Integrand size = 44, antiderivative size = 31 \[ \int \frac {1}{320} e^{-x} \left (-10000+10400 x+\left (400 x-204 x^2\right ) \log \left (x^2\right )+\left (-3 x^2+x^3\right ) \log ^2\left (x^2\right )\right ) \, dx=5+\frac {1}{4} \left (-4-\frac {1}{5} e^{-x} x \left (-25+\frac {1}{4} x \log \left (x^2\right )\right )^2\right ) \] Output:
4-1/20*x*(1/4*x*ln(x^2)-25)^2/exp(x)
Time = 0.07 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.65 \[ \int \frac {1}{320} e^{-x} \left (-10000+10400 x+\left (400 x-204 x^2\right ) \log \left (x^2\right )+\left (-3 x^2+x^3\right ) \log ^2\left (x^2\right )\right ) \, dx=-\frac {1}{320} e^{-x} x \left (-100+x \log \left (x^2\right )\right )^2 \] Input:
Integrate[(-10000 + 10400*x + (400*x - 204*x^2)*Log[x^2] + (-3*x^2 + x^3)* Log[x^2]^2)/(320*E^x),x]
Output:
-1/320*(x*(-100 + x*Log[x^2])^2)/E^x
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.
\(\displaystyle \int \frac {1}{320} e^{-x} \left (\left (400 x-204 x^2\right ) \log \left (x^2\right )+\left (x^3-3 x^2\right ) \log ^2\left (x^2\right )+10400 x-10000\right ) \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{320} \int -e^{-x} \left (\left (3 x^2-x^3\right ) \log ^2\left (x^2\right )-4 \left (100 x-51 x^2\right ) \log \left (x^2\right )-10400 x+10000\right )dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{320} \int e^{-x} \left (\left (3 x^2-x^3\right ) \log ^2\left (x^2\right )-4 \left (100 x-51 x^2\right ) \log \left (x^2\right )-10400 x+10000\right )dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {1}{320} \int \left (-e^{-x} (x-3) x^2 \log ^2\left (x^2\right )+4 e^{-x} x (51 x-100) \log \left (x^2\right )+10000 e^{-x}-10400 e^{-x} x\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{320} \left (-3 \int e^{-x} x^2 \log ^2\left (x^2\right )dx+\int e^{-x} x^3 \log ^2\left (x^2\right )dx-16 \operatorname {ExpIntegralEi}(-x)+204 e^{-x} x^2 \log \left (x^2\right )+8 e^{-x} x \log \left (x^2\right )+8 e^{-x} \log \left (x^2\right )-9992 e^{-x} x+24 e^{-x}\right )\) |
Input:
Int[(-10000 + 10400*x + (400*x - 204*x^2)*Log[x^2] + (-3*x^2 + x^3)*Log[x^ 2]^2)/(320*E^x),x]
Output:
$Aborted
Time = 0.84 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97
method | result | size |
parallelrisch | \(-\frac {\left (x^{3} \ln \left (x^{2}\right )^{2}-200 x^{2} \ln \left (x^{2}\right )+10000 x \right ) {\mathrm e}^{-x}}{320}\) | \(30\) |
risch | \(-\frac {\ln \left (x \right )^{2} {\mathrm e}^{-x} x^{3}}{80}+\frac {i x^{2} \left (x \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-2 x \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+x \pi \operatorname {csgn}\left (i x^{2}\right )^{3}-200 i\right ) {\mathrm e}^{-x} \ln \left (x \right )}{160}+\frac {\left (-40000 x -400 i \pi \,x^{2} \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )+800 i \pi \,x^{2} \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}-400 i \pi \,x^{2} \operatorname {csgn}\left (i x^{2}\right )^{3}+\pi ^{2} x^{3} \operatorname {csgn}\left (i x \right )^{4} \operatorname {csgn}\left (i x^{2}\right )^{2}-4 \pi ^{2} x^{3} \operatorname {csgn}\left (i x \right )^{3} \operatorname {csgn}\left (i x^{2}\right )^{3}+6 \pi ^{2} x^{3} \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )^{4}-4 \pi ^{2} x^{3} \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{5}+\pi ^{2} x^{3} \operatorname {csgn}\left (i x^{2}\right )^{6}\right ) {\mathrm e}^{-x}}{1280}\) | \(254\) |
