Integrand size = 363, antiderivative size = 33 \[ \int \frac {6 x^2+6 e^x x^2+\left (-2 x^2+e^x \left (-2 x^2+x^3\right )\right ) \log \left (x^2\right )+\left (16-3 x+e^x \left (16+x-3 x^2\right )\right ) \log ^2\left (x^2\right )+\left (-2 x^2-2 e^x x^2+\left (x^2+e^x x^2\right ) \log \left (x^2\right )+\left (-4-4 e^x\right ) \log ^2\left (x^2\right )\right ) \log \left (x+e^x x\right )}{9 x^4+9 e^x x^4+\left (72 x^2-54 x^3+e^x \left (72 x^2-54 x^3\right )\right ) \log \left (x^2\right )+\left (144-216 x+81 x^2+e^x \left (144-216 x+81 x^2\right )\right ) \log ^2\left (x^2\right )+\left (-6 x^4-6 e^x x^4+\left (-48 x^2+36 x^3+e^x \left (-48 x^2+36 x^3\right )\right ) \log \left (x^2\right )+\left (-96+144 x-54 x^2+e^x \left (-96+144 x-54 x^2\right )\right ) \log ^2\left (x^2\right )\right ) \log \left (x+e^x x\right )+\left (x^4+e^x x^4+\left (8 x^2-6 x^3+e^x \left (8 x^2-6 x^3\right )\right ) \log \left (x^2\right )+\left (16-24 x+9 x^2+e^x \left (16-24 x+9 x^2\right )\right ) \log ^2\left (x^2\right )\right ) \log ^2\left (x+e^x x\right )} \, dx=\frac {x}{\left (4-3 x+\frac {x^2}{\log \left (x^2\right )}\right ) \left (3-\log \left (x+e^x x\right )\right )} \] Output:
x/(4+x^2/ln(x^2)-3*x)/(3-ln(exp(x)*x+x))
Time = 0.15 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int \frac {6 x^2+6 e^x x^2+\left (-2 x^2+e^x \left (-2 x^2+x^3\right )\right ) \log \left (x^2\right )+\left (16-3 x+e^x \left (16+x-3 x^2\right )\right ) \log ^2\left (x^2\right )+\left (-2 x^2-2 e^x x^2+\left (x^2+e^x x^2\right ) \log \left (x^2\right )+\left (-4-4 e^x\right ) \log ^2\left (x^2\right )\right ) \log \left (x+e^x x\right )}{9 x^4+9 e^x x^4+\left (72 x^2-54 x^3+e^x \left (72 x^2-54 x^3\right )\right ) \log \left (x^2\right )+\left (144-216 x+81 x^2+e^x \left (144-216 x+81 x^2\right )\right ) \log ^2\left (x^2\right )+\left (-6 x^4-6 e^x x^4+\left (-48 x^2+36 x^3+e^x \left (-48 x^2+36 x^3\right )\right ) \log \left (x^2\right )+\left (-96+144 x-54 x^2+e^x \left (-96+144 x-54 x^2\right )\right ) \log ^2\left (x^2\right )\right ) \log \left (x+e^x x\right )+\left (x^4+e^x x^4+\left (8 x^2-6 x^3+e^x \left (8 x^2-6 x^3\right )\right ) \log \left (x^2\right )+\left (16-24 x+9 x^2+e^x \left (16-24 x+9 x^2\right )\right ) \log ^2\left (x^2\right )\right ) \log ^2\left (x+e^x x\right )} \, dx=-\frac {x \log \left (x^2\right )}{\left (-3+\log \left (\left (1+e^x\right ) x\right )\right ) \left (x^2+(4-3 x) \log \left (x^2\right )\right )} \] Input:
Integrate[(6*x^2 + 6*E^x*x^2 + (-2*x^2 + E^x*(-2*x^2 + x^3))*Log[x^2] + (1 6 - 3*x + E^x*(16 + x - 3*x^2))*Log[x^2]^2 + (-2*x^2 - 2*E^x*x^2 + (x^2 + E^x*x^2)*Log[x^2] + (-4 - 4*E^x)*Log[x^2]^2)*Log[x + E^x*x])/(9*x^4 + 9*E^ x*x^4 + (72*x^2 - 54*x^3 + E^x*(72*x^2 - 54*x^3))*Log[x^2] + (144 - 216*x + 81*x^2 + E^x*(144 - 216*x + 81*x^2))*Log[x^2]^2 + (-6*x^4 - 6*E^x*x^4 + (-48*x^2 + 36*x^3 + E^x*(-48*x^2 + 36*x^3))*Log[x^2] + (-96 + 144*x - 54*x ^2 + E^x*(-96 + 144*x - 54*x^2))*Log[x^2]^2)*Log[x + E^x*x] + (x^4 + E^x*x ^4 + (8*x^2 - 6*x^3 + E^x*(8*x^2 - 6*x^3))*Log[x^2] + (16 - 24*x + 9*x^2 + E^x*(16 - 24*x + 9*x^2))*Log[x^2]^2)*Log[x + E^x*x]^2),x]
