\(\int \frac {(-30 x-24 x^2-6 x^3+e^{4 x} (-10 x-8 x^2-2 x^3)+(10 x+8 x^2+2 x^3) \log (x)) \log (\frac {-4+8 x+4 x^2}{2+x})+(4-6 x-8 x^2-2 x^3+e^{4 x} (-16 x+24 x^2+32 x^3+8 x^4)) \log ^2(\frac {-4+8 x+4 x^2}{2+x})}{54 x-81 x^2-108 x^3-27 x^4+e^{4 x} (54 x-81 x^2-108 x^3-27 x^4)+e^{8 x} (18 x-27 x^2-36 x^3-9 x^4)+e^{12 x} (2 x-3 x^2-4 x^3-x^4)+(-54 x+81 x^2+108 x^3+27 x^4+e^{8 x} (-6 x+9 x^2+12 x^3+3 x^4)+e^{4 x} (-36 x+54 x^2+72 x^3+18 x^4)) \log (x)+(18 x-27 x^2-36 x^3-9 x^4+e^{4 x} (6 x-9 x^2-12 x^3-3 x^4)) \log ^2(x)+(-2 x+3 x^2+4 x^3+x^4) \log ^3(x)} \, dx\) [8716]

Optimal result

Integrand size = 370, antiderivative size = 28 \[ \int \frac {\left (-30 x-24 x^2-6 x^3+e^{4 x} \left (-10 x-8 x^2-2 x^3\right )+\left (10 x+8 x^2+2 x^3\right ) \log (x)\right ) \log \left (\frac {-4+8 x+4 x^2}{2+x}\right )+\left (4-6 x-8 x^2-2 x^3+e^{4 x} \left (-16 x+24 x^2+32 x^3+8 x^4\right )\right ) \log ^2\left (\frac {-4+8 x+4 x^2}{2+x}\right )}{54 x-81 x^2-108 x^3-27 x^4+e^{4 x} \left (54 x-81 x^2-108 x^3-27 x^4\right )+e^{8 x} \left (18 x-27 x^2-36 x^3-9 x^4\right )+e^{12 x} \left (2 x-3 x^2-4 x^3-x^4\right )+\left (-54 x+81 x^2+108 x^3+27 x^4+e^{8 x} \left (-6 x+9 x^2+12 x^3+3 x^4\right )+e^{4 x} \left (-36 x+54 x^2+72 x^3+18 x^4\right )\right ) \log (x)+\left (18 x-27 x^2-36 x^3-9 x^4+e^{4 x} \left (6 x-9 x^2-12 x^3-3 x^4\right )\right ) \log ^2(x)+\left (-2 x+3 x^2+4 x^3+x^4\right ) \log ^3(x)} \, dx=\frac {\log ^2\left (4 \left (x-\frac {1}{2+x}\right )\right )}{\left (3+e^{4 x}-\log (x)\right )^2} \]

[Out]

ln(4*x-4/(2+x))^2/(3+exp(4*x)-ln(x))^2

Rubi [F(-1)]

Timed out. \[ \int \frac {\left (-30 x-24 x^2-6 x^3+e^{4 x} \left (-10 x-8 x^2-2 x^3\right )+\left (10 x+8 x^2+2 x^3\right ) \log (x)\right ) \log \left (\frac {-4+8 x+4 x^2}{2+x}\right )+\left (4-6 x-8 x^2-2 x^3+e^{4 x} \left (-16 x+24 x^2+32 x^3+8 x^4\right )\right ) \log ^2\left (\frac {-4+8 x+4 x^2}{2+x}\right )}{54 x-81 x^2-108 x^3-27 x^4+e^{4 x} \left (54 x-81 x^2-108 x^3-27 x^4\right )+e^{8 x} \left (18 x-27 x^2-36 x^3-9 x^4\right )+e^{12 x} \left (2 x-3 x^2-4 x^3-x^4\right )+\left (-54 x+81 x^2+108 x^3+27 x^4+e^{8 x} \left (-6 x+9 x^2+12 x^3+3 x^4\right )+e^{4 x} \left (-36 x+54 x^2+72 x^3+18 x^4\right )\right ) \log (x)+\left (18 x-27 x^2-36 x^3-9 x^4+e^{4 x} \left (6 x-9 x^2-12 x^3-3 x^4\right )\right ) \log ^2(x)+\left (-2 x+3 x^2+4 x^3+x^4\right ) \log ^3(x)} \, dx=\text {\$Aborted} \]

