3.101.19 \(\int \frac {-9+e^{2 x^2} (-4+16 e)-8 x-4 x^2+e^{x^2} (13+e (-72-32 x)+8 x-10 x^2)+e (81+72 x+16 x^2)}{81 x^2+16 e^{2 x^2} x^2+72 x^3+16 x^4+e^{x^2} (-72 x^2-32 x^3)} \, dx\)

Optimal. Leaf size=34 \[ \frac {-e+x-\frac {x}{x-5 \left (x+\frac {x}{1-e^{x^2}+x}\right )}}{x} \]

________________________________________________________________________________________

Rubi [F]  time = 1.66, 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+e^{2 x^2} (-4+16 e)-8 x-4 x^2+e^{x^2} \left (13+e (-72-32 x)+8 x-10 x^2\right )+e \left (81+72 x+16 x^2\right )}{81 x^2+16 e^{2 x^2} x^2+72 x^3+16 x^4+e^{x^2} \left (-72 x^2-32 x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-9 + E^(2*x^2)*(-4 + 16*E) - 8*x - 4*x^2 + E^x^2*(13 + E*(-72 - 32*x) + 8*x - 10*x^2) + E*(81 + 72*x + 16
*x^2))/(81*x^2 + 16*E^(2*x^2)*x^2 + 72*x^3 + 16*x^4 + E^x^2*(-72*x^2 - 32*x^3)),x]

[Out]

(1 - 4*E)/(4*x) - (45*Defer[Int][(-9 + 4*E^x^2 - 4*x)^(-2), x])/2 - (5*Defer[Int][(-9 + 4*E^x^2 - 4*x)^(-1), x
])/2 - (5*Defer[Int][1/((-9 + 4*E^x^2 - 4*x)*x^2), x])/4 - 10*Defer[Int][x/(-9 + 4*E^x^2 - 4*x)^2, x] + 5*Defe
r[Int][1/(x*(9 - 4*E^x^2 + 4*x)^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-9+e^{2 x^2} (-4+16 e)-8 x-4 x^2+e^{x^2} \left (13+e (-72-32 x)+8 x-10 x^2\right )+e \left (81+72 x+16 x^2\right )}{x^2 \left (9-4 e^{x^2}+4 x\right )^2} \, dx\\ &=\int \left (\frac {-1+4 e}{4 x^2}-\frac {5 \left (1+2 x^2\right )}{4 \left (-9+4 e^{x^2}-4 x\right ) x^2}-\frac {5 \left (-2+9 x+4 x^2\right )}{2 x \left (9-4 e^{x^2}+4 x\right )^2}\right ) \, dx\\ &=\frac {1-4 e}{4 x}-\frac {5}{4} \int \frac {1+2 x^2}{\left (-9+4 e^{x^2}-4 x\right ) x^2} \, dx-\frac {5}{2} \int \frac {-2+9 x+4 x^2}{x \left (9-4 e^{x^2}+4 x\right )^2} \, dx\\ &=\frac {1-4 e}{4 x}-\frac {5}{4} \int \left (\frac {2}{-9+4 e^{x^2}-4 x}+\frac {1}{\left (-9+4 e^{x^2}-4 x\right ) x^2}\right ) \, dx-\frac {5}{2} \int \left (\frac {9}{\left (-9+4 e^{x^2}-4 x\right )^2}+\frac {4 x}{\left (-9+4 e^{x^2}-4 x\right )^2}-\frac {2}{x \left (9-4 e^{x^2}+4 x\right )^2}\right ) \, dx\\ &=\frac {1-4 e}{4 x}-\frac {5}{4} \int \frac {1}{\left (-9+4 e^{x^2}-4 x\right ) x^2} \, dx-\frac {5}{2} \int \frac {1}{-9+4 e^{x^2}-4 x} \, dx+5 \int \frac {1}{x \left (9-4 e^{x^2}+4 x\right )^2} \, dx-10 \int \frac {x}{\left (-9+4 e^{x^2}-4 x\right )^2} \, dx-\frac {45}{2} \int \frac {1}{\left (-9+4 e^{x^2}-4 x\right )^2} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.06, size = 28, normalized size = 0.82 \begin {gather*} \frac {1-4 e+\frac {5}{-9+4 e^{x^2}-4 x}}{4 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-9 + E^(2*x^2)*(-4 + 16*E) - 8*x - 4*x^2 + E^x^2*(13 + E*(-72 - 32*x) + 8*x - 10*x^2) + E*(81 + 72*
x + 16*x^2))/(81*x^2 + 16*E^(2*x^2)*x^2 + 72*x^3 + 16*x^4 + E^x^2*(-72*x^2 - 32*x^3)),x]

[Out]

(1 - 4*E + 5/(-9 + 4*E^x^2 - 4*x))/(4*x)

________________________________________________________________________________________

fricas [A]  time = 0.64, size = 45, normalized size = 1.32 \begin {gather*} -\frac {{\left (4 \, x + 9\right )} e - {\left (4 \, e - 1\right )} e^{\left (x^{2}\right )} - x - 1}{4 \, x^{2} - 4 \, x e^{\left (x^{2}\right )} + 9 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*exp(1)-4)*exp(x^2)^2+((-32*x-72)*exp(1)-10*x^2+8*x+13)*exp(x^2)+(16*x^2+72*x+81)*exp(1)-4*x^2-8
*x-9)/(16*x^2*exp(x^2)^2+(-32*x^3-72*x^2)*exp(x^2)+16*x^4+72*x^3+81*x^2),x, algorithm="fricas")

