3.27.27 \(\int \frac {30+30 e^4+e^{e-x^2} (5+10 x^2)}{36 e^{10}+e^{2+2 e-2 x^2}+e^6 (72+12 x)+e^2 (36+12 x+x^2)+e^{e-x^2} (12 e^6+e^2 (12+2 x))} \, dx\)

Optimal. Leaf size=25 \[ \frac {5 x}{e^2 \left (6+6 e^4+e^{e-x^2}+x\right )} \]

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Rubi [F]  time = 3.72, 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 {30+30 e^4+e^{e-x^2} \left (5+10 x^2\right )}{36 e^{10}+e^{2+2 e-2 x^2}+e^6 (72+12 x)+e^2 \left (36+12 x+x^2\right )+e^{e-x^2} \left (12 e^6+e^2 (12+2 x)\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(30 + 30*E^4 + E^(E - x^2)*(5 + 10*x^2))/(36*E^10 + E^(2 + 2*E - 2*x^2) + E^6*(72 + 12*x) + E^2*(36 + 12*x
+ x^2) + E^(E - x^2)*(12*E^6 + E^2*(12 + 2*x))),x]

[Out]

5*Defer[Int][E^(-2 + E + x^2)/(E^E + 6*E^x^2*(1 + E^4) + E^x^2*x)^2, x] + 10*Defer[Int][(E^(-2 + E + x^2)*x^2)
/(E^E + 6*E^x^2*(1 + E^4) + E^x^2*x)^2, x] - 30*(1 + E^4)*Defer[Int][E^(-2 + E + x^2)/((6 + 6*E^4 + x)*(E^E +
6*E^x^2*(1 + E^4) + E^x^2*x)^2), x] + 30*(1 + E^4)*Defer[Int][E^(-2 + x^2)/((6 + 6*E^4 + x)*(E^E + 6*E^x^2*(1
+ E^4) + E^x^2*x)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {5 e^{-2+x^2} \left (6 e^{x^2} \left (1+e^4\right )+e^e \left (1+2 x^2\right )\right )}{\left (e^e+6 e^{4+x^2}+e^{x^2} (6+x)\right )^2} \, dx\\ &=5 \int \frac {e^{-2+x^2} \left (6 e^{x^2} \left (1+e^4\right )+e^e \left (1+2 x^2\right )\right )}{\left (e^e+6 e^{4+x^2}+e^{x^2} (6+x)\right )^2} \, dx\\ &=5 \int \left (\frac {6 e^{-2+x^2} \left (1+e^4\right )}{\left (6+6 e^4+x\right ) \left (e^e+6 e^{x^2} \left (1+e^4\right )+e^{x^2} x\right )}+\frac {e^{-2+e+x^2} x \left (1+12 \left (1+e^4\right ) x+2 x^2\right )}{\left (6+6 e^4+x\right ) \left (e^e+6 e^{x^2} \left (1+e^4\right )+e^{x^2} x\right )^2}\right ) \, dx\\ &=5 \int \frac {e^{-2+e+x^2} x \left (1+12 \left (1+e^4\right ) x+2 x^2\right )}{\left (6+6 e^4+x\right ) \left (e^e+6 e^{x^2} \left (1+e^4\right )+e^{x^2} x\right )^2} \, dx+\left (30 \left (1+e^4\right )\right ) \int \frac {e^{-2+x^2}}{\left (6+6 e^4+x\right ) \left (e^e+6 e^{x^2} \left (1+e^4\right )+e^{x^2} x\right )} \, dx\\ &=5 \int \left (\frac {e^{-2+e+x^2}}{\left (e^e+6 e^{x^2} \left (1+e^4\right )+e^{x^2} x\right )^2}+\frac {2 e^{-2+e+x^2} x^2}{\left (e^e+6 e^{x^2} \left (1+e^4\right )+e^{x^2} x\right )^2}+\frac {6 e^{-2+e+x^2} \left (-1-e^4\right )}{\left (6+6 e^4+x\right ) \left (e^e+6 e^{x^2} \left (1+e^4\right )+e^{x^2} x\right )^2}\right ) \, dx+\left (30 \left (1+e^4\right )\right ) \int \frac {e^{-2+x^2}}{\left (6+6 e^4+x\right ) \left (e^e+6 e^{x^2} \left (1+e^4\right )+e^{x^2} x\right )} \, dx\\ &=5 \int \frac {e^{-2+e+x^2}}{\left (e^e+6 e^{x^2} \left (1+e^4\right )+e^{x^2} x\right )^2} \, dx+10 \int \frac {e^{-2+e+x^2} x^2}{\left (e^e+6 e^{x^2} \left (1+e^4\right )+e^{x^2} x\right )^2} \, dx-\left (30 \left (1+e^4\right )\right ) \int \frac {e^{-2+e+x^2}}{\left (6+6 e^4+x\right ) \left (e^e+6 e^{x^2} \left (1+e^4\right )+e^{x^2} x\right )^2} \, dx+\left (30 \left (1+e^4\right )\right ) \int \frac {e^{-2+x^2}}{\left (6+6 e^4+x\right ) \left (e^e+6 e^{x^2} \left (1+e^4\right )+e^{x^2} x\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.96, size = 49, normalized size = 1.96 \begin {gather*} -\frac {5 \left (e^e+6 e^{x^2}+6 e^{4+x^2}\right )}{e^2 \left (e^e+6 e^{4+x^2}+e^{x^2} (6+x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(30 + 30*E^4 + E^(E - x^2)*(5 + 10*x^2))/(36*E^10 + E^(2 + 2*E - 2*x^2) + E^6*(72 + 12*x) + E^2*(36
+ 12*x + x^2) + E^(E - x^2)*(12*E^6 + E^2*(12 + 2*x))),x]

[Out]

(-5*(E^E + 6*E^x^2 + 6*E^(4 + x^2)))/(E^2*(E^E + 6*E^(4 + x^2) + E^x^2*(6 + x)))

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fricas [A]  time = 0.59, size = 26, normalized size = 1.04 \begin {gather*} \frac {5 \, x}{{\left (x + 6\right )} e^{2} + 6 \, e^{6} + e^{\left (-x^{2} + e + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x^2+5)*exp(exp(1)-x^2)+30*exp(4)+30)/(exp(2)*exp(exp(1)-x^2)^2+(12*exp(2)*exp(4)+(