\(\int \frac {-5 x^2+e^x (5 x-x^2+x^3)+(5 x^2-5 e^x x^2) \log (\frac {x}{3})+(-12 x+4 x^2+(60-20 x) \log (\frac {x}{3})) \log (3 x^2)+(-15+11 x-x^2-15 \log (\frac {x}{3})) \log ^2(3 x^2)}{e^{2 x} x^2-2 e^x x^3+x^4+(6 x^2-2 x^3+e^x (-6 x+2 x^2)) \log ^2(3 x^2)+(9-6 x+x^2) \log ^4(3 x^2)} \, dx\) [8040]

Optimal result

Integrand size = 172, antiderivative size = 40 \[ \int \frac {-5 x^2+e^x \left (5 x-x^2+x^3\right )+\left (5 x^2-5 e^x x^2\right ) \log \left (\frac {x}{3}\right )+\left (-12 x+4 x^2+(60-20 x) \log \left (\frac {x}{3}\right )\right ) \log \left (3 x^2\right )+\left (-15+11 x-x^2-15 \log \left (\frac {x}{3}\right )\right ) \log ^2\left (3 x^2\right )}{e^{2 x} x^2-2 e^x x^3+x^4+\left (6 x^2-2 x^3+e^x \left (-6 x+2 x^2\right )\right ) \log ^2\left (3 x^2\right )+\left (9-6 x+x^2\right ) \log ^4\left (3 x^2\right )} \, dx=\frac {-x+5 \log \left (\frac {x}{3}\right )}{e^x-x-\frac {(3-x) \log ^2\left (3 x^2\right )}{x}} \]

[Out]

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

Rubi [F]

\[ \int \frac {-5 x^2+e^x \left (5 x-x^2+x^3\right )+\left (5 x^2-5 e^x x^2\right ) \log \left (\frac {x}{3}\right )+\left (-12 x+4 x^2+(60-20 x) \log \left (\frac {x}{3}\right )\right ) \log \left (3 x^2\right )+\left (-15+11 x-x^2-15 \log \left (\frac {x}{3}\right )\right ) \log ^2\left (3 x^2\right )}{e^{2 x} x^2-2 e^x x^3+x^4+\left (6 x^2-2 x^3+e^x \left (-6 x+2 x^2\right )\right ) \log ^2\left (3 x^2\right )+\left (9-6 x+x^2\right ) \log ^4\left (3 x^2\right )} \, dx=\int \frac {-5 x^2+e^x \left (5 x-x^2+x^3\right )+\left (5 x^2-5 e^x x^2\right ) \log \left (\frac {x}{3}\right )+\left (-12 x+4 x^2+(60-20 x) \log \left (\frac {x}{3}\right )\right ) \log \left (3 x^2\right )+\left (-15+11 x-x^2-15 \log \left (\frac {x}{3}\right )\right ) \log ^2\left (3 x^2\right )}{e^{2 x} x^2-2 e^x x^3+x^4+\left (6 x^2-2 x^3+e^x \left (-6 x+2 x^2\right )\right ) \log ^2\left (3 x^2\right )+\left (9-6 x+x^2\right ) \log ^4\left (3 x^2\right )} \, dx \]

[In]

Int[(-5*x^2 + E^x*(5*x - x^2 + x^3) + (5*x^2 - 5*E^x*x^2)*Log[x/3] + (-12*x + 4*x^2 + (60 - 20*x)*Log[x/3])*Lo
g[3*x^2] + (-15 + 11*x - x^2 - 15*Log[x/3])*Log[3*x^2]^2)/(E^(2*x)*x^2 - 2*E^x*x^3 + x^4 + (6*x^2 - 2*x^3 + E^
x*(-6*x + 2*x^2))*Log[3*x^2]^2 + (9 - 6*x + x^2)*Log[3*x^2]^4),x]

[Out]

