3.64.92 \(\int \frac {1134+40 e^9+e^6 (360-120 x)-1080 x+360 x^2-40 x^3+1080 e^{3 x^2} x^3+e^3 (1080-720 x+120 x^2)+e^{2 x^2} (3240 x^2+1080 e^3 x^2-1080 x^3)+e^{x^2} (-162+3240 x+360 e^6 x-2484 x^2+360 x^3+e^3 (2160 x-720 x^2))}{27+e^9+e^6 (9-3 x)-27 x+9 x^2-x^3+27 e^{3 x^2} x^3+e^3 (27-18 x+3 x^2)+e^{2 x^2} (81 x^2+27 e^3 x^2-27 x^3)+e^{x^2} (81 x+9 e^6 x-54 x^2+9 x^3+e^3 (54 x-18 x^2))} \, dx\)

Optimal. Leaf size=28 \[ 40 x+\frac {3}{\left (\frac {1}{3} \left (3+e^3-x\right )+e^{x^2} x\right )^2} \]

________________________________________________________________________________________

Rubi [F]  time = 2.36, 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 {1134+40 e^9+e^6 (360-120 x)-1080 x+360 x^2-40 x^3+1080 e^{3 x^2} x^3+e^3 \left (1080-720 x+120 x^2\right )+e^{2 x^2} \left (3240 x^2+1080 e^3 x^2-1080 x^3\right )+e^{x^2} \left (-162+3240 x+360 e^6 x-2484 x^2+360 x^3+e^3 \left (2160 x-720 x^2\right )\right )}{27+e^9+e^6 (9-3 x)-27 x+9 x^2-x^3+27 e^{3 x^2} x^3+e^3 \left (27-18 x+3 x^2\right )+e^{2 x^2} \left (81 x^2+27 e^3 x^2-27 x^3\right )+e^{x^2} \left (81 x+9 e^6 x-54 x^2+9 x^3+e^3 \left (54 x-18 x^2\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(1134 + 40*E^9 + E^6*(360 - 120*x) - 1080*x + 360*x^2 - 40*x^3 + 1080*E^(3*x^2)*x^3 + E^3*(1080 - 720*x +
120*x^2) + E^(2*x^2)*(3240*x^2 + 1080*E^3*x^2 - 1080*x^3) + E^x^2*(-162 + 3240*x + 360*E^6*x - 2484*x^2 + 360*
x^3 + E^3*(2160*x - 720*x^2)))/(27 + E^9 + E^6*(9 - 3*x) - 27*x + 9*x^2 - x^3 + 27*E^(3*x^2)*x^3 + E^3*(27 - 1
8*x + 3*x^2) + E^(2*x^2)*(81*x^2 + 27*E^3*x^2 - 27*x^3) + E^x^2*(81*x + 9*E^6*x - 54*x^2 + 9*x^3 + E^3*(54*x -
18*x^2))),x]

[Out]

