^2*ln(2)-x-ln(x)-x^2*(ln(2)-2*ln(x)+ln(x^2+1/2*x+2)+1/2*I*Pi*csgn(I*x^2)*(-csgn(
I*x^2)+csgn(I*x))^2-1/2*I*Pi*csgn(I/x^2*(x^2+1/2*x+2))*(-csgn(I/x^2*(x^2+1/2*x+2))+csgn(I/x^2))*(-csgn(I/x^2*(
x^2+1/2*x+2))+csgn(I*(x^2+1/2*x+2))))+ln(x+x^(-x^2)*x^(-2*x^2)*2^(x^2)*(x^2+1/2*x+2)^(x^2)*exp(1/2*x*(I*x*Pi*c
sgn(I*x^2)^3-2*I*x*Pi*csgn(I*x)*csgn(I*x^2)^2+I*x*Pi*csgn(I*x^2)*csgn(I*x)^2-I*x*Pi*csgn(I/x^2*(x^2+1/2*x+2))^
3+I*x*Pi*csgn(I/x^2*(x^2+1/2*x+2))^2*csgn(I/x^2)+I*x*Pi*csgn(I/x^2*(x^2+1/2*x+2))^2*csgn(I*(x^2+1/2*x+2))-I*x*
Pi*csgn(I/x^2*(x^2+1/2*x+2))*csgn(I/x^2)*csgn(I*(x^2+1/2*x+2))-2)))

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maxima [A]  time = 0.70, size = 41, normalized size = 1.14 \begin {gather*} -{\left (3 \, x^{2} + 1\right )} \log \relax (x) - 2 \, x + \log \left (x e^{\left (3 \, x^{2} \log \relax (x) + x\right )} + {\left (2 \, x^{2} + x + 4\right )}^{\left (x^{2}\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*x^4-2*x^3-8*x^2)*log(x)+(4*x^4+2*x^3+8*x^2)*log((2*x^2+x+4)/x^2)-2*x^4-6*x^3-16*x^2-9*x-4)*exp
(-x^2*log(x)+x^2*log((2*x^2+x+4)/x^2)-x)-2*x^4-x^3-4*x^2)/((2*x^3+x^2+4*x)*exp(-x^2*log(x)+x^2*log((2*x^2+x+4)
/x^2)-x)+2*x^4+x^3+4*x^2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.84, size = 49, normalized size = 1.36 \begin {gather*} \ln \left (\frac {x^{x^2+1}+{\mathrm {e}}^{-x}\,{\left (\frac {1}{x^2}\right )}^{x^2}\,{\left (2\,x^2+x+4\right )}^{x^2}}{x^{x^2}}\right )-x-\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x^2*log((x + 2*x^2 + 4)/x^2) - x^2*log(x) - x)*(9*x - log((x + 2*x^2 + 4)/x^2)*(8*x^2 + 2*x^3 + 4*x^
4) + log(x)*(8*x^2 + 2*x^3 + 4*x^4) + 16*x^2 + 6*x^3 + 2*x^4 + 4) + 4*x^2 + x^3 + 2*x^4)/(exp(x^2*log((x + 2*x
^2 + 4)/x^2) - x^2*log(x) - x)*(4*x + x^2 + 2*x^3) + 4*x^2 + x^3 + 2*x^4),x)

[Out]

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

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sympy [A]  time = 1.16, size = 34, normalized size = 0.94 \begin {gather*} - x - \log {\relax (x )} + \log {\left (x + e^{- x^{2} \log {\relax (x )} + x^{2} \log {\left (\frac {2 x^{2} + x + 4}{x^{2}} \right )} - x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*x**4-2*x**3-8*x**2)*ln(x)+(4*x**4+2*x**3+8*x**2)*ln((2*x**2+x+4)/x**2)-2*x**4-6*x**3-16*x**2-9
*x-4)*exp(-x**2*ln(x)+x**2*ln((2*x**2+x+4)/x**2)-x)-2*x**4-x**3-4*x**2)/((2*x**3+x**2+4*x)*exp(-x**2*ln(x)+x**
2*ln((2*x**2+x+4)/x**2)-x)+2*x**4+x**3+4*x**2),x)

[Out]

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

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