-Defer[Int][x^3/((E^x - x)*x + (-3 + x)*Log[3*x^2]^2)^2, x] + Defer[Int][x^4/((E^x - x)*x + (-3 + x)*Log[3*x^2
]^2)^2, x] + 5*Defer[Int][(x^2*Log[x/3])/((E^x - x)*x + (-3 + x)*Log[3*x^2]^2)^2, x] - 5*Defer[Int][(x^3*Log[x
/3])/((E^x - x)*x + (-3 + x)*Log[3*x^2]^2)^2, x] - 12*Defer[Int][(x*Log[3*x^2])/((E^x - x)*x + (-3 + x)*Log[3*
x^2]^2)^2, x] + 4*Defer[Int][(x^2*Log[3*x^2])/((E^x - x)*x + (-3 + x)*Log[3*x^2]^2)^2, x] + 60*Defer[Int][(Log
[x/3]*Log[3*x^2])/((E^x - x)*x + (-3 + x)*Log[3*x^2]^2)^2, x] - 20*Defer[Int][(x*Log[x/3]*Log[3*x^2])/((E^x -
x)*x + (-3 + x)*Log[3*x^2]^2)^2, x] + 3*Defer[Int][(x*Log[3*x^2]^2)/((E^x - x)*x + (-3 + x)*Log[3*x^2]^2)^2, x
] + 3*Defer[Int][(x^2*Log[3*x^2]^2)/((E^x - x)*x + (-3 + x)*Log[3*x^2]^2)^2, x] - Defer[Int][(x^3*Log[3*x^2]^2
)/((E^x - x)*x + (-3 + x)*Log[3*x^2]^2)^2, x] - 15*Defer[Int][(Log[x/3]*Log[3*x^2]^2)/((E^x - x)*x + (-3 + x)*
Log[3*x^2]^2)^2, x] - 15*Defer[Int][(x*Log[x/3]*Log[3*x^2]^2)/((E^x - x)*x + (-3 + x)*Log[3*x^2]^2)^2, x] + 5*
Defer[Int][(x^2*Log[x/3]*Log[3*x^2]^2)/((E^x - x)*x + (-3 + x)*Log[3*x^2]^2)^2, x] + 5*Defer[Int][((E^x - x)*x
+ (-3 + x)*Log[3*x^2]^2)^(-1), x] + Defer[Int][x/(-(E^x*x) + x^2 + 3*Log[3*x^2]^2 - x*Log[3*x^2]^2), x] - Def
er[Int][x^2/(-(E^x*x) + x^2 + 3*Log[3*x^2]^2 - x*Log[3*x^2]^2), x] + 5*Defer[Int][(x*Log[x/3])/(-(E^x*x) + x^2
+ 3*Log[3*x^2]^2 - x*Log[3*x^2]^2), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {x \left (-5 x+e^x \left (5-x+x^2\right )\right )+4 (-3+x) x \log \left (3 x^2\right )-\left (15-11 x+x^2\right ) \log ^2\left (3 x^2\right )-5 \log \left (\frac {x}{3}\right ) \left (\left (-1+e^x\right ) x^2+4 (-3+x) \log \left (3 x^2\right )+3 \log ^2\left (3 x^2\right )\right )}{\left (\left (e^x-x\right ) x+(-3+x) \log ^2\left (3 x^2\right )\right )^2} \, dx \\ & = \int \left (-\frac {5-x+x^2-5 x \log \left (\frac {x}{3}\right )}{-e^x x+x^2+3 \log ^2\left (3 x^2\right )-x \log ^2\left (3 x^2\right )}+\frac {\left (x-5 \log \left (\frac {x}{3}\right )\right ) \left (-x^2+x^3-12 \log \left (3 x^2\right )+4 x \log \left (3 x^2\right )+3 \log ^2\left (3 x^2\right )+3 x \log ^2\left (3 x^2\right )-x^2 \log ^2\left (3 x^2\right )\right )}{\left (-e^x x+x^2+3 \log ^2\left (3 x^2\right )-x \log ^2\left (3 x^2\right )\right )^2}\right ) \, dx \\ & = -\int \frac {5-x+x^2-5 x \log \left (\frac {x}{3}\right )}{-e^x x+x^2+3 \log ^2\left (3 x^2\right )-x \log ^2\left (3 x^2\right )} \, dx+\int \frac {\left (x-5 \log \left (\frac {x}{3}\right )\right ) \left (-x^2+x^3-12 \log \left (3 x^2\right )+4 x \log \left (3 x^2\right )+3 \log ^2\left (3 x^2\right )+3 x \log ^2\left (3 x^2\right )-x^2 \log ^2\left (3 x^2\right )\right )}{\left (-e^x x+x^2+3 \log ^2\left (3 x^2\right )-x \log ^2\left (3 x^2\right )\right )^2} \, dx \\ & = \int \frac {\left (x-5 \log \left (\frac {x}{3}\right )\right ) \left ((-1+x) x^2+4 (-3+x) \log \left (3 x^2\right )+\left (3+3 x-x^2\right ) \log ^2\left (3 x^2\right )\right )}{\left (\left (e^x-x\right ) x+(-3+x) \log ^2\left (3 x^2\right )\right )^2} \, dx-\int \left (-\frac {x}{-e^x x+x^2+3 \log ^2\left (3 x^2\right )-x \log ^2\left (3 x^2\right )}+\frac {x^2}{-e^x x+x^2+3 \log ^2\left (3 x^2\right )-x \log ^2\left (3 x^2\right )}-\frac {5 x \log \left (\frac {x}{3}\right )}{-e^x x+x^2+3 \log ^2\left (3 x^2\right )-x \log ^2\left (3 x^2\right )}-\frac {5}{e^x x-x^2-3 \log ^2\left (3 x^2\right )+x \log ^2\left (3 x^2\right )}\right ) \, dx \\ & = 5 \int \frac {x \log \left (\frac {x}{3}\right )}{-e^x x+x^2+3 \log ^2\left (3 x^2\right )-x \log ^2\left (3 x^2\right )} \, dx+5 \int \frac {1}{e^x x-x^2-3 \log ^2\left (3 x^2\right )+x \log ^2\left (3 x^2\right )} \, dx+\int \frac {x}{-e^x x+x^2+3 \log ^2\left (3 x^2\right )-x \log ^2\left (3 x^2\right )} \, dx-\int \frac {x^2}{-e^x x+x^2+3 \log ^2\left (3 x^2\right )-x \log ^2\left (3 x^2\right )} \, dx+\int \left (-\frac {x^3}{\left (-e^x x+x^2+3 \log ^2\left (3 x^2\right )-x \log ^2\left (3 x^2\right )\right )^2}+\frac {x^4}{\left (-e^x x+x^2+3 \log ^2\left (3 x^2\right )-x \log ^2\left (3 x^2\right )\right )^2}+\frac {5 x^2 \log \left (\frac {x}{3}\right )}{\left (-e^x x+x^2+3 \log ^2\left (3 x^2\right )-x \log ^2\left (3 x^2\right )\right )^2}-\frac {5 x^3 \log \left (\frac {x}{3}\right )}{\left (-e^x x+x^2+3 \log ^2\left (3 x^2\right )-x \log ^2\left (3 x^2\right )\right )^2}-\frac {12 x \log \left (3 x^2\right )}{\left (-e^x x+x^2+3 \log ^2\left (3 x^2\right )-x \log ^2\left (3 x^2\right )\right )^2}+\frac {4 x^2 \log \left (3 x^2\right )}{\left (-e^x x+x^2+3 \log ^2\left (3 x^2\right )-x \log ^2\left (3 x^2\right )\right )^2}-\frac {20 x \log \left (\frac {x}{3}\right ) \log \left (3 x^2\right )}{\left (-e^x x+x^2+3 \log ^2\left (3 x^2\right )-x \log ^2\left (3 x^2\right )\right )^2}+\frac {3 x \log ^2\left (3 x^2\right )}{\left (-e^x x+x^2+3 \log ^2\left (3 x^2\right )-x \log ^2\left (3 x^2\right )\right )^2}+\frac {3 x^2 \log ^2\left (3 x^2\right )}{\left (-e^x x+x^2+3 \log ^2\left (3 x^2\right )-x \log ^2\left (3 x^2\right )\right )^2}-\frac {x^3 \log ^2\left (3 x^2\right )}{\left (-e^x x+x^2+3 \log ^2\left (3 x^2\right )-x \log ^2\left (3 x^2\right )\right )^2}-\frac {15 x \log \left (\frac {x}{3}\right ) \log ^2\left (3 x^2\right )}{\left (-e^x x+x^2+3 \log ^2\left (3 x^2\right )-x \log ^2\left (3 x^2\right )\right )^2}+\frac {5 x^2 \log \left (\frac {x}{3}\right ) \log ^2\left (3 x^2\right )}{\left (-e^x x+x^2+3 \log ^2\left (3 x^2\right )-x \log ^2\left (3 x^2\right )\right )^2}+\frac {60 \log \left (\frac {x}{3}\right ) \log \left (3 x^2\right )}{\left (e^x x-x^2-3 \log ^2\left (3 x^2\right )+x \log ^2\left (3 x^2\right )\right )^2}-\frac {15 \log \left (\frac {x}{3}\right ) \log ^2\left (3 x^2\right )}{\left (e^x x-x^2-3 \log ^2\left (3 x^2\right )+x \log ^2\left (3 x^2\right )\right )^2}\right ) \, dx \\ & = 3 \int \frac {x \log ^2\left (3 x^2\right )}{\left (-e^x x+x^2+3 \log ^2\left (3 x^2\right )-x \log ^2\left (3 x^2\right )\right )^2} \, dx+3 \int \frac {x^2 \log ^2\left (3 x^2\right )}{\left (-e^x x+x^2+3 \log ^2\left (3 x^2\right )-x \log ^2\left (3 x^2\right )\right )^2} \, dx+4 \int \frac {x^2 \log \left (3 x^2\right )}{\left (-e^x x+x^2+3 \log ^2\left (3 x^2\right )-x \log ^2\left (3 x^2\right )\right )^2} \, dx+5 \int \frac {1}{\left (e^x-x\right ) x+(-3+x) \log ^2\left (3 x^2\right )} \, dx+5 \int \frac {x^2 \log \left (\frac {x}{3}\right )}{\left (-e^x x+x^2+3 \log ^2\left (3 x^2\right )-x \log ^2\left (3 x^2\right )\right )^2} \, dx-5 \int \frac {x^3 \log \left (\frac {x}{3}\right )}{\left (-e^x x+x^2+3 \log ^2\left (3 x^2\right )-x \log ^2\left (3 x^2\right )\right )^2} \, dx+5 \int \frac {x^2 \log \left (\frac {x}{3}\right ) \log ^2\left (3 x^2\right )}{\left (-e^x x+x^2+3 \log ^2\left (3 x^2\right )-x \log ^2\left (3 x^2\right )\right )^2} \, dx+5 \int \frac {x \log \left (\frac {x}{3}\right )}{-e^x x+x^2+3 \log ^2\left (3 x^2\right )-x \log ^2\left (3 x^2\right )} \, dx-12 \int \frac {x \log \left (3 x^2\right )}{\left (-e^x x+x^2+3 \log ^2\left (3 x^2\right )-x \log ^2\left (3 x^2\right )\right )^2} \, dx-15 \int \frac {x \log \left (\frac {x}{3}\right ) \log ^2\left (3 x^2\right )}{\left (-e^x x+x^2+3 \log ^2\left (3 x^2\right )-x \log ^2\left (3 x^2\right )\right )^2} \, dx-15 \int \frac {\log \left (\frac {x}{3}\right ) \log ^2\left (3 x^2\right )}{\left (e^x x-x^2-3 \log ^2\left (3 x^2\right )+x \log ^2\left (3 x^2\right )\right )^2} \, dx-20 \int \frac {x \log \left (\frac {x}{3}\right ) \log \left (3 x^2\right )}{\left (-e^x x+x^2+3 \log ^2\left (3 x^2\right )-x \log ^2\left (3 x^2\right )\right )^2} \, dx+60 \int \frac {\log \left (\frac {x}{3}\right ) \log \left (3 x^2\right )}{\left (e^x x-x^2-3 \log ^2\left (3 x^2\right )+x \log ^2\left (3 x^2\right )\right )^2} \, dx-\int \frac {x^3}{\left (-e^x x+x^2+3 \log ^2\left (3 x^2\right )-x \log ^2\left (3 x^2\right )\right )^2} \, dx+\int \frac {x^4}{\left (-e^x x+x^2+3 \log ^2\left (3 x^2\right )-x \log ^2\left (3 x^2\right )\right )^2} \, dx-\int \frac {x^3 \log ^2\left (3 x^2\right )}{\left (-e^x x+x^2+3 \log ^2\left (3 x^2\right )-x \log ^2\left (3 x^2\right )\right )^2} \, dx+\int \frac {x}{-e^x x+x^2+3 \log ^2\left (3 x^2\right )-x \log ^2\left (3 x^2\right )} \, dx-\int \frac {x^2}{-e^x x+x^2+3 \log ^2\left (3 x^2\right )-x \log ^2\left (3 x^2\right )} \, dx \\ & = 3 \int \frac {x \log ^2\left (3 x^2\right )}{\left (\left (e^x-x\right ) x+(-3+x) \log ^2\left (3 x^2\right )\right )^2} \, dx+3 \int \frac {x^2 \log ^2\left (3 x^2\right )}{\left (\left (e^x-x\right ) x+(-3+x) \log ^2\left (3 x^2\right )\right )^2} \, dx+4 \int \frac {x^2 \log \left (3 x^2\right )}{\left (\left (e^x-x\right ) x+(-3+x) \log ^2\left (3 x^2\right )\right )^2} \, dx+5 \int \frac {x^2 \log \left (\frac {x}{3}\right )}{\left (\left (e^x-x\right ) x+(-3+x) \log ^2\left (3 x^2\right )\right )^2} \, dx-5 \int \frac {x^3 \log \left (\frac {x}{3}\right )}{\left (\left (e^x-x\right ) x+(-3+x) \log ^2\left (3 x^2\right )\right )^2} \, dx+5 \int \frac {x^2 \log \left (\frac {x}{3}\right ) \log ^2\left (3 x^2\right )}{\left (\left (e^x-x\right ) x+(-3+x) \log ^2\left (3 x^2\right )\right )^2} \, dx+5 \int \frac {1}{\left (e^x-x\right ) x+(-3+x) \log ^2\left (3 x^2\right )} \, dx+5 \int \frac {x \log \left (\frac {x}{3}\right )}{-e^x x+x^2+3 \log ^2\left (3 x^2\right )-x \log ^2\left (3 x^2\right )} \, dx-12 \int \frac {x \log \left (3 x^2\right )}{\left (\left (e^x-x\right ) x+(-3+x) \log ^2\left (3 x^2\right )\right )^2} \, dx-15 \int \frac {\log \left (\frac {x}{3}\right ) \log ^2\left (3 x^2\right )}{\left (\left (e^x-x\right ) x+(-3+x) \log ^2\left (3 x^2\right )\right )^2} \, dx-15 \int \frac {x \log \left (\frac {x}{3}\right ) \log ^2\left (3 x^2\right )}{\left (\left (e^x-x\right ) x+(-3+x) \log ^2\left (3 x^2\right )\right )^2} \, dx-20 \int \frac {x \log \left (\frac {x}{3}\right ) \log \left (3 x^2\right )}{\left (\left (e^x-x\right ) x+(-3+x) \log ^2\left (3 x^2\right )\right )^2} \, dx+60 \int \frac {\log \left (\frac {x}{3}\right ) \log \left (3 x^2\right )}{\left (\left (e^x-x\right ) x+(-3+x) \log ^2\left (3 x^2\right )\right )^2} \, dx-\int \frac {x^3}{\left (\left (e^x-x\right ) x+(-3+x) \log ^2\left (3 x^2\right )\right )^2} \, dx+\int \frac {x^4}{\left (\left (e^x-x\right ) x+(-3+x) \log ^2\left (3 x^2\right )\right )^2} \, dx-\int \frac {x^3 \log ^2\left (3 x^2\right )}{\left (\left (e^x-x\right ) x+(-3+x) \log ^2\left (3 x^2\right )\right )^2} \, dx+\int \frac {x}{-e^x x+x^2+3 \log ^2\left (3 x^2\right )-x \log ^2\left (3 x^2\right )} \, dx-\int \frac {x^2}{-e^x x+x^2+3 \log ^2\left (3 x^2\right )-x \log ^2\left (3 x^2\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.92 \[ \int \frac {-5 x^2+e^x \left (5 x-x^2+x^3\right )+\left (5 x^2-5 e^x x^2\right ) \log \left (\frac {x}{3}\right )+\left (-12 x+4 x^2+(60-20 x) \log \left (\frac {x}{3}\right )\right ) \log \left (3 x^2\right )+\left (-15+11 x-x^2-15 \log \left (\frac {x}{3}\right )\right ) \log ^2\left (3 x^2\right )}{e^{2 x} x^2-2 e^x x^3+x^4+\left (6 x^2-2 x^3+e^x \left (-6 x+2 x^2\right )\right ) \log ^2\left (3 x^2\right )+\left (9-6 x+x^2\right ) \log ^4\left (3 x^2\right )} \, dx=\frac {x \left (x-5 \log \left (\frac {x}{3}\right )\right )}{x \left (-e^x+x\right )-(-3+x) \log ^2\left (3 x^2\right )} \]

