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

Optimal. Leaf size=32 \[ x-\frac {1}{4} e^{-1-\frac {2-x}{3-e^{x^2}+x}} x^2 \]

________________________________________________________________________________________

Rubi [F]  time = 11.01, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {9 x+e^{2 x^2} x+6 x^2+x^3+e^{x^2} \left (-6 x-2 x^2\right )+\exp \left (\frac {5-e^{x^2}+\left (-3+e^{x^2}-x\right ) \log \left (\frac {x^2}{4}\right )}{-3+e^{x^2}-x}\right ) \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 \end {gather*}

Verification is not applicable to the result.

[In]

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]

[Out]

x - Defer[Int][x/E^((-5 + E^x^2)/(-3 + E^x^2 - x)), x]/2 - Defer[Int][x^2/(E^((-5 + E^x^2)/(-3 + E^x^2 - x))*(
-3 + E^x^2 - x)^2), x]/2 + (13*Defer[Int][x^3/(E^((-5 + E^x^2)/(-3 + E^x^2 - x))*(-3 + E^x^2 - x)^2), x])/4 -
Defer[Int][x^4/(E^((-5 + E^x^2)/(-3 + E^x^2 - x))*(-3 + E^x^2 - x)^2), x]/2 - Defer[Int][x^5/(E^((-5 + E^x^2)/
(-3 + E^x^2 - x))*(-3 + E^x^2 - x)^2), x]/2 - Defer[Int][x^2/(E^((-5 + E^x^2)/(-3 + E^x^2 - x))*(3 - E^x^2 + x
)), x]/4 - Defer[Int][x^3/(E^((-5 + E^x^2)/(-3 + E^x^2 - x))*(3 - E^x^2 + x)), x] + Defer[Int][x^4/(E^((-5 + E
^x^2)/(-3 + E^x^2 - x))*(3 - E^x^2 + x)), x]/2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {9 x+e^{2 x^2} x+6 x^2+x^3-2 e^{x^2} x (3+x)-\frac {1}{4} e^{-\frac {-5+e^{x^2}}{-3+e^{x^2}-x}} x^2 \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 )}{x \left (3-e^{x^2}+x\right )^2} \, dx\\ &=\int \left (\frac {1}{2} e^{-\frac {e^{x^2}}{-3+e^{x^2}-x}} \left (2 e^{\frac {e^{x^2}}{-3+e^{x^2}-x}}-e^{\frac {5}{-3+e^{x^2}-x}} x\right )+\frac {e^{-\frac {-5+e^{x^2}}{-3+e^{x^2}-x}} x^2 \left (-1-4 x+2 x^2\right )}{4 \left (3-e^{x^2}+x\right )}+\frac {e^{-\frac {-5+e^{x^2}}{-3+e^{x^2}-x}} x^2 \left (-2+13 x-2 x^2-2 x^3\right )}{4 \left (3-e^{x^2}+x\right )^2}\right ) \, dx\\ &=\frac {1}{4} \int \frac {e^{-\frac {-5+e^{x^2}}{-3+e^{x^2}-x}} x^2 \left (-1-4 x+2 x^2\right )}{3-e^{x^2}+x} \, dx+\frac {1}{4} \int \frac {e^{-\frac {-5+e^{x^2}}{-3+e^{x^2}-x}} x^2 \left (-2+13 x-2 x^2-2 x^3\right )}{\left (3-e^{x^2}+x\right )^2} \, dx+\frac {1}{2} \int e^{-\frac {e^{x^2}}{-3+e^{x^2}-x}} \left (2 e^{\frac {e^{x^2}}{-3+e^{x^2}-x}}-e^{\frac {5}{-3+e^{x^2}-x}} x\right ) \, dx\\ &=\frac {1}{4} \int \left (-\frac {2 e^{-\frac {-5+e^{x^2}}{-3+e^{x^2}-x}} x^2}{\left (-3+e^{x^2}-x\right )^2}+\frac {13 e^{-\frac {-5+e^{x^2}}{-3+e^{x^2}-x}} x^3}{\left (-3+e^{x^2}-x\right )^2}-\frac {2 e^{-\frac {-5+e^{x^2}}{-3+e^{x^2}-x}} x^4}{\left (-3+e^{x^2}-x\right )^2}-\frac {2 e^{-\frac {-5+e^{x^2}}{-3+e^{x^2}-x}} x^5}{\left (-3+e^{x^2}-x\right )^2}\right ) \, dx+\frac {1}{4} \int \left (-\frac {e^{-\frac {-5+e^{x^2}}{-3+e^{x^2}-x}} x^2}{3-e^{x^2}+x}-\frac {4 e^{-\frac {-5+e^{x^2}}{-3+e^{x^2}-x}} x^3}{3-e^{x^2}+x}+\frac {2 e^{-\frac {-5+e^{x^2}}{-3+e^{x^2}-x}} x^4}{3-e^{x^2}+x}\right ) \, dx+\frac {1}{2} \int \left (2-e^{-\frac {-5+e^{x^2}}{-3+e^{x^2}-x}} x\right ) \, dx\\ &=x-\frac {1}{4} \int \frac {e^{-\frac {-5+e^{x^2}}{-3+e^{x^2}-x}} x^2}{3-e^{x^2}+x} \, dx-\frac {1}{2} \int e^{-\frac {-5+e^{x^2}}{-3+e^{x^2}-x}} x \, dx-\frac {1}{2} \int \frac {e^{-\frac {-5+e^{x^2}}{-3+e^{x^2}-x}} x^2}{\left (-3+e^{x^2}-x\right )^2} \, dx-\frac {1}{2} \int \frac {e^{-\frac {-5+e^{x^2}}{-3+e^{x^2}-x}} x^4}{\left (-3+e^{x^2}-x\right )^2} \, dx-\frac {1}{2} \int \frac {e^{-\frac {-5+e^{x^2}}{-3+e^{x^2}-x}} x^5}{\left (-3+e^{x^2}-x\right )^2} \, dx+\frac {1}{2} \int \frac {e^{-\frac {-5+e^{x^2}}{-3+e^{x^2}-x}} x^4}{3-e^{x^2}+x} \, dx+\frac {13}{4} \int \frac {e^{-\frac {-5+e^{x^2}}{-3+e^{x^2}-x}} x^3}{\left (-3+e^{x^2}-x\right )^2} \, dx-\int \frac {e^{-\frac {-5+e^{x^2}}{-3+e^{x^2}-x}} x^3}{3-e^{x^2}+x} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.23, size = 31, normalized size = 0.97 \begin {gather*} x-\frac {1}{4} e^{-1+\frac {2-x}{-3+e^{x^2}-x}} x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