Output:
-((x*Log[x^2])/((-3 + Log[(1 + E^x)*x])*(x^2 + (4 - 3*x)*Log[x^2])))
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {6 e^x x^2+6 x^2+\left (e^x \left (-3 x^2+x+16\right )-3 x+16\right ) \log ^2\left (x^2\right )+\left (-2 e^x x^2-2 x^2+\left (-4 e^x-4\right ) \log ^2\left (x^2\right )+\left (e^x x^2+x^2\right ) \log \left (x^2\right )\right ) \log \left (e^x x+x\right )+\left (e^x \left (x^3-2 x^2\right )-2 x^2\right ) \log \left (x^2\right )}{9 e^x x^4+9 x^4+\left (81 x^2+e^x \left (81 x^2-216 x+144\right )-216 x+144\right ) \log ^2\left (x^2\right )+\left (-54 x^3+72 x^2+e^x \left (72 x^2-54 x^3\right )\right ) \log \left (x^2\right )+\left (e^x x^4+x^4+\left (9 x^2+e^x \left (9 x^2-24 x+16\right )-24 x+16\right ) \log ^2\left (x^2\right )+\left (-6 x^3+8 x^2+e^x \left (8 x^2-6 x^3\right )\right ) \log \left (x^2\right )\right ) \log ^2\left (e^x x+x\right )+\left (-6 e^x x^4-6 x^4+\left (-54 x^2+e^x \left (-54 x^2+144 x-96\right )+144 x-96\right ) \log ^2\left (x^2\right )+\left (36 x^3-48 x^2+e^x \left (36 x^3-48 x^2\right )\right ) \log \left (x^2\right )\right ) \log \left (e^x x+x\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {6 \left (e^x+1\right ) x^2+\left (e^x \left (-3 x^2+x+16\right )-3 x+16\right ) \log ^2\left (x^2\right )+\left (e^x+1\right ) \log \left (\left (e^x+1\right ) x\right ) \left (-2 x^2-4 \log ^2\left (x^2\right )+x^2 \log \left (x^2\right )\right )+\left (e^x (x-2)-2\right ) x^2 \log \left (x^2\right )}{\left (e^x+1\right ) \left (3-\log \left (\left (e^x+1\right ) x\right )\right )^2 \left (x^2+(4-3 x) \log \left (x^2\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {6 x^2-3 x^2 \log ^2\left (x^2\right )+x \log ^2\left (x^2\right )-4 \log \left (\left (e^x+1\right ) x\right ) \log ^2\left (x^2\right )+16 \log ^2\left (x^2\right )-2 x^2 \log \left (\left (e^x+1\right ) x\right )+x^2 \log \left (\left (e^x+1\right ) x\right ) \log \left (x^2\right )-2 x^2 \log \left (x^2\right )+x^3 \log \left (x^2\right )}{\left (\log \left (\left (e^x+1\right ) x\right )-3\right )^2 \left (x^2-3 x \log \left (x^2\right )+4 \log \left (x^2\right )\right )^2}-\frac {x \log \left (x^2\right )}{\left (e^x+1\right ) \left (\log \left (\left (e^x+1\right ) x\right )-3\right )^2 \left (x^2-3 x \log \left (x^2\right )+4 \log \left (x^2\right )\right )}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \left (\frac {6 x^2-3 x^2 \log ^2\left (x^2\right )+x \log ^2\left (x^2\right )-4 \log \left (\left (e^x+1\right ) x\right ) \log ^2\left (x^2\right )+16 \log ^2\left (x^2\right )-2 x^2 \log \left (\left (e^x+1\right ) x\right )+x^2 \log \left (\left (e^x+1\right ) x\right ) \log \left (x^2\right )-2 x^2 \log \left (x^2\right )+x^3 \log \left (x^2\right )}{\left (\log \left (\left (e^x+1\right ) x\right )-3\right )^2 \left (x^2-3 x \log \left (x^2\right )+4 \log \left (x^2\right )\right )^2}-\frac {x \log \left (x^2\right )}{\left (e^x+1\right ) \left (\log \left (\left (e^x+1\right ) x\right )-3\right )^2 \left (x^2-3 x \log \left (x^2\right )+4 \log \left (x^2\right )\right )}\right )dx\) |
Input:
Int[(6*x^2 + 6*E^x*x^2 + (-2*x^2 + E^x*(-2*x^2 + x^3))*Log[x^2] + (16 - 3* x + E^x*(16 + x - 3*x^2))*Log[x^2]^2 + (-2*x^2 - 2*E^x*x^2 + (x^2 + E^x*x^ 2)*Log[x^2] + (-4 - 4*E^x)*Log[x^2]^2)*Log[x + E^x*x])/(9*x^4 + 9*E^x*x^4 + (72*x^2 - 54*x^3 + E^x*(72*x^2 - 54*x^3))*Log[x^2] + (144 - 216*x + 81*x ^2 + E^x*(144 - 216*x + 81*x^2))*Log[x^2]^2 + (-6*x^4 - 6*E^x*x^4 + (-48*x ^2 + 36*x^3 + E^x*(-48*x^2 + 36*x^3))*Log[x^2] + (-96 + 144*x - 54*x^2 + E ^x*(-96 + 144*x - 54*x^2))*Log[x^2]^2)*Log[x + E^x*x] + (x^4 + E^x*x^4 + ( 8*x^2 - 6*x^3 + E^x*(8*x^2 - 6*x^3))*Log[x^2] + (16 - 24*x + 9*x^2 + E^x*( 16 - 24*x + 9*x^2))*Log[x^2]^2)*Log[x + E^x*x]^2),x]
Output:
$Aborted
Time = 57.86 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.15
| method | result | size |
| parallelrisch | \(-\frac {x \ln \left (x^{2}\right )}{\left (x^{2}-3 x \ln \left (x^{2}\right )+4 \ln \left (x^{2}\right )\right ) \left (\ln \left (x \left ({\mathrm e}^{x}+1\right )\right )-3\right )}\) | \(38\) |
| risch | \(\frac {2 i x \left (\pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-2 \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+\pi \operatorname {csgn}\left (i x^{2}\right )^{3}+4 i \ln \left (x \right )\right )}{\left (3 \pi x \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-6 \pi x \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+3 \pi x \operatorname {csgn}\left (i x^{2}\right )^{3}-4 \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )+8 \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}-4 \pi \operatorname {csgn}\left (i x^{2}\right )^{3}-2 i x^{2}+12 i \ln \left (x \right ) x -16 i \ln \left (x \right )\right ) \left (\pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i \left ({\mathrm e}^{x}+1\right )\right ) \operatorname {csgn}\left (i x \left ({\mathrm e}^{x}+1\right )\right )-\pi \,\operatorname {csgn}\left (i x \right ) {\operatorname {csgn}\left (i x \left ({\mathrm e}^{x}+1\right )\right )}^{2}-\pi \,\operatorname {csgn}\left (i \left ({\mathrm e}^{x}+1\right )\right ) {\operatorname {csgn}\left (i x \left ({\mathrm e}^{x}+1\right )\right )}^{2}+\pi {\operatorname {csgn}\left (i x \left ({\mathrm e}^{x}+1\right )\right )}^{3}+2 i \ln \left (x \right )+2 i \ln \left ({\mathrm e}^{x}+1\right )-6 i\right )}\) | \(266\) |
Input:
int((((-4*exp(x)-4)*ln(x^2)^2+(exp(x)*x^2+x^2)*ln(x^2)-2*exp(x)*x^2-2*x^2) *ln(exp(x)*x+x)+((-3*x^2+x+16)*exp(x)-3*x+16)*ln(x^2)^2+((x^3-2*x^2)*exp(x )-2*x^2)*ln(x^2)+6*exp(x)*x^2+6*x^2)/((((9*x^2-24*x+16)*exp(x)+9*x^2-24*x+ 16)*ln(x^2)^2+((-6*x^3+8*x^2)*exp(x)-6*x^3+8*x^2)*ln(x^2)+exp(x)*x^4+x^4)* ln(exp(x)*x+x)^2+(((-54*x^2+144*x-96)*exp(x)-54*x^2+144*x-96)*ln(x^2)^2+(( 36*x^3-48*x^2)*exp(x)+36*x^3-48*x^2)*ln(x^2)-6*exp(x)*x^4-6*x^4)*ln(exp(x) *x+x)+((81*x^2-216*x+144)*exp(x)+81*x^2-216*x+144)*ln(x^2)^2+((-54*x^3+72* x^2)*exp(x)-54*x^3+72*x^2)*ln(x^2)+9*exp(x)*x^4+9*x^4),x,method=_RETURNVER BOSE)
Output:
-x*ln(x^2)/(x^2-3*x*ln(x^2)+4*ln(x^2))/(ln(x*(exp(x)+1))-3)