[In]

Int[((-30*x - 24*x^2 - 6*x^3 + E^(4*x)*(-10*x - 8*x^2 - 2*x^3) + (10*x + 8*x^2 + 2*x^3)*Log[x])*Log[(-4 + 8*x
+ 4*x^2)/(2 + x)] + (4 - 6*x - 8*x^2 - 2*x^3 + E^(4*x)*(-16*x + 24*x^2 + 32*x^3 + 8*x^4))*Log[(-4 + 8*x + 4*x^
2)/(2 + x)]^2)/(54*x - 81*x^2 - 108*x^3 - 27*x^4 + E^(4*x)*(54*x - 81*x^2 - 108*x^3 - 27*x^4) + E^(8*x)*(18*x
- 27*x^2 - 36*x^3 - 9*x^4) + E^(12*x)*(2*x - 3*x^2 - 4*x^3 - x^4) + (-54*x + 81*x^2 + 108*x^3 + 27*x^4 + E^(8*
x)*(-6*x + 9*x^2 + 12*x^3 + 3*x^4) + E^(4*x)*(-36*x + 54*x^2 + 72*x^3 + 18*x^4))*Log[x] + (18*x - 27*x^2 - 36*
x^3 - 9*x^4 + E^(4*x)*(6*x - 9*x^2 - 12*x^3 - 3*x^4))*Log[x]^2 + (-2*x + 3*x^2 + 4*x^3 + x^4)*Log[x]^3),x]

[Out]

$Aborted

Rubi steps Aborted

Mathematica [F]

\[ \int \frac {\left (-30 x-24 x^2-6 x^3+e^{4 x} \left (-10 x-8 x^2-2 x^3\right )+\left (10 x+8 x^2+2 x^3\right ) \log (x)\right ) \log \left (\frac {-4+8 x+4 x^2}{2+x}\right )+\left (4-6 x-8 x^2-2 x^3+e^{4 x} \left (-16 x+24 x^2+32 x^3+8 x^4\right )\right ) \log ^2\left (\frac {-4+8 x+4 x^2}{2+x}\right )}{54 x-81 x^2-108 x^3-27 x^4+e^{4 x} \left (54 x-81 x^2-108 x^3-27 x^4\right )+e^{8 x} \left (18 x-27 x^2-36 x^3-9 x^4\right )+e^{12 x} \left (2 x-3 x^2-4 x^3-x^4\right )+\left (-54 x+81 x^2+108 x^3+27 x^4+e^{8 x} \left (-6 x+9 x^2+12 x^3+3 x^4\right )+e^{4 x} \left (-36 x+54 x^2+72 x^3+18 x^4\right )\right ) \log (x)+\left (18 x-27 x^2-36 x^3-9 x^4+e^{4 x} \left (6 x-9 x^2-12 x^3-3 x^4\right )\right ) \log ^2(x)+\left (-2 x+3 x^2+4 x^3+x^4\right ) \log ^3(x)} \, dx=\int \frac {\left (-30 x-24 x^2-6 x^3+e^{4 x} \left (-10 x-8 x^2-2 x^3\right )+\left (10 x+8 x^2+2 x^3\right ) \log (x)\right ) \log \left (\frac {-4+8 x+4 x^2}{2+x}\right )+\left (4-6 x-8 x^2-2 x^3+e^{4 x} \left (-16 x+24 x^2+32 x^3+8 x^4\right )\right ) \log ^2\left (\frac {-4+8 x+4 x^2}{2+x}\right )}{54 x-81 x^2-108 x^3-27 x^4+e^{4 x} \left (54 x-81 x^2-108 x^3-27 x^4\right )+e^{8 x} \left (18 x-27 x^2-36 x^3-9 x^4\right )+e^{12 x} \left (2 x-3 x^2-4 x^3-x^4\right )+\left (-54 x+81 x^2+108 x^3+27 x^4+e^{8 x} \left (-6 x+9 x^2+12 x^3+3 x^4\right )+e^{4 x} \left (-36 x+54 x^2+72 x^3+18 x^4\right )\right ) \log (x)+\left (18 x-27 x^2-36 x^3-9 x^4+e^{4 x} \left (6 x-9 x^2-12 x^3-3 x^4\right )\right ) \log ^2(x)+\left (-2 x+3 x^2+4 x^3+x^4\right ) \log ^3(x)} \, dx \]

