\(\int \frac {e^{\sqrt [4]{e}} (-4+x)-4 x-2 x^2+3 x^3}{4 x^2+8 x^3+4 x^4+e^{\sqrt [4]{e}} (4 x+4 x^2)+(e^{\sqrt [4]{e}} x^2+x^3+x^4) \log (\frac {1}{2} (e^{\sqrt [4]{e}} x+x^2+x^3))} \, dx\) [6666]

Optimal result

Integrand size = 101, antiderivative size = 26 \[ \int \frac {e^{\sqrt [4]{e}} (-4+x)-4 x-2 x^2+3 x^3}{4 x^2+8 x^3+4 x^4+e^{\sqrt [4]{e}} \left (4 x+4 x^2\right )+\left (e^{\sqrt [4]{e}} x^2+x^3+x^4\right ) \log \left (\frac {1}{2} \left (e^{\sqrt [4]{e}} x+x^2+x^3\right )\right )} \, dx=\log \left (4+\frac {4}{x}+\log \left (\frac {1}{2} x \left (e^{\sqrt [4]{e}}+x+x^2\right )\right )\right ) \]

[Out]

ln(4/x+ln(1/2*x*(exp(exp(1/4))+x+x^2))+4)

Rubi [F]

\[ \int \frac {e^{\sqrt [4]{e}} (-4+x)-4 x-2 x^2+3 x^3}{4 x^2+8 x^3+4 x^4+e^{\sqrt [4]{e}} \left (4 x+4 x^2\right )+\left (e^{\sqrt [4]{e}} x^2+x^3+x^4\right ) \log \left (\frac {1}{2} \left (e^{\sqrt [4]{e}} x+x^2+x^3\right )\right )} \, dx=\int \frac {e^{\sqrt [4]{e}} (-4+x)-4 x-2 x^2+3 x^3}{4 x^2+8 x^3+4 x^4+e^{\sqrt [4]{e}} \left (4 x+4 x^2\right )+\left (e^{\sqrt [4]{e}} x^2+x^3+x^4\right ) \log \left (\frac {1}{2} \left (e^{\sqrt [4]{e}} x+x^2+x^3\right )\right )} \, dx \]

[In]

Int[(E^E^(1/4)*(-4 + x) - 4*x - 2*x^2 + 3*x^3)/(4*x^2 + 8*x^3 + 4*x^4 + E^E^(1/4)*(4*x + 4*x^2) + (E^E^(1/4)*x
^2 + x^3 + x^4)*Log[(E^E^(1/4)*x + x^2 + x^3)/2]),x]

[Out]

