Integrand size = 159, antiderivative size = 32 \[ \int \frac {9 x+e^{2 x^2} x+6 x^2+x^3+e^{x^2} \left (-6 x-2 x^2\right )+e^{\frac {5-e^{x^2}+\left (-3+e^{x^2}-x\right ) \log \left (\frac {x^2}{4}\right )}{-3+e^{x^2}-x}} \left (-18-2 e^{2 x^2}-17 x-2 x^2+e^{x^2} \left (12+5 x+4 x^2-2 x^3\right )\right )}{9 x+e^{2 x^2} x+6 x^2+x^3+e^{x^2} \left (-6 x-2 x^2\right )} \, dx=x-\frac {1}{4} e^{-1-\frac {2-x}{3-e^{x^2}+x}} x^2 \] Output:
x-exp(-1-(2-x)/(x+3-exp(x^2))+ln(1/4*x^2))
Time = 0.15 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \frac {9 x+e^{2 x^2} x+6 x^2+x^3+e^{x^2} \left (-6 x-2 x^2\right )+e^{\frac {5-e^{x^2}+\left (-3+e^{x^2}-x\right ) \log \left (\frac {x^2}{4}\right )}{-3+e^{x^2}-x}} \left (-18-2 e^{2 x^2}-17 x-2 x^2+e^{x^2} \left (12+5 x+4 x^2-2 x^3\right )\right )}{9 x+e^{2 x^2} x+6 x^2+x^3+e^{x^2} \left (-6 x-2 x^2\right )} \, dx=x-\frac {1}{4} e^{-1+\frac {2-x}{-3+e^{x^2}-x}} x^2 \] Input:
Integrate[(9*x + E^(2*x^2)*x + 6*x^2 + x^3 + E^x^2*(-6*x - 2*x^2) + E^((5 - E^x^2 + (-3 + E^x^2 - x)*Log[x^2/4])/(-3 + E^x^2 - x))*(-18 - 2*E^(2*x^2 ) - 17*x - 2*x^2 + E^x^2*(12 + 5*x + 4*x^2 - 2*x^3)))/(9*x + E^(2*x^2)*x + 6*x^2 + x^3 + E^x^2*(-6*x - 2*x^2)),x]
Output:
x - (E^(-1 + (2 - x)/(-3 + E^x^2 - x))*x^2)/4
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.
\(\displaystyle \int \frac {\left (-2 x^2-2 e^{2 x^2}+e^{x^2} \left (-2 x^3+4 x^2+5 x+12\right )-17 x-18\right ) \exp \left (\frac {-e^{x^2}+\left (e^{x^2}-x-3\right ) \log \left (\frac {x^2}{4}\right )+5}{e^{x^2}-x-3}\right )+x^3+6 x^2+e^{2 x^2} x+e^{x^2} \left (-2 x^2-6 x\right )+9 x}{x^3+6 x^2+e^{2 x^2} x+e^{x^2} \left (-2 x^2-6 x\right )+9 x} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {\left (-2 x^2-2 e^{2 x^2}+e^{x^2} \left (-2 x^3+4 x^2+5 x+12\right )-17 x-18\right ) \exp \left (\frac {-e^{x^2}+\left (e^{x^2}-x-3\right ) \log \left (\frac {x^2}{4}\right )+5}{e^{x^2}-x-3}\right )+x^3+6 x^2+e^{2 x^2} x+e^{x^2} \left (-2 x^2-6 x\right )+9 x}{x \left (-e^{x^2}+x+3\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {x^2}{\left (e^{x^2}-x-3\right )^2}+\frac {6 x}{\left (e^{x^2}-x-3\right )^2}-\frac {2 e^{x^2} (x+3)}{\left (e^{x^2}-x-3\right )^2}+\frac {e^{2 x^2}}{\left (e^{x^2}-x-3\right )^2}+\frac {9}{\left (e^{x^2}-x-3\right )^2}+\frac {e^{-\frac {e^{x^2}-5}{e^{x^2}-x-3}} \left (4 e^{x^2} x^2-2 x^2+5 e^{x^2} x+12 e^{x^2}-2 e^{2 x^2}-2 e^{x^2} x^3-17 x-18\right ) x}{4 \left (-e^{x^2}+x+3\right )^2}\right )dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \left (\frac {x^2}{\left (e^{x^2}-x-3\right )^2}+\frac {6 x}{\left (e^{x^2}-x-3\right )^2}-\frac {2 e^{x^2} (x+3)}{\left (e^{x^2}-x-3\right )^2}+\frac {e^{2 x^2}}{\left (e^{x^2}-x-3\right )^2}+\frac {9}{\left (e^{x^2}-x-3\right )^2}+\frac {e^{-\frac {e^{x^2}-5}{e^{x^2}-x-3}} \left (4 e^{x^2} x^2-2 x^2+5 e^{x^2} x+12 e^{x^2}-2 e^{2 x^2}-2 e^{x^2} x^3-17 x-18\right ) x}{4 \left (-e^{x^2}+x+3\right )^2}\right )dx\) |
Input:
Int[(9*x + E^(2*x^2)*x + 6*x^2 + x^3 + E^x^2*(-6*x - 2*x^2) + E^((5 - E^x^ 2 + (-3 + E^x^2 - x)*Log[x^2/4])/(-3 + E^x^2 - x))*(-18 - 2*E^(2*x^2) - 17 *x - 2*x^2 + E^x^2*(12 + 5*x + 4*x^2 - 2*x^3)))/(9*x + E^(2*x^2)*x + 6*x^2 + x^3 + E^x^2*(-6*x - 2*x^2)),x]
Output:
$Aborted
Time = 3.40 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.34
method | result | size |
parallelrisch | \(-3+x -{\mathrm e}^{\frac {\left ({\mathrm e}^{x^{2}}-3-x \right ) \ln \left (\frac {x^{2}}{4}\right )+5-{\mathrm e}^{x^{2}}}{{\mathrm e}^{x^{2}}-3-x}}\) | \(43\) |
risch | \(x -{\mathrm e}^{-\frac {-2 i \pi x \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+2 i {\mathrm e}^{x^{2}} \pi \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right )+3 i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+i x \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )^{2}+3 i \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )^{2}+i x \pi \operatorname {csgn}\left (i x^{2}\right )^{3}-i {\mathrm e}^{x^{2}} \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )^{2}-i {\mathrm e}^{x^{2}} \pi \operatorname {csgn}\left (i x^{2}\right )^{3}-6 i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}-4 \ln \left (2\right ) {\mathrm e}^{x^{2}}+4 \,{\mathrm e}^{x^{2}} \ln \left (x \right )+4 x \ln \left (2\right )-4 x \ln \left (x \right )-2 \,{\mathrm e}^{x^{2}}+12 \ln \left (2\right )-12 \ln \left (x \right )+10}{2 \left (x +3-{\mathrm e}^{x^{2}}\right )}}\) | \(223\) |
Input:
int(((-2*exp(x^2)^2+(-2*x^3+4*x^2+5*x+12)*exp(x^2)-2*x^2-17*x-18)*exp(((ex p(x^2)-3-x)*ln(1/4*x^2)+5-exp(x^2))/(exp(x^2)-3-x))+x*exp(x^2)^2+(-2*x^2-6 *x)*exp(x^2)+x^3+6*x^2+9*x)/(x*exp(x^2)^2+(-2*x^2-6*x)*exp(x^2)+x^3+6*x^2+ 9*x),x,method=_RETURNVERBOSE)
Output:
-3+x-exp(((exp(x^2)-3-x)*ln(1/4*x^2)+5-exp(x^2))/(exp(x^2)-3-x))
Time = 0.10 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.22 \[ \int \frac {9 x+e^{2 x^2} x+6 x^2+x^3+e^{x^2} \left (-6 x-2 x^2\right )+e^{\frac {5-e^{x^2}+\left (-3+e^{x^2}-x\right ) \log \left (\frac {x^2}{4}\right )}{-3+e^{x^2}-x}} \left (-18-2 e^{2 x^2}-17 x-2 x^2+e^{x^2} \left (12+5 x+4 x^2-2 x^3\right )\right )}{9 x+e^{2 x^2} x+6 x^2+x^3+e^{x^2} \left (-6 x-2 x^2\right )} \, dx=x - e^{\left (\frac {{\left (x - e^{\left (x^{2}\right )} + 3\right )} \log \left (\frac {1}{4} \, x^{2}\right ) + e^{\left (x^{2}\right )} - 5}{x - e^{\left (x^{2}\right )} + 3}\right )} \] Input:
integrate(((-2*exp(x^2)^2+(-2*x^3+4*x^2+5*x+12)*exp(x^2)-2*x^2-17*x-18)*ex p(((exp(x^2)-3-x)*log(1/4*x^2)+5-exp(x^2))/(exp(x^2)-3-x))+x*exp(x^2)^2+(- 2*x^2-6*x)*exp(x^2)+x^3+6*x^2+9*x)/(x*exp(x^2)^2+(-2*x^2-6*x)*exp(x^2)+x^3 +6*x^2+9*x),x, algorithm="fricas")
Output:
x - e^(((x - e^(x^2) + 3)*log(1/4*x^2) + e^(x^2) - 5)/(x - e^(x^2) + 3))
Time = 0.61 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \frac {9 x+e^{2 x^2} x+6 x^2+x^3+e^{x^2} \left (-6 x-2 x^2\right )+e^{\frac {5-e^{x^2}+\left (-3+e^{x^2}-x\right ) \log \left (\frac {x^2}{4}\right )}{-3+e^{x^2}-x}} \left (-18-2 e^{2 x^2}-17 x-2 x^2+e^{x^2} \left (12+5 x+4 x^2-2 x^3\right )\right )}{9 x+e^{2 x^2} x+6 x^2+x^3+e^{x^2} \left (-6 x-2 x^2\right )} \, dx=x - e^{\frac {\left (- x + e^{x^{2}} - 3\right ) \log {\left (\frac {x^{2}}{4} \right )} - e^{x^{2}} + 5}{- x + e^{x^{2}} - 3}} \] Input:
integrate(((-2*exp(x**2)**2+(-2*x**3+4*x**2+5*x+12)*exp(x**2)-2*x**2-17*x- 18)*exp(((exp(x**2)-3-x)*ln(1/4*x**2)+5-exp(x**2))/(exp(x**2)-3-x))+x*exp( x**2)**2+(-2*x**2-6*x)*exp(x**2)+x**3+6*x**2+9*x)/(x*exp(x**2)**2+(-2*x**2 -6*x)*exp(x**2)+x**3+6*x**2+9*x),x)
Output:
x - exp(((-x + exp(x**2) - 3)*log(x**2/4) - exp(x**2) + 5)/(-x + exp(x**2) - 3))