[In]

Integrate[((-30*x - 24*x^2 - 6*x^3 + E^(4*x)*(-10*x - 8*x^2 - 2*x^3) + (10*x + 8*x^2 + 2*x^3)*Log[x])*Log[(-4
+ 8*x + 4*x^2)/(2 + x)] + (4 - 6*x - 8*x^2 - 2*x^3 + E^(4*x)*(-16*x + 24*x^2 + 32*x^3 + 8*x^4))*Log[(-4 + 8*x
+ 4*x^2)/(2 + x)]^2)/(54*x - 81*x^2 - 108*x^3 - 27*x^4 + E^(4*x)*(54*x - 81*x^2 - 108*x^3 - 27*x^4) + E^(8*x)*
(18*x - 27*x^2 - 36*x^3 - 9*x^4) + E^(12*x)*(2*x - 3*x^2 - 4*x^3 - x^4) + (-54*x + 81*x^2 + 108*x^3 + 27*x^4 +
E^(8*x)*(-6*x + 9*x^2 + 12*x^3 + 3*x^4) + E^(4*x)*(-36*x + 54*x^2 + 72*x^3 + 18*x^4))*Log[x] + (18*x - 27*x^2
- 36*x^3 - 9*x^4 + E^(4*x)*(6*x - 9*x^2 - 12*x^3 - 3*x^4))*Log[x]^2 + (-2*x + 3*x^2 + 4*x^3 + x^4)*Log[x]^3),
x]

[Out]

Integrate[((-30*x - 24*x^2 - 6*x^3 + E^(4*x)*(-10*x - 8*x^2 - 2*x^3) + (10*x + 8*x^2 + 2*x^3)*Log[x])*Log[(-4
+ 8*x + 4*x^2)/(2 + x)] + (4 - 6*x - 8*x^2 - 2*x^3 + E^(4*x)*(-16*x + 24*x^2 + 32*x^3 + 8*x^4))*Log[(-4 + 8*x
+ 4*x^2)/(2 + x)]^2)/(54*x - 81*x^2 - 108*x^3 - 27*x^4 + E^(4*x)*(54*x - 81*x^2 - 108*x^3 - 27*x^4) + E^(8*x)*
(18*x - 27*x^2 - 36*x^3 - 9*x^4) + E^(12*x)*(2*x - 3*x^2 - 4*x^3 - x^4) + (-54*x + 81*x^2 + 108*x^3 + 27*x^4 +
E^(8*x)*(-6*x + 9*x^2 + 12*x^3 + 3*x^4) + E^(4*x)*(-36*x + 54*x^2 + 72*x^3 + 18*x^4))*Log[x] + (18*x - 27*x^2
- 36*x^3 - 9*x^4 + E^(4*x)*(6*x - 9*x^2 - 12*x^3 - 3*x^4))*Log[x]^2 + (-2*x + 3*x^2 + 4*x^3 + x^4)*Log[x]^3),
x]