[In]

Integrate[(-5*x^2 + E^x*(5*x - x^2 + x^3) + (5*x^2 - 5*E^x*x^2)*Log[x/3] + (-12*x + 4*x^2 + (60 - 20*x)*Log[x/
3])*Log[3*x^2] + (-15 + 11*x - x^2 - 15*Log[x/3])*Log[3*x^2]^2)/(E^(2*x)*x^2 - 2*E^x*x^3 + x^4 + (6*x^2 - 2*x^
3 + E^x*(-6*x + 2*x^2))*Log[3*x^2]^2 + (9 - 6*x + x^2)*Log[3*x^2]^4),x]

[Out]

(x*(x - 5*Log[x/3]))/(x*(-E^x + x) - (-3 + x)*Log[3*x^2]^2)

Maple [A] (verified)

Time = 48.68 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.20

method	result	size
parallelrisch	\(\frac {-120 x \ln \left (\frac {x}{3}\right )+24 x^{2}}{-24 x \ln \left (3 x^{2}\right )^{2}+24 x^{2}-24 \,{\mathrm e}^{x} x +72 \ln \left (3 x^{2}\right )^{2}}\)	\(48\)
risch	\(\frac {-20 x \ln \left (x \right )+20 x \ln \left (3\right )+4 x^{2}}{-4 x \ln \left (3\right )^{2}+48 \ln \left (3\right ) \ln \left (x \right )-16 x \ln \left (x \right )^{2}-4 \,{\mathrm e}^{x} x -12 i \ln \left (3\right ) \pi \operatorname {csgn}\left (i x^{2}\right )^{3}-24 i \ln \left (x \right ) \pi \operatorname {csgn}\left (i x^{2}\right )^{3}-4 x \,\pi ^{2} \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{5}+x \,\pi ^{2} \operatorname {csgn}\left (i x \right )^{4} \operatorname {csgn}\left (i x^{2}\right )^{2}-4 x \,\pi ^{2} \operatorname {csgn}\left (i x \right )^{3} \operatorname {csgn}\left (i x^{2}\right )^{3}+6 x \,\pi ^{2} \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )^{4}+12 \ln \left (3\right )^{2}-16 x \ln \left (3\right ) \ln \left (x \right )+48 \ln \left (x \right )^{2}+4 x^{2}+8 i x \ln \left (x \right ) \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-16 i x \ln \left (x \right ) \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+4 i x \ln \left (3\right ) \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-8 i x \ln \left (3\right ) \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+4 i x \ln \left (3\right ) \pi \operatorname {csgn}\left (i x^{2}\right )^{3}-24 i \ln \left (x \right ) \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )+8 i x \ln \left (x \right ) \pi \operatorname {csgn}\left (i x^{2}\right )^{3}-12 i \ln \left (3\right ) \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )+24 i \ln \left (3\right ) \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}-3 \pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{6}+12 \pi ^{2} \operatorname {csgn}\left (i x \right )^{3} \operatorname {csgn}\left (i x^{2}\right )^{3}-18 \pi ^{2} \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )^{4}+12 \pi ^{2} \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{5}+x \,\pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{6}-3 \pi ^{2} \operatorname {csgn}\left (i x \right )^{4} \operatorname {csgn}\left (i x^{2}\right )^{2}+48 i \ln \left (x \right ) \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}}\)	\(492\)

[In]

int(((-15*ln(1/3*x)-x^2+11*x-15)*ln(3*x^2)^2+((-20*x+60)*ln(1/3*x)+4*x^2-12*x)*ln(3*x^2)+(-5*exp(x)*x^2+5*x^2)
*ln(1/3*x)+(x^3-x^2+5*x)*exp(x)-5*x^2)/((x^2-6*x+9)*ln(3*x^2)^4+((2*x^2-6*x)*exp(x)-2*x^3+6*x^2)*ln(3*x^2)^2+e
xp(x)^2*x^2-2*exp(x)*x^3+x^4),x,method=_RETURNVERBOSE)

[Out]

1/24*(-120*x*ln(1/3*x)+24*x^2)/(-x*ln(3*x^2)^2+x^2-exp(x)*x+3*ln(3*x^2)^2)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.40 \[ \int \frac {-5 x^2+e^x \left (5 x-x^2+x^3\right )+\left (5 x^2-5 e^x x^2\right ) \log \left (\frac {x}{3}\right )+\left (-12 x+4 x^2+(60-20 x) \log \left (\frac {x}{3}\right )\right ) \log \left (3 x^2\right )+\left (-15+11 x-x^2-15 \log \left (\frac {x}{3}\right )\right ) \log ^2\left (3 x^2\right )}{e^{2 x} x^2-2 e^x x^3+x^4+\left (6 x^2-2 x^3+e^x \left (-6 x+2 x^2\right )\right ) \log ^2\left (3 x^2\right )+\left (9-6 x+x^2\right ) \log ^4\left (3 x^2\right )} \, dx=-\frac {x^{2} - 5 \, x \log \left (\frac {1}{3} \, x\right )}{9 \, {\left (x - 3\right )} \log \left (3\right )^{2} + 12 \, {\left (x - 3\right )} \log \left (3\right ) \log \left (\frac {1}{3} \, x\right ) + 4 \, {\left (x - 3\right )} \log \left (\frac {1}{3} \, x\right )^{2} - x^{2} + x e^{x}} \]

