\(\int \frac {48 x^3+3 x^5+(-144 x^2-9 x^4) \log (x)+(144 x+9 x^3) \log ^2(x)+(-48-3 x^2) \log ^3(x)+e^{\frac {2 (3 x^2+3 x^4-6 x^3 \log (x)+3 x^2 \log ^2(x))}{x^2-2 x \log (x)+\log ^2(x)}} (12 x^2+x^3+12 x^5+(-15 x^2-36 x^4) \log (x)+(3 x+36 x^3) \log ^2(x)+(-1-12 x^2) \log ^3(x))+e^{\frac {3 x^2+3 x^4-6 x^3 \log (x)+3 x^2 \log ^2(x)}{x^2-2 x \log (x)+\log ^2(x)}} (12 x^3+4 x^4+12 x^6+(-24 x^3-36 x^5) \log (x)+(12 x^2+36 x^4) \log ^2(x)+(-4 x-12 x^3) \log ^3(x))}{-16 x^3+48 x^2 \log (x)-48 x \log ^2(x)+16 \log ^3(x)} \, dx\) [7778]

Optimal result

Integrand size = 287, antiderivative size = 32 \[ \int \frac {48 x^3+3 x^5+\left (-144 x^2-9 x^4\right ) \log (x)+\left (144 x+9 x^3\right ) \log ^2(x)+\left (-48-3 x^2\right ) \log ^3(x)+e^{\frac {2 \left (3 x^2+3 x^4-6 x^3 \log (x)+3 x^2 \log ^2(x)\right )}{x^2-2 x \log (x)+\log ^2(x)}} \left (12 x^2+x^3+12 x^5+\left (-15 x^2-36 x^4\right ) \log (x)+\left (3 x+36 x^3\right ) \log ^2(x)+\left (-1-12 x^2\right ) \log ^3(x)\right )+e^{\frac {3 x^2+3 x^4-6 x^3 \log (x)+3 x^2 \log ^2(x)}{x^2-2 x \log (x)+\log ^2(x)}} \left (12 x^3+4 x^4+12 x^6+\left (-24 x^3-36 x^5\right ) \log (x)+\left (12 x^2+36 x^4\right ) \log ^2(x)+\left (-4 x-12 x^3\right ) \log ^3(x)\right )}{-16 x^3+48 x^2 \log (x)-48 x \log ^2(x)+16 \log ^3(x)} \, dx=x-x \left (4+\frac {1}{16} \left (e^{3 x \left (x+\frac {x}{(x-\log (x))^2}\right )}+x\right )^2\right ) \]

[Out]

x-(1/16*(exp(3*x*(x+x/(x-ln(x))^2))+x)^2+4)*x

Rubi [F]

\[ \int \frac {48 x^3+3 x^5+\left (-144 x^2-9 x^4\right ) \log (x)+\left (144 x+9 x^3\right ) \log ^2(x)+\left (-48-3 x^2\right ) \log ^3(x)+e^{\frac {2 \left (3 x^2+3 x^4-6 x^3 \log (x)+3 x^2 \log ^2(x)\right )}{x^2-2 x \log (x)+\log ^2(x)}} \left (12 x^2+x^3+12 x^5+\left (-15 x^2-36 x^4\right ) \log (x)+\left (3 x+36 x^3\right ) \log ^2(x)+\left (-1-12 x^2\right ) \log ^3(x)\right )+e^{\frac {3 x^2+3 x^4-6 x^3 \log (x)+3 x^2 \log ^2(x)}{x^2-2 x \log (x)+\log ^2(x)}} \left (12 x^3+4 x^4+12 x^6+\left (-24 x^3-36 x^5\right ) \log (x)+\left (12 x^2+36 x^4\right ) \log ^2(x)+\left (-4 x-12 x^3\right ) \log ^3(x)\right )}{-16 x^3+48 x^2 \log (x)-48 x \log ^2(x)+16 \log ^3(x)} \, dx=\int \frac {48 x^3+3 x^5+\left (-144 x^2-9 x^4\right ) \log (x)+\left (144 x+9 x^3\right ) \log ^2(x)+\left (-48-3 x^2\right ) \log ^3(x)+\exp \left (\frac {2 \left (3 x^2+3 x^4-6 x^3 \log (x)+3 x^2 \log ^2(x)\right )}{x^2-2 x \log (x)+\log ^2(x)}\right ) \left (12 x^2+x^3+12 x^5+\left (-15 x^2-36 x^4\right ) \log (x)+\left (3 x+36 x^3\right ) \log ^2(x)+\left (-1-12 x^2\right ) \log ^3(x)\right )+\exp \left (\frac {3 x^2+3 x^4-6 x^3 \log (x)+3 x^2 \log ^2(x)}{x^2-2 x \log (x)+\log ^2(x)}\right ) \left (12 x^3+4 x^4+12 x^6+\left (-24 x^3-36 x^5\right ) \log (x)+\left (12 x^2+36 x^4\right ) \log ^2(x)+\left (-4 x-12 x^3\right ) \log ^3(x)\right )}{-16 x^3+48 x^2 \log (x)-48 x \log ^2(x)+16 \log ^3(x)} \, dx \]

[In]

Int[(48*x^3 + 3*x^5 + (-144*x^2 - 9*x^4)*Log[x] + (144*x + 9*x^3)*Log[x]^2 + (-48 - 3*x^2)*Log[x]^3 + E^((2*(3
*x^2 + 3*x^4 - 6*x^3*Log[x] + 3*x^2*Log[x]^2))/(x^2 - 2*x*Log[x] + Log[x]^2))*(12*x^2 + x^3 + 12*x^5 + (-15*x^
2 - 36*x^4)*Log[x] + (3*x + 36*x^3)*Log[x]^2 + (-1 - 12*x^2)*Log[x]^3) + E^((3*x^2 + 3*x^4 - 6*x^3*Log[x] + 3*
x^2*Log[x]^2)/(x^2 - 2*x*Log[x] + Log[x]^2))*(12*x^3 + 4*x^4 + 12*x^6 + (-24*x^3 - 36*x^5)*Log[x] + (12*x^2 +
36*x^4)*Log[x]^2 + (-4*x - 12*x^3)*Log[x]^3))/(-16*x^3 + 48*x^2*Log[x] - 48*x*Log[x]^2 + 16*Log[x]^3),x]