3*Defer[Int][(4 + 4*x + x*Log[(x*(E^E^(1/4) + x + x^2))/2])^(-1), x] - ((4*I)*E^E^(1/4)*Defer[Int][1/((-1 + I*
Sqrt[-1 + 4*E^E^(1/4)] - 2*x)*(4 + 4*x + x*Log[(x*(E^E^(1/4) + x + x^2))/2])), x])/Sqrt[-1 + 4*E^E^(1/4)] - 4*
Defer[Int][1/(x*(4 + 4*x + x*Log[(x*(E^E^(1/4) + x + x^2))/2])), x] - (1 + I/Sqrt[-1 + 4*E^E^(1/4)])*Defer[Int
][1/((1 - I*Sqrt[-1 + 4*E^E^(1/4)] + 2*x)*(4 + 4*x + x*Log[(x*(E^E^(1/4) + x + x^2))/2])), x] - ((4*I)*E^E^(1/
4)*Defer[Int][1/((1 + I*Sqrt[-1 + 4*E^E^(1/4)] + 2*x)*(4 + 4*x + x*Log[(x*(E^E^(1/4) + x + x^2))/2])), x])/Sqr
t[-1 + 4*E^E^(1/4)] - (1 - I/Sqrt[-1 + 4*E^E^(1/4)])*Defer[Int][1/((1 + I*Sqrt[-1 + 4*E^E^(1/4)] + 2*x)*(4 + 4
*x + x*Log[(x*(E^E^(1/4) + x + x^2))/2])), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {-4 e^{\sqrt [4]{e}}-\left (4-e^{\sqrt [4]{e}}\right ) x-2 x^2+3 x^3}{x \left (e^{\sqrt [4]{e}}+x+x^2\right ) \left (4+4 x+x \log \left (\frac {1}{2} x \left (e^{\sqrt [4]{e}}+x+x^2\right )\right )\right )} \, dx \\ & = \int \left (\frac {3}{4+4 x+x \log \left (\frac {1}{2} x \left (e^{\sqrt [4]{e}}+x+x^2\right )\right )}-\frac {4}{x \left (4+4 x+x \log \left (\frac {1}{2} x \left (e^{\sqrt [4]{e}}+x+x^2\right )\right )\right )}-\frac {2 e^{\sqrt [4]{e}}+x}{\left (e^{\sqrt [4]{e}}+x+x^2\right ) \left (4+4 x+x \log \left (\frac {1}{2} x \left (e^{\sqrt [4]{e}}+x+x^2\right )\right )\right )}\right ) \, dx \\ & = 3 \int \frac {1}{4+4 x+x \log \left (\frac {1}{2} x \left (e^{\sqrt [4]{e}}+x+x^2\right )\right )} \, dx-4 \int \frac {1}{x \left (4+4 x+x \log \left (\frac {1}{2} x \left (e^{\sqrt [4]{e}}+x+x^2\right )\right )\right )} \, dx-\int \frac {2 e^{\sqrt [4]{e}}+x}{\left (e^{\sqrt [4]{e}}+x+x^2\right ) \left (4+4 x+x \log \left (\frac {1}{2} x \left (e^{\sqrt [4]{e}}+x+x^2\right )\right )\right )} \, dx \\ & = 3 \int \frac {1}{4+4 x+x \log \left (\frac {1}{2} x \left (e^{\sqrt [4]{e}}+x+x^2\right )\right )} \, dx-4 \int \frac {1}{x \left (4+4 x+x \log \left (\frac {1}{2} x \left (e^{\sqrt [4]{e}}+x+x^2\right )\right )\right )} \, dx-\int \left (\frac {2 e^{\sqrt [4]{e}}}{\left (e^{\sqrt [4]{e}}+x+x^2\right ) \left (4+4 x+x \log \left (\frac {1}{2} x \left (e^{\sqrt [4]{e}}+x+x^2\right )\right )\right )}+\frac {x}{\left (e^{\sqrt [4]{e}}+x+x^2\right ) \left (4+4 x+x \log \left (\frac {1}{2} x \left (e^{\sqrt [4]{e}}+x+x^2\right )\right )\right )}\right ) \, dx \\ & = 3 \int \frac {1}{4+4 x+x \log \left (\frac {1}{2} x \left (e^{\sqrt [4]{e}}+x+x^2\right )\right )} \, dx-4 \int \frac {1}{x \left (4+4 x+x \log \left (\frac {1}{2} x \left (e^{\sqrt [4]{e}}+x+x^2\right )\right )\right )} \, dx-\left (2 e^{\sqrt [4]{e}}\right ) \int \frac {1}{\left (e^{\sqrt [4]{e}}+x+x^2\right ) \left (4+4 x+x \log \left (\frac {1}{2} x \left (e^{\sqrt [4]{e}}+x+x^2\right )\right )\right )} \, dx-\int \frac {x}{\left (e^{\sqrt [4]{e}}+x+x^2\right ) \left (4+4 x+x \log \left (\frac {1}{2} x \left (e^{\sqrt [4]{e}}+x+x^2\right )\right )\right )} \, dx \\ & = 3 \int \frac {1}{4+4 x+x \log \left (\frac {1}{2} x \left (e^{\sqrt [4]{e}}+x+x^2\right )\right )} \, dx-4 \int \frac {1}{x \left (4+4 x+x \log \left (\frac {1}{2} x \left (e^{\sqrt [4]{e}}+x+x^2\right )\right )\right )} \, dx-\left (2 e^{\sqrt [4]{e}}\right ) \int \left (\frac {2 i}{\sqrt {-1+4 e^{\sqrt [4]{e}}} \left (-1+i \sqrt {-1+4 e^{\sqrt [4]{e}}}-2 x\right ) \left (4+4 x+x \log \left (\frac {1}{2} x \left (e^{\sqrt [4]{e}}+x+x^2\right )\right )\right )}+\frac {2 i}{\sqrt {-1+4 e^{\sqrt [4]{e}}} \left (1+i \sqrt {-1+4 e^{\sqrt [4]{e}}}+2 x\right ) \left (4+4 x+x \log \left (\frac {1}{2} x \left (e^{\sqrt [4]{e}}+x+x^2\right )\right )\right )}\right ) \, dx-\int \left (\frac {1+\frac {i}{\sqrt {-1+4 e^{\sqrt [4]{e}}}}}{\left (1-i \sqrt {-1+4 e^{\sqrt [4]{e}}}+2 x\right ) \left (4+4 x+x \log \left (\frac {1}{2} x \left (e^{\sqrt [4]{e}}+x+x^2\right )\right )\right )}+\frac {1-\frac {i}{\sqrt {-1+4 e^{\sqrt [4]{e}}}}}{\left (1+i \sqrt {-1+4 e^{\sqrt [4]{e}}}+2 x\right ) \left (4+4 x+x \log \left (\frac {1}{2} x \left (e^{\sqrt [4]{e}}+x+x^2\right )\right )\right )}\right ) \, dx \\ & = 3 \int \frac {1}{4+4 x+x \log \left (\frac {1}{2} x \left (e^{\sqrt [4]{e}}+x+x^2\right )\right )} \, dx-4 \int \frac {1}{x \left (4+4 x+x \log \left (\frac {1}{2} x \left (e^{\sqrt [4]{e}}+x+x^2\right )\right )\right )} \, dx-\frac {\left (4 i e^{\sqrt [4]{e}}\right ) \int \frac {1}{\left (-1+i \sqrt {-1+4 e^{\sqrt [4]{e}}}-2 x\right ) \left (4+4 x+x \log \left (\frac {1}{2} x \left (e^{\sqrt [4]{e}}+x+x^2\right )\right )\right )} \, dx}{\sqrt {-1+4 e^{\sqrt [4]{e}}}}-\frac {\left (4 i e^{\sqrt [4]{e}}\right ) \int \frac {1}{\left (1+i \sqrt {-1+4 e^{\sqrt [4]{e}}}+2 x\right ) \left (4+4 x+x \log \left (\frac {1}{2} x \left (e^{\sqrt [4]{e}}+x+x^2\right )\right )\right )} \, dx}{\sqrt {-1+4 e^{\sqrt [4]{e}}}}-\left (1-\frac {i}{\sqrt {-1+4 e^{\sqrt [4]{e}}}}\right ) \int \frac {1}{\left (1+i \sqrt {-1+4 e^{\sqrt [4]{e}}}+2 x\right ) \left (4+4 x+x \log \left (\frac {1}{2} x \left (e^{\sqrt [4]{e}}+x+x^2\right )\right )\right )} \, dx-\left (1+\frac {i}{\sqrt {-1+4 e^{\sqrt [4]{e}}}}\right ) \int \frac {1}{\left (1-i \sqrt {-1+4 e^{\sqrt [4]{e}}}+2 x\right ) \left (4+4 x+x \log \left (\frac {1}{2} x \left (e^{\sqrt [4]{e}}+x+x^2\right )\right )\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 5.06 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19 \[ \int \frac {e^{\sqrt [4]{e}} (-4+x)-4 x-2 x^2+3 x^3}{4 x^2+8 x^3+4 x^4+e^{\sqrt [4]{e}} \left (4 x+4 x^2\right )+\left (e^{\sqrt [4]{e}} x^2+x^3+x^4\right ) \log \left (\frac {1}{2} \left (e^{\sqrt [4]{e}} x+x^2+x^3\right )\right )} \, dx=-\log (x)+\log \left (4+4 x+x \log \left (\frac {1}{2} x \left (e^{\sqrt [4]{e}}+x+x^2\right )\right )\right ) \]