Maple [A] (verified)

Time = 179.06 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.86

method	result	size
parallelrisch	\(\frac {\ln \left (\frac {4 x^{2}+8 x -4}{2+x}\right )^{2}}{\ln \left (x \right )^{2}-2 \ln \left (x \right ) {\mathrm e}^{4 x}+{\mathrm e}^{8 x}-6 \ln \left (x \right )+6 \,{\mathrm e}^{4 x}+9}\)	\(52\)
risch	\(\text {Expression too large to display}\)	\(870\)

[In]

int((((8*x^4+32*x^3+24*x^2-16*x)*exp(4*x)-2*x^3-8*x^2-6*x+4)*ln((4*x^2+8*x-4)/(2+x))^2+((2*x^3+8*x^2+10*x)*ln(
x)+(-2*x^3-8*x^2-10*x)*exp(4*x)-6*x^3-24*x^2-30*x)*ln((4*x^2+8*x-4)/(2+x)))/((x^4+4*x^3+3*x^2-2*x)*ln(x)^3+((-
3*x^4-12*x^3-9*x^2+6*x)*exp(4*x)-9*x^4-36*x^3-27*x^2+18*x)*ln(x)^2+((3*x^4+12*x^3+9*x^2-6*x)*exp(4*x)^2+(18*x^
4+72*x^3+54*x^2-36*x)*exp(4*x)+27*x^4+108*x^3+81*x^2-54*x)*ln(x)+(-x^4-4*x^3-3*x^2+2*x)*exp(4*x)^3+(-9*x^4-36*
x^3-27*x^2+18*x)*exp(4*x)^2+(-27*x^4-108*x^3-81*x^2+54*x)*exp(4*x)-27*x^4-108*x^3-81*x^2+54*x),x,method=_RETUR
NVERBOSE)

[Out]

ln(4*(x^2+2*x-1)/(2+x))^2/(ln(x)^2-2*ln(x)*exp(4*x)+exp(4*x)^2-6*ln(x)+6*exp(4*x)+9)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.86 \[ \int \frac {\left (-30 x-24 x^2-6 x^3+e^{4 x} \left (-10 x-8 x^2-2 x^3\right )+\left (10 x+8 x^2+2 x^3\right ) \log (x)\right ) \log \left (\frac {-4+8 x+4 x^2}{2+x}\right )+\left (4-6 x-8 x^2-2 x^3+e^{4 x} \left (-16 x+24 x^2+32 x^3+8 x^4\right )\right ) \log ^2\left (\frac {-4+8 x+4 x^2}{2+x}\right )}{54 x-81 x^2-108 x^3-27 x^4+e^{4 x} \left (54 x-81 x^2-108 x^3-27 x^4\right )+e^{8 x} \left (18 x-27 x^2-36 x^3-9 x^4\right )+e^{12 x} \left (2 x-3 x^2-4 x^3-x^4\right )+\left (-54 x+81 x^2+108 x^3+27 x^4+e^{8 x} \left (-6 x+9 x^2+12 x^3+3 x^4\right )+e^{4 x} \left (-36 x+54 x^2+72 x^3+18 x^4\right )\right ) \log (x)+\left (18 x-27 x^2-36 x^3-9 x^4+e^{4 x} \left (6 x-9 x^2-12 x^3-3 x^4\right )\right ) \log ^2(x)+\left (-2 x+3 x^2+4 x^3+x^4\right ) \log ^3(x)} \, dx=-\frac {\log \left (\frac {4 \, {\left (x^{2} + 2 \, x - 1\right )}}{x + 2}\right )^{2}}{2 \, {\left (e^{\left (4 \, x\right )} + 3\right )} \log \left (x\right ) - \log \left (x\right )^{2} - e^{\left (8 \, x\right )} - 6 \, e^{\left (4 \, x\right )} - 9} \]