[Out]

-3*x - x^3/16 - (3*Defer[Int][E^((6*x^2*(1 + x^2 + Log[x]^2))/(x - Log[x])^2)*x^(2 - (12*x^3)/(x - Log[x])^2),
x])/4 - Defer[Int][E^((3*x^2*(1 + x^2 + Log[x]^2))/(x - Log[x])^2)*x^(1 - (6*x^3)/(x - Log[x])^2), x]/4 - (3*
Defer[Int][E^((3*x^2*(1 + x^2 + Log[x]^2))/(x - Log[x])^2)*x^(3 - (6*x^3)/(x - Log[x])^2), x])/4 - Defer[Int][
E^((6*x^2*(1 + x^2 + Log[x]^2))/(x - Log[x])^2)/x^((12*x^3)/(x - Log[x])^2), x]/16 - (3*Defer[Int][(E^((6*x^2*
(1 + x^2 + Log[x]^2))/(x - Log[x])^2)*x^(2 - (12*x^3)/(x - Log[x])^2))/(x - Log[x])^3, x])/4 + (3*Defer[Int][(
E^((6*x^2*(1 + x^2 + Log[x]^2))/(x - Log[x])^2)*x^(3 - (12*x^3)/(x - Log[x])^2))/(x - Log[x])^3, x])/4 - (3*De
fer[Int][(E^((3*x^2*(1 + x^2 + Log[x]^2))/(x - Log[x])^2)*x^(3 - (6*x^3)/(x - Log[x])^2))/(x - Log[x])^3, x])/
4 + (3*Defer[Int][(E^((3*x^2*(1 + x^2 + Log[x]^2))/(x - Log[x])^2)*x^(4 - (6*x^3)/(x - Log[x])^2))/(x - Log[x]
)^3, x])/4 - (3*Defer[Int][(E^((6*x^2*(1 + x^2 + Log[x]^2))/(x - Log[x])^2)*x^(2 - (12*x^3)/(x - Log[x])^2))/(
x - Log[x])^2, x])/4 - (3*Defer[Int][(E^((3*x^2*(1 + x^2 + Log[x]^2))/(x - Log[x])^2)*x^(3 - (6*x^3)/(x - Log[
x])^2))/(x - Log[x])^2, x])/4

