\(\int \frac {450 x-360 x^2+9 x^3-24 x^4-2 x^5+e^x (-450+360 x+15 x^3+2 x^4)+(-225 x^2+360 x^3-114 x^4-24 x^5-x^6+e^x (225 x-360 x^2+114 x^3+24 x^4+x^5)+(-9 e^x x^3+9 x^4) \log (-e^x+x)) \log ^2(\frac {225-360 x+114 x^2+24 x^3+x^4-9 x^2 \log (-e^x+x)}{9 x^2})}{(225 x^2-360 x^3+114 x^4+24 x^5+x^6+e^x (-225 x+360 x^2-114 x^3-24 x^4-x^5)+(9 e^x x^3-9 x^4) \log (-e^x+x)) \log ^2(\frac {225-360 x+114 x^2+24 x^3+x^4-9 x^2 \log (-e^x+x)}{9 x^2})} \, dx\) [136]

Optimal result

Integrand size = 279, antiderivative size = 39 \[ \int \frac {450 x-360 x^2+9 x^3-24 x^4-2 x^5+e^x \left (-450+360 x+15 x^3+2 x^4\right )+\left (-225 x^2+360 x^3-114 x^4-24 x^5-x^6+e^x \left (225 x-360 x^2+114 x^3+24 x^4+x^5\right )+\left (-9 e^x x^3+9 x^4\right ) \log \left (-e^x+x\right )\right ) \log ^2\left (\frac {225-360 x+114 x^2+24 x^3+x^4-9 x^2 \log \left (-e^x+x\right )}{9 x^2}\right )}{\left (225 x^2-360 x^3+114 x^4+24 x^5+x^6+e^x \left (-225 x+360 x^2-114 x^3-24 x^4-x^5\right )+\left (9 e^x x^3-9 x^4\right ) \log \left (-e^x+x\right )\right ) \log ^2\left (\frac {225-360 x+114 x^2+24 x^3+x^4-9 x^2 \log \left (-e^x+x\right )}{9 x^2}\right )} \, dx=-x+\frac {1}{\log \left (\left (2-\frac {5-\left (2+\frac {x}{3}\right ) x}{x}\right )^2-\log \left (-e^x+x\right )\right )} \]

[Out]

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

Rubi [F]

\[ \int \frac {450 x-360 x^2+9 x^3-24 x^4-2 x^5+e^x \left (-450+360 x+15 x^3+2 x^4\right )+\left (-225 x^2+360 x^3-114 x^4-24 x^5-x^6+e^x \left (225 x-360 x^2+114 x^3+24 x^4+x^5\right )+\left (-9 e^x x^3+9 x^4\right ) \log \left (-e^x+x\right )\right ) \log ^2\left (\frac {225-360 x+114 x^2+24 x^3+x^4-9 x^2 \log \left (-e^x+x\right )}{9 x^2}\right )}{\left (225 x^2-360 x^3+114 x^4+24 x^5+x^6+e^x \left (-225 x+360 x^2-114 x^3-24 x^4-x^5\right )+\left (9 e^x x^3-9 x^4\right ) \log \left (-e^x+x\right )\right ) \log ^2\left (\frac {225-360 x+114 x^2+24 x^3+x^4-9 x^2 \log \left (-e^x+x\right )}{9 x^2}\right )} \, dx=\int \frac {450 x-360 x^2+9 x^3-24 x^4-2 x^5+e^x \left (-450+360 x+15 x^3+2 x^4\right )+\left (-225 x^2+360 x^3-114 x^4-24 x^5-x^6+e^x \left (225 x-360 x^2+114 x^3+24 x^4+x^5\right )+\left (-9 e^x x^3+9 x^4\right ) \log \left (-e^x+x\right )\right ) \log ^2\left (\frac {225-360 x+114 x^2+24 x^3+x^4-9 x^2 \log \left (-e^x+x\right )}{9 x^2}\right )}{\left (225 x^2-360 x^3+114 x^4+24 x^5+x^6+e^x \left (-225 x+360 x^2-114 x^3-24 x^4-x^5\right )+\left (9 e^x x^3-9 x^4\right ) \log \left (-e^x+x\right )\right ) \log ^2\left (\frac {225-360 x+114 x^2+24 x^3+x^4-9 x^2 \log \left (-e^x+x\right )}{9 x^2}\right )} \, dx \]

[In]

