\(\int \frac {2 x+x^2+x^3+e^{8 x^4} (-8+4 x^2) \log ^3(\frac {x}{2+x+x^2})+e^{8 x^4} (-64 x^4-32 x^5-32 x^6) \log ^4(\frac {x}{2+x+x^2})}{2 x+x^2+x^3} \, dx\) [7715]

Optimal result

Integrand size = 88, antiderivative size = 24 \[ \int \frac {2 x+x^2+x^3+e^{8 x^4} \left (-8+4 x^2\right ) \log ^3\left (\frac {x}{2+x+x^2}\right )+e^{8 x^4} \left (-64 x^4-32 x^5-32 x^6\right ) \log ^4\left (\frac {x}{2+x+x^2}\right )}{2 x+x^2+x^3} \, dx=x-e^{8 x^4} \log ^4\left (\frac {x}{2+x+x^2}\right ) \]

[Out]

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

Rubi [F]

\[ \int \frac {2 x+x^2+x^3+e^{8 x^4} \left (-8+4 x^2\right ) \log ^3\left (\frac {x}{2+x+x^2}\right )+e^{8 x^4} \left (-64 x^4-32 x^5-32 x^6\right ) \log ^4\left (\frac {x}{2+x+x^2}\right )}{2 x+x^2+x^3} \, dx=\int \frac {2 x+x^2+x^3+e^{8 x^4} \left (-8+4 x^2\right ) \log ^3\left (\frac {x}{2+x+x^2}\right )+e^{8 x^4} \left (-64 x^4-32 x^5-32 x^6\right ) \log ^4\left (\frac {x}{2+x+x^2}\right )}{2 x+x^2+x^3} \, dx \]

[In]

Int[(2*x + x^2 + x^3 + E^(8*x^4)*(-8 + 4*x^2)*Log[x/(2 + x + x^2)]^3 + E^(8*x^4)*(-64*x^4 - 32*x^5 - 32*x^6)*L
og[x/(2 + x + x^2)]^4)/(2*x + x^2 + x^3),x]

[Out]