Rubi steps \begin{align*} \text {integral}& = \int \frac {-48 x^3-3 x^5-\left (-144 x^2-9 x^4\right ) \log (x)-\left (144 x+9 x^3\right ) \log ^2(x)-\left (-48-3 x^2\right ) \log ^3(x)-\exp \left (\frac {2 \left (3 x^2+3 x^4-6 x^3 \log (x)+3 x^2 \log ^2(x)\right )}{x^2-2 x \log (x)+\log ^2(x)}\right ) \left (12 x^2+x^3+12 x^5+\left (-15 x^2-36 x^4\right ) \log (x)+\left (3 x+36 x^3\right ) \log ^2(x)+\left (-1-12 x^2\right ) \log ^3(x)\right )-\exp \left (\frac {3 x^2+3 x^4-6 x^3 \log (x)+3 x^2 \log ^2(x)}{x^2-2 x \log (x)+\log ^2(x)}\right ) \left (12 x^3+4 x^4+12 x^6+\left (-24 x^3-36 x^5\right ) \log (x)+\left (12 x^2+36 x^4\right ) \log ^2(x)+\left (-4 x-12 x^3\right ) \log ^3(x)\right )}{16 (x-\log (x))^3} \, dx \\ & = \frac {1}{16} \int \frac {-48 x^3-3 x^5-\left (-144 x^2-9 x^4\right ) \log (x)-\left (144 x+9 x^3\right ) \log ^2(x)-\left (-48-3 x^2\right ) \log ^3(x)-\exp \left (\frac {2 \left (3 x^2+3 x^4-6 x^3 \log (x)+3 x^2 \log ^2(x)\right )}{x^2-2 x \log (x)+\log ^2(x)}\right ) \left (12 x^2+x^3+12 x^5+\left (-15 x^2-36 x^4\right ) \log (x)+\left (3 x+36 x^3\right ) \log ^2(x)+\left (-1-12 x^2\right ) \log ^3(x)\right )-\exp \left (\frac {3 x^2+3 x^4-6 x^3 \log (x)+3 x^2 \log ^2(x)}{x^2-2 x \log (x)+\log ^2(x)}\right ) \left (12 x^3+4 x^4+12 x^6+\left (-24 x^3-36 x^5\right ) \log (x)+\left (12 x^2+36 x^4\right ) \log ^2(x)+\left (-4 x-12 x^3\right ) \log ^3(x)\right )}{(x-\log (x))^3} \, dx \\ & = \frac {1}{16} \int \left (-\frac {48 x^3}{(x-\log (x))^3}-\frac {3 x^5}{(x-\log (x))^3}+\frac {9 x^2 \left (16+x^2\right ) \log (x)}{(x-\log (x))^3}-\frac {9 x \left (16+x^2\right ) \log ^2(x)}{(x-\log (x))^3}+\frac {3 \left (16+x^2\right ) \log ^3(x)}{(x-\log (x))^3}-\frac {e^{\frac {6 x^2 \left (1+x^2+\log ^2(x)\right )}{(x-\log (x))^2}} x^{-\frac {12 x^3}{(x-\log (x))^2}} \left (12 x^2+x^3+12 x^5-15 x^2 \log (x)-36 x^4 \log (x)+3 x \log ^2(x)+36 x^3 \log ^2(x)-\log ^3(x)-12 x^2 \log ^3(x)\right )}{(x-\log (x))^3}-\frac {4 e^{\frac {3 x^2 \left (1+x^2+\log ^2(x)\right )}{(x-\log (x))^2}} x^{1-\frac {6 x^3}{(x-\log (x))^2}} \left (3 x^2+x^3+3 x^5-6 x^2 \log (x)-9 x^4 \log (x)+3 x \log ^2(x)+9 x^3 \log ^2(x)-\log ^3(x)-3 x^2 \log ^3(x)\right )}{(x-\log (x))^3}\right ) \, dx \\ & = -\left (\frac {1}{16} \int \frac {e^{\frac {6 x^2 \left (1+x^2+\log ^2(x)\right )}{(x-\log (x))^2}} x^{-\frac {12 x^3}{(x-\log (x))^2}} \left (12 x^2+x^3+12 x^5-15 x^2 \log (x)-36 x^4 \log (x)+3 x \log ^2(x)+36 x^3 \log ^2(x)-\log ^3(x)-12 x^2 \log ^3(x)\right )}{(x-\log (x))^3} \, dx\right )-\frac {3}{16} \int \frac {x^5}{(x-\log (x))^3} \, dx+\frac {3}{16} \int \frac {\left (16+x^2\right ) \log ^3(x)}{(x-\log (x))^3} \, dx-\frac {1}{4} \int \frac {e^{\frac {3 x^2 \left (1+x^2+\log ^2(x)\right )}{(x-\log (x))^2}} x^{1-\frac {6 x^3}{(x-\log (x))^2}} \left (3 x^2+x^3+3 x^5-6 x^2 \log (x)-9 x^4 \log (x)+3 x \log ^2(x)+9 x^3 \log ^2(x)-\log ^3(x)-3 x^2 \log ^3(x)\right )}{(x-\log (x))^3} \, dx+\frac {9}{16} \int \frac {x^2 \left (16+x^2\right ) \log (x)}{(x-\log (x))^3} \, dx-\frac {9}{16} \int \frac {x \left (16+x^2\right ) \log ^2(x)}{(x-\log (x))^3} \, dx-3 \int \frac {x^3}{(x-\log (x))^3} \, dx \\ & = -\left (\frac {1}{16} \int \left (12 e^{\frac {6 x^2 \left (1+x^2+\log ^2(x)\right )}{(x-\log (x))^2}} x^{2-\frac {12 x^3}{(x-\log (x))^2}}+e^{\frac {6 x^2 \left (1+x^2+\log ^2(x)\right )}{(x-\log (x))^2}} x^{-\frac {12 x^3}{(x-\log (x))^2}}-\frac {12 e^{\frac {6 x^2 \left (1+x^2+\log ^2(x)\right )}{(x-\log (x))^2}} (-1+x) x^{2-\frac {12 x^3}{(x-\log (x))^2}}}{(x-\log (x))^3}+\frac {12 e^{\frac {6 x^2 \left (1+x^2+\log ^2(x)\right )}{(x-\log (x))^2}} x^{2-\frac {12 x^3}{(x-\log (x))^2}}}{(x-\log (x))^2}\right ) \, dx\right )+\frac {3}{16} \int \left (-16-x^2+\frac {x^3 \left (16+x^2\right )}{(x-\log (x))^3}-\frac {3 x^2 \left (16+x^2\right )}{(x-\log (x))^2}+\frac {3 x \left (16+x^2\right )}{x-\log (x)}\right ) \, dx-\frac {3}{16} \int \frac {x^5}{(x-\log (x))^3} \, dx-\frac {1}{4} \int \left (e^{\frac {3 x^2 \left (1+x^2+\log ^2(x)\right )}{(x-\log (x))^2}} x^{1-\frac {6 x^3}{(x-\log (x))^2}}+3 e^{\frac {3 x^2 \left (1+x^2+\log ^2(x)\right )}{(x-\log (x))^2}} x^{3-\frac {6 x^3}{(x-\log (x))^2}}-\frac {3 e^{\frac {3 x^2 \left (1+x^2+\log ^2(x)\right )}{(x-\log (x))^2}} (-1+x) x^{3-\frac {6 x^3}{(x-\log (x))^2}}}{(x-\log (x))^3}+\frac {3 e^{\frac {3 x^2 \left (1+x^2+\log ^2(x)\right )}{(x-\log (x))^2}} x^{3-\frac {6 x^3}{(x-\log (x))^2}}}{(x-\log (x))^2}\right ) \, dx+\frac {9}{16} \int \left (\frac {x^3 \left (16+x^2\right )}{(x-\log (x))^3}-\frac {x^2 \left (16+x^2\right )}{(x-\log (x))^2}\right ) \, dx-\frac {9}{16} \int \left (\frac {x^3 \left (16+x^2\right )}{(x-\log (x))^3}-\frac {2 x^2 \left (16+x^2\right )}{(x-\log (x))^2}+\frac {x \left (16+x^2\right )}{x-\log (x)}\right ) \, dx-3 \int \frac {x^3}{(x-\log (x))^3} \, dx \\ & = -3 x-\frac {x^3}{16}-\frac {1}{16} \int e^{\frac {6 x^2 \left (1+x^2+\log ^2(x)\right )}{(x-\log (x))^2}} x^{-\frac {12 x^3}{(x-\log (x))^2}} \, dx-\frac {3}{16} \int \frac {x^5}{(x-\log (x))^3} \, dx+\frac {3}{16} \int \frac {x^3 \left (16+x^2\right )}{(x-\log (x))^3} \, dx-\frac {1}{4} \int e^{\frac {3 x^2 \left (1+x^2+\log ^2(x)\right )}{(x-\log (x))^2}} x^{1-\frac {6 x^3}{(x-\log (x))^2}} \, dx-2 \left (\frac {9}{16} \int \frac {x^2 \left (16+x^2\right )}{(x-\log (x))^2} \, dx\right )-\frac {3}{4} \int e^{\frac {6 x^2 \left (1+x^2+\log ^2(x)\right )}{(x-\log (x))^2}} x^{2-\frac {12 x^3}{(x-\log (x))^2}} \, dx-\frac {3}{4} \int e^{\frac {3 x^2 \left (1+x^2+\log ^2(x)\right )}{(x-\log (x))^2}} x^{3-\frac {6 x^3}{(x-\log (x))^2}} \, dx+\frac {3}{4} \int \frac {e^{\frac {6 x^2 \left (1+x^2+\log ^2(x)\right )}{(x-\log (x))^2}} (-1+x) x^{2-\frac {12 x^3}{(x-\log (x))^2}}}{(x-\log (x))^3} \, dx+\frac {3}{4} \int \frac {e^{\frac {3 x^2 \left (1+x^2+\log ^2(x)\right )}{(x-\log (x))^2}} (-1+x) x^{3-\frac {6 x^3}{(x-\log (x))^2}}}{(x-\log (x))^3} \, dx-\frac {3}{4} \int \frac {e^{\frac {6 x^2 \left (1+x^2+\log ^2(x)\right )}{(x-\log (x))^2}} x^{2-\frac {12 x^3}{(x-\log (x))^2}}}{(x-\log (x))^2} \, dx-\frac {3}{4} \int \frac {e^{\frac {3 x^2 \left (1+x^2+\log ^2(x)\right )}{(x-\log (x))^2}} x^{3-\frac {6 x^3}{(x-\log (x))^2}}}{(x-\log (x))^2} \, dx+\frac {9}{8} \int \frac {x^2 \left (16+x^2\right )}{(x-\log (x))^2} \, dx-3 \int \frac {x^3}{(x-\log (x))^3} \, dx \\ & = -3 x-\frac {x^3}{16}-\frac {1}{16} \int e^{\frac {6 x^2 \left (1+x^2+\log ^2(x)\right )}{(x-\log (x))^2}} x^{-\frac {12 x^3}{(x-\log (x))^2}} \, dx+\frac {3}{16} \int \left (\frac {16 x^3}{(x-\log (x))^3}+\frac {x^5}{(x-\log (x))^3}\right ) \, dx-\frac {3}{16} \int \frac {x^5}{(x-\log (x))^3} \, dx-\frac {1}{4} \int e^{\frac {3 x^2 \left (1+x^2+\log ^2(x)\right )}{(x-\log (x))^2}} x^{1-\frac {6 x^3}{(x-\log (x))^2}} \, dx-2 \left (\frac {9}{16} \int \left (\frac {16 x^2}{(x-\log (x))^2}+\frac {x^4}{(x-\log (x))^2}\right ) \, dx\right )-\frac {3}{4} \int e^{\frac {6 x^2 \left (1+x^2+\log ^2(x)\right )}{(x-\log (x))^2}} x^{2-\frac {12 x^3}{(x-\log (x))^2}} \, dx-\frac {3}{4} \int e^{\frac {3 x^2 \left (1+x^2+\log ^2(x)\right )}{(x-\log (x))^2}} x^{3-\frac {6 x^3}{(x-\log (x))^2}} \, dx+\frac {3}{4} \int \left (-\frac {e^{\frac {6 x^2 \left (1+x^2+\log ^2(x)\right )}{(x-\log (x))^2}} x^{2-\frac {12 x^3}{(x-\log (x))^2}}}{(x-\log (x))^3}+\frac {e^{\frac {6 x^2 \left (1+x^2+\log ^2(x)\right )}{(x-\log (x))^2}} x^{3-\frac {12 x^3}{(x-\log (x))^2}}}{(x-\log (x))^3}\right ) \, dx+\frac {3}{4} \int \left (-\frac {e^{\frac {3 x^2 \left (1+x^2+\log ^2(x)\right )}{(x-\log (x))^2}} x^{3-\frac {6 x^3}{(x-\log (x))^2}}}{(x-\log (x))^3}+\frac {e^{\frac {3 x^2 \left (1+x^2+\log ^2(x)\right )}{(x-\log (x))^2}} x^{4-\frac {6 x^3}{(x-\log (x))^2}}}{(x-\log (x))^3}\right ) \, dx-\frac {3}{4} \int \frac {e^{\frac {6 x^2 \left (1+x^2+\log ^2(x)\right )}{(x-\log (x))^2}} x^{2-\frac {12 x^3}{(x-\log (x))^2}}}{(x-\log (x))^2} \, dx-\frac {3}{4} \int \frac {e^{\frac {3 x^2 \left (1+x^2+\log ^2(x)\right )}{(x-\log (x))^2}} x^{3-\frac {6 x^3}{(x-\log (x))^2}}}{(x-\log (x))^2} \, dx+\frac {9}{8} \int \left (\frac {16 x^2}{(x-\log (x))^2}+\frac {x^4}{(x-\log (x))^2}\right ) \, dx-3 \int \frac {x^3}{(x-\log (x))^3} \, dx \\ & = -3 x-\frac {x^3}{16}-\frac {1}{16} \int e^{\frac {6 x^2 \left (1+x^2+\log ^2(x)\right )}{(x-\log (x))^2}} x^{-\frac {12 x^3}{(x-\log (x))^2}} \, dx-\frac {1}{4} \int e^{\frac {3 x^2 \left (1+x^2+\log ^2(x)\right )}{(x-\log (x))^2}} x^{1-\frac {6 x^3}{(x-\log (x))^2}} \, dx-\frac {3}{4} \int e^{\frac {6 x^2 \left (1+x^2+\log ^2(x)\right )}{(x-\log (x))^2}} x^{2-\frac {12 x^3}{(x-\log (x))^2}} \, dx-\frac {3}{4} \int e^{\frac {3 x^2 \left (1+x^2+\log ^2(x)\right )}{(x-\log (x))^2}} x^{3-\frac {6 x^3}{(x-\log (x))^2}} \, dx-\frac {3}{4} \int \frac {e^{\frac {6 x^2 \left (1+x^2+\log ^2(x)\right )}{(x-\log (x))^2}} x^{2-\frac {12 x^3}{(x-\log (x))^2}}}{(x-\log (x))^3} \, dx+\frac {3}{4} \int \frac {e^{\frac {6 x^2 \left (1+x^2+\log ^2(x)\right )}{(x-\log (x))^2}} x^{3-\frac {12 x^3}{(x-\log (x))^2}}}{(x-\log (x))^3} \, dx-\frac {3}{4} \int \frac {e^{\frac {3 x^2 \left (1+x^2+\log ^2(x)\right )}{(x-\log (x))^2}} x^{3-\frac {6 x^3}{(x-\log (x))^2}}}{(x-\log (x))^3} \, dx+\frac {3}{4} \int \frac {e^{\frac {3 x^2 \left (1+x^2+\log ^2(x)\right )}{(x-\log (x))^2}} x^{4-\frac {6 x^3}{(x-\log (x))^2}}}{(x-\log (x))^3} \, dx-\frac {3}{4} \int \frac {e^{\frac {6 x^2 \left (1+x^2+\log ^2(x)\right )}{(x-\log (x))^2}} x^{2-\frac {12 x^3}{(x-\log (x))^2}}}{(x-\log (x))^2} \, dx-\frac {3}{4} \int \frac {e^{\frac {3 x^2 \left (1+x^2+\log ^2(x)\right )}{(x-\log (x))^2}} x^{3-\frac {6 x^3}{(x-\log (x))^2}}}{(x-\log (x))^2} \, dx+\frac {9}{8} \int \frac {x^4}{(x-\log (x))^2} \, dx-2 \left (\frac {9}{16} \int \frac {x^4}{(x-\log (x))^2} \, dx+9 \int \frac {x^2}{(x-\log (x))^2} \, dx\right )+18 \int \frac {x^2}{(x-\log (x))^2} \, dx \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(70\) vs. \(2(32)=64\).