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]

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.89, size = 39, normalized size = 1.22 \begin {gather*} 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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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)*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")

[Out]

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

________________________________________________________________________________________

giac [A]  time = 5.79, size = 33, normalized size = 1.03 \begin {gather*} -\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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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)*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")

[Out]

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

________________________________________________________________________________________

maple [C]  time = 0.25, size = 223, normalized size = 6.97

method	result	size
risch	\(x -{\mathrm e}^{-\frac {i \pi x \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )+i \pi x \mathrm {csgn}\left (i x^{2}\right )^{3}+3 i \pi \mathrm {csgn}\left (i x^{2}\right )^{3}+3 i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x \right )^{2}-i {\mathrm e}^{x^{2}} \pi \mathrm {csgn}\left (i x^{2}\right )^{3}-2 i \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}-6 i \pi \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i x \right )-i {\mathrm e}^{x^{2}} \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x \right )^{2}+2 i {\mathrm e}^{x^{2}} \pi \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i x \right )+4 \,{\mathrm e}^{x^{2}} \ln \relax (x )-4 x \ln \relax (x )-4 \ln \relax (2) {\mathrm e}^{x^{2}}+4 x \ln \relax (2)-12 \ln \relax (x )+12 \ln \relax (2)-2 \,{\mathrm e}^{x^{2}}+10}{2 \left (x +3-{\mathrm e}^{x^{2}}\right )}}\)	\(223\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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(((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,method=_RETURNVERBOSE)

[Out]

x-exp(-1/2*(I*Pi*x*csgn(I*x)^2*csgn(I*x^2)+I*Pi*x*csgn(I*x^2)^3+3*I*Pi*csgn(I*x^2)^3+3*I*Pi*csgn(I*x)^2*csgn(I
*x^2)-I*exp(x^2)*Pi*csgn(I*x^2)^3-2*I*Pi*x*csgn(I*x)*csgn(I*x^2)^2-6*I*Pi*csgn(I*x)*csgn(I*x^2)^2-I*exp(x^2)*P
i*csgn(I*x^2)*csgn(I*x)^2+2*I*exp(x^2)*Pi*csgn(I*x^2)^2*csgn(I*x)+4*exp(x^2)*ln(x)-4*x*ln(x)-4*ln(2)*exp(x^2)+
4*x*ln(2)-12*ln(x)+12*ln(2)-2*exp(x^2)+10)/(x+3-exp(x^2)))

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} x - \int \frac {{\left (2 \, x^{3} + 17 \, x^{2} + 2 \, x e^{\left (2 \, x^{2}\right )} + {\left (2 \, x^{4} - 4 \, x^{3} - 5 \, x^{2} - 12 \, x\right )} e^{\left (x^{2}\right )} + 18 \, x\right )} e^{\left (\frac {e^{\left (x^{2}\right )}}{x - e^{\left (x^{2}\right )} + 3} - \frac {5}{x - e^{\left (x^{2}\right )} + 3}\right )}}{4 \, {\left (x^{2} - 2 \, {\left (x + 3\right )} e^{\left (x^{2}\right )} + 6 \, x + e^{\left (2 \, x^{2}\right )} + 9\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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)*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")

[Out]

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)

________________________________________________________________________________________

mupad [B]  time = 9.09, size = 145, normalized size = 4.53 \begin {gather*} 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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 1.29, size = 32, normalized size = 1.00 \begin {gather*} 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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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)

[Out]

x - exp(((-x + exp(x**2) - 3)*log(x**2/4) - exp(x**2) + 5)/(-x + exp(x**2) - 3))

________________________________________________________________________________________