40*x + 108*Defer[Int][x^2/(-3*(1 + E^3/3) + x - 3*E^x^2*x)^3, x] + 54*(3 + E^3)*Defer[Int][1/(x*(3*(1 + E^3/3)
- x + 3*E^x^2*x)^3), x] + 108*(3 + E^3)*Defer[Int][x/(3*(1 + E^3/3) - x + 3*E^x^2*x)^3, x] - 54*Defer[Int][1/
(x*(3*(1 + E^3/3) - x + 3*E^x^2*x)^2), x] - 108*Defer[Int][x/(3*(1 + E^3/3) - x + 3*E^x^2*x)^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (567 \left (1+\frac {20 e^9}{567}\right )-60 e^6 (-3+x)+60 e^3 (-3+x)^2-540 x+180 e^{6+x^2} x-360 e^{3+x^2} (-3+x) x+180 x^2+540 e^{3+2 x^2} x^2-540 e^{2 x^2} (-3+x) x^2-20 x^3+540 e^{3 x^2} x^3+9 e^{x^2} \left (-9+180 x-138 x^2+20 x^3\right )\right )}{\left (3 \left (1+\frac {e^3}{3}\right )-x+3 e^{x^2} x\right )^3} \, dx\\ &=2 \int \frac {567 \left (1+\frac {20 e^9}{567}\right )-60 e^6 (-3+x)+60 e^3 (-3+x)^2-540 x+180 e^{6+x^2} x-360 e^{3+x^2} (-3+x) x+180 x^2+540 e^{3+2 x^2} x^2-540 e^{2 x^2} (-3+x) x^2-20 x^3+540 e^{3 x^2} x^3+9 e^{x^2} \left (-9+180 x-138 x^2+20 x^3\right )}{\left (3 \left (1+\frac {e^3}{3}\right )-x+3 e^{x^2} x\right )^3} \, dx\\ &=2 \int \left (20+\frac {27 \left (-1-2 x^2\right )}{x \left (3 \left (1+\frac {e^3}{3}\right )-x+3 e^{x^2} x\right )^2}+\frac {27 \left (3+e^3+2 \left (3+e^3\right ) x^2-2 x^3\right )}{x \left (3 \left (1+\frac {e^3}{3}\right )-x+3 e^{x^2} x\right )^3}\right ) \, dx\\ &=40 x+54 \int \frac {-1-2 x^2}{x \left (3 \left (1+\frac {e^3}{3}\right )-x+3 e^{x^2} x\right )^2} \, dx+54 \int \frac {3+e^3+2 \left (3+e^3\right ) x^2-2 x^3}{x \left (3 \left (1+\frac {e^3}{3}\right )-x+3 e^{x^2} x\right )^3} \, dx\\ &=40 x+54 \int \left (\frac {2 x^2}{\left (-3 \left (1+\frac {e^3}{3}\right )+x-3 e^{x^2} x\right )^3}+\frac {3+e^3}{x \left (3 \left (1+\frac {e^3}{3}\right )-x+3 e^{x^2} x\right )^3}+\frac {2 \left (3+e^3\right ) x}{\left (3 \left (1+\frac {e^3}{3}\right )-x+3 e^{x^2} x\right )^3}\right ) \, dx+54 \int \left (-\frac {1}{x \left (3 \left (1+\frac {e^3}{3}\right )-x+3 e^{x^2} x\right )^2}-\frac {2 x}{\left (3 \left (1+\frac {e^3}{3}\right )-x+3 e^{x^2} x\right )^2}\right ) \, dx\\ &=40 x-54 \int \frac {1}{x \left (3 \left (1+\frac {e^3}{3}\right )-x+3 e^{x^2} x\right )^2} \, dx+108 \int \frac {x^2}{\left (-3 \left (1+\frac {e^3}{3}\right )+x-3 e^{x^2} x\right )^3} \, dx-108 \int \frac {x}{\left (3 \left (1+\frac {e^3}{3}\right )-x+3 e^{x^2} x\right )^2} \, dx+\left (54 \left (3+e^3\right )\right ) \int \frac {1}{x \left (3 \left (1+\frac {e^3}{3}\right )-x+3 e^{x^2} x\right )^3} \, dx+\left (108 \left (3+e^3\right )\right ) \int \frac {x}{\left (3 \left (1+\frac {e^3}{3}\right )-x+3 e^{x^2} x\right )^3} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.08, size = 28, normalized size = 1.00 \begin {gather*} 2 \left (20 x+\frac {27}{2 \left (3+e^3-x+3 e^{x^2} x\right )^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1134 + 40*E^9 + E^6*(360 - 120*x) - 1080*x + 360*x^2 - 40*x^3 + 1080*E^(3*x^2)*x^3 + E^3*(1080 - 72
0*x + 120*x^2) + E^(2*x^2)*(3240*x^2 + 1080*E^3*x^2 - 1080*x^3) + E^x^2*(-162 + 3240*x + 360*E^6*x - 2484*x^2
+ 360*x^3 + E^3*(2160*x - 720*x^2)))/(27 + E^9 + E^6*(9 - 3*x) - 27*x + 9*x^2 - x^3 + 27*E^(3*x^2)*x^3 + E^3*(
27 - 18*x + 3*x^2) + E^(2*x^2)*(81*x^2 + 27*E^3*x^2 - 27*x^3) + E^x^2*(81*x + 9*E^6*x - 54*x^2 + 9*x^3 + E^3*(
54*x - 18*x^2))),x]

[Out]

2*(20*x + 27/(2*(3 + E^3 - x + 3*E^x^2*x)^2))