x + ((8*I)*Defer[Int][(E^(8*x^4)*Log[x/(2 + x + x^2)]^3)/(-1 + I*Sqrt[7] - 2*x), x])/Sqrt[7] - 4*Defer[Int][(E
^(8*x^4)*Log[x/(2 + x + x^2)]^3)/x, x] + (8*(7 + I*Sqrt[7])*Defer[Int][(E^(8*x^4)*Log[x/(2 + x + x^2)]^3)/(1 -
I*Sqrt[7] + 2*x), x])/7 + ((8*I)*Defer[Int][(E^(8*x^4)*Log[x/(2 + x + x^2)]^3)/(1 + I*Sqrt[7] + 2*x), x])/Sqr
t[7] + (8*(7 - I*Sqrt[7])*Defer[Int][(E^(8*x^4)*Log[x/(2 + x + x^2)]^3)/(1 + I*Sqrt[7] + 2*x), x])/7 - 32*Defe
r[Int][E^(8*x^4)*x^3*Log[x/(2 + x + x^2)]^4, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {2 x+x^2+x^3+e^{8 x^4} \left (-8+4 x^2\right ) \log ^3\left (\frac {x}{2+x+x^2}\right )+e^{8 x^4} \left (-64 x^4-32 x^5-32 x^6\right ) \log ^4\left (\frac {x}{2+x+x^2}\right )}{x \left (2+x+x^2\right )} \, dx \\ & = \int \left (1+\frac {4 e^{8 x^4} \left (-2+x^2\right ) \log ^3\left (\frac {x}{2+x+x^2}\right )}{x \left (2+x+x^2\right )}-32 e^{8 x^4} x^3 \log ^4\left (\frac {x}{2+x+x^2}\right )\right ) \, dx \\ & = x+4 \int \frac {e^{8 x^4} \left (-2+x^2\right ) \log ^3\left (\frac {x}{2+x+x^2}\right )}{x \left (2+x+x^2\right )} \, dx-32 \int e^{8 x^4} x^3 \log ^4\left (\frac {x}{2+x+x^2}\right ) \, dx \\ & = x+4 \int \left (-\frac {e^{8 x^4} \log ^3\left (\frac {x}{2+x+x^2}\right )}{x}+\frac {e^{8 x^4} (1+2 x) \log ^3\left (\frac {x}{2+x+x^2}\right )}{2+x+x^2}\right ) \, dx-32 \int e^{8 x^4} x^3 \log ^4\left (\frac {x}{2+x+x^2}\right ) \, dx \\ & = x-4 \int \frac {e^{8 x^4} \log ^3\left (\frac {x}{2+x+x^2}\right )}{x} \, dx+4 \int \frac {e^{8 x^4} (1+2 x) \log ^3\left (\frac {x}{2+x+x^2}\right )}{2+x+x^2} \, dx-32 \int e^{8 x^4} x^3 \log ^4\left (\frac {x}{2+x+x^2}\right ) \, dx \\ & = x-4 \int \frac {e^{8 x^4} \log ^3\left (\frac {x}{2+x+x^2}\right )}{x} \, dx+4 \int \left (\frac {e^{8 x^4} \log ^3\left (\frac {x}{2+x+x^2}\right )}{2+x+x^2}+\frac {2 e^{8 x^4} x \log ^3\left (\frac {x}{2+x+x^2}\right )}{2+x+x^2}\right ) \, dx-32 \int e^{8 x^4} x^3 \log ^4\left (\frac {x}{2+x+x^2}\right ) \, dx \\ & = x-4 \int \frac {e^{8 x^4} \log ^3\left (\frac {x}{2+x+x^2}\right )}{x} \, dx+4 \int \frac {e^{8 x^4} \log ^3\left (\frac {x}{2+x+x^2}\right )}{2+x+x^2} \, dx+8 \int \frac {e^{8 x^4} x \log ^3\left (\frac {x}{2+x+x^2}\right )}{2+x+x^2} \, dx-32 \int e^{8 x^4} x^3 \log ^4\left (\frac {x}{2+x+x^2}\right ) \, dx \\ & = x-4 \int \frac {e^{8 x^4} \log ^3\left (\frac {x}{2+x+x^2}\right )}{x} \, dx+4 \int \left (\frac {2 i e^{8 x^4} \log ^3\left (\frac {x}{2+x+x^2}\right )}{\sqrt {7} \left (-1+i \sqrt {7}-2 x\right )}+\frac {2 i e^{8 x^4} \log ^3\left (\frac {x}{2+x+x^2}\right )}{\sqrt {7} \left (1+i \sqrt {7}+2 x\right )}\right ) \, dx+8 \int \left (\frac {\left (1+\frac {i}{\sqrt {7}}\right ) e^{8 x^4} \log ^3\left (\frac {x}{2+x+x^2}\right )}{1-i \sqrt {7}+2 x}+\frac {\left (1-\frac {i}{\sqrt {7}}\right ) e^{8 x^4} \log ^3\left (\frac {x}{2+x+x^2}\right )}{1+i \sqrt {7}+2 x}\right ) \, dx-32 \int e^{8 x^4} x^3 \log ^4\left (\frac {x}{2+x+x^2}\right ) \, dx \\ & = x-4 \int \frac {e^{8 x^4} \log ^3\left (\frac {x}{2+x+x^2}\right )}{x} \, dx-32 \int e^{8 x^4} x^3 \log ^4\left (\frac {x}{2+x+x^2}\right ) \, dx+\frac {(8 i) \int \frac {e^{8 x^4} \log ^3\left (\frac {x}{2+x+x^2}\right )}{-1+i \sqrt {7}-2 x} \, dx}{\sqrt {7}}+\frac {(8 i) \int \frac {e^{8 x^4} \log ^3\left (\frac {x}{2+x+x^2}\right )}{1+i \sqrt {7}+2 x} \, dx}{\sqrt {7}}+\frac {1}{7} \left (8 \left (7-i \sqrt {7}\right )\right ) \int \frac {e^{8 x^4} \log ^3\left (\frac {x}{2+x+x^2}\right )}{1+i \sqrt {7}+2 x} \, dx+\frac {1}{7} \left (8 \left (7+i \sqrt {7}\right )\right ) \int \frac {e^{8 x^4} \log ^3\left (\frac {x}{2+x+x^2}\right )}{1-i \sqrt {7}+2 x} \, dx \\ \end{align*}