Time = 0.48 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.19 \[ \int \frac {48 x^3+3 x^5+\left (-144 x^2-9 x^4\right ) \log (x)+\left (144 x+9 x^3\right ) \log ^2(x)+\left (-48-3 x^2\right ) \log ^3(x)+e^{\frac {2 \left (3 x^2+3 x^4-6 x^3 \log (x)+3 x^2 \log ^2(x)\right )}{x^2-2 x \log (x)+\log ^2(x)}} \left (12 x^2+x^3+12 x^5+\left (-15 x^2-36 x^4\right ) \log (x)+\left (3 x+36 x^3\right ) \log ^2(x)+\left (-1-12 x^2\right ) \log ^3(x)\right )+e^{\frac {3 x^2+3 x^4-6 x^3 \log (x)+3 x^2 \log ^2(x)}{x^2-2 x \log (x)+\log ^2(x)}} \left (12 x^3+4 x^4+12 x^6+\left (-24 x^3-36 x^5\right ) \log (x)+\left (12 x^2+36 x^4\right ) \log ^2(x)+\left (-4 x-12 x^3\right ) \log ^3(x)\right )}{-16 x^3+48 x^2 \log (x)-48 x \log ^2(x)+16 \log ^3(x)} \, dx=-\frac {1}{16} x \left (48+2 e^{3 x^2 \left (1+\frac {1}{(x-\log (x))^2}\right )} x+x^2+e^{\frac {6 x^2 \left (1+x^2+\log ^2(x)\right )}{(x-\log (x))^2}} x^{-\frac {12 x^3}{(x-\log (x))^2}}\right ) \]