________________________________________________________________________________________

fricas [B]  time = 0.72, size = 113, normalized size = 4.04 \begin {gather*} \frac {360 \, x^{3} e^{\left (2 \, x^{2}\right )} + 40 \, x^{3} - 240 \, x^{2} + 40 \, x e^{6} - 80 \, {\left (x^{2} - 3 \, x\right )} e^{3} - 240 \, {\left (x^{3} - x^{2} e^{3} - 3 \, x^{2}\right )} e^{\left (x^{2}\right )} + 360 \, x + 27}{9 \, x^{2} e^{\left (2 \, x^{2}\right )} + x^{2} - 2 \, {\left (x - 3\right )} e^{3} - 6 \, {\left (x^{2} - x e^{3} - 3 \, x\right )} e^{\left (x^{2}\right )} - 6 \, x + e^{6} + 9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1080*x^3*exp(x^2)^3+(1080*x^2*exp(3)-1080*x^3+3240*x^2)*exp(x^2)^2+(360*x*exp(3)^2+(-720*x^2+2160*x
)*exp(3)+360*x^3-2484*x^2+3240*x-162)*exp(x^2)+40*exp(3)^3+(-120*x+360)*exp(3)^2+(120*x^2-720*x+1080)*exp(3)-4
0*x^3+360*x^2-1080*x+1134)/(27*x^3*exp(x^2)^3+(27*x^2*exp(3)-27*x^3+81*x^2)*exp(x^2)^2+(9*x*exp(3)^2+(-18*x^2+
54*x)*exp(3)+9*x^3-54*x^2+81*x)*exp(x^2)+exp(3)^3+(-3*x+9)*exp(3)^2+(3*x^2-18*x+27)*exp(3)-x^3+9*x^2-27*x+27),
x, algorithm="fricas")

[Out]

(360*x^3*e^(2*x^2) + 40*x^3 - 240*x^2 + 40*x*e^6 - 80*(x^2 - 3*x)*e^3 - 240*(x^3 - x^2*e^3 - 3*x^2)*e^(x^2) +
360*x + 27)/(9*x^2*e^(2*x^2) + x^2 - 2*(x - 3)*e^3 - 6*(x^2 - x*e^3 - 3*x)*e^(x^2) - 6*x + e^6 + 9)

________________________________________________________________________________________

giac [B]  time = 0.95, size = 130, normalized size = 4.64 \begin {gather*} \frac {360 \, x^{3} e^{\left (2 \, x^{2}\right )} - 240 \, x^{3} e^{\left (x^{2}\right )} + 40 \, x^{3} - 80 \, x^{2} e^{3} + 240 \, x^{2} e^{\left (x^{2} + 3\right )} + 720 \, x^{2} e^{\left (x^{2}\right )} - 240 \, x^{2} + 40 \, x e^{6} + 240 \, x e^{3} + 360 \, x + 27}{9 \, x^{2} e^{\left (2 \, x^{2}\right )} - 6 \, x^{2} e^{\left (x^{2}\right )} + x^{2} - 2 \, x e^{3} + 6 \, x e^{\left (x^{2} + 3\right )} + 18 \, x e^{\left (x^{2}\right )} - 6 \, x + e^{6} + 6 \, e^{3} + 9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1080*x^3*exp(x^2)^3+(1080*x^2*exp(3)-1080*x^3+3240*x^2)*exp(x^2)^2+(360*x*exp(3)^2+(-720*x^2+2160*x
)*exp(3)+360*x^3-2484*x^2+3240*x-162)*exp(x^2)+40*exp(3)^3+(-120*x+360)*exp(3)^2+(120*x^2-720*x+1080)*exp(3)-4
0*x^3+360*x^2-1080*x+1134)/(27*x^3*exp(x^2)^3+(27*x^2*exp(3)-27*x^3+81*x^2)*exp(x^2)^2+(9*x*exp(3)^2+(-18*x^2+
54*x)*exp(3)+9*x^3-54*x^2+81*x)*exp(x^2)+exp(3)^3+(-3*x+9)*exp(3)^2+(3*x^2-18*x+27)*exp(3)-x^3+9*x^2-27*x+27),
x, algorithm="giac")

[Out]

(360*x^3*e^(2*x^2) - 240*x^3*e^(x^2) + 40*x^3 - 80*x^2*e^3 + 240*x^2*e^(x^2 + 3) + 720*x^2*e^(x^2) - 240*x^2 +
40*x*e^6 + 240*x*e^3 + 360*x + 27)/(9*x^2*e^(2*x^2) - 6*x^2*e^(x^2) + x^2 - 2*x*e^3 + 6*x*e^(x^2 + 3) + 18*x*
e^(x^2) - 6*x + e^6 + 6*e^3 + 9)