Time = 0.10 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.48 \[ \int \frac {6 x^2+6 e^x x^2+\left (-2 x^2+e^x \left (-2 x^2+x^3\right )\right ) \log \left (x^2\right )+\left (16-3 x+e^x \left (16+x-3 x^2\right )\right ) \log ^2\left (x^2\right )+\left (-2 x^2-2 e^x x^2+\left (x^2+e^x x^2\right ) \log \left (x^2\right )+\left (-4-4 e^x\right ) \log ^2\left (x^2\right )\right ) \log \left (x+e^x x\right )}{9 x^4+9 e^x x^4+\left (72 x^2-54 x^3+e^x \left (72 x^2-54 x^3\right )\right ) \log \left (x^2\right )+\left (144-216 x+81 x^2+e^x \left (144-216 x+81 x^2\right )\right ) \log ^2\left (x^2\right )+\left (-6 x^4-6 e^x x^4+\left (-48 x^2+36 x^3+e^x \left (-48 x^2+36 x^3\right )\right ) \log \left (x^2\right )+\left (-96+144 x-54 x^2+e^x \left (-96+144 x-54 x^2\right )\right ) \log ^2\left (x^2\right )\right ) \log \left (x+e^x x\right )+\left (x^4+e^x x^4+\left (8 x^2-6 x^3+e^x \left (8 x^2-6 x^3\right )\right ) \log \left (x^2\right )+\left (16-24 x+9 x^2+e^x \left (16-24 x+9 x^2\right )\right ) \log ^2\left (x^2\right )\right ) \log ^2\left (x+e^x x\right )} \, dx=\frac {x \log \left (x^{2}\right )}{3 \, x^{2} - 3 \, {\left (3 \, x - 4\right )} \log \left (x^{2}\right ) - {\left (x^{2} - {\left (3 \, x - 4\right )} \log \left (x^{2}\right )\right )} \log \left (x e^{x} + x\right )} \] Input:
integrate((((-4*exp(x)-4)*log(x^2)^2+(exp(x)*x^2+x^2)*log(x^2)-2*exp(x)*x^ 2-2*x^2)*log(exp(x)*x+x)+((-3*x^2+x+16)*exp(x)-3*x+16)*log(x^2)^2+((x^3-2* x^2)*exp(x)-2*x^2)*log(x^2)+6*exp(x)*x^2+6*x^2)/((((9*x^2-24*x+16)*exp(x)+ 9*x^2-24*x+16)*log(x^2)^2+((-6*x^3+8*x^2)*exp(x)-6*x^3+8*x^2)*log(x^2)+exp (x)*x^4+x^4)*log(exp(x)*x+x)^2+(((-54*x^2+144*x-96)*exp(x)-54*x^2+144*x-96 )*log(x^2)^2+((36*x^3-48*x^2)*exp(x)+36*x^3-48*x^2)*log(x^2)-6*exp(x)*x^4- 6*x^4)*log(exp(x)*x+x)+((81*x^2-216*x+144)*exp(x)+81*x^2-216*x+144)*log(x^ 2)^2+((-54*x^3+72*x^2)*exp(x)-54*x^3+72*x^2)*log(x^2)+9*exp(x)*x^4+9*x^4), x, algorithm="fricas")
Output:
x*log(x^2)/(3*x^2 - 3*(3*x - 4)*log(x^2) - (x^2 - (3*x - 4)*log(x^2))*log( x*e^x + x))
Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (24) = 48\).
Time = 0.34 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.64 \[ \int \frac {6 x^2+6 e^x x^2+\left (-2 x^2+e^x \left (-2 x^2+x^3\right )\right ) \log \left (x^2\right )+\left (16-3 x+e^x \left (16+x-3 x^2\right )\right ) \log ^2\left (x^2\right )+\left (-2 x^2-2 e^x x^2+\left (x^2+e^x x^2\right ) \log \left (x^2\right )+\left (-4-4 e^x\right ) \log ^2\left (x^2\right )\right ) \log \left (x+e^x x\right )}{9 x^4+9 e^x x^4+\left (72 x^2-54 x^3+e^x \left (72 x^2-54 x^3\right )\right ) \log \left (x^2\right )+\left (144-216 x+81 x^2+e^x \left (144-216 x+81 x^2\right )\right ) \log ^2\left (x^2\right )+\left (-6 x^4-6 e^x x^4+\left (-48 x^2+36 x^3+e^x \left (-48 x^2+36 x^3\right )\right ) \log \left (x^2\right )+\left (-96+144 x-54 x^2+e^x \left (-96+144 x-54 x^2\right )\right ) \log ^2\left (x^2\right )\right ) \log \left (x+e^x x\right )+\left (x^4+e^x x^4+\left (8 x^2-6 x^3+e^x \left (8 x^2-6 x^3\right )\right ) \log \left (x^2\right )+\left (16-24 x+9 x^2+e^x \left (16-24 x+9 x^2\right )\right ) \log ^2\left (x^2\right )\right ) \log ^2\left (x+e^x x\right )} \, dx=- \frac {x \log {\left (x^{2} \right )}}{- 3 x^{2} + 9 x \log {\left (x^{2} \right )} + \left (x^{2} - 3 x \log {\left (x^{2} \right )} + 4 \log {\left (x^{2} \right )}\right ) \log {\left (x e^{x} + x \right )} - 12 \log {\left (x^{2} \right )}} \] Input:
integrate((((-4*exp(x)-4)*ln(x**2)**2+(exp(x)*x**2+x**2)*ln(x**2)-2*exp(x) *x**2-2*x**2)*ln(exp(x)*x+x)+((-3*x**2+x+16)*exp(x)-3*x+16)*ln(x**2)**2+(( x**3-2*x**2)*exp(x)-2*x**2)*ln(x**2)+6*exp(x)*x**2+6*x**2)/((((9*x**2-24*x +16)*exp(x)+9*x**2-24*x+16)*ln(x**2)**2+((-6*x**3+8*x**2)*exp(x)-6*x**3+8* x**2)*ln(x**2)+exp(x)*x**4+x**4)*ln(exp(x)*x+x)**2+(((-54*x**2+144*x-96)*e xp(x)-54*x**2+144*x-96)*ln(x**2)**2+((36*x**3-48*x**2)*exp(x)+36*x**3-48*x **2)*ln(x**2)-6*exp(x)*x**4-6*x**4)*ln(exp(x)*x+x)+((81*x**2-216*x+144)*ex p(x)+81*x**2-216*x+144)*ln(x**2)**2+((-54*x**3+72*x**2)*exp(x)-54*x**3+72* x**2)*ln(x**2)+9*exp(x)*x**4+9*x**4),x)
Output:
-x*log(x**2)/(-3*x**2 + 9*x*log(x**2) + (x**2 - 3*x*log(x**2) + 4*log(x**2 ))*log(x*exp(x) + x) - 12*log(x**2))
Time = 0.82 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.70 \[ \int \frac {6 x^2+6 e^x x^2+\left (-2 x^2+e^x \left (-2 x^2+x^3\right )\right ) \log \left (x^2\right )+\left (16-3 x+e^x \left (16+x-3 x^2\right )\right ) \log ^2\left (x^2\right )+\left (-2 x^2-2 e^x x^2+\left (x^2+e^x x^2\right ) \log \left (x^2\right )+\left (-4-4 e^x\right ) \log ^2\left (x^2\right )\right ) \log \left (x+e^x x\right )}{9 x^4+9 e^x x^4+\left (72 x^2-54 x^3+e^x \left (72 x^2-54 x^3\right )\right ) \log \left (x^2\right )+\left (144-216 x+81 x^2+e^x \left (144-216 x+81 x^2\right )\right ) \log ^2\left (x^2\right )+\left (-6 x^4-6 e^x x^4+\left (-48 x^2+36 x^3+e^x \left (-48 x^2+36 x^3\right )\right ) \log \left (x^2\right )+\left (-96+144 x-54 x^2+e^x \left (-96+144 x-54 x^2\right )\right ) \log ^2\left (x^2\right )\right ) \log \left (x+e^x x\right )+\left (x^4+e^x x^4+\left (8 x^2-6 x^3+e^x \left (8 x^2-6 x^3\right )\right ) \log \left (x^2\right )+\left (16-24 x+9 x^2+e^x \left (16-24 x+9 x^2\right )\right ) \log ^2\left (x^2\right )\right ) \log ^2\left (x+e^x x\right )} \, dx=\frac {2 \, x \log \left (x\right )}{2 \, {\left (3 \, x - 4\right )} \log \left (x\right )^{2} + 3 \, x^{2} - {\left (x^{2} + 18 \, x - 24\right )} \log \left (x\right ) - {\left (x^{2} - 2 \, {\left (3 \, x - 4\right )} \log \left (x\right )\right )} \log \left (e^{x} + 1\right )} \] Input:
integrate((((-4*exp(x)-4)*log(x^2)^2+(exp(x)*x^2+x^2)*log(x^2)-2*exp(x)*x^ 2-2*x^2)*log(exp(x)*x+x)+((-3*x^2+x+16)*exp(x)-3*x+16)*log(x^2)^2+((x^3-2* x^2)*exp(x)-2*x^2)*log(x^2)+6*exp(x)*x^2+6*x^2)/((((9*x^2-24*x+16)*exp(x)+ 9*x^2-24*x+16)*log(x^2)^2+((-6*x^3+8*x^2)*exp(x)-6*x^3+8*x^2)*log(x^2)+exp (x)*x^4+x^4)*log(exp(x)*x+x)^2+(((-54*x^2+144*x-96)*exp(x)-54*x^2+144*x-96 )*log(x^2)^2+((36*x^3-48*x^2)*exp(x)+36*x^3-48*x^2)*log(x^2)-6*exp(x)*x^4- 6*x^4)*log(exp(x)*x+x)+((81*x^2-216*x+144)*exp(x)+81*x^2-216*x+144)*log(x^ 2)^2+((-54*x^3+72*x^2)*exp(x)-54*x^3+72*x^2)*log(x^2)+9*exp(x)*x^4+9*x^4), x, algorithm="maxima")
Output:
2*x*log(x)/(2*(3*x - 4)*log(x)^2 + 3*x^2 - (x^2 + 18*x - 24)*log(x) - (x^2 - 2*(3*x - 4)*log(x))*log(e^x + 1))
Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (31) = 62\).