Int[(450*x - 360*x^2 + 9*x^3 - 24*x^4 - 2*x^5 + E^x*(-450 + 360*x + 15*x^3 + 2*x^4) + (-225*x^2 + 360*x^3 - 11
4*x^4 - 24*x^5 - x^6 + E^x*(225*x - 360*x^2 + 114*x^3 + 24*x^4 + x^5) + (-9*E^x*x^3 + 9*x^4)*Log[-E^x + x])*Lo
g[(225 - 360*x + 114*x^2 + 24*x^3 + x^4 - 9*x^2*Log[-E^x + x])/(9*x^2)]^2)/((225*x^2 - 360*x^3 + 114*x^4 + 24*
x^5 + x^6 + E^x*(-225*x + 360*x^2 - 114*x^3 - 24*x^4 - x^5) + (9*E^x*x^3 - 9*x^4)*Log[-E^x + x])*Log[(225 - 36
0*x + 114*x^2 + 24*x^3 + x^4 - 9*x^2*Log[-E^x + x])/(9*x^2)]^2),x]

[Out]

-x - 360*Defer[Int][1/((225 - 360*x + 114*x^2 + 24*x^3 + x^4 - 9*x^2*Log[-E^x + x])*Log[(-15 + 12*x + x^2)^2/(
9*x^2) - Log[-E^x + x]]^2), x] + 450*Defer[Int][1/(x*(225 - 360*x + 114*x^2 + 24*x^3 + x^4 - 9*x^2*Log[-E^x +
x])*Log[(-15 + 12*x + x^2)^2/(9*x^2) - Log[-E^x + x]]^2), x] - 15*Defer[Int][x^2/((225 - 360*x + 114*x^2 + 24*
x^3 + x^4 - 9*x^2*Log[-E^x + x])*Log[(-15 + 12*x + x^2)^2/(9*x^2) - Log[-E^x + x]]^2), x] - 9*Defer[Int][x^2/(
(E^x - x)*(225 - 360*x + 114*x^2 + 24*x^3 + x^4 - 9*x^2*Log[-E^x + x])*Log[(-15 + 12*x + x^2)^2/(9*x^2) - Log[
-E^x + x]]^2), x] - 2*Defer[Int][x^3/((225 - 360*x + 114*x^2 + 24*x^3 + x^4 - 9*x^2*Log[-E^x + x])*Log[(-15 +
12*x + x^2)^2/(9*x^2) - Log[-E^x + x]]^2), x] + 9*Defer[Int][x^3/((E^x - x)*(225 - 360*x + 114*x^2 + 24*x^3 +
x^4 - 9*x^2*Log[-E^x + x])*Log[(-15 + 12*x + x^2)^2/(9*x^2) - Log[-E^x + x]]^2), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {-450 x+360 x^2-9 x^3+24 x^4+2 x^5-e^x \left (-450+360 x+15 x^3+2 x^4\right )-\left (e^x-x\right ) x \left (\left (-15+12 x+x^2\right )^2-9 x^2 \log \left (-e^x+x\right )\right ) \log ^2\left (\frac {\left (-15+12 x+x^2\right )^2}{9 x^2}-\log \left (-e^x+x\right )\right )}{\left (e^x-x\right ) x \left (\left (-15+12 x+x^2\right )^2-9 x^2 \log \left (-e^x+x\right )\right ) \log ^2\left (\frac {\left (-15+12 x+x^2\right )^2}{9 x^2}-\log \left (-e^x+x\right )\right )} \, dx \\ & = \int \left (\frac {9 (-1+x) x^2}{\left (e^x-x\right ) \left (225-360 x+114 x^2+24 x^3+x^4-9 x^2 \log \left (-e^x+x\right )\right ) \log ^2\left (\frac {\left (-15+12 x+x^2\right )^2}{9 x^2}-\log \left (-e^x+x\right )\right )}+\frac {450-360 x-15 x^3-2 x^4-225 x \log ^2\left (\frac {\left (-15+12 x+x^2\right )^2}{9 x^2}-\log \left (-e^x+x\right )\right )+360 x^2 \log ^2\left (\frac {\left (-15+12 x+x^2\right )^2}{9 x^2}-\log \left (-e^x+x\right )\right )-114 x^3 \log ^2\left (\frac {\left (-15+12 x+x^2\right )^2}{9 x^2}-\log \left (-e^x+x\right )\right )-24 x^4 \log ^2\left (\frac {\left (-15+12 x+x^2\right )^2}{9 x^2}-\log \left (-e^x+x\right )\right )-x^5 \log ^2\left (\frac {\left (-15+12 x+x^2\right )^2}{9 x^2}-\log \left (-e^x+x\right )\right )+9 x^3 \log \left (-e^x+x\right ) \log ^2\left (\frac {\left (-15+12 x+x^2\right )^2}{9 x^2}-\log \left (-e^x+x\right )\right )}{x \left (225-360 x+114 x^2+24 x^3+x^4-9 x^2 \log \left (-e^x+x\right )\right ) \log ^2\left (\frac {\left (-15+12 x+x^2\right )^2}{9 x^2}-\log \left (-e^x+x\right )\right )}\right ) \, dx \\ & = 9 \int \frac {(-1+x) x^2}{\left (e^x-x\right ) \left (225-360 x+114 x^2+24 x^3+x^4-9 x^2 \log \left (-e^x+x\right )\right ) \log ^2\left (\frac {\left (-15+12 x+x^2\right )^2}{9 x^2}-\log \left (-e^x+x\right )\right )} \, dx+\int \frac {450-360 x-15 x^3-2 x^4-225 x \log ^2\left (\frac {\left (-15+12 x+x^2\right )^2}{9 x^2}-\log \left (-e^x+x\right )\right )+360 x^2 \log ^2\left (\frac {\left (-15+12 x+x^2\right )^2}{9 x^2}-\log \left (-e^x+x\right )\right )-114 x^3 \log ^2\left (\frac {\left (-15+12 x+x^2\right )^2}{9 x^2}-\log \left (-e^x+x\right )\right )-24 x^4 \log ^2\left (\frac {\left (-15+12 x+x^2\right )^2}{9 x^2}-\log \left (-e^x+x\right )\right )-x^5 \log ^2\left (\frac {\left (-15+12 x+x^2\right )^2}{9 x^2}-\log \left (-e^x+x\right )\right )+9 x^3 \log \left (-e^x+x\right ) \log ^2\left (\frac {\left (-15+12 x+x^2\right )^2}{9 x^2}-\log \left (-e^x+x\right )\right )}{x \left (225-360 x+114 x^2+24 x^3+x^4-9 x^2 \log \left (-e^x+x\right )\right ) \log ^2\left (\frac {\left (-15+12 x+x^2\right )^2}{9 x^2}-\log \left (-e^x+x\right )\right )} \, dx \\ & = 9 \int \left (-\frac {x^2}{\left (e^x-x\right ) \left (225-360 x+114 x^2+24 x^3+x^4-9 x^2 \log \left (-e^x+x\right )\right ) \log ^2\left (\frac {\left (-15+12 x+x^2\right )^2}{9 x^2}-\log \left (-e^x+x\right )\right )}+\frac {x^3}{\left (e^x-x\right ) \left (225-360 x+114 x^2+24 x^3+x^4-9 x^2 \log \left (-e^x+x\right )\right ) \log ^2\left (\frac {\left (-15+12 x+x^2\right )^2}{9 x^2}-\log \left (-e^x+x\right )\right )}\right ) \, dx+\int \frac {450-360 x-15 x^3-2 x^4-x \left (\left (-15+12 x+x^2\right )^2-9 x^2 \log \left (-e^x+x\right )\right ) \log ^2\left (\frac {\left (-15+12 x+x^2\right )^2}{9 x^2}-\log \left (-e^x+x\right )\right )}{x \left (\left (-15+12 x+x^2\right )^2-9 x^2 \log \left (-e^x+x\right )\right ) \log ^2\left (\frac {\left (-15+12 x+x^2\right )^2}{9 x^2}-\log \left (-e^x+x\right )\right )} \, dx \\ & = -\left (9 \int \frac {x^2}{\left (e^x-x\right ) \left (225-360 x+114 x^2+24 x^3+x^4-9 x^2 \log \left (-e^x+x\right )\right ) \log ^2\left (\frac {\left (-15+12 x+x^2\right )^2}{9 x^2}-\log \left (-e^x+x\right )\right )} \, dx\right )+9 \int \frac {x^3}{\left (e^x-x\right ) \left (225-360 x+114 x^2+24 x^3+x^4-9 x^2 \log \left (-e^x+x\right )\right ) \log ^2\left (\frac {\left (-15+12 x+x^2\right )^2}{9 x^2}-\log \left (-e^x+x\right )\right )} \, dx+\int \left (-1+\frac {450-360 x-15 x^3-2 x^4}{x \left (225-360 x+114 x^2+24 x^3+x^4-9 x^2 \log \left (-e^x+x\right )\right ) \log ^2\left (\frac {\left (-15+12 x+x^2\right )^2}{9 x^2}-\log \left (-e^x+x\right )\right )}\right ) \, dx \\ & = -x-9 \int \frac {x^2}{\left (e^x-x\right ) \left (225-360 x+114 x^2+24 x^3+x^4-9 x^2 \log \left (-e^x+x\right )\right ) \log ^2\left (\frac {\left (-15+12 x+x^2\right )^2}{9 x^2}-\log \left (-e^x+x\right )\right )} \, dx+9 \int \frac {x^3}{\left (e^x-x\right ) \left (225-360 x+114 x^2+24 x^3+x^4-9 x^2 \log \left (-e^x+x\right )\right ) \log ^2\left (\frac {\left (-15+12 x+x^2\right )^2}{9 x^2}-\log \left (-e^x+x\right )\right )} \, dx+\int \frac {450-360 x-15 x^3-2 x^4}{x \left (225-360 x+114 x^2+24 x^3+x^4-9 x^2 \log \left (-e^x+x\right )\right ) \log ^2\left (\frac {\left (-15+12 x+x^2\right )^2}{9 x^2}-\log \left (-e^x+x\right )\right )} \, dx \\ & = -x-9 \int \frac {x^2}{\left (e^x-x\right ) \left (225-360 