________________________________________________________________________________________

maple [A]  time = 0.52, size = 23, normalized size = 0.82

method	result	size
risch	\(40 x +\frac {27}{\left (3 \,{\mathrm e}^{x^{2}} x +{\mathrm e}^{3}-x +3\right )^{2}}\)	\(23\)
norman	\(\frac {\left (-120 \,{\mathrm e}^{6}-720 \,{\mathrm e}^{3}-1080\right ) x +\left (-240 \,{\mathrm e}^{3}-720\right ) x^{2} {\mathrm e}^{x^{2}}+\left (720 \,{\mathrm e}^{3}+2160\right ) x^{2} {\mathrm e}^{2 x^{2}}+\left (480 \,{\mathrm e}^{6}+2880 \,{\mathrm e}^{3}+4320\right ) x \,{\mathrm e}^{x^{2}}+40 x^{3}-240 x^{3} {\mathrm e}^{x^{2}}+360 x^{3} {\mathrm e}^{2 x^{2}}+2187+80 \,{\mathrm e}^{9}+720 \,{\mathrm e}^{6}+2160 \,{\mathrm e}^{3}}{\left (3 \,{\mathrm e}^{x^{2}} x +{\mathrm e}^{3}-x +3\right )^{2}}\)	\(123\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1080*x^3*exp(x^2)^3+(1080*x^2*exp(3)-1080*x^3+3240*x^2)*exp(x^2)^2+(360*x*exp(3)^2+(-720*x^2+2160*x)*exp(
3)+360*x^3-2484*x^2+3240*x-162)*exp(x^2)+40*exp(3)^3+(-120*x+360)*exp(3)^2+(120*x^2-720*x+1080)*exp(3)-40*x^3+
360*x^2-1080*x+1134)/(27*x^3*exp(x^2)^3+(27*x^2*exp(3)-27*x^3+81*x^2)*exp(x^2)^2+(9*x*exp(3)^2+(-18*x^2+54*x)*
exp(3)+9*x^3-54*x^2+81*x)*exp(x^2)+exp(3)^3+(-3*x+9)*exp(3)^2+(3*x^2-18*x+27)*exp(3)-x^3+9*x^2-27*x+27),x,meth
od=_RETURNVERBOSE)

[Out]

40*x+27/(3*exp(x^2)*x+exp(3)-x+3)^2

________________________________________________________________________________________

maxima [B]  time = 0.43, size = 106, normalized size = 3.79 \begin {gather*} \frac {360 \, x^{3} e^{\left (2 \, x^{2}\right )} + 40 \, x^{3} - 80 \, x^{2} {\left (e^{3} + 3\right )} + 40 \, x {\left (e^{6} + 6 \, e^{3} + 9\right )} - 240 \, {\left (x^{3} - x^{2} {\left (e^{3} + 3\right )}\right )} e^{\left (x^{2}\right )} + 27}{9 \, x^{2} e^{\left (2 \, x^{2}\right )} + x^{2} - 2 \, x {\left (e^{3} + 3\right )} - 6 \, {\left (x^{2} - x {\left (e^{3} + 3\right )}\right )} e^{\left (x^{2}\right )} + e^{6} + 6 \, e^{3} + 9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1080*x^3*exp(x^2)^3+(1080*x^2*exp(3)-1080*x^3+3240*x^2)*exp(x^2)^2+(360*x*exp(3)^2+(-720*x^2+2160*x
)*exp(3)+360*x^3-2484*x^2+3240*x-162)*exp(x^2)+40*exp(3)^3+(-120*x+360)*exp(3)^2+(120*x^2-720*x+1080)*exp(3)-4
0*x^3+360*x^2-1080*x+1134)/(27*x^3*exp(x^2)^3+(27*x^2*exp(3)-27*x^3+81*x^2)*exp(x^2)^2+(9*x*exp(3)^2+(-18*x^2+
54*x)*exp(3)+9*x^3-54*x^2+81*x)*exp(x^2)+exp(3)^3+(-3*x+9)*exp(3)^2+(3*x^2-18*x+27)*exp(3)-x^3+9*x^2-27*x+27),
x, algorithm="maxima")

[Out]