[In]

integrate((((8*x^4+32*x^3+24*x^2-16*x)*exp(4*x)-2*x^3-8*x^2-6*x+4)*log((4*x^2+8*x-4)/(2+x))^2+((2*x^3+8*x^2+10
*x)*log(x)+(-2*x^3-8*x^2-10*x)*exp(4*x)-6*x^3-24*x^2-30*x)*log((4*x^2+8*x-4)/(2+x)))/((x^4+4*x^3+3*x^2-2*x)*lo
g(x)^3+((-3*x^4-12*x^3-9*x^2+6*x)*exp(4*x)-9*x^4-36*x^3-27*x^2+18*x)*log(x)^2+((3*x^4+12*x^3+9*x^2-6*x)*exp(4*
x)^2+(18*x^4+72*x^3+54*x^2-36*x)*exp(4*x)+27*x^4+108*x^3+81*x^2-54*x)*log(x)+(-x^4-4*x^3-3*x^2+2*x)*exp(4*x)^3
+(-9*x^4-36*x^3-27*x^2+18*x)*exp(4*x)^2+(-27*x^4-108*x^3-81*x^2+54*x)*exp(4*x)-27*x^4-108*x^3-81*x^2+54*x),x,
algorithm="fricas")

[Out]

-log(4*(x^2 + 2*x - 1)/(x + 2))^2/(2*(e^(4*x) + 3)*log(x) - log(x)^2 - e^(8*x) - 6*e^(4*x) - 9)

Sympy [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.57 \[ \int \frac {\left (-30 x-24 x^2-6 x^3+e^{4 x} \left (-10 x-8 x^2-2 x^3\right )+\left (10 x+8 x^2+2 x^3\right ) \log (x)\right ) \log \left (\frac {-4+8 x+4 x^2}{2+x}\right )+\left (4-6 x-8 x^2-2 x^3+e^{4 x} \left (-16 x+24 x^2+32 x^3+8 x^4\right )\right ) \log ^2\left (\frac {-4+8 x+4 x^2}{2+x}\right )}{54 x-81 x^2-108 x^3-27 x^4+e^{4 x} \left (54 x-81 x^2-108 x^3-27 x^4\right )+e^{8 x} \left (18 x-27 x^2-36 x^3-9 x^4\right )+e^{12 x} \left (2 x-3 x^2-4 x^3-x^4\right )+\left (-54 x+81 x^2+108 x^3+27 x^4+e^{8 x} \left (-6 x+9 x^2+12 x^3+3 x^4\right )+e^{4 x} \left (-36 x+54 x^2+72 x^3+18 x^4\right )\right ) \log (x)+\left (18 x-27 x^2-36 x^3-9 x^4+e^{4 x} \left (6 x-9 x^2-12 x^3-3 x^4\right )\right ) \log ^2(x)+\left (-2 x+3 x^2+4 x^3+x^4\right ) \log ^3(x)} \, dx=\frac {\log {\left (\frac {4 x^{2} + 8 x - 4}{x + 2} \right )}^{2}}{\left (6 - 2 \log {\left (x \right )}\right ) e^{4 x} + e^{8 x} + \log {\left (x \right )}^{2} - 6 \log {\left (x \right )} + 9} \]