[In]

integrate(((-15*log(1/3*x)-x^2+11*x-15)*log(3*x^2)^2+((-20*x+60)*log(1/3*x)+4*x^2-12*x)*log(3*x^2)+(-5*exp(x)*
x^2+5*x^2)*log(1/3*x)+(x^3-x^2+5*x)*exp(x)-5*x^2)/((x^2-6*x+9)*log(3*x^2)^4+((2*x^2-6*x)*exp(x)-2*x^3+6*x^2)*l
og(3*x^2)^2+exp(x)^2*x^2-2*exp(x)*x^3+x^4),x, algorithm="fricas")

[Out]

-(x^2 - 5*x*log(1/3*x))/(9*(x - 3)*log(3)^2 + 12*(x - 3)*log(3)*log(1/3*x) + 4*(x - 3)*log(1/3*x)^2 - x^2 + x*
e^x)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (26) = 52\).

Time = 0.27 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.88 \[ \int \frac {-5 x^2+e^x \left (5 x-x^2+x^3\right )+\left (5 x^2-5 e^x x^2\right ) \log \left (\frac {x}{3}\right )+\left (-12 x+4 x^2+(60-20 x) \log \left (\frac {x}{3}\right )\right ) \log \left (3 x^2\right )+\left (-15+11 x-x^2-15 \log \left (\frac {x}{3}\right )\right ) \log ^2\left (3 x^2\right )}{e^{2 x} x^2-2 e^x x^3+x^4+\left (6 x^2-2 x^3+e^x \left (-6 x+2 x^2\right )\right ) \log ^2\left (3 x^2\right )+\left (9-6 x+x^2\right ) \log ^4\left (3 x^2\right )} \, dx=\frac {- x^{2} + 5 x \log {\left (\frac {x}{3} \right )}}{- x^{2} + x e^{x} + 4 x \log {\left (\frac {x}{3} \right )}^{2} + 12 x \log {\left (3 \right )} \log {\left (\frac {x}{3} \right )} + 9 x \log {\left (3 \right )}^{2} - 12 \log {\left (\frac {x}{3} \right )}^{2} - 36 \log {\left (3 \right )} \log {\left (\frac {x}{3} \right )} - 27 \log {\left (3 \right )}^{2}} \]

[In]

integrate(((-15*ln(1/3*x)-x**2+11*x-15)*ln(3*x**2)**2+((-20*x+60)*ln(1/3*x)+4*x**2-12*x)*ln(3*x**2)+(-5*exp(x)
*x**2+5*x**2)*ln(1/3*x)+(x**3-x**2+5*x)*exp(x)-5*x**2)/((x**2-6*x+9)*ln(3*x**2)**4+((2*x**2-6*x)*exp(x)-2*x**3
+6*x**2)*ln(3*x**2)**2+exp(x)**2*x**2-2*exp(x)*x**3+x**4),x)

[Out]

(-x**2 + 5*x*log(x/3))/(-x**2 + x*exp(x) + 4*x*log(x/3)**2 + 12*x*log(3)*log(x/3) + 9*x*log(3)**2 - 12*log(x/3
)**2 - 36*log(3)*log(x/3) - 27*log(3)**2)

Maxima [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.55 \[ \int \frac {-5 x^2+e^x \left (5 x-x^2+x^3\right )+\left (5 x^2-5 e^x x^2\right ) \log \left (\frac {x}{3}\right )+\left (-12 x+4 x^2+(60-20 x) \log \left (\frac {x}{3}\right )\right ) \log \left (3 x^2\right )+\left (-15+11 x-x^2-15 \log \left (\frac {x}{3}\right )\right ) \log ^2\left (3 x^2\right )}{e^{2 x} x^2-2 e^x x^3+x^4+\left (6 x^2-2 x^3+e^x \left (-6 x+2 x^2\right )\right ) \log ^2\left (3 x^2\right )+\left (9-6 x+x^2\right ) \log ^4\left (3 x^2\right )} \, dx=-\frac {x^{2} + 5 \, x \log \left (3\right ) - 5 \, x \log \left (x\right )}{x \log \left (3\right )^{2} + 4 \, {\left (x - 3\right )} \log \left (x\right )^{2} - x^{2} + x e^{x} - 3 \, \log \left (3\right )^{2} + 4 \, {\left (x \log \left (3\right ) - 3 \, \log \left (3\right )\right )} \log \left (x\right )} \]

[In]

integrate(((-15*log(1/3*x)-x^2+11*x-15)*log(3*x^2)^2+((-20*x+60)*log(1/3*x)+4*x^2-12*x)*log(3*x^2)+(-5*exp(x)*
x^2+5*x^2)*log(1/3*x)+(x^3-x^2+5*x)*exp(x)-5*x^2)/((x^2-6*x+9)*log(3*x^2)^4+((2*x^2-6*x)*exp(x)-2*x^3+6*x^2)*l
og(3*x^2)^2+exp(x)^2*x^2-2*exp(x)*x^3+x^4),x, algorithm="maxima")

[Out]

-(x^2 + 5*x*log(3) - 5*x*log(x))/(x*log(3)^2 + 4*(x - 3)*log(x)^2 - x^2 + x*e^x - 3*log(3)^2 + 4*(x*log(3) - 3
*log(3))*log(x))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1267 vs. \(2 (33) = 66\).