[In]

Integrate[(E^E^(1/4)*(-4 + x) - 4*x - 2*x^2 + 3*x^3)/(4*x^2 + 8*x^3 + 4*x^4 + E^E^(1/4)*(4*x + 4*x^2) + (E^E^(
1/4)*x^2 + x^3 + x^4)*Log[(E^E^(1/4)*x + x^2 + x^3)/2]),x]

[Out]

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

Maple [A] (verified)

Time = 1.36 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00

method	result	size
parallelrisch	\(-\ln \left (x \right )+\ln \left (x \ln \left (\frac {x \left ({\mathrm e}^{{\mathrm e}^{\frac {1}{4}}}+x +x^{2}\right )}{2}\right )+4 x +4\right )\)	\(26\)
risch	\(\ln \left (\ln \left (\frac {x \,{\mathrm e}^{{\mathrm e}^{\frac {1}{4}}}}{2}+\frac {x^{3}}{2}+\frac {x^{2}}{2}\right )+\frac {4+4 x}{x}\right )\)	\(29\)
norman	\(-\ln \left (x \right )+\ln \left (x \ln \left (\frac {x \,{\mathrm e}^{{\mathrm e}^{\frac {1}{4}}}}{2}+\frac {x^{3}}{2}+\frac {x^{2}}{2}\right )+4 x +4\right )\)	\(32\)

[In]

int(((x-4)*exp(exp(1/4))+3*x^3-2*x^2-4*x)/((x^2*exp(exp(1/4))+x^4+x^3)*ln(1/2*x*exp(exp(1/4))+1/2*x^3+1/2*x^2)
+(4*x^2+4*x)*exp(exp(1/4))+4*x^4+8*x^3+4*x^2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15 \[ \int \frac {e^{\sqrt [4]{e}} (-4+x)-4 x-2 x^2+3 x^3}{4 x^2+8 x^3+4 x^4+e^{\sqrt [4]{e}} \left (4 x+4 x^2\right )+\left (e^{\sqrt [4]{e}} x^2+x^3+x^4\right ) \log \left (\frac {1}{2} \left (e^{\sqrt [4]{e}} x+x^2+x^3\right )\right )} \, dx=\log \left (\frac {x \log \left (\frac {1}{2} \, x^{3} + \frac {1}{2} \, x^{2} + \frac {1}{2} \, x e^{\left (e^{\frac {1}{4}}\right )}\right ) + 4 \, x + 4}{x}\right ) \]

[In]

integrate(((x-4)*exp(exp(1/4))+3*x^3-2*x^2-4*x)/((x^2*exp(exp(1/4))+x^4+x^3)*log(1/2*x*exp(exp(1/4))+1/2*x^3+1
/2*x^2)+(4*x^2+4*x)*exp(exp(1/4))+4*x^4+8*x^3+4*x^2),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {e^{\sqrt [4]{e}} (-4+x)-4 x-2 x^2+3 x^3}{4 x^2+8 x^3+4 x^4+e^{\sqrt [4]{e}} \left (4 x+4 x^2\right )+\left (e^{\sqrt [4]{e}} x^2+x^3+x^4\right ) \log \left (\frac {1}{2} \left (e^{\sqrt [4]{e}} x+x^2+x^3\right )\right )} \, dx=\log {\left (\log {\left (\frac {x^{3}}{2} + \frac {x^{2}}{2} + \frac {x e^{e^{\frac {1}{4}}}}{2} \right )} + \frac {4 x + 4}{x} \right )} \]