[In]

integrate((((8*x**4+32*x**3+24*x**2-16*x)*exp(4*x)-2*x**3-8*x**2-6*x+4)*ln((4*x**2+8*x-4)/(2+x))**2+((2*x**3+8
*x**2+10*x)*ln(x)+(-2*x**3-8*x**2-10*x)*exp(4*x)-6*x**3-24*x**2-30*x)*ln((4*x**2+8*x-4)/(2+x)))/((x**4+4*x**3+
3*x**2-2*x)*ln(x)**3+((-3*x**4-12*x**3-9*x**2+6*x)*exp(4*x)-9*x**4-36*x**3-27*x**2+18*x)*ln(x)**2+((3*x**4+12*
x**3+9*x**2-6*x)*exp(4*x)**2+(18*x**4+72*x**3+54*x**2-36*x)*exp(4*x)+27*x**4+108*x**3+81*x**2-54*x)*ln(x)+(-x*
*4-4*x**3-3*x**2+2*x)*exp(4*x)**3+(-9*x**4-36*x**3-27*x**2+18*x)*exp(4*x)**2+(-27*x**4-108*x**3-81*x**2+54*x)*
exp(4*x)-27*x**4-108*x**3-81*x**2+54*x),x)

[Out]

log((4*x**2 + 8*x - 4)/(x + 2))**2/((6 - 2*log(x))*exp(4*x) + exp(8*x) + log(x)**2 - 6*log(x) + 9)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (27) = 54\).

Time = 0.97 (sec) , antiderivative size = 86, normalized size of antiderivative = 3.07 \[ \int \frac {\left (-30 x-24 x^2-6 x^3+e^{4 x} \left (-10 x-8 x^2-2 x^3\right )+\left (10 x+8 x^2+2 x^3\right ) \log (x)\right ) \log \left (\frac {-4+8 x+4 x^2}{2+x}\right )+\left (4-6 x-8 x^2-2 x^3+e^{4 x} \left (-16 x+24 x^2+32 x^3+8 x^4\right )\right ) \log ^2\left (\frac {-4+8 x+4 x^2}{2+x}\right )}{54 x-81 x^2-108 x^3-27 x^4+e^{4 x} \left (54 x-81 x^2-108 x^3-27 x^4\right )+e^{8 x} \left (18 x-27 x^2-36 x^3-9 x^4\right )+e^{12 x} \left (2 x-3 x^2-4 x^3-x^4\right )+\left (-54 x+81 x^2+108 x^3+27 x^4+e^{8 x} \left (-6 x+9 x^2+12 x^3+3 x^4\right )+e^{4 x} \left (-36 x+54 x^2+72 x^3+18 x^4\right )\right ) \log (x)+\left (18 x-27 x^2-36 x^3-9 x^4+e^{4 x} \left (6 x-9 x^2-12 x^3-3 x^4\right )\right ) \log ^2(x)+\left (-2 x+3 x^2+4 x^3+x^4\right ) \log ^3(x)} \, dx=-\frac {4 \, \log \left (2\right )^{2} + 2 \, {\left (2 \, \log \left (2\right ) - \log \left (x + 2\right )\right )} \log \left (x^{2} + 2 \, x - 1\right ) + \log \left (x^{2} + 2 \, x - 1\right )^{2} - 4 \, \log \left (2\right ) \log \left (x + 2\right ) + \log \left (x + 2\right )^{2}}{2 \, {\left (\log \left (x\right ) - 3\right )} e^{\left (4 \, x\right )} - \log \left (x\right )^{2} - e^{\left (8 \, x\right )} + 6 \, \log \left (x\right ) - 9} \]

[In]

integrate((((8*x^4+32*x^3+24*x^2-16*x)*exp(4*x)-2*x^3-8*x^2-6*x+4)*log((4*x^2+8*x-4)/(2+x))^2+((2*x^3+8*x^2+10
*x)*log(x)+(-2*x^3-8*x^2-10*x)*exp(4*x)-6*x^3-24*x^2-30*x)*log((4*x^2+8*x-4)/(2+x)))/((x^4+4*x^3+3*x^2-2*x)*lo
g(x)^3+((-3*x^4-12*x^3-9*x^2+6*x)*exp(4*x)-9*x^4-36*x^3-27*x^2+18*x)*log(x)^2+((3*x^4+12*x^3+9*x^2-6*x)*exp(4*
x)^2+(18*x^4+72*x^3+54*x^2-36*x)*exp(4*x)+27*x^4+108*x^3+81*x^2-54*x)*log(x)+(-x^4-4*x^3-3*x^2+2*x)*exp(4*x)^3
+(-9*x^4-36*x^3-27*x^2+18*x)*exp(4*x)^2+(-27*x^4-108*x^3-81*x^2+54*x)*exp(4*x)-27*x^4-108*x^3-81*x^2+54*x),x,
algorithm="maxima")