[Out]

-((4*x + 9)*e - (4*e - 1)*e^(x^2) - x - 1)/(4*x^2 - 4*x*e^(x^2) + 9*x)

________________________________________________________________________________________

giac [A]  time = 0.14, size = 46, normalized size = 1.35 \begin {gather*} -\frac {4 \, x e - x + 9 \, e - 4 \, e^{\left (x^{2} + 1\right )} + e^{\left (x^{2}\right )} - 1}{4 \, x^{2} - 4 \, x e^{\left (x^{2}\right )} + 9 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*exp(1)-4)*exp(x^2)^2+((-32*x-72)*exp(1)-10*x^2+8*x+13)*exp(x^2)+(16*x^2+72*x+81)*exp(1)-4*x^2-8
*x-9)/(16*x^2*exp(x^2)^2+(-32*x^3-72*x^2)*exp(x^2)+16*x^4+72*x^3+81*x^2),x, algorithm="giac")

[Out]

-(4*x*e - x + 9*e - 4*e^(x^2 + 1) + e^(x^2) - 1)/(4*x^2 - 4*x*e^(x^2) + 9*x)

________________________________________________________________________________________

maple [A]  time = 0.09, size = 32, normalized size = 0.94

method	result	size
risch	\(-\frac {{\mathrm e}}{x}+\frac {1}{4 x}-\frac {5}{4 x \left (4 x -4 \,{\mathrm e}^{x^{2}}+9\right )}\)	\(32\)
norman	\(\frac {\left (4 \,{\mathrm e}-1\right ) {\mathrm e}^{x^{2}}+\left (-4 \,{\mathrm e}+1\right ) x +1-9 \,{\mathrm e}}{x \left (4 x -4 \,{\mathrm e}^{x^{2}}+9\right )}\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((16*exp(1)-4)*exp(x^2)^2+((-32*x-72)*exp(1)-10*x^2+8*x+13)*exp(x^2)+(16*x^2+72*x+81)*exp(1)-4*x^2-8*x-9)/
(16*x^2*exp(x^2)^2+(-32*x^3-72*x^2)*exp(x^2)+16*x^4+72*x^3+81*x^2),x,method=_RETURNVERBOSE)

[Out]

-exp(1)/x+1/4/x-5/4/x/(4*x-4*exp(x^2)+9)

________________________________________________________________________________________

maxima [A]  time = 0.47, size = 46, normalized size = 1.35 \begin {gather*} -\frac {x {\left (4 \, e - 1\right )} - {\left (4 \, e - 1\right )} e^{\left (x^{2}\right )} + 9 \, e - 1}{4 \, x^{2} - 4 \, x e^{\left (x^{2}\right )} + 9 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*exp(1)-4)*exp(x^2)^2+((-32*x-72)*exp(1)-10*x^2+8*x+13)*exp(x^2)+(16*x^2+72*x+81)*exp(1)-4*x^2-8
*x-9)/(16*x^2*exp(x^2)^2+(-32*x^3-72*x^2)*exp(x^2)+16*x^4+72*x^3+81*x^2),x, algorithm="maxima")

[Out]

-(x*(4*e - 1) - (4*e - 1)*e^(x^2) + 9*e - 1)/(4*x^2 - 4*x*e^(x^2) + 9*x)

________________________________________________________________________________________

mupad [B]  time = 8.00, size = 59, normalized size = 1.74 \begin {gather*} \frac {x^2\,\left (\frac {16\,\mathrm {e}}{9}-\frac {4}{9}\right )-9\,\mathrm {e}+{\mathrm {e}}^{x^2}\,\left (4\,\mathrm {e}-1\right )-x\,{\mathrm {e}}^{x^2}\,\left (\frac {16\,\mathrm {e}}{9}-\frac {4}{9}\right )+1}{9\,x-4\,x\,{\mathrm {e}}^{x^2}+4\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(8*x - exp(x^2)*(8*x - 10*x^2 - exp(1)*(32*x + 72) + 13) - exp(2*x^2)*(16*exp(1) - 4) - exp(1)*(72*x + 16
*x^2 + 81) + 4*x^2 + 9)/(16*x^2*exp(2*x^2) - exp(x^2)*(72*x^2 + 32*x^3) + 81*x^2 + 72*x^3 + 16*x^4),x)

[Out]

(x^2*((16*exp(1))/9 - 4/9) - 9*exp(1) + exp(x^2)*(4*exp(1) - 1) - x*exp(x^2)*((16*exp(1))/9 - 4/9) + 1)/(9*x -
4*x*exp(x^2) + 4*x^2)

________________________________________________________________________________________

sympy [A]  time = 0.17, size = 26, normalized size = 0.76 \begin {gather*} \frac {5}{- 16 x^{2} + 16 x e^{x^{2}} - 36 x} - \frac {- \frac {1}{4} + e}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*exp(1)-4)*exp(x**2)**2+((-32*x-72)*exp(1)-10*x**2+8*x+13)*exp(x**2)+(16*x**2+72*x+81)*exp(1)-4*
x**2-8*x-9)/(16*x**2*exp(x**2)**2+(-32*x**3-72*x**2)*exp(x**2)+16*x**4+72*x**3+81*x**2),x)

[Out]

5/(-16*x**2 + 16*x*exp(x**2) - 36*x) - (-1/4 + E)/x

________________________________________________________________________________________