Time = 157.09 (sec) , antiderivative size = 1267, normalized size of antiderivative = 31.68 \[ \int \frac {-5 x^2+e^x \left (5 x-x^2+x^3\right )+\left (5 x^2-5 e^x x^2\right ) \log \left (\frac {x}{3}\right )+\left (-12 x+4 x^2+(60-20 x) \log \left (\frac {x}{3}\right )\right ) \log \left (3 x^2\right )+\left (-15+11 x-x^2-15 \log \left (\frac {x}{3}\right )\right ) \log ^2\left (3 x^2\right )}{e^{2 x} x^2-2 e^x x^3+x^4+\left (6 x^2-2 x^3+e^x \left (-6 x+2 x^2\right )\right ) \log ^2\left (3 x^2\right )+\left (9-6 x+x^2\right ) \log ^4\left (3 x^2\right )} \, dx=\text {Too large to display} \]

[In]

integrate(((-15*log(1/3*x)-x^2+11*x-15)*log(3*x^2)^2+((-20*x+60)*log(1/3*x)+4*x^2-12*x)*log(3*x^2)+(-5*exp(x)*
x^2+5*x^2)*log(1/3*x)+(x^3-x^2+5*x)*exp(x)-5*x^2)/((x^2-6*x+9)*log(3*x^2)^4+((2*x^2-6*x)*exp(x)-2*x^3+6*x^2)*l
og(3*x^2)^2+exp(x)^2*x^2-2*exp(x)*x^3+x^4),x, algorithm="giac")

[Out]

-(2*x^7*e^(2*x) - 3*x^7*e^x + 10*x^6*e^(2*x)*log(3) - 19*x^6*e^x*log(3) - 30*x^5*e^x*log(3)^2 - 10*x^6*e^(2*x)
*log(x) + 7*x^6*e^x*log(x) - 60*x^5*e^x*log(3)*log(x) + x^7 - 13*x^6*e^(2*x) + 38*x^6*e^x + 5*x^6*log(3) - 60*
x^5*e^(2*x)*log(3) + 175*x^5*e^x*log(3) + 270*x^4*e^x*log(3)^2 - 5*x^6*log(x) + 60*x^5*e^(2*x)*log(x) - 55*x^5
*e^x*log(x) + 540*x^4*e^x*log(3)*log(x) - 28*x^6 + 10*x^5*e^(2*x) - 106*x^5*e^x - 140*x^5*log(3) + 45*x^4*e^(2
*x)*log(3) - 324*x^4*e^x*log(3) - 810*x^3*e^x*log(3)^2 + 140*x^5*log(x) - 45*x^4*e^(2*x)*log(x) - 168*x^4*e^x*
log(x) - 1620*x^3*e^x*log(3)*log(x) + 180*x^5 + 51*x^4*e^(2*x) - 27*x^4*e^x + 900*x^4*log(3) + 135*x^3*e^(2*x)
*log(3) - 504*x^3*e^x*log(3) + 810*x^2*e^x*log(3)^2 - 900*x^4*log(x) - 135*x^3*e^(2*x)*log(x) + 1692*x^3*e^x*l
og(x) + 1620*x^2*e^x*log(3)*log(x) - 432*x^4 - 60*x^3*e^(2*x) + 342*x^3*e^x - 2160*x^3*log(3) + 45*x^2*e^(2*x)
*log(3) + 1512*x^2*e^x*log(3) + 2160*x^3*log(x) - 45*x^2*e^(2*x)*log(x) - 3456*x^2*e^x*log(x) + 432*x^3 + 45*x
^2*e^(2*x) - 432*x^2*e^x + 2160*x^2*log(3) - 1620*x*e^x*log(3) - 2160*x^2*log(x) + 3240*x*e^x*log(x))/(x^6*e^(
2*x)*log(3)^2 - 2*x^6*e^x*log(3)^2 + 4*x^6*e^(2*x)*log(3)*log(x) - 8*x^6*e^x*log(3)*log(x) + 4*x^6*e^(2*x)*log
(x)^2 - 8*x^6*e^x*log(x)^2 - x^7*e^(2*x) + 2*x^7*e^x + x^6*log(3)^2 - 9*x^5*e^(2*x)*log(3)^2 + 24*x^5*e^x*log(
3)^2 + 4*x^6*log(3)*log(x) - 36*x^5*e^(2*x)*log(3)*log(x) + 96*x^5*e^x*log(3)*log(x) + 4*x^6*log(x)^2 - 36*x^5
*e^(2*x)*log(x)^2 + 96*x^5*e^x*log(x)^2 - x^7 + x^6*e^(3*x) + 4*x^6*e^(2*x) - 17*x^6*e^x - 31*x^5*log(3)^2 + 2
1*x^4*e^(2*x)*log(3)^2 - 68*x^4*e^x*log(3)^2 - 124*x^5*log(3)*log(x) + 84*x^4*e^(2*x)*log(3)*log(x) - 272*x^4*
e^x*log(3)*log(x) - 124*x^5*log(x)^2 + 84*x^4*e^(2*x)*log(x)^2 - 272*x^4*e^x*log(x)^2 + 28*x^6 - 6*x^5*e^(3*x)
+ 15*x^5*e^(2*x) - 14*x^5*e^x + 264*x^4*log(3)^2 + 9*x^3*e^(2*x)*log(3)^2 - 138*x^3*e^x*log(3)^2 + 1056*x^4*l
og(3)*log(x) + 36*x^3*e^(2*x)*log(3)*log(x) - 552*x^3*e^x*log(3)*log(x) + 1056*x^4*log(x)^2 + 36*x^3*e^(2*x)*l
og(x)^2 - 552*x^3*e^x*log(x)^2 - 180*x^5 + 3*x^4*e^(3*x) - 32*x^4*e^(2*x) + 360*x^4*e^x - 972*x^3*log(3)^2 - 4
5*x^2*e^(2*x)*log(3)^2 + 972*x^2*e^x*log(3)^2 - 3888*x^3*log(3)*log(x) - 180*x^2*e^(2*x)*log(3)*log(x) + 3888*
x^2*e^x*log(3)*log(x) - 3888*x^3*log(x)^2 - 180*x^2*e^(2*x)*log(x)^2 + 3888*x^2*e^x*log(x)^2 + 432*x^4 + 18*x^
3*e^(3*x) - 189*x^3*e^(2*x) - 864*x^3*e^x + 1728*x^2*log(3)^2 - 27*x*e^(2*x)*log(3)^2 - 1728*x*e^x*log(3)^2 +
6912*x^2*log(3)*log(x) - 108*x*e^(2*x)*log(3)*log(x) - 6912*x*e^x*log(3)*log(x) + 6912*x^2*log(x)^2 - 108*x*e^
(2*x)*log(x)^2 - 6912*x*e^x*log(x)^2 - 432*x^3 + 9*x^2*e^(3*x) + 432*x^2*e^(2*x) + 864*x^2*e^x - 1296*x*log(3)
^2 + 1296*e^x*log(3)^2 - 5184*x*log(3)*log(x) + 5184*e^x*log(3)*log(x) - 5184*x*log(x)^2 + 5184*e^x*log(x)^2 -
432*x*e^(2*x))