x+114 x^2+24 x^3+x^4-9 x^2 \log \left (-e^x+x\right )\right ) \log ^2\left (\frac {\left (-15+12 x+x^2\right )^2}{9 x^2}-\log \left (-e^x+x\right )\right )} \, dx+9 \int \frac {x^3}{\left (e^x-x\right ) \left (225-360 x+114 x^2+24 x^3+x^4-9 x^2 \log \left (-e^x+x\right )\right ) \log ^2\left (\frac {\left (-15+12 x+x^2\right )^2}{9 x^2}-\log \left (-e^x+x\right )\right )} \, dx+\int \left (-\frac {360}{\left (225-360 x+114 x^2+24 x^3+x^4-9 x^2 \log \left (-e^x+x\right )\right ) \log ^2\left (\frac {\left (-15+12 x+x^2\right )^2}{9 x^2}-\log \left (-e^x+x\right )\right )}+\frac {450}{x \left (225-360 x+114 x^2+24 x^3+x^4-9 x^2 \log \left (-e^x+x\right )\right ) \log ^2\left (\frac {\left (-15+12 x+x^2\right )^2}{9 x^2}-\log \left (-e^x+x\right )\right )}-\frac {15 x^2}{\left (225-360 x+114 x^2+24 x^3+x^4-9 x^2 \log \left (-e^x+x\right )\right ) \log ^2\left (\frac {\left (-15+12 x+x^2\right )^2}{9 x^2}-\log \left (-e^x+x\right )\right )}-\frac {2 x^3}{\left (225-360 x+114 x^2+24 x^3+x^4-9 x^2 \log \left (-e^x+x\right )\right ) \log ^2\left (\frac {\left (-15+12 x+x^2\right )^2}{9 x^2}-\log \left (-e^x+x\right )\right )}\right ) \, dx \\ & = -x-2 \int \frac {x^3}{\left (225-360 x+114 x^2+24 x^3+x^4-9 x^2 \log \left (-e^x+x\right )\right ) \log ^2\left (\frac {\left (-15+12 x+x^2\right )^2}{9 x^2}-\log \left (-e^x+x\right )\right )} \, dx-9 \int \frac {x^2}{\left (e^x-x\right ) \left (225-360 x+114 x^2+24 x^3+x^4-9 x^2 \log \left (-e^x+x\right )\right ) \log ^2\left (\frac {\left (-15+12 x+x^2\right )^2}{9 x^2}-\log \left (-e^x+x\right )\right )} \, dx+9 \int \frac {x^3}{\left (e^x-x\right ) \left (225-360 x+114 x^2+24 x^3+x^4-9 x^2 \log \left (-e^x+x\right )\right ) \log ^2\left (\frac {\left (-15+12 x+x^2\right )^2}{9 x^2}-\log \left (-e^x+x\right )\right )} \, dx-15 \int \frac {x^2}{\left (225-360 x+114 x^2+24 x^3+x^4-9 x^2 \log \left (-e^x+x\right )\right ) \log ^2\left (\frac {\left (-15+12 x+x^2\right )^2}{9 x^2}-\log \left (-e^x+x\right )\right )} \, dx-360 \int \frac {1}{\left (225-360 x+114 x^2+24 x^3+x^4-9 x^2 \log \left (-e^x+x\right )\right ) \log ^2\left (\frac {\left (-15+12 x+x^2\right )^2}{9 x^2}-\log \left (-e^x+x\right )\right )} \, dx+450 \int \frac {1}{x \left (225-360 x+114 x^2+24 x^3+x^4-9 x^2 \log \left (-e^x+x\right )\right ) \log ^2\left (\frac {\left (-15+12 x+x^2\right )^2}{9 x^2}-\log \left (-e^x+x\right )\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.90 \[ \int \frac {450 x-360 x^2+9 x^3-24 x^4-2 x^5+e^x \left (-450+360 x+15 x^3+2 x^4\right )+\left (-225 x^2+360 x^3-114 x^4-24 x^5-x^6+e^x \left (225 x-360 x^2+114 x^3+24 x^4+x^5\right )+\left (-9 e^x x^3+9 x^4\right ) \log \left (-e^x+x\right )\right ) \log ^2\left (\frac {225-360 x+114 x^2+24 x^3+x^4-9 x^2 \log \left (-e^x+x\right )}{9 x^2}\right )}{\left (225 x^2-360 x^3+114 x^4+24 x^5+x^6+e^x \left (-225 x+360 x^2-114 x^3-24 x^4-x^5\right )+\left (9 e^x x^3-9 x^4\right ) \log \left (-e^x+x\right )\right ) \log ^2\left (\frac {225-360 x+114 x^2+24 x^3+x^4-9 x^2 \log \left (-e^x+x\right )}{9 x^2}\right )} \, dx=-x+\frac {1}{\log \left (\frac {\left (-15+12 x+x^2\right )^2}{9 x^2}-\log \left (-e^x+x\right )\right )} \]