Mathematica [F]

\[ \int \frac {2 x+x^2+x^3+e^{8 x^4} \left (-8+4 x^2\right ) \log ^3\left (\frac {x}{2+x+x^2}\right )+e^{8 x^4} \left (-64 x^4-32 x^5-32 x^6\right ) \log ^4\left (\frac {x}{2+x+x^2}\right )}{2 x+x^2+x^3} \, dx=\int \frac {2 x+x^2+x^3+e^{8 x^4} \left (-8+4 x^2\right ) \log ^3\left (\frac {x}{2+x+x^2}\right )+e^{8 x^4} \left (-64 x^4-32 x^5-32 x^6\right ) \log ^4\left (\frac {x}{2+x+x^2}\right )}{2 x+x^2+x^3} \, dx \]

[In]

Integrate[(2*x + x^2 + x^3 + E^(8*x^4)*(-8 + 4*x^2)*Log[x/(2 + x + x^2)]^3 + E^(8*x^4)*(-64*x^4 - 32*x^5 - 32*
x^6)*Log[x/(2 + x + x^2)]^4)/(2*x + x^2 + x^3),x]

[Out]

Integrate[(2*x + x^2 + x^3 + E^(8*x^4)*(-8 + 4*x^2)*Log[x/(2 + x + x^2)]^3 + E^(8*x^4)*(-64*x^4 - 32*x^5 - 32*
x^6)*Log[x/(2 + x + x^2)]^4)/(2*x + x^2 + x^3), x]

Maple [A] (verified)

Time = 2.56 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.12

method	result	size
parallelrisch	\(-{\mathrm e}^{8 x^{4}} \ln \left (\frac {x}{x^{2}+x +2}\right )^{4}-\frac {5}{2}+x\)	\(27\)
risch	\(\text {Expression too large to display}\)	\(4181\)

[In]

int(((-32*x^6-32*x^5-64*x^4)*exp(2*x^4)^4*ln(x/(x^2+x+2))^4+(4*x^2-8)*exp(2*x^4)^4*ln(x/(x^2+x+2))^3+x^3+x^2+2
*x)/(x^3+x^2+2*x),x,method=_RETURNVERBOSE)

[Out]

-exp(2*x^4)^4*ln(x/(x^2+x+2))^4-5/2+x

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96 \[ \int \frac {2 x+x^2+x^3+e^{8 x^4} \left (-8+4 x^2\right ) \log ^3\left (\frac {x}{2+x+x^2}\right )+e^{8 x^4} \left (-64 x^4-32 x^5-32 x^6\right ) \log ^4\left (\frac {x}{2+x+x^2}\right )}{2 x+x^2+x^3} \, dx=-e^{\left (8 \, x^{4}\right )} \log \left (\frac {x}{x^{2} + x + 2}\right )^{4} + x \]

[In]

integrate(((-32*x^6-32*x^5-64*x^4)*exp(2*x^4)^4*log(x/(x^2+x+2))^4+(4*x^2-8)*exp(2*x^4)^4*log(x/(x^2+x+2))^3+x
^3+x^2+2*x)/(x^3+x^2+2*x),x, algorithm="fricas")

[Out]

-e^(8*x^4)*log(x/(x^2 + x + 2))^4 + x

Sympy [A] (verification not implemented)

Time = 11.31 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79 \[ \int \frac {2 x+x^2+x^3+e^{8 x^4} \left (-8+4 x^2\right ) \log ^3\left (\frac {x}{2+x+x^2}\right )+e^{8 x^4} \left (-64 x^4-32 x^5-32 x^6\right ) \log ^4\left (\frac {x}{2+x+x^2}\right )}{2 x+x^2+x^3} \, dx=x - e^{8 x^{4}} \log {\left (\frac {x}{x^{2} + x + 2} \right )}^{4} \]

[In]

integrate(((-32*x**6-32*x**5-64*x**4)*exp(2*x**4)**4*ln(x/(x**2+x+2))**4+(4*x**2-8)*exp(2*x**4)**4*ln(x/(x**2+
x+2))**3+x**3+x**2+2*x)/(x**3+x**2+2*x),x)

[Out]

x - exp(8*x**4)*log(x/(x**2 + x + 2))**4

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (23) = 46\).