[In]

integrate(((x-4)*exp(exp(1/4))+3*x**3-2*x**2-4*x)/((x**2*exp(exp(1/4))+x**4+x**3)*ln(1/2*x*exp(exp(1/4))+1/2*x
**3+1/2*x**2)+(4*x**2+4*x)*exp(exp(1/4))+4*x**4+8*x**3+4*x**2),x)

[Out]

log(log(x**3/2 + x**2/2 + x*exp(exp(1/4))/2) + (4*x + 4)/x)

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19 \[ \int \frac {e^{\sqrt [4]{e}} (-4+x)-4 x-2 x^2+3 x^3}{4 x^2+8 x^3+4 x^4+e^{\sqrt [4]{e}} \left (4 x+4 x^2\right )+\left (e^{\sqrt [4]{e}} x^2+x^3+x^4\right ) \log \left (\frac {1}{2} \left (e^{\sqrt [4]{e}} x+x^2+x^3\right )\right )} \, dx=\log \left (-\frac {x {\left (\log \left (2\right ) - 4\right )} - x \log \left (x^{2} + x + e^{\left (e^{\frac {1}{4}}\right )}\right ) - x \log \left (x\right ) - 4}{x}\right ) \]

[In]

integrate(((x-4)*exp(exp(1/4))+3*x^3-2*x^2-4*x)/((x^2*exp(exp(1/4))+x^4+x^3)*log(1/2*x*exp(exp(1/4))+1/2*x^3+1
/2*x^2)+(4*x^2+4*x)*exp(exp(1/4))+4*x^4+8*x^3+4*x^2),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19 \[ \int \frac {e^{\sqrt [4]{e}} (-4+x)-4 x-2 x^2+3 x^3}{4 x^2+8 x^3+4 x^4+e^{\sqrt [4]{e}} \left (4 x+4 x^2\right )+\left (e^{\sqrt [4]{e}} x^2+x^3+x^4\right ) \log \left (\frac {1}{2} \left (e^{\sqrt [4]{e}} x+x^2+x^3\right )\right )} \, dx=\log \left (x \log \left (\frac {1}{2} \, x^{3} + \frac {1}{2} \, x^{2} + \frac {1}{2} \, x e^{\left (e^{\frac {1}{4}}\right )}\right ) + 4 \, x + 4\right ) - \log \left (x\right ) \]

[In]

integrate(((x-4)*exp(exp(1/4))+3*x^3-2*x^2-4*x)/((x^2*exp(exp(1/4))+x^4+x^3)*log(1/2*x*exp(exp(1/4))+1/2*x^3+1
/2*x^2)+(4*x^2+4*x)*exp(exp(1/4))+4*x^4+8*x^3+4*x^2),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 15.28 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\sqrt [4]{e}} (-4+x)-4 x-2 x^2+3 x^3}{4 x^2+8 x^3+4 x^4+e^{\sqrt [4]{e}} \left (4 x+4 x^2\right )+\left (e^{\sqrt [4]{e}} x^2+x^3+x^4\right ) \log \left (\frac {1}{2} \left (e^{\sqrt [4]{e}} x+x^2+x^3\right )\right )} \, dx=\ln \left (\ln \left (\frac {x^3}{2}+\frac {x^2}{2}+\frac {{\mathrm {e}}^{{\mathrm {e}}^{1/4}}\,x}{2}\right )+\frac {4}{x}+4\right ) \]

[In]

int(-(4*x - exp(exp(1/4))*(x - 4) + 2*x^2 - 3*x^3)/(log((x*exp(exp(1/4)))/2 + x^2/2 + x^3/2)*(x^2*exp(exp(1/4)
) + x^3 + x^4) + exp(exp(1/4))*(4*x + 4*x^2) + 4*x^2 + 8*x^3 + 4*x^4),x)

[Out]

log(log((x*exp(exp(1/4)))/2 + x^2/2 + x^3/2) + 4/x + 4)