Mupad [F(-1)]

Timed out. \[ \int \frac {-5 x^2+e^x \left (5 x-x^2+x^3\right )+\left (5 x^2-5 e^x x^2\right ) \log \left (\frac {x}{3}\right )+\left (-12 x+4 x^2+(60-20 x) \log \left (\frac {x}{3}\right )\right ) \log \left (3 x^2\right )+\left (-15+11 x-x^2-15 \log \left (\frac {x}{3}\right )\right ) \log ^2\left (3 x^2\right )}{e^{2 x} x^2-2 e^x x^3+x^4+\left (6 x^2-2 x^3+e^x \left (-6 x+2 x^2\right )\right ) \log ^2\left (3 x^2\right )+\left (9-6 x+x^2\right ) \log ^4\left (3 x^2\right )} \, dx=\int -\frac {{\ln \left (3\,x^2\right )}^2\,\left (15\,\ln \left (\frac {x}{3}\right )-11\,x+x^2+15\right )+\ln \left (3\,x^2\right )\,\left (12\,x-4\,x^2+\ln \left (\frac {x}{3}\right )\,\left (20\,x-60\right )\right )+\ln \left (\frac {x}{3}\right )\,\left (5\,x^2\,{\mathrm {e}}^x-5\,x^2\right )-{\mathrm {e}}^x\,\left (x^3-x^2+5\,x\right )+5\,x^2}{{\ln \left (3\,x^2\right )}^4\,\left (x^2-6\,x+9\right )-2\,x^3\,{\mathrm {e}}^x+x^2\,{\mathrm {e}}^{2\,x}+x^4-{\ln \left (3\,x^2\right )}^2\,\left ({\mathrm {e}}^x\,\left (6\,x-2\,x^2\right )-6\,x^2+2\,x^3\right )} \,d x \]

[In]

int(-(log(3*x^2)^2*(15*log(x/3) - 11*x + x^2 + 15) + log(3*x^2)*(12*x - 4*x^2 + log(x/3)*(20*x - 60)) + log(x/
3)*(5*x^2*exp(x) - 5*x^2) - exp(x)*(5*x - x^2 + x^3) + 5*x^2)/(log(3*x^2)^4*(x^2 - 6*x + 9) - 2*x^3*exp(x) + x
^2*exp(2*x) + x^4 - log(3*x^2)^2*(exp(x)*(6*x - 2*x^2) - 6*x^2 + 2*x^3)),x)

[Out]

int(-(log(3*x^2)^2*(15*log(x/3) - 11*x + x^2 + 15) + log(3*x^2)*(12*x - 4*x^2 + log(x/3)*(20*x - 60)) + log(x/
3)*(5*x^2*exp(x) - 5*x^2) - exp(x)*(5*x - x^2 + x^3) + 5*x^2)/(log(3*x^2)^4*(x^2 - 6*x + 9) - 2*x^3*exp(x) + x
^2*exp(2*x) + x^4 - log(3*x^2)^2*(exp(x)*(6*x - 2*x^2) - 6*x^2 + 2*x^3)), x)