3.4.15 \(\int \frac {9 x+e^{2 x^2} x+6 x^2+x^3+e^{x^2} (-6 x-2 x^2)+e^{\frac {5-e^{x^2}+(-3+e^{x^2}-x) \log (\frac {x^2}{4})}{-3+e^{x^2}-x}} (-18-2 e^{2 x^2}-17 x-2 x^2+e^{x^2} (12+5 x+4 x^2-2 x^3))}{9 x+e^{2 x^2} x+6 x^2+x^3+e^{x^2} (-6 x-2 x^2)} \, dx\) [315]

3.4.15.1 Optimal result

Integrand size = 159, antiderivative size = 32 \begin {dmath*} \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 \end {dmath*}

output

x-exp(-1-(2-x)/(x+3-exp(x^2))+ln(1/4*x^2))
 

3.4.15.2 Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \begin {dmath*} \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 \end {dmath*}

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
 

3.4.15.3 Rubi [F]

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

\(\displaystyle \int \left (\frac {x^2}{\left (e^{x^2}-x-3\right )^2}+\frac {6 x}{\left (e^{x^2}-x-3\right )^2}-\frac {2 e^{x^2} (x+3)}{\left (e^{x^2}-x-3\right )^2}+\frac {e^{2 x^2}}{\left (e^{x^2}-x-3\right )^2}+\frac {9}{\left (e^{x^2}-x-3\right )^2}+\frac {e^{-\frac {e^{x^2}-5}{e^{x^2}-x-3}} \left (4 e^{x^2} x^2-2 x^2+5 e^{x^2} x+12 e^{x^2}-2 e^{2 x^2}-2 e^{x^2} x^3-17 x-18\right ) x}{4 \left (-e^{x^2}+x+3\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 9 \int \frac {1}{\left (-x+e^{x^2}-3\right )^2}dx-6 \int \frac {e^{x^2}}{\left (-x+e^{x^2}-3\right )^2}dx+\int \frac {e^{2 x^2}}{\left (-x+e^{x^2}-3\right )^2}dx-\frac {1}{2} \int e^{-\frac {-5+e^{x^2}}{-x+e^{x^2}-3}} xdx+6 \int \frac {x}{\left (-x+e^{x^2}-3\right )^2}dx-2 \int \frac {e^{x^2} x}{\left (-x+e^{x^2}-3\right )^2}dx+\int \frac {x^2}{\left (-x+e^{x^2}-3\right )^2}dx-\frac {1}{2} \int \frac {e^{-\frac {-5+e^{x^2}}{-x+e^{x^2}-3}} x^2}{\left (-x+e^{x^2}-3\right )^2}dx-\frac {1}{4} \int \frac {e^{-\frac {-5+e^{x^2}}{-x+e^{x^2}-3}} x^2}{x-e^{x^2}+3}dx-\frac {1}{2} \int \frac {e^{-\frac {-5+e^{x^2}}{-x+e^{x^2}-3}} x^5}{\left (-x+e^{x^2}-3\right )^2}dx-\frac {1}{2} \int \frac {e^{-\frac {-5+e^{x^2}}{-x+e^{x^2}-3}} x^4}{\left (-x+e^{x^2}-3\right )^2}dx+\frac {1}{2} \int \frac {e^{-\frac {-5+e^{x^2}}{-x+e^{x^2}-3}} x^4}{x-e^{x^2}+3}dx+\frac {13}{4} \int \frac {e^{-\frac {-5+e^{x^2}}{-x+e^{x^2}-3}} x^3}{\left (-x+e^{x^2}-3\right )^2}dx-\int \frac {e^{-\frac {-5+e^{x^2}}{-x+e^{x^2}-3}} x^3}{x-e^{x^2}+3}dx\)

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
 

3.4.15.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7292

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =! 
= u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.4.15.4 Maple [A] (verified)

Time = 3.04 (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	\(x -{\mathrm e}^{-\frac {i \pi x \operatorname {csgn}\left (i x^{2}\right )^{3}+3 i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}-i {\mathrm e}^{x^{2}} \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )^{2}+2 i {\mathrm e}^{x^{2}} \pi \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right )-i {\mathrm e}^{x^{2}} \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+i \pi x \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )+3 i \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )^{2}-6 i \pi \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right )-2 i \pi x \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+4 \,{\mathrm e}^{x^{2}} \ln \left (x \right )-4 x \ln \left (x \right )-4 \ln \left (2\right ) {\mathrm e}^{x^{2}}+4 x \ln \left (2\right )-12 \ln \left (x \right )+12 \ln \left (2\right )-2 \,{\mathrm e}^{x^{2}}+10}{2 \left (x +3-{\mathrm e}^{x^{2}}\right )}}\)	\(223\)

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))
 

3.4.15.5 Fricas [A] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.22 \begin {dmath*} \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 )} \end {dmath*}

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=\
 

output

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

3.4.15.6 Sympy [A] (verification not implemented)

Time = 0.59 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \begin {dmath*} \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}} \end {dmath*}

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))
 

3.4.15.7 Maxima [F]

\begin {dmath*} \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 } \end {dmath*}

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=\
 

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)
 

3.4.15.8 Giac [A] (verification not implemented)

Time = 1.70 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.03 \begin {dmath*} \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 \end {dmath*}

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=\
 

output

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

3.4.15.9 Mupad [B] (verification not implemented)

Time = 12.94 (sec) , antiderivative size = 145, normalized size of antiderivative = 4.53 \begin {dmath*} \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}}} \end {dmath*}

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)))