\[ \int \frac {9 x+e^{2 x^2} x+6 x^2+x^3+e^{x^2} \left (-6 x-2 x^2\right )+e^{\frac {5-e^{x^2}+\left (-3+e^{x^2}-x\right ) \log \left (\frac {x^2}{4}\right )}{-3+e^{x^2}-x}} \left (-18-2 e^{2 x^2}-17 x-2 x^2+e^{x^2} \left (12+5 x+4 x^2-2 x^3\right )\right )}{9 x+e^{2 x^2} x+6 x^2+x^3+e^{x^2} \left (-6 x-2 x^2\right )} \, dx=\int { \frac {x^{3} + 6 \, x^{2} + x e^{\left (2 \, x^{2}\right )} - 2 \, {\left (x^{2} + 3 \, x\right )} e^{\left (x^{2}\right )} - {\left (2 \, x^{2} + {\left (2 \, x^{3} - 4 \, x^{2} - 5 \, x - 12\right )} e^{\left (x^{2}\right )} + 17 \, x + 2 \, e^{\left (2 \, x^{2}\right )} + 18\right )} e^{\left (\frac {{\left (x - e^{\left (x^{2}\right )} + 3\right )} \log \left (\frac {1}{4} \, x^{2}\right ) + e^{\left (x^{2}\right )} - 5}{x - e^{\left (x^{2}\right )} + 3}\right )} + 9 \, x}{x^{3} + 6 \, x^{2} + x e^{\left (2 \, x^{2}\right )} - 2 \, {\left (x^{2} + 3 \, x\right )} e^{\left (x^{2}\right )} + 9 \, x} \,d x } \] Input:
integrate(((-2*exp(x^2)^2+(-2*x^3+4*x^2+5*x+12)*exp(x^2)-2*x^2-17*x-18)*ex p(((exp(x^2)-3-x)*log(1/4*x^2)+5-exp(x^2))/(exp(x^2)-3-x))+x*exp(x^2)^2+(- 2*x^2-6*x)*exp(x^2)+x^3+6*x^2+9*x)/(x*exp(x^2)^2+(-2*x^2-6*x)*exp(x^2)+x^3 +6*x^2+9*x),x, algorithm="maxima")
Output:
x - integrate(1/4*(2*x^3 + 17*x^2 + 2*x*e^(2*x^2) + (2*x^4 - 4*x^3 - 5*x^2 - 12*x)*e^(x^2) + 18*x)*e^(e^(x^2)/(x - e^(x^2) + 3) - 5/(x - e^(x^2) + 3 ))/(x^2 - 2*(x + 3)*e^(x^2) + 6*x + e^(2*x^2) + 9), x)
Time = 1.33 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.03 \[ \int \frac {9 x+e^{2 x^2} x+6 x^2+x^3+e^{x^2} \left (-6 x-2 x^2\right )+e^{\frac {5-e^{x^2}+\left (-3+e^{x^2}-x\right ) \log \left (\frac {x^2}{4}\right )}{-3+e^{x^2}-x}} \left (-18-2 e^{2 x^2}-17 x-2 x^2+e^{x^2} \left (12+5 x+4 x^2-2 x^3\right )\right )}{9 x+e^{2 x^2} x+6 x^2+x^3+e^{x^2} \left (-6 x-2 x^2\right )} \, dx=-\frac {1}{4} \, x^{2} e^{\left (\frac {5 \, x - 2 \, e^{\left (x^{2}\right )}}{3 \, {\left (x - e^{\left (x^{2}\right )} + 3\right )}} - \frac {5}{3}\right )} + x \] Input:
integrate(((-2*exp(x^2)^2+(-2*x^3+4*x^2+5*x+12)*exp(x^2)-2*x^2-17*x-18)*ex p(((exp(x^2)-3-x)*log(1/4*x^2)+5-exp(x^2))/(exp(x^2)-3-x))+x*exp(x^2)^2+(- 2*x^2-6*x)*exp(x^2)+x^3+6*x^2+9*x)/(x*exp(x^2)^2+(-2*x^2-6*x)*exp(x^2)+x^3 +6*x^2+9*x),x, algorithm="giac")
Output:
-1/4*x^2*e^(1/3*(5*x - 2*e^(x^2))/(x - e^(x^2) + 3) - 5/3) + x