[In]

Integrate[(450*x - 360*x^2 + 9*x^3 - 24*x^4 - 2*x^5 + E^x*(-450 + 360*x + 15*x^3 + 2*x^4) + (-225*x^2 + 360*x^
3 - 114*x^4 - 24*x^5 - x^6 + E^x*(225*x - 360*x^2 + 114*x^3 + 24*x^4 + x^5) + (-9*E^x*x^3 + 9*x^4)*Log[-E^x +
x])*Log[(225 - 360*x + 114*x^2 + 24*x^3 + x^4 - 9*x^2*Log[-E^x + x])/(9*x^2)]^2)/((225*x^2 - 360*x^3 + 114*x^4
+ 24*x^5 + x^6 + E^x*(-225*x + 360*x^2 - 114*x^3 - 24*x^4 - x^5) + (9*E^x*x^3 - 9*x^4)*Log[-E^x + x])*Log[(22
5 - 360*x + 114*x^2 + 24*x^3 + x^4 - 9*x^2*Log[-E^x + x])/(9*x^2)]^2),x]

[Out]

-x + Log[(-15 + 12*x + x^2)^2/(9*x^2) - Log[-E^x + x]]^(-1)

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.05 (sec) , antiderivative size = 337, normalized size of antiderivative = 8.64

\[-x -\frac {2 i}{\pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-2 \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+\pi \operatorname {csgn}\left (i x^{2}\right )^{3}-\pi \,\operatorname {csgn}\left (\frac {i}{x^{2}}\right ) \operatorname {csgn}\left (i \left (225+x^{4}+24 x^{3}-\left (9 \ln \left (x -{\mathrm e}^{x}\right )-114\right ) x^{2}-360 x \right )\right ) \operatorname {csgn}\left (\frac {i \left (225+x^{4}+24 x^{3}-\left (9 \ln \left (x -{\mathrm e}^{x}\right )-114\right ) x^{2}-360 x \right )}{x^{2}}\right )+\pi \,\operatorname {csgn}\left (\frac {i}{x^{2}}\right ) {\operatorname {csgn}\left (\frac {i \left (225+x^{4}+24 x^{3}-\left (9 \ln \left (x -{\mathrm e}^{x}\right )-114\right ) x^{2}-360 x \right )}{x^{2}}\right )}^{2}+\pi \,\operatorname {csgn}\left (i \left (225+x^{4}+24 x^{3}-\left (9 \ln \left (x -{\mathrm e}^{x}\right )-114\right ) x^{2}-360 x \right )\right ) {\operatorname {csgn}\left (\frac {i \left (225+x^{4}+24 x^{3}-\left (9 \ln \left (x -{\mathrm e}^{x}\right )-114\right ) x^{2}-360 x \right )}{x^{2}}\right )}^{2}-\pi {\operatorname {csgn}\left (\frac {i \left (225+x^{4}+24 x^{3}-\left (9 \ln \left (x -{\mathrm e}^{x}\right )-114\right ) x^{2}-360 x \right )}{x^{2}}\right )}^{3}+4 i \ln \left (3\right )+4 i \ln \left (x \right )-2 i \ln \left (225+x^{4}+24 x^{3}+\left (-9 \ln \left (x -{\mathrm e}^{x}\right )+114\right ) x^{2}-360 x \right )}\]

[In]

int((((-9*exp(x)*x^3+9*x^4)*ln(x-exp(x))+(x^5+24*x^4+114*x^3-360*x^2+225*x)*exp(x)-x^6-24*x^5-114*x^4+360*x^3-
225*x^2)*ln(1/9*(-9*x^2*ln(x-exp(x))+x^4+24*x^3+114*x^2-360*x+225)/x^2)^2+(2*x^4+15*x^3+360*x-450)*exp(x)-2*x^
5-24*x^4+9*x^3-360*x^2+450*x)/((9*exp(x)*x^3-9*x^4)*ln(x-exp(x))+(-x^5-24*x^4-114*x^3+360*x^2-225*x)*exp(x)+x^
6+24*x^5+114*x^4-360*x^3+225*x^2)/ln(1/9*(-9*x^2*ln(x-exp(x))+x^4+24*x^3+114*x^2-360*x+225)/x^2)^2,x)

[Out]

-x-2*I/(Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2+Pi*csgn(I*x^2)^3-Pi*csgn(I/x^2)*csgn(I*(225+x^
4+24*x^3-(9*ln(x-exp(x))-114)*x^2-360*x))*csgn(I/x^2*(225+x^4+24*x^3-(9*ln(x-exp(x))-114)*x^2-360*x))+Pi*csgn(
I/x^2)*csgn(I/x^2*(225+x^4+24*x^3-(9*ln(x-exp(x))-114)*x^2-360*x))^2+Pi*csgn(I*(225+x^4+24*x^3-(9*ln(x-exp(x))
-114)*x^2-360*x))*csgn(I/x^2*(225+x^4+24*x^3-(9*ln(x-exp(x))-114)*x^2-360*x))^2-Pi*csgn(I/x^2*(225+x^4+24*x^3-
(9*ln(x-exp(x))-114)*x^2-360*x))^3+4*I*ln(3)+4*I*ln(x)-2*I*ln(225+x^4+24*x^3+(-9*ln(x-exp(x))+114)*x^2-360*x))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (34) = 68\).