[Out]

-(4*log(2)^2 + 2*(2*log(2) - log(x + 2))*log(x^2 + 2*x - 1) + log(x^2 + 2*x - 1)^2 - 4*log(2)*log(x + 2) + log
(x + 2)^2)/(2*(log(x) - 3)*e^(4*x) - log(x)^2 - e^(8*x) + 6*log(x) - 9)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (27) = 54\).

Time = 1.04 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.61 \[ \int \frac {\left (-30 x-24 x^2-6 x^3+e^{4 x} \left (-10 x-8 x^2-2 x^3\right )+\left (10 x+8 x^2+2 x^3\right ) \log (x)\right ) \log \left (\frac {-4+8 x+4 x^2}{2+x}\right )+\left (4-6 x-8 x^2-2 x^3+e^{4 x} \left (-16 x+24 x^2+32 x^3+8 x^4\right )\right ) \log ^2\left (\frac {-4+8 x+4 x^2}{2+x}\right )}{54 x-81 x^2-108 x^3-27 x^4+e^{4 x} \left (54 x-81 x^2-108 x^3-27 x^4\right )+e^{8 x} \left (18 x-27 x^2-36 x^3-9 x^4\right )+e^{12 x} \left (2 x-3 x^2-4 x^3-x^4\right )+\left (-54 x+81 x^2+108 x^3+27 x^4+e^{8 x} \left (-6 x+9 x^2+12 x^3+3 x^4\right )+e^{4 x} \left (-36 x+54 x^2+72 x^3+18 x^4\right )\right ) \log (x)+\left (18 x-27 x^2-36 x^3-9 x^4+e^{4 x} \left (6 x-9 x^2-12 x^3-3 x^4\right )\right ) \log ^2(x)+\left (-2 x+3 x^2+4 x^3+x^4\right ) \log ^3(x)} \, dx=-\frac {\log \left (4 \, x^{2} + 8 \, x - 4\right )^{2} - 2 \, \log \left (4 \, x^{2} + 8 \, x - 4\right ) \log \left (x + 2\right ) + \log \left (x + 2\right )^{2}}{2 \, e^{\left (4 \, x\right )} \log \left (x\right ) - \log \left (x\right )^{2} - e^{\left (8 \, x\right )} - 6 \, e^{\left (4 \, x\right )} + 6 \, \log \left (x\right ) - 9} \]

[In]

integrate((((8*x^4+32*x^3+24*x^2-16*x)*exp(4*x)-2*x^3-8*x^2-6*x+4)*log((4*x^2+8*x-4)/(2+x))^2+((2*x^3+8*x^2+10
*x)*log(x)+(-2*x^3-8*x^2-10*x)*exp(4*x)-6*x^3-24*x^2-30*x)*log((4*x^2+8*x-4)/(2+x)))/((x^4+4*x^3+3*x^2-2*x)*lo
g(x)^3+((-3*x^4-12*x^3-9*x^2+6*x)*exp(4*x)-9*x^4-36*x^3-27*x^2+18*x)*log(x)^2+((3*x^4+12*x^3+9*x^2-6*x)*exp(4*
x)^2+(18*x^4+72*x^3+54*x^2-36*x)*exp(4*x)+27*x^4+108*x^3+81*x^2-54*x)*log(x)+(-x^4-4*x^3-3*x^2+2*x)*exp(4*x)^3
+(-9*x^4-36*x^3-27*x^2+18*x)*exp(4*x)^2+(-27*x^4-108*x^3-81*x^2+54*x)*exp(4*x)-27*x^4-108*x^3-81*x^2+54*x),x,
algorithm="giac")