Time = 0.78 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.09 \[ \int \frac {6 x^2+6 e^x x^2+\left (-2 x^2+e^x \left (-2 x^2+x^3\right )\right ) \log \left (x^2\right )+\left (16-3 x+e^x \left (16+x-3 x^2\right )\right ) \log ^2\left (x^2\right )+\left (-2 x^2-2 e^x x^2+\left (x^2+e^x x^2\right ) \log \left (x^2\right )+\left (-4-4 e^x\right ) \log ^2\left (x^2\right )\right ) \log \left (x+e^x x\right )}{9 x^4+9 e^x x^4+\left (72 x^2-54 x^3+e^x \left (72 x^2-54 x^3\right )\right ) \log \left (x^2\right )+\left (144-216 x+81 x^2+e^x \left (144-216 x+81 x^2\right )\right ) \log ^2\left (x^2\right )+\left (-6 x^4-6 e^x x^4+\left (-48 x^2+36 x^3+e^x \left (-48 x^2+36 x^3\right )\right ) \log \left (x^2\right )+\left (-96+144 x-54 x^2+e^x \left (-96+144 x-54 x^2\right )\right ) \log ^2\left (x^2\right )\right ) \log \left (x+e^x x\right )+\left (x^4+e^x x^4+\left (8 x^2-6 x^3+e^x \left (8 x^2-6 x^3\right )\right ) \log \left (x^2\right )+\left (16-24 x+9 x^2+e^x \left (16-24 x+9 x^2\right )\right ) \log ^2\left (x^2\right )\right ) \log ^2\left (x+e^x x\right )} \, dx=-\frac {2 \, x \log \left (x\right )}{x^{2} \log \left (x\right ) - 6 \, x \log \left (x\right )^{2} + x^{2} \log \left (e^{x} + 1\right ) - 6 \, x \log \left (x\right ) \log \left (e^{x} + 1\right ) - 3 \, x^{2} + 18 \, x \log \left (x\right ) + 8 \, \log \left (x\right )^{2} + 8 \, \log \left (x\right ) \log \left (e^{x} + 1\right ) - 24 \, \log \left (x\right )} \] Input:
integrate((((-4*exp(x)-4)*log(x^2)^2+(exp(x)*x^2+x^2)*log(x^2)-2*exp(x)*x^ 2-2*x^2)*log(exp(x)*x+x)+((-3*x^2+x+16)*exp(x)-3*x+16)*log(x^2)^2+((x^3-2* x^2)*exp(x)-2*x^2)*log(x^2)+6*exp(x)*x^2+6*x^2)/((((9*x^2-24*x+16)*exp(x)+ 9*x^2-24*x+16)*log(x^2)^2+((-6*x^3+8*x^2)*exp(x)-6*x^3+8*x^2)*log(x^2)+exp (x)*x^4+x^4)*log(exp(x)*x+x)^2+(((-54*x^2+144*x-96)*exp(x)-54*x^2+144*x-96 )*log(x^2)^2+((36*x^3-48*x^2)*exp(x)+36*x^3-48*x^2)*log(x^2)-6*exp(x)*x^4- 6*x^4)*log(exp(x)*x+x)+((81*x^2-216*x+144)*exp(x)+81*x^2-216*x+144)*log(x^ 2)^2+((-54*x^3+72*x^2)*exp(x)-54*x^3+72*x^2)*log(x^2)+9*exp(x)*x^4+9*x^4), x, algorithm="giac")
Output:
-2*x*log(x)/(x^2*log(x) - 6*x*log(x)^2 + x^2*log(e^x + 1) - 6*x*log(x)*log (e^x + 1) - 3*x^2 + 18*x*log(x) + 8*log(x)^2 + 8*log(x)*log(e^x + 1) - 24* log(x))
Timed out. \[ \int \frac {6 x^2+6 e^x x^2+\left (-2 x^2+e^x \left (-2 x^2+x^3\right )\right ) \log \left (x^2\right )+\left (16-3 x+e^x \left (16+x-3 x^2\right )\right ) \log ^2\left (x^2\right )+\left (-2 x^2-2 e^x x^2+\left (x^2+e^x x^2\right ) \log \left (x^2\right )+\left (-4-4 e^x\right ) \log ^2\left (x^2\right )\right ) \log \left (x+e^x x\right )}{9 x^4+9 e^x x^4+\left (72 x^2-54 x^3+e^x \left (72 x^2-54 x^3\right )\right ) \log \left (x^2\right )+\left (144-216 x+81 x^2+e^x \left (144-216 x+81 x^2\right )\right ) \log ^2\left (x^2\right )+\left (-6 x^4-6 e^x x^4+\left (-48 x^2+36 x^3+e^x \left (-48 x^2+36 x^3\right )\right ) \log \left (x^2\right )+\left (-96+144 x-54 x^2+e^x \left (-96+144 x-54 x^2\right )\right ) \log ^2\left (x^2\right )\right ) \log \left (x+e^x x\right )+\left (x^4+e^x x^4+\left (8 x^2-6 x^3+e^x \left (8 x^2-6 x^3\right )\right ) \log \left (x^2\right )+\left (16-24 x+9 x^2+e^x \left (16-24 x+9 x^2\right )\right ) \log ^2\left (x^2\right )\right ) \log ^2\left (x+e^x x\right )} \, dx=\int \frac {6\,x^2\,{\mathrm {e}}^x-\ln \left (x^2\right )\,\left ({\mathrm {e}}^x\,\left (2\,x^2-x^3\right )+2\,x^2\right )+{\ln \left (x^2\right )}^2\,\left ({\mathrm {e}}^x\,\left (-3\,x^2+x+16\right )-3\,x+16\right )-\ln \left (x+x\,{\mathrm {e}}^x\right )\,\left (2\,x^2\,{\mathrm {e}}^x+{\ln \left (x^2\right )}^2\,\left (4\,{\mathrm {e}}^x+4\right )+2\,x^2-\ln \left (x^2\right )\,\left (x^2\,{\mathrm {e}}^x+x^2\right )\right )+6\,x^2}{9\,x^4\,{\mathrm {e}}^x+{\ln \left (x^2\right )}^2\,\left ({\mathrm {e}}^x\,\left (81\,x^2-216\,x+144\right )-216\,x+81\,x^2+144\right )+\ln \left (x^2\right )\,\left ({\mathrm {e}}^x\,\left (72\,x^2-54\,x^3\right )+72\,x^2-54\,x^3\right )-\ln \left (x+x\,{\mathrm {e}}^x\right )\,\left (6\,x^4\,{\mathrm {e}}^x+{\ln \left (x^2\right )}^2\,\left ({\mathrm {e}}^x\,\left (54\,x^2-144\,x+96\right )-144\,x+54\,x^2+96\right )+\ln \left (x^2\right )\,\left ({\mathrm {e}}^x\,\left (48\,x^2-36\,x^3\right )+48\,x^2-36\,x^3\right )+6\,x^4\right )+{\ln \left (x+x\,{\mathrm {e}}^x\right )}^2\,\left (x^4\,{\mathrm {e}}^x+{\ln \left (x^2\right )}^2\,\left ({\mathrm {e}}^x\,\left (9\,x^2-24\,x+16\right )-24\,x+9\,x^2+16\right )+\ln \left (x^2\right )\,\left ({\mathrm {e}}^x\,\left (8\,x^2-6\,x^3\right )+8\,x^2-6\,x^3\right )+x^4\right )+9\,x^4} \,d x \] Input:
int((6*x^2*exp(x) - log(x^2)*(exp(x)*(2*x^2 - x^3) + 2*x^2) + log(x^2)^2*( exp(x)*(x - 3*x^2 + 16) - 3*x + 16) - log(x + x*exp(x))*(2*x^2*exp(x) + lo g(x^2)^2*(4*exp(x) + 4) + 2*x^2 - log(x^2)*(x^2*exp(x) + x^2)) + 6*x^2)/(9 *x^4*exp(x) + log(x^2)^2*(exp(x)*(81*x^2 - 216*x + 144) - 216*x + 81*x^2 + 144) + log(x^2)*(exp(x)*(72*x^2 - 54*x^3) + 72*x^2 - 54*x^3) - log(x + x* exp(x))*(6*x^4*exp(x) + log(x^2)^2*(exp(x)*(54*x^2 - 144*x + 96) - 144*x + 54*x^2 + 96) + log(x^2)*(exp(x)*(48*x^2 - 36*x^3) + 48*x^2 - 36*x^3) + 6* x^4) + log(x + x*exp(x))^2*(x^4*exp(x) + log(x^2)^2*(exp(x)*(9*x^2 - 24*x + 16) - 24*x + 9*x^2 + 16) + log(x^2)*(exp(x)*(8*x^2 - 6*x^3) + 8*x^2 - 6* x^3) + x^4) + 9*x^4),x)
Output:
int((6*x^2*exp(x) - log(x^2)*(exp(x)*(2*x^2 - x^3) + 2*x^2) + log(x^2)^2*( exp(x)*(x - 3*x^2 + 16) - 3*x + 16) - log(x + x*exp(x))*(2*x^2*exp(x) + lo g(x^2)^2*(4*exp(x) + 4) + 2*x^2 - log(x^2)*(x^2*exp(x) + x^2)) + 6*x^2)/(9 *x^4*exp(x) + log(x^2)^2*(exp(x)*(81*x^2 - 216*x + 144) - 216*x + 81*x^2 + 144) + log(x^2)*(exp(x)*(72*x^2 - 54*x^3) + 72*x^2 - 54*x^3) - log(x + x* exp(x))*(6*x^4*exp(x) + log(x^2)^2*(exp(x)*(54*x^2 - 144*x + 96) - 144*x + 54*x^2 + 96) + log(x^2)*(exp(x)*(48*x^2 - 36*x^3) + 48*x^2 - 36*x^3) + 6* x^4) + log(x + x*exp(x))^2*(x^4*exp(x) + log(x^2)^2*(exp(x)*(9*x^2 - 24*x + 16) - 24*x + 9*x^2 + 16) + log(x^2)*(exp(x)*(8*x^2 - 6*x^3) + 8*x^2 - 6* x^3) + x^4) + 9*x^4), x)
Time = 0.30 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.09 \[ \int \frac {6 x^2+6 e^x x^2+\left (-2 x^2+e^x \left (-2 x^2+x^3\right )\right ) \log \left (x^2\right )+\left (16-3 x+e^x \left (16+x-3 x^2\right )\right ) \log ^2\left (x^2\right )+\left (-2 x^2-2 e^x x^2+\left (x^2+e^x x^2\right ) \log \left (x^2\right )+\left (-4-4 e^x\right ) \log ^2\left (x^2\right )\right ) \log \left (x+e^x x\right )}{9 x^4+9 e^x x^4+\left (72 x^2-54 x^3+e^x \left (72 x^2-54 x^3\right )\right ) \log \left (x^2\right )+\left (144-216 x+81 x^2+e^x \left (144-216 x+81 x^2\right )\right ) \log ^2\left (x^2\right )+\left (-6 x^4-6 e^x x^4+\left (-48 x^2+36 x^3+e^x \left (-48 x^2+36 x^3\right )\right ) \log \left (x^2\right )+\left (-96+144 x-54 x^2+e^x \left (-96+144 x-54 x^2\right )\right ) \log ^2\left (x^2\right )\right ) \log \left (x+e^x x\right )+\left (x^4+e^x x^4+\left (8 x^2-6 x^3+e^x \left (8 x^2-6 x^3\right )\right ) \log \left (x^2\right )+\left (16-24 x+9 x^2+e^x \left (16-24 x+9 x^2\right )\right ) \log ^2\left (x^2\right )\right ) \log ^2\left (x+e^x x\right )} \, dx=\frac {\mathrm {log}\left (x^{2}\right ) x}{3 \,\mathrm {log}\left (x^{2}\right ) \mathrm {log}\left (e^{x} x +x \right ) x -4 \,\mathrm {log}\left (x^{2}\right ) \mathrm {log}\left (e^{x} x +x \right )-9 \,\mathrm {log}\left (x^{2}\right ) x +12 \,\mathrm {log}\left (x^{2}\right )-\mathrm {log}\left (e^{x} x +x \right ) x^{2}+3 x^{2}} \] Input:
int((((-4*exp(x)-4)*log(x^2)^2+(exp(x)*x^2+x^2)*log(x^2)-2*exp(x)*x^2-2*x^ 2)*log(exp(x)*x+x)+((-3*x^2+x+16)*exp(x)-3*x+16)*log(x^2)^2+((x^3-2*x^2)*e xp(x)-2*x^2)*log(x^2)+6*exp(x)*x^2+6*x^2)/((((9*x^2-24*x+16)*exp(x)+9*x^2- 24*x+16)*log(x^2)^2+((-6*x^3+8*x^2)*exp(x)-6*x^3+8*x^2)*log(x^2)+exp(x)*x^ 4+x^4)*log(exp(x)*x+x)^2+(((-54*x^2+144*x-96)*exp(x)-54*x^2+144*x-96)*log( x^2)^2+((36*x^3-48*x^2)*exp(x)+36*x^3-48*x^2)*log(x^2)-6*exp(x)*x^4-6*x^4) *log(exp(x)*x+x)+((81*x^2-216*x+144)*exp(x)+81*x^2-216*x+144)*log(x^2)^2+( (-54*x^3+72*x^2)*exp(x)-54*x^3+72*x^2)*log(x^2)+9*exp(x)*x^4+9*x^4),x)
Output:
(log(x**2)*x)/(3*log(x**2)*log(e**x*x + x)*x - 4*log(x**2)*log(e**x*x + x) - 9*log(x**2)*x + 12*log(x**2) - log(e**x*x + x)*x**2 + 3*x**2)