Time = 3.89 (sec) , antiderivative size = 145, normalized size of antiderivative = 4.53 \[ \int \frac {9 x+e^{2 x^2} x+6 x^2+x^3+e^{x^2} \left (-6 x-2 x^2\right )+e^{\frac {5-e^{x^2}+\left (-3+e^{x^2}-x\right ) \log \left (\frac {x^2}{4}\right )}{-3+e^{x^2}-x}} \left (-18-2 e^{2 x^2}-17 x-2 x^2+e^{x^2} \left (12+5 x+4 x^2-2 x^3\right )\right )}{9 x+e^{2 x^2} x+6 x^2+x^3+e^{x^2} \left (-6 x-2 x^2\right )} \, dx=x-\frac {2^{\frac {2\,{\mathrm {e}}^{x^2}}{x-{\mathrm {e}}^{x^2}+3}}\,{\mathrm {e}}^{-\frac {5}{x-{\mathrm {e}}^{x^2}+3}}\,{\mathrm {e}}^{\frac {{\mathrm {e}}^{x^2}}{x-{\mathrm {e}}^{x^2}+3}}\,{\left (x^2\right )}^{\frac {3}{x-{\mathrm {e}}^{x^2}+3}}\,{\left (x^2\right )}^{\frac {x}{x-{\mathrm {e}}^{x^2}+3}}}{2^{\frac {2\,x}{x-{\mathrm {e}}^{x^2}+3}}\,2^{\frac {6}{x-{\mathrm {e}}^{x^2}+3}}\,{\left (x^2\right )}^{\frac {{\mathrm {e}}^{x^2}}{x-{\mathrm {e}}^{x^2}+3}}} \] Input:
int((9*x - exp(x^2)*(6*x + 2*x^2) + x*exp(2*x^2) - exp((exp(x^2) + log(x^2 /4)*(x - exp(x^2) + 3) - 5)/(x - exp(x^2) + 3))*(17*x + 2*exp(2*x^2) - exp (x^2)*(5*x + 4*x^2 - 2*x^3 + 12) + 2*x^2 + 18) + 6*x^2 + x^3)/(9*x - exp(x ^2)*(6*x + 2*x^2) + x*exp(2*x^2) + 6*x^2 + x^3),x)
Output:
x - (2^((2*exp(x^2))/(x - exp(x^2) + 3))*exp(-5/(x - exp(x^2) + 3))*exp(ex p(x^2)/(x - exp(x^2) + 3))*(x^2)^(3/(x - exp(x^2) + 3))*(x^2)^(x/(x - exp( x^2) + 3)))/(2^((2*x)/(x - exp(x^2) + 3))*2^(6/(x - exp(x^2) + 3))*(x^2)^( exp(x^2)/(x - exp(x^2) + 3)))
Time = 45.74 (sec) , antiderivative size = 84, normalized size of antiderivative = 2.62 \[ \int \frac {9 x+e^{2 x^2} x+6 x^2+x^3+e^{x^2} \left (-6 x-2 x^2\right )+e^{\frac {5-e^{x^2}+\left (-3+e^{x^2}-x\right ) \log \left (\frac {x^2}{4}\right )}{-3+e^{x^2}-x}} \left (-18-2 e^{2 x^2}-17 x-2 x^2+e^{x^2} \left (12+5 x+4 x^2-2 x^3\right )\right )}{9 x+e^{2 x^2} x+6 x^2+x^3+e^{x^2} \left (-6 x-2 x^2\right )} \, dx=\frac {4 e^{\frac {x}{e^{x^{2}}-x -3}} e x -12 e^{\frac {x}{e^{x^{2}}-x -3}} e -e^{\frac {2}{e^{x^{2}}-x -3}} x^{2}}{4 e^{\frac {x}{e^{x^{2}}-x -3}} e} \] Input:
int(((-2*exp(x^2)^2+(-2*x^3+4*x^2+5*x+12)*exp(x^2)-2*x^2-17*x-18)*exp(((ex p(x^2)-3-x)*log(1/4*x^2)+5-exp(x^2))/(exp(x^2)-3-x))+x*exp(x^2)^2+(-2*x^2- 6*x)*exp(x^2)+x^3+6*x^2+9*x)/(x*exp(x^2)^2+(-2*x^2-6*x)*exp(x^2)+x^3+6*x^2 +9*x),x)
Output:
(4*e**(x/(e**(x**2) - x - 3))*e*x - 12*e**(x/(e**(x**2) - x - 3))*e - e**( 2/(e**(x**2) - x - 3))*x**2)/(4*e**(x/(e**(x**2) - x - 3))*e)