Time = 0.25 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.05 \[ \int \frac {450 x-360 x^2+9 x^3-24 x^4-2 x^5+e^x \left (-450+360 x+15 x^3+2 x^4\right )+\left (-225 x^2+360 x^3-114 x^4-24 x^5-x^6+e^x \left (225 x-360 x^2+114 x^3+24 x^4+x^5\right )+\left (-9 e^x x^3+9 x^4\right ) \log \left (-e^x+x\right )\right ) \log ^2\left (\frac {225-360 x+114 x^2+24 x^3+x^4-9 x^2 \log \left (-e^x+x\right )}{9 x^2}\right )}{\left (225 x^2-360 x^3+114 x^4+24 x^5+x^6+e^x \left (-225 x+360 x^2-114 x^3-24 x^4-x^5\right )+\left (9 e^x x^3-9 x^4\right ) \log \left (-e^x+x\right )\right ) \log ^2\left (\frac {225-360 x+114 x^2+24 x^3+x^4-9 x^2 \log \left (-e^x+x\right )}{9 x^2}\right )} \, dx=-\frac {x \log \left (\frac {x^{4} + 24 \, x^{3} - 9 \, x^{2} \log \left (x - e^{x}\right ) + 114 \, x^{2} - 360 \, x + 225}{9 \, x^{2}}\right ) - 1}{\log \left (\frac {x^{4} + 24 \, x^{3} - 9 \, x^{2} \log \left (x - e^{x}\right ) + 114 \, x^{2} - 360 \, x + 225}{9 \, x^{2}}\right )} \]

[In]

integrate((((-9*exp(x)*x^3+9*x^4)*log(x-exp(x))+(x^5+24*x^4+114*x^3-360*x^2+225*x)*exp(x)-x^6-24*x^5-114*x^4+3
60*x^3-225*x^2)*log(1/9*(-9*x^2*log(x-exp(x))+x^4+24*x^3+114*x^2-360*x+225)/x^2)^2+(2*x^4+15*x^3+360*x-450)*ex
p(x)-2*x^5-24*x^4+9*x^3-360*x^2+450*x)/((9*exp(x)*x^3-9*x^4)*log(x-exp(x))+(-x^5-24*x^4-114*x^3+360*x^2-225*x)
*exp(x)+x^6+24*x^5+114*x^4-360*x^3+225*x^2)/log(1/9*(-9*x^2*log(x-exp(x))+x^4+24*x^3+114*x^2-360*x+225)/x^2)^2
,x, algorithm="fricas")

[Out]

-(x*log(1/9*(x^4 + 24*x^3 - 9*x^2*log(x - e^x) + 114*x^2 - 360*x + 225)/x^2) - 1)/log(1/9*(x^4 + 24*x^3 - 9*x^
2*log(x - e^x) + 114*x^2 - 360*x + 225)/x^2)

Sympy [A] (verification not implemented)

Time = 57.62 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.05 \[ \int \frac {450 x-360 x^2+9 x^3-24 x^4-2 x^5+e^x \left (-450+360 x+15 x^3+2 x^4\right )+\left (-225 x^2+360 x^3-114 x^4-24 x^5-x^6+e^x \left (225 x-360 x^2+114 x^3+24 x^4+x^5\right )+\left (-9 e^x x^3+9 x^4\right ) \log \left (-e^x+x\right )\right ) \log ^2\left (\frac {225-360 x+114 x^2+24 x^3+x^4-9 x^2 \log \left (-e^x+x\right )}{9 x^2}\right )}{\left (225 x^2-360 x^3+114 x^4+24 x^5+x^6+e^x \left (-225 x+360 x^2-114 x^3-24 x^4-x^5\right )+\left (9 e^x x^3-9 x^4\right ) \log \left (-e^x+x\right )\right ) \log ^2\left (\frac {225-360 x+114 x^2+24 x^3+x^4-9 x^2 \log \left (-e^x+x\right )}{9 x^2}\right )} \, dx=- x + \frac {1}{\log {\left (\frac {\frac {x^{4}}{9} + \frac {8 x^{3}}{3} - x^{2} \log {\left (x - e^{x} \right )} + \frac {38 x^{2}}{3} - 40 x + 25}{x^{2}} \right )}} \]