orering | \(-\frac {x \left (3 x^{4}+288 x^{3}+6619 x^{2}-13900 x +15000\right ) \left (\left (x^{3}-3 x^{2}\right ) \ln \left (x^{2}\right )^{2}+\left (-204 x^{2}+400 x \right ) \ln \left (x^{2}\right )+10400 x -10000\right ) {\mathrm e}^{-x}}{320 \left (x^{5}+94 x^{4}+2069 x^{3}-6427 x^{2}+13800 x -15000\right )}-\frac {3 x^{2} \left (x^{3}+98 x^{2}+2350 x -2500\right ) \left (\frac {\left (\left (3 x^{2}-6 x \right ) \ln \left (x^{2}\right )^{2}+\frac {4 \left (x^{3}-3 x^{2}\right ) \ln \left (x^{2}\right )}{x}+\left (-408 x +400\right ) \ln \left (x^{2}\right )+\frac {-408 x^{2}+800 x}{x}+10400\right ) {\mathrm e}^{-x}}{320}-\frac {\left (\left (x^{3}-3 x^{2}\right ) \ln \left (x^{2}\right )^{2}+\left (-204 x^{2}+400 x \right ) \ln \left (x^{2}\right )+10400 x -10000\right ) {\mathrm e}^{-x}}{320}\right )}{x^{5}+94 x^{4}+2069 x^{3}-6427 x^{2}+13800 x -15000}-\frac {\left (x^{2}+100 x +2500\right ) x^{3} \left (\frac {\left (\left (6 x -6\right ) \ln \left (x^{2}\right )^{2}+\frac {8 \left (3 x^{2}-6 x \right ) \ln \left (x^{2}\right )}{x}+\frac {8 x^{3}-24 x^{2}}{x^{2}}-\frac {4 \left (x^{3}-3 x^{2}\right ) \ln \left (x^{2}\right )}{x^{2}}-408 \ln \left (x^{2}\right )+\frac {-1632 x +1600}{x}-\frac {2 \left (-204 x^{2}+400 x \right )}{x^{2}}\right ) {\mathrm e}^{-x}}{320}-\frac {\left (\left (3 x^{2}-6 x \right ) \ln \left (x^{2}\right )^{2}+\frac {4 \left (x^{3}-3 x^{2}\right ) \ln \left (x^{2}\right )}{x}+\left (-408 x +400\right ) \ln \left (x^{2}\right )+\frac {-408 x^{2}+800 x}{x}+10400\right ) {\mathrm e}^{-x}}{160}+\frac {\left (\left (x^{3}-3 x^{2}\right ) \ln \left (x^{2}\right )^{2}+\left (-204 x^{2}+400 x \right ) \ln \left (x^{2}\right )+10400 x -10000\right ) {\mathrm e}^{-x}}{320}\right )}{x^{5}+94 x^{4}+2069 x^{3}-6427 x^{2}+13800 x -15000}\) | \(485\) |
Input:
int(1/320*((x^3-3*x^2)*ln(x^2)^2+(-204*x^2+400*x)*ln(x^2)+10400*x-10000)/e xp(x),x,method=_RETURNVERBOSE)
Output:
-1/320*(x^3*ln(x^2)^2-200*x^2*ln(x^2)+10000*x)/exp(x)
Time = 0.07 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16 \[ \int \frac {1}{320} e^{-x} \left (-10000+10400 x+\left (400 x-204 x^2\right ) \log \left (x^2\right )+\left (-3 x^2+x^3\right ) \log ^2\left (x^2\right )\right ) \, dx=-\frac {1}{320} \, x^{3} e^{\left (-x\right )} \log \left (x^{2}\right )^{2} + \frac {5}{8} \, x^{2} e^{\left (-x\right )} \log \left (x^{2}\right ) - \frac {125}{4} \, x e^{\left (-x\right )} \] Input:
integrate(1/320*((x^3-3*x^2)*log(x^2)^2+(-204*x^2+400*x)*log(x^2)+10400*x- 10000)/exp(x),x, algorithm="fricas")
Output:
-1/320*x^3*e^(-x)*log(x^2)^2 + 5/8*x^2*e^(-x)*log(x^2) - 125/4*x*e^(-x)
Time = 0.14 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {1}{320} e^{-x} \left (-10000+10400 x+\left (400 x-204 x^2\right ) \log \left (x^2\right )+\left (-3 x^2+x^3\right ) \log ^2\left (x^2\right )\right ) \, dx=\frac {\left (- x^{3} \log {\left (x^{2} \right )}^{2} + 200 x^{2} \log {\left (x^{2} \right )} - 10000 x\right ) e^{- x}}{320} \] Input:
integrate(1/320*((x**3-3*x**2)*ln(x**2)**2+(-204*x**2+400*x)*ln(x**2)+1040 0*x-10000)/exp(x),x)
Output:
(-x**3*log(x**2)**2 + 200*x**2*log(x**2) - 10000*x)*exp(-x)/320
Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (19) = 38\).