Time = 0.38 (sec) , antiderivative size = 90, normalized size of antiderivative = 3.75 \[ \int \frac {2 x+x^2+x^3+e^{8 x^4} \left (-8+4 x^2\right ) \log ^3\left (\frac {x}{2+x+x^2}\right )+e^{8 x^4} \left (-64 x^4-32 x^5-32 x^6\right ) \log ^4\left (\frac {x}{2+x+x^2}\right )}{2 x+x^2+x^3} \, dx=-e^{\left (8 \, x^{4}\right )} \log \left (x^{2} + x + 2\right )^{4} + 4 \, e^{\left (8 \, x^{4}\right )} \log \left (x^{2} + x + 2\right )^{3} \log \left (x\right ) - 6 \, e^{\left (8 \, x^{4}\right )} \log \left (x^{2} + x + 2\right )^{2} \log \left (x\right )^{2} + 4 \, e^{\left (8 \, x^{4}\right )} \log \left (x^{2} + x + 2\right ) \log \left (x\right )^{3} - e^{\left (8 \, x^{4}\right )} \log \left (x\right )^{4} + x \]

[In]

integrate(((-32*x^6-32*x^5-64*x^4)*exp(2*x^4)^4*log(x/(x^2+x+2))^4+(4*x^2-8)*exp(2*x^4)^4*log(x/(x^2+x+2))^3+x
^3+x^2+2*x)/(x^3+x^2+2*x),x, algorithm="maxima")

[Out]

-e^(8*x^4)*log(x^2 + x + 2)^4 + 4*e^(8*x^4)*log(x^2 + x + 2)^3*log(x) - 6*e^(8*x^4)*log(x^2 + x + 2)^2*log(x)^
2 + 4*e^(8*x^4)*log(x^2 + x + 2)*log(x)^3 - e^(8*x^4)*log(x)^4 + x

Giac [A] (verification not implemented)

none

Time = 0.51 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96 \[ \int \frac {2 x+x^2+x^3+e^{8 x^4} \left (-8+4 x^2\right ) \log ^3\left (\frac {x}{2+x+x^2}\right )+e^{8 x^4} \left (-64 x^4-32 x^5-32 x^6\right ) \log ^4\left (\frac {x}{2+x+x^2}\right )}{2 x+x^2+x^3} \, dx=-e^{\left (8 \, x^{4}\right )} \log \left (\frac {x}{x^{2} + x + 2}\right )^{4} + x \]

[In]

integrate(((-32*x^6-32*x^5-64*x^4)*exp(2*x^4)^4*log(x/(x^2+x+2))^4+(4*x^2-8)*exp(2*x^4)^4*log(x/(x^2+x+2))^3+x
^3+x^2+2*x)/(x^3+x^2+2*x),x, algorithm="giac")

[Out]

-e^(8*x^4)*log(x/(x^2 + x + 2))^4 + x

Mupad [B] (verification not implemented)

Time = 14.06 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96 \[ \int \frac {2 x+x^2+x^3+e^{8 x^4} \left (-8+4 x^2\right ) \log ^3\left (\frac {x}{2+x+x^2}\right )+e^{8 x^4} \left (-64 x^4-32 x^5-32 x^6\right ) \log ^4\left (\frac {x}{2+x+x^2}\right )}{2 x+x^2+x^3} \, dx=x-{\ln \left (\frac {x}{x^2+x+2}\right )}^4\,{\mathrm {e}}^{8\,x^4} \]

[In]

int((2*x + x^2 + x^3 + log(x/(x + x^2 + 2))^3*exp(8*x^4)*(4*x^2 - 8) - log(x/(x + x^2 + 2))^4*exp(8*x^4)*(64*x
^4 + 32*x^5 + 32*x^6))/(2*x + x^2 + x^3),x)

[Out]

x - log(x/(x + x^2 + 2))^4*exp(8*x^4)