[In]

integrate((((-9*exp(x)*x**3+9*x**4)*ln(x-exp(x))+(x**5+24*x**4+114*x**3-360*x**2+225*x)*exp(x)-x**6-24*x**5-11
4*x**4+360*x**3-225*x**2)*ln(1/9*(-9*x**2*ln(x-exp(x))+x**4+24*x**3+114*x**2-360*x+225)/x**2)**2+(2*x**4+15*x*
*3+360*x-450)*exp(x)-2*x**5-24*x**4+9*x**3-360*x**2+450*x)/((9*exp(x)*x**3-9*x**4)*ln(x-exp(x))+(-x**5-24*x**4
-114*x**3+360*x**2-225*x)*exp(x)+x**6+24*x**5+114*x**4-360*x**3+225*x**2)/ln(1/9*(-9*x**2*ln(x-exp(x))+x**4+24
*x**3+114*x**2-360*x+225)/x**2)**2,x)

[Out]

-x + 1/log((x**4/9 + 8*x**3/3 - x**2*log(x - exp(x)) + 38*x**2/3 - 40*x + 25)/x**2)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (34) = 68\).

Time = 0.52 (sec) , antiderivative size = 92, normalized size of antiderivative = 2.36 \[ \int \frac {450 x-360 x^2+9 x^3-24 x^4-2 x^5+e^x \left (-450+360 x+15 x^3+2 x^4\right )+\left (-225 x^2+360 x^3-114 x^4-24 x^5-x^6+e^x \left (225 x-360 x^2+114 x^3+24 x^4+x^5\right )+\left (-9 e^x x^3+9 x^4\right ) \log \left (-e^x+x\right )\right ) \log ^2\left (\frac {225-360 x+114 x^2+24 x^3+x^4-9 x^2 \log \left (-e^x+x\right )}{9 x^2}\right )}{\left (225 x^2-360 x^3+114 x^4+24 x^5+x^6+e^x \left (-225 x+360 x^2-114 x^3-24 x^4-x^5\right )+\left (9 e^x x^3-9 x^4\right ) \log \left (-e^x+x\right )\right ) \log ^2\left (\frac {225-360 x+114 x^2+24 x^3+x^4-9 x^2 \log \left (-e^x+x\right )}{9 x^2}\right )} \, dx=-\frac {2 \, x \log \left (3\right ) - x \log \left (x^{4} + 24 \, x^{3} - 9 \, x^{2} \log \left (x - e^{x}\right ) + 114 \, x^{2} - 360 \, x + 225\right ) + 2 \, x \log \left (x\right ) + 1}{2 \, \log \left (3\right ) - \log \left (x^{4} + 24 \, x^{3} - 9 \, x^{2} \log \left (x - e^{x}\right ) + 114 \, x^{2} - 360 \, x + 225\right ) + 2 \, \log \left (x\right )} \]

[In]

integrate((((-9*exp(x)*x^3+9*x^4)*log(x-exp(x))+(x^5+24*x^4+114*x^3-360*x^2+225*x)*exp(x)-x^6-24*x^5-114*x^4+3
60*x^3-225*x^2)*log(1/9*(-9*x^2*log(x-exp(x))+x^4+24*x^3+114*x^2-360*x+225)/x^2)^2+(2*x^4+15*x^3+360*x-450)*ex
p(x)-2*x^5-24*x^4+9*x^3-360*x^2+450*x)/((9*exp(x)*x^3-9*x^4)*log(x-exp(x))+(-x^5-24*x^4-114*x^3+360*x^2-225*x)
*exp(x)+x^6+24*x^5+114*x^4-360*x^3+225*x^2)/log(1/9*(-9*x^2*log(x-exp(x))+x^4+24*x^3+114*x^2-360*x+225)/x^2)^2
,x, algorithm="maxima")

[Out]

-(2*x*log(3) - x*log(x^4 + 24*x^3 - 9*x^2*log(x - e^x) + 114*x^2 - 360*x + 225) + 2*x*log(x) + 1)/(2*log(3) -
log(x^4 + 24*x^3 - 9*x^2*log(x - e^x) + 114*x^2 - 360*x + 225) + 2*log(x))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (34) = 68\).