[In]

Integrate[(48*x^3 + 3*x^5 + (-144*x^2 - 9*x^4)*Log[x] + (144*x + 9*x^3)*Log[x]^2 + (-48 - 3*x^2)*Log[x]^3 + E^
((2*(3*x^2 + 3*x^4 - 6*x^3*Log[x] + 3*x^2*Log[x]^2))/(x^2 - 2*x*Log[x] + Log[x]^2))*(12*x^2 + x^3 + 12*x^5 + (
-15*x^2 - 36*x^4)*Log[x] + (3*x + 36*x^3)*Log[x]^2 + (-1 - 12*x^2)*Log[x]^3) + E^((3*x^2 + 3*x^4 - 6*x^3*Log[x
] + 3*x^2*Log[x]^2)/(x^2 - 2*x*Log[x] + Log[x]^2))*(12*x^3 + 4*x^4 + 12*x^6 + (-24*x^3 - 36*x^5)*Log[x] + (12*
x^2 + 36*x^4)*Log[x]^2 + (-4*x - 12*x^3)*Log[x]^3))/(-16*x^3 + 48*x^2*Log[x] - 48*x*Log[x]^2 + 16*Log[x]^3),x]

[Out]

-1/16*(x*(48 + 2*E^(3*x^2*(1 + (x - Log[x])^(-2)))*x + x^2 + E^((6*x^2*(1 + x^2 + Log[x]^2))/(x - Log[x])^2)/x
^((12*x^3)/(x - Log[x])^2)))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(73\) vs. \(2(29)=58\).

Time = 3.16 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.31

method	result	size
risch	\(-\frac {x^{3}}{16}-3 x -\frac {x \,{\mathrm e}^{\frac {6 x^{2} \left (\ln \left (x \right )^{2}-2 x \ln \left (x \right )+x^{2}+1\right )}{\left (\ln \left (x \right )-x \right )^{2}}}}{16}-\frac {x^{2} {\mathrm e}^{\frac {3 x^{2} \left (\ln \left (x \right )^{2}-2 x \ln \left (x \right )+x^{2}+1\right )}{\left (\ln \left (x \right )-x \right )^{2}}}}{8}\)	\(74\)
parallelrisch	\(-\frac {x^{3}}{16}-3 x -\frac {x \,{\mathrm e}^{\frac {6 x^{2} \ln \left (x \right )^{2}-12 x^{3} \ln \left (x \right )+6 x^{4}+6 x^{2}}{\ln \left (x \right )^{2}-2 x \ln \left (x \right )+x^{2}}}}{16}-\frac {x^{2} {\mathrm e}^{\frac {3 x^{2} \ln \left (x \right )^{2}-6 x^{3} \ln \left (x \right )+3 x^{4}+3 x^{2}}{\ln \left (x \right )^{2}-2 x \ln \left (x \right )+x^{2}}}}{8}\)	\(108\)

[In]

int((((-12*x^2-1)*ln(x)^3+(36*x^3+3*x)*ln(x)^2+(-36*x^4-15*x^2)*ln(x)+12*x^5+x^3+12*x^2)*exp((3*x^2*ln(x)^2-6*
x^3*ln(x)+3*x^4+3*x^2)/(ln(x)^2-2*x*ln(x)+x^2))^2+((-12*x^3-4*x)*ln(x)^3+(36*x^4+12*x^2)*ln(x)^2+(-36*x^5-24*x
^3)*ln(x)+12*x^6+4*x^4+12*x^3)*exp((3*x^2*ln(x)^2-6*x^3*ln(x)+3*x^4+3*x^2)/(ln(x)^2-2*x*ln(x)+x^2))+(-3*x^2-48
)*ln(x)^3+(9*x^3+144*x)*ln(x)^2+(-9*x^4-144*x^2)*ln(x)+3*x^5+48*x^3)/(16*ln(x)^3-48*x*ln(x)^2+48*x^2*ln(x)-16*
x^3),x,method=_RETURNVERBOSE)

[Out]

-1/16*x^3-3*x-1/16*x*exp(6*x^2*(ln(x)^2-2*x*ln(x)+x^2+1)/(ln(x)-x)^2)-1/8*x^2*exp(3*x^2*(ln(x)^2-2*x*ln(x)+x^2
+1)/(ln(x)-x)^2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (27) = 54\).