[Out]

-(log(4*x^2 + 8*x - 4)^2 - 2*log(4*x^2 + 8*x - 4)*log(x + 2) + log(x + 2)^2)/(2*e^(4*x)*log(x) - log(x)^2 - e^
(8*x) - 6*e^(4*x) + 6*log(x) - 9)

Mupad [B] (verification not implemented)

Time = 14.55 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.79 \[ \int \frac {\left (-30 x-24 x^2-6 x^3+e^{4 x} \left (-10 x-8 x^2-2 x^3\right )+\left (10 x+8 x^2+2 x^3\right ) \log (x)\right ) \log \left (\frac {-4+8 x+4 x^2}{2+x}\right )+\left (4-6 x-8 x^2-2 x^3+e^{4 x} \left (-16 x+24 x^2+32 x^3+8 x^4\right )\right ) \log ^2\left (\frac {-4+8 x+4 x^2}{2+x}\right )}{54 x-81 x^2-108 x^3-27 x^4+e^{4 x} \left (54 x-81 x^2-108 x^3-27 x^4\right )+e^{8 x} \left (18 x-27 x^2-36 x^3-9 x^4\right )+e^{12 x} \left (2 x-3 x^2-4 x^3-x^4\right )+\left (-54 x+81 x^2+108 x^3+27 x^4+e^{8 x} \left (-6 x+9 x^2+12 x^3+3 x^4\right )+e^{4 x} \left (-36 x+54 x^2+72 x^3+18 x^4\right )\right ) \log (x)+\left (18 x-27 x^2-36 x^3-9 x^4+e^{4 x} \left (6 x-9 x^2-12 x^3-3 x^4\right )\right ) \log ^2(x)+\left (-2 x+3 x^2+4 x^3+x^4\right ) \log ^3(x)} \, dx=\frac {{\ln \left (\frac {4\,x^2+8\,x-4}{x+2}\right )}^2}{{\ln \left (x\right )}^2+\left (-2\,{\mathrm {e}}^{4\,x}-6\right )\,\ln \left (x\right )+6\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{8\,x}+9} \]

[In]

int((log((8*x + 4*x^2 - 4)/(x + 2))^2*(6*x - exp(4*x)*(24*x^2 - 16*x + 32*x^3 + 8*x^4) + 8*x^2 + 2*x^3 - 4) +
log((8*x + 4*x^2 - 4)/(x + 2))*(30*x + exp(4*x)*(10*x + 8*x^2 + 2*x^3) + 24*x^2 + 6*x^3 - log(x)*(10*x + 8*x^2
+ 2*x^3)))/(exp(12*x)*(3*x^2 - 2*x + 4*x^3 + x^4) - log(x)*(exp(8*x)*(9*x^2 - 6*x + 12*x^3 + 3*x^4) - 54*x +
exp(4*x)*(54*x^2 - 36*x + 72*x^3 + 18*x^4) + 81*x^2 + 108*x^3 + 27*x^4) - 54*x - log(x)^3*(3*x^2 - 2*x + 4*x^3
+ x^4) + exp(8*x)*(27*x^2 - 18*x + 36*x^3 + 9*x^4) + exp(4*x)*(81*x^2 - 54*x + 108*x^3 + 27*x^4) + 81*x^2 + 1
08*x^3 + 27*x^4 + log(x)^2*(exp(4*x)*(9*x^2 - 6*x + 12*x^3 + 3*x^4) - 18*x + 27*x^2 + 36*x^3 + 9*x^4)),x)

[Out]

log((8*x + 4*x^2 - 4)/(x + 2))^2/(6*exp(4*x) + exp(8*x) - log(x)*(2*exp(4*x) + 6) + log(x)^2 + 9)