Time = 2.26 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.26 \[ \int \frac {450 x-360 x^2+9 x^3-24 x^4-2 x^5+e^x \left (-450+360 x+15 x^3+2 x^4\right )+\left (-225 x^2+360 x^3-114 x^4-24 x^5-x^6+e^x \left (225 x-360 x^2+114 x^3+24 x^4+x^5\right )+\left (-9 e^x x^3+9 x^4\right ) \log \left (-e^x+x\right )\right ) \log ^2\left (\frac {225-360 x+114 x^2+24 x^3+x^4-9 x^2 \log \left (-e^x+x\right )}{9 x^2}\right )}{\left (225 x^2-360 x^3+114 x^4+24 x^5+x^6+e^x \left (-225 x+360 x^2-114 x^3-24 x^4-x^5\right )+\left (9 e^x x^3-9 x^4\right ) \log \left (-e^x+x\right )\right ) \log ^2\left (\frac {225-360 x+114 x^2+24 x^3+x^4-9 x^2 \log \left (-e^x+x\right )}{9 x^2}\right )} \, dx=-\frac {x \log \left (x^{4} + 24 \, x^{3} - 9 \, x^{2} \log \left (x - e^{x}\right ) + 114 \, x^{2} - 360 \, x + 225\right ) - x \log \left (9 \, x^{2}\right ) - 1}{\log \left (x^{4} + 24 \, x^{3} - 9 \, x^{2} \log \left (x - e^{x}\right ) + 114 \, x^{2} - 360 \, x + 225\right ) - \log \left (9 \, x^{2}\right )} \]

[In]

integrate((((-9*exp(x)*x^3+9*x^4)*log(x-exp(x))+(x^5+24*x^4+114*x^3-360*x^2+225*x)*exp(x)-x^6-24*x^5-114*x^4+3
60*x^3-225*x^2)*log(1/9*(-9*x^2*log(x-exp(x))+x^4+24*x^3+114*x^2-360*x+225)/x^2)^2+(2*x^4+15*x^3+360*x-450)*ex
p(x)-2*x^5-24*x^4+9*x^3-360*x^2+450*x)/((9*exp(x)*x^3-9*x^4)*log(x-exp(x))+(-x^5-24*x^4-114*x^3+360*x^2-225*x)
*exp(x)+x^6+24*x^5+114*x^4-360*x^3+225*x^2)/log(1/9*(-9*x^2*log(x-exp(x))+x^4+24*x^3+114*x^2-360*x+225)/x^2)^2
,x, algorithm="giac")

[Out]

-(x*log(x^4 + 24*x^3 - 9*x^2*log(x - e^x) + 114*x^2 - 360*x + 225) - x*log(9*x^2) - 1)/(log(x^4 + 24*x^3 - 9*x
^2*log(x - e^x) + 114*x^2 - 360*x + 225) - log(9*x^2))

Mupad [B] (verification not implemented)

Time = 8.75 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.08 \[ \int \frac {450 x-360 x^2+9 x^3-24 x^4-2 x^5+e^x \left (-450+360 x+15 x^3+2 x^4\right )+\left (-225 x^2+360 x^3-114 x^4-24 x^5-x^6+e^x \left (225 x-360 x^2+114 x^3+24 x^4+x^5\right )+\left (-9 e^x x^3+9 x^4\right ) \log \left (-e^x+x\right )\right ) \log ^2\left (\frac {225-360 x+114 x^2+24 x^3+x^4-9 x^2 \log \left (-e^x+x\right )}{9 x^2}\right )}{\left (225 x^2-360 x^3+114 x^4+24 x^5+x^6+e^x \left (-225 x+360 x^2-114 x^3-24 x^4-x^5\right )+\left (9 e^x x^3-9 x^4\right ) \log \left (-e^x+x\right )\right ) \log ^2\left (\frac {225-360 x+114 x^2+24 x^3+x^4-9 x^2 \log \left (-e^x+x\right )}{9 x^2}\right )} \, dx=\frac {1}{\ln \left (\frac {114\,x^2-9\,x^2\,\ln \left (x-{\mathrm {e}}^x\right )-360\,x+24\,x^3+x^4+225}{9\,x^2}\right )}-x \]

[In]

int(-(log(((38*x^2)/3 - x^2*log(x - exp(x)) - 40*x + (8*x^3)/3 + x^4/9 + 25)/x^2)^2*(log(x - exp(x))*(9*x^3*ex
p(x) - 9*x^4) + 225*x^2 - 360*x^3 + 114*x^4 + 24*x^5 + x^6 - exp(x)*(225*x - 360*x^2 + 114*x^3 + 24*x^4 + x^5)
) - 450*x + 360*x^2 - 9*x^3 + 24*x^4 + 2*x^5 - exp(x)*(360*x + 15*x^3 + 2*x^4 - 450))/(log(((38*x^2)/3 - x^2*l
og(x - exp(x)) - 40*x + (8*x^3)/3 + x^4/9 + 25)/x^2)^2*(log(x - exp(x))*(9*x^3*exp(x) - 9*x^4) + 225*x^2 - 360
*x^3 + 114*x^4 + 24*x^5 + x^6 - exp(x)*(225*x - 360*x^2 + 114*x^3 + 24*x^4 + x^5))),x)

[Out]

1/log((114*x^2 - 9*x^2*log(x - exp(x)) - 360*x + 24*x^3 + x^4 + 225)/(9*x^2)) - x