Time = 0.26 (sec) , antiderivative size = 97, normalized size of antiderivative = 3.03 \[ \int \frac {48 x^3+3 x^5+\left (-144 x^2-9 x^4\right ) \log (x)+\left (144 x+9 x^3\right ) \log ^2(x)+\left (-48-3 x^2\right ) \log ^3(x)+e^{\frac {2 \left (3 x^2+3 x^4-6 x^3 \log (x)+3 x^2 \log ^2(x)\right )}{x^2-2 x \log (x)+\log ^2(x)}} \left (12 x^2+x^3+12 x^5+\left (-15 x^2-36 x^4\right ) \log (x)+\left (3 x+36 x^3\right ) \log ^2(x)+\left (-1-12 x^2\right ) \log ^3(x)\right )+e^{\frac {3 x^2+3 x^4-6 x^3 \log (x)+3 x^2 \log ^2(x)}{x^2-2 x \log (x)+\log ^2(x)}} \left (12 x^3+4 x^4+12 x^6+\left (-24 x^3-36 x^5\right ) \log (x)+\left (12 x^2+36 x^4\right ) \log ^2(x)+\left (-4 x-12 x^3\right ) \log ^3(x)\right )}{-16 x^3+48 x^2 \log (x)-48 x \log ^2(x)+16 \log ^3(x)} \, dx=-\frac {1}{16} \, x^{3} - \frac {1}{8} \, x^{2} e^{\left (\frac {3 \, {\left (x^{4} - 2 \, x^{3} \log \left (x\right ) + x^{2} \log \left (x\right )^{2} + x^{2}\right )}}{x^{2} - 2 \, x \log \left (x\right ) + \log \left (x\right )^{2}}\right )} - \frac {1}{16} \, x e^{\left (\frac {6 \, {\left (x^{4} - 2 \, x^{3} \log \left (x\right ) + x^{2} \log \left (x\right )^{2} + x^{2}\right )}}{x^{2} - 2 \, x \log \left (x\right ) + \log \left (x\right )^{2}}\right )} - 3 \, x \]

[In]

integrate((((-12*x^2-1)*log(x)^3+(36*x^3+3*x)*log(x)^2+(-36*x^4-15*x^2)*log(x)+12*x^5+x^3+12*x^2)*exp((3*x^2*l
og(x)^2-6*x^3*log(x)+3*x^4+3*x^2)/(log(x)^2-2*x*log(x)+x^2))^2+((-12*x^3-4*x)*log(x)^3+(36*x^4+12*x^2)*log(x)^
2+(-36*x^5-24*x^3)*log(x)+12*x^6+4*x^4+12*x^3)*exp((3*x^2*log(x)^2-6*x^3*log(x)+3*x^4+3*x^2)/(log(x)^2-2*x*log
(x)+x^2))+(-3*x^2-48)*log(x)^3+(9*x^3+144*x)*log(x)^2+(-9*x^4-144*x^2)*log(x)+3*x^5+48*x^3)/(16*log(x)^3-48*x*
log(x)^2+48*x^2*log(x)-16*x^3),x, algorithm="fricas")

[Out]

-1/16*x^3 - 1/8*x^2*e^(3*(x^4 - 2*x^3*log(x) + x^2*log(x)^2 + x^2)/(x^2 - 2*x*log(x) + log(x)^2)) - 1/16*x*e^(
6*(x^4 - 2*x^3*log(x) + x^2*log(x)^2 + x^2)/(x^2 - 2*x*log(x) + log(x)^2)) - 3*x

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (24) = 48\).

Time = 48.17 (sec) , antiderivative size = 109, normalized size of antiderivative = 3.41 \[ \int \frac {48 x^3+3 x^5+\left (-144 x^2-9 x^4\right ) \log (x)+\left (144 x+9 x^3\right ) \log ^2(x)+\left (-48-3 x^2\right ) \log ^3(x)+e^{\frac {2 \left (3 x^2+3 x^4-6 x^3 \log (x)+3 x^2 \log ^2(x)\right )}{x^2-2 x \log (x)+\log ^2(x)}} \left (12 x^2+x^3+12 x^5+\left (-15 x^2-36 x^4\right ) \log (x)+\left (3 x+36 x^3\right ) \log ^2(x)+\left (-1-12 x^2\right ) \log ^3(x)\right )+e^{\frac {3 x^2+3 x^4-6 x^3 \log (x)+3 x^2 \log ^2(x)}{x^2-2 x \log (x)+\log ^2(x)}} \left (12 x^3+4 x^4+12 x^6+\left (-24 x^3-36 x^5\right ) \log (x)+\left (12 x^2+36 x^4\right ) \log ^2(x)+\left (-4 x-12 x^3\right ) \log ^3(x)\right )}{-16 x^3+48 x^2 \log (x)-48 x \log ^2(x)+16 \log ^3(x)} \, dx=- \frac {x^{3}}{16} - \frac {x^{2} e^{\frac {3 x^{4} - 6 x^{3} \log {\left (x \right )} + 3 x^{2} \log {\left (x \right )}^{2} + 3 x^{2}}{x^{2} - 2 x \log {\left (x \right )} + \log {\left (x \right )}^{2}}}}{8} - \frac {x e^{\frac {2 \cdot \left (3 x^{4} - 6 x^{3} \log {\left (x \right )} + 3 x^{2} \log {\left (x \right )}^{2} + 3 x^{2}\right )}{x^{2} - 2 x \log {\left (x \right )} + \log {\left (x \right )}^{2}}}}{16} - 3 x \]