Time = 0.09 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.35 \[ \int \frac {1}{320} e^{-x} \left (-10000+10400 x+\left (400 x-204 x^2\right ) \log \left (x^2\right )+\left (-3 x^2+x^3\right ) \log ^2\left (x^2\right )\right ) \, dx=-\frac {1}{80} \, {\left (x^{3} \log \left (x\right )^{2} - 100 \, x^{2} \log \left (x\right ) - 100 \, x - 100\right )} e^{\left (-x\right )} - \frac {65}{2} \, {\left (x + 1\right )} e^{\left (-x\right )} + \frac {125}{4} \, e^{\left (-x\right )} \] Input:
integrate(1/320*((x^3-3*x^2)*log(x^2)^2+(-204*x^2+400*x)*log(x^2)+10400*x- 10000)/exp(x),x, algorithm="maxima")
Output:
-1/80*(x^3*log(x)^2 - 100*x^2*log(x) - 100*x - 100)*e^(-x) - 65/2*(x + 1)* e^(-x) + 125/4*e^(-x)
Time = 0.12 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16 \[ \int \frac {1}{320} e^{-x} \left (-10000+10400 x+\left (400 x-204 x^2\right ) \log \left (x^2\right )+\left (-3 x^2+x^3\right ) \log ^2\left (x^2\right )\right ) \, dx=-\frac {1}{320} \, x^{3} e^{\left (-x\right )} \log \left (x^{2}\right )^{2} + \frac {5}{8} \, x^{2} e^{\left (-x\right )} \log \left (x^{2}\right ) - \frac {125}{4} \, x e^{\left (-x\right )} \] Input:
integrate(1/320*((x^3-3*x^2)*log(x^2)^2+(-204*x^2+400*x)*log(x^2)+10400*x- 10000)/exp(x),x, algorithm="giac")
Output:
-1/320*x^3*e^(-x)*log(x^2)^2 + 5/8*x^2*e^(-x)*log(x^2) - 125/4*x*e^(-x)
Time = 1.78 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.55 \[ \int \frac {1}{320} e^{-x} \left (-10000+10400 x+\left (400 x-204 x^2\right ) \log \left (x^2\right )+\left (-3 x^2+x^3\right ) \log ^2\left (x^2\right )\right ) \, dx=-\frac {x\,{\mathrm {e}}^{-x}\,{\left (x\,\ln \left (x^2\right )-100\right )}^2}{320} \] Input:
int(exp(-x)*((65*x)/2 + (log(x^2)*(400*x - 204*x^2))/320 - (log(x^2)^2*(3* x^2 - x^3))/320 - 125/4),x)
Output:
-(x*exp(-x)*(x*log(x^2) - 100)^2)/320
Time = 0.21 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.90 \[ \int \frac {1}{320} e^{-x} \left (-10000+10400 x+\left (400 x-204 x^2\right ) \log \left (x^2\right )+\left (-3 x^2+x^3\right ) \log ^2\left (x^2\right )\right ) \, dx=\frac {x \left (-\mathrm {log}\left (x^{2}\right )^{2} x^{2}+200 \,\mathrm {log}\left (x^{2}\right ) x -10000\right )}{320 e^{x}} \] Input:
int(1/320*((x^3-3*x^2)*log(x^2)^2+(-204*x^2+400*x)*log(x^2)+10400*x-10000) /exp(x),x)
Output:
(x*( - log(x**2)**2*x**2 + 200*log(x**2)*x - 10000))/(320*e**x)