(360*x^3*e^(2*x^2) + 40*x^3 - 80*x^2*(e^3 + 3) + 40*x*(e^6 + 6*e^3 + 9) - 240*(x^3 - x^2*(e^3 + 3))*e^(x^2) +
27)/(9*x^2*e^(2*x^2) + x^2 - 2*x*(e^3 + 3) - 6*(x^2 - x*(e^3 + 3))*e^(x^2) + e^6 + 6*e^3 + 9)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {40\,{\mathrm {e}}^9-1080\,x+{\mathrm {e}}^3\,\left (120\,x^2-720\,x+1080\right )+{\mathrm {e}}^{2\,x^2}\,\left (1080\,x^2\,{\mathrm {e}}^3+3240\,x^2-1080\,x^3\right )+{\mathrm {e}}^{x^2}\,\left (3240\,x+{\mathrm {e}}^3\,\left (2160\,x-720\,x^2\right )+360\,x\,{\mathrm {e}}^6-2484\,x^2+360\,x^3-162\right )+1080\,x^3\,{\mathrm {e}}^{3\,x^2}+360\,x^2-40\,x^3-{\mathrm {e}}^6\,\left (120\,x-360\right )+1134}{{\mathrm {e}}^9-27\,x+{\mathrm {e}}^3\,\left (3\,x^2-18\,x+27\right )+{\mathrm {e}}^{2\,x^2}\,\left (27\,x^2\,{\mathrm {e}}^3+81\,x^2-27\,x^3\right )+{\mathrm {e}}^{x^2}\,\left (81\,x+{\mathrm {e}}^3\,\left (54\,x-18\,x^2\right )+9\,x\,{\mathrm {e}}^6-54\,x^2+9\,x^3\right )+27\,x^3\,{\mathrm {e}}^{3\,x^2}+9\,x^2-x^3-{\mathrm {e}}^6\,\left (3\,x-9\right )+27} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((40*exp(9) - 1080*x + exp(3)*(120*x^2 - 720*x + 1080) + exp(2*x^2)*(1080*x^2*exp(3) + 3240*x^2 - 1080*x^3)
+ exp(x^2)*(3240*x + exp(3)*(2160*x - 720*x^2) + 360*x*exp(6) - 2484*x^2 + 360*x^3 - 162) + 1080*x^3*exp(3*x^
2) + 360*x^2 - 40*x^3 - exp(6)*(120*x - 360) + 1134)/(exp(9) - 27*x + exp(3)*(3*x^2 - 18*x + 27) + exp(2*x^2)*
(27*x^2*exp(3) + 81*x^2 - 27*x^3) + exp(x^2)*(81*x + exp(3)*(54*x - 18*x^2) + 9*x*exp(6) - 54*x^2 + 9*x^3) + 2
7*x^3*exp(3*x^2) + 9*x^2 - x^3 - exp(6)*(3*x - 9) + 27),x)

[Out]

int((40*exp(9) - 1080*x + exp(3)*(120*x^2 - 720*x + 1080) + exp(2*x^2)*(1080*x^2*exp(3) + 3240*x^2 - 1080*x^3)
+ exp(x^2)*(3240*x + exp(3)*(2160*x - 720*x^2) + 360*x*exp(6) - 2484*x^2 + 360*x^3 - 162) + 1080*x^3*exp(3*x^
2) + 360*x^2 - 40*x^3 - exp(6)*(120*x - 360) + 1134)/(exp(9) - 27*x + exp(3)*(3*x^2 - 18*x + 27) + exp(2*x^2)*
(27*x^2*exp(3) + 81*x^2 - 27*x^3) + exp(x^2)*(81*x + exp(3)*(54*x - 18*x^2) + 9*x*exp(6) - 54*x^2 + 9*x^3) + 2
7*x^3*exp(3*x^2) + 9*x^2 - x^3 - exp(6)*(3*x - 9) + 27), x)

________________________________________________________________________________________

sympy [B]  time = 0.35, size = 60, normalized size = 2.14 \begin {gather*} 40 x + \frac {27}{9 x^{2} e^{2 x^{2}} + x^{2} - 2 x e^{3} - 6 x + \left (- 6 x^{2} + 18 x + 6 x e^{3}\right ) e^{x^{2}} + 9 + 6 e^{3} + e^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1080*x**3*exp(x**2)**3+(1080*x**2*exp(3)-1080*x**3+3240*x**2)*exp(x**2)**2+(360*x*exp(3)**2+(-720*x
**2+2160*x)*exp(3)+360*x**3-2484*x**2+3240*x-162)*exp(x**2)+40*exp(3)**3+(-120*x+360)*exp(3)**2+(120*x**2-720*
x+1080)*exp(3)-40*x**3+360*x**2-1080*x+1134)/(27*x**3*exp(x**2)**3+(27*x**2*exp(3)-27*x**3+81*x**2)*exp(x**2)*
*2+(9*x*exp(3)**2+(-18*x**2+54*x)*exp(3)+9*x**3-54*x**2+81*x)*exp(x**2)+exp(3)**3+(-3*x+9)*exp(3)**2+(3*x**2-1
8*x+27)*exp(3)-x**3+9*x**2-27*x+27),x)

[Out]

40*x + 27/(9*x**2*exp(2*x**2) + x**2 - 2*x*exp(3) - 6*x + (-6*x**2 + 18*x + 6*x*exp(3))*exp(x**2) + 9 + 6*exp(
3) + exp(6))

________________________________________________________________________________________