[In]

integrate((((-12*x**2-1)*ln(x)**3+(36*x**3+3*x)*ln(x)**2+(-36*x**4-15*x**2)*ln(x)+12*x**5+x**3+12*x**2)*exp((3
*x**2*ln(x)**2-6*x**3*ln(x)+3*x**4+3*x**2)/(ln(x)**2-2*x*ln(x)+x**2))**2+((-12*x**3-4*x)*ln(x)**3+(36*x**4+12*
x**2)*ln(x)**2+(-36*x**5-24*x**3)*ln(x)+12*x**6+4*x**4+12*x**3)*exp((3*x**2*ln(x)**2-6*x**3*ln(x)+3*x**4+3*x**
2)/(ln(x)**2-2*x*ln(x)+x**2))+(-3*x**2-48)*ln(x)**3+(9*x**3+144*x)*ln(x)**2+(-9*x**4-144*x**2)*ln(x)+3*x**5+48
*x**3)/(16*ln(x)**3-48*x*ln(x)**2+48*x**2*ln(x)-16*x**3),x)

[Out]

-x**3/16 - x**2*exp((3*x**4 - 6*x**3*log(x) + 3*x**2*log(x)**2 + 3*x**2)/(x**2 - 2*x*log(x) + log(x)**2))/8 -
x*exp(2*(3*x**4 - 6*x**3*log(x) + 3*x**2*log(x)**2 + 3*x**2)/(x**2 - 2*x*log(x) + log(x)**2))/16 - 3*x

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (27) = 54\).

Time = 0.30 (sec) , antiderivative size = 114, normalized size of antiderivative = 3.56 \[ \int \frac {48 x^3+3 x^5+\left (-144 x^2-9 x^4\right ) \log (x)+\left (144 x+9 x^3\right ) \log ^2(x)+\left (-48-3 x^2\right ) \log ^3(x)+e^{\frac {2 \left (3 x^2+3 x^4-6 x^3 \log (x)+3 x^2 \log ^2(x)\right )}{x^2-2 x \log (x)+\log ^2(x)}} \left (12 x^2+x^3+12 x^5+\left (-15 x^2-36 x^4\right ) \log (x)+\left (3 x+36 x^3\right ) \log ^2(x)+\left (-1-12 x^2\right ) \log ^3(x)\right )+e^{\frac {3 x^2+3 x^4-6 x^3 \log (x)+3 x^2 \log ^2(x)}{x^2-2 x \log (x)+\log ^2(x)}} \left (12 x^3+4 x^4+12 x^6+\left (-24 x^3-36 x^5\right ) \log (x)+\left (12 x^2+36 x^4\right ) \log ^2(x)+\left (-4 x-12 x^3\right ) \log ^3(x)\right )}{-16 x^3+48 x^2 \log (x)-48 x \log ^2(x)+16 \log ^3(x)} \, dx=-\frac {1}{16} \, {\left (2 \, x^{2} e^{\left (3 \, x^{2} + \frac {3 \, \log \left (x\right )^{2}}{x^{2} - 2 \, x \log \left (x\right ) + \log \left (x\right )^{2}} - \frac {6 \, \log \left (x\right )}{x - \log \left (x\right )} + 3\right )} + x e^{\left (6 \, x^{2} + \frac {6 \, \log \left (x\right )^{2}}{x^{2} - 2 \, x \log \left (x\right ) + \log \left (x\right )^{2}} + 6\right )} + \frac {x^{3} + 48 \, x}{x^{\frac {12}{x - \log \left (x\right )}}}\right )} x^{\frac {12}{x - \log \left (x\right )}} \]

[In]

integrate((((-12*x^2-1)*log(x)^3+(36*x^3+3*x)*log(x)^2+(-36*x^4-15*x^2)*log(x)+12*x^5+x^3+12*x^2)*exp((3*x^2*l
og(x)^2-6*x^3*log(x)+3*x^4+3*x^2)/(log(x)^2-2*x*log(x)+x^2))^2+((-12*x^3-4*x)*log(x)^3+(36*x^4+12*x^2)*log(x)^
2+(-36*x^5-24*x^3)*log(x)+12*x^6+4*x^4+12*x^3)*exp((3*x^2*log(x)^2-6*x^3*log(x)+3*x^4+3*x^2)/(log(x)^2-2*x*log
(x)+x^2))+(-3*x^2-48)*log(x)^3+(9*x^3+144*x)*log(x)^2+(-9*x^4-144*x^2)*log(x)+3*x^5+48*x^3)/(16*log(x)^3-48*x*
log(x)^2+48*x^2*log(x)-16*x^3),x, algorithm="maxima")

[Out]

-1/16*(2*x^2*e^(3*x^2 + 3*log(x)^2/(x^2 - 2*x*log(x) + log(x)^2) - 6*log(x)/(x - log(x)) + 3) + x*e^(6*x^2 + 6
*log(x)^2/(x^2 - 2*x*log(x) + log(x)^2) + 6) + (x^3 + 48*x)/x^(12/(x - log(x))))*x^(12/(x - log(x)))

Giac [F(-2)]

Exception generated. \[ \int \frac {48 x^3+3 x^5+\left (-144 x^2-9 x^4\right ) \log (x)+\left (144 x+9 x^3\right ) \log ^2(x)+\left (-48-3 x^2\right ) \log ^3(x)+e^{\frac {2 \left (3 x^2+3 x^4-6 x^3 \log (x)+3 x^2 \log ^2(x)\right )}{x^2-2 x \log (x)+\log ^2(x)}} \left (12 x^2+x^3+12 x^5+\left (-15 x^2-36 x^4\right ) \log (x)+\left (3 x+36 x^3\right ) \log ^2(x)+\left (-1-12 x^2\right ) \log ^3(x)\right )+e^{\frac {3 x^2+3 x^4-6 x^3 \log (x)+3 x^2 \log ^2(x)}{x^2-2 x \log (x)+\log ^2(x)}} \left (12 x^3+4 x^4+12 x^6+\left (-24 x^3-36 x^5\right ) \log (x)+\left (12 x^2+36 x^4\right ) \log ^2(x)+\left (-4 x-12 x^3\right ) \log ^3(x)\right )}{-16 x^3+48 x^2 \log (x)-48 x \log ^2(x)+16 \log ^3(x)} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((((-12*x^2-1)*log(x)^3+(36*x^3+3*x)*log(x)^2+(-36*x^4-15*x^2)*log(x)+12*x^5+x^3+12*x^2)*exp((3*x^2*l
og(x)^2-6*x^3*log(x)+3*x^4+3*x^2)/(log(x)^2-2*x*log(x)+x^2))^2+((-12*x^3-4*x)*log(x)^3+(36*x^4+12*x^2)*log(x)^
2+(-36*x^5-24*x^3)*log(x)+12*x^6+4*x^4+12*x^3)*exp((3*x^2*log(x)^2-6*x^3*log(x)+3*x^4+3*x^2)/(log(x)^2-2*x*log
(x)+x^2))+(-3*x^2-48)*log(x)^3+(9*x^3+144*x)*log(x)^2+(-9*x^4-144*x^2)*log(x)+3*x^5+48*x^3)/(16*log(x)^3-48*x*
log(x)^2+48*x^2*log(x)-16*x^3),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Unable to divide, perhaps due to rounding error%%%{62208,[2,38]%%%}+%%%{-808704,[2,37]%%%}+%%%{4852224,[2,3
6]%%%}+%%%{

Mupad [B] (verification not implemented)

Time = 13.15 (sec) , antiderivative size = 197, normalized size of antiderivative = 6.16 \[ \int \frac {48 x^3+3 x^5+\left (-144 x^2-9 x^4\right ) \log (x)+\left (144 x+9 x^3\right ) \log ^2(x)+\left (-48-3 x^2\right ) \log ^3(x)+e^{\frac {2 \left (3 x^2+3 x^4-6 x^3 \log (x)+3 x^2 \log ^2(x)\right )}{x^2-2 x \log (x)+\log ^2(x)}} \left (12 x^2+x^3+12 x^5+\left (-15 x^2-36 x^4\right ) \log (x)+\left (3 x+36 x^3\right ) \log ^2(x)+\left (-1-12 x^2\right ) \log ^3(x)\right )+e^{\frac {3 x^2+3 x^4-6 x^3 \log (x)+3 x^2 \log ^2(x)}{x^2-2 x \log (x)+\log ^2(x)}} \left (12 x^3+4 x^4+12 x^6+\left (-24 x^3-36 x^5\right ) \log (x)+\left (12 x^2+36 x^4\right ) \log ^2(x)+\left (-4 x-12 x^3\right ) \log ^3(x)\right )}{-16 x^3+48 x^2 \log (x)-48 x \log ^2(x)+16 \log ^3(x)} \, dx=-3\,x-\frac {x^3}{16}-\frac {x^2\,{\mathrm {e}}^{\frac {3\,x^2}{x^2-2\,x\,\ln \left (x\right )+{\ln \left (x\right )}^2}+\frac {3\,x^4}{x^2-2\,x\,\ln \left (x\right )+{\ln \left (x\right )}^2}+\frac {3\,x^2\,{\ln \left (x\right )}^2}{x^2-2\,x\,\ln \left (x\right )+{\ln \left (x\right )}^2}}}{8\,x^{\frac {6\,x^3}{x^2-2\,x\,\ln \left (x\right )+{\ln \left (x\right )}^2}}}-\frac {x\,{\mathrm {e}}^{\frac {6\,x^2}{x^2-2\,x\,\ln \left (x\right )+{\ln \left (x\right )}^2}+\frac {6\,x^4}{x^2-2\,x\,\ln \left (x\right )+{\ln \left (x\right )}^2}+\frac {6\,x^2\,{\ln \left (x\right )}^2}{x^2-2\,x\,\ln \left (x\right )+{\ln \left (x\right )}^2}}}{16\,x^{\frac {12\,x^3}{x^2-2\,x\,\ln \left (x\right )+{\ln \left (x\right )}^2}}} \]

[In]

int(-(log(x)^2*(144*x + 9*x^3) - log(x)*(144*x^2 + 9*x^4) + exp((2*(3*x^2*log(x)^2 - 6*x^3*log(x) + 3*x^2 + 3*
x^4))/(log(x)^2 - 2*x*log(x) + x^2))*(log(x)^2*(3*x + 36*x^3) - log(x)*(15*x^2 + 36*x^4) - log(x)^3*(12*x^2 +
1) + 12*x^2 + x^3 + 12*x^5) - log(x)^3*(3*x^2 + 48) + exp((3*x^2*log(x)^2 - 6*x^3*log(x) + 3*x^2 + 3*x^4)/(log
(x)^2 - 2*x*log(x) + x^2))*(log(x)^2*(12*x^2 + 36*x^4) - log(x)*(24*x^3 + 36*x^5) - log(x)^3*(4*x + 12*x^3) +
12*x^3 + 4*x^4 + 12*x^6) + 48*x^3 + 3*x^5)/(48*x*log(x)^2 - 48*x^2*log(x) - 16*log(x)^3 + 16*x^3),x)

[Out]

- 3*x - x^3/16 - (x^2*exp((3*x^2)/(log(x)^2 - 2*x*log(x) + x^2) + (3*x^4)/(log(x)^2 - 2*x*log(x) + x^2) + (3*x
^2*log(x)^2)/(log(x)^2 - 2*x*log(x) + x^2)))/(8*x^((6*x^3)/(log(x)^2 - 2*x*log(x) + x^2))) - (x*exp((6*x^2)/(l
og(x)^2 - 2*x*log(x) + x^2) + (6*x^4)/(log(x)^2 - 2*x*log(x) + x^2) + (6*x^2*log(x)^2)/(log(x)^2 - 2*x*log(x)
+ x^2)))/(16*x^((12*x^3)/(log(x)^2 - 2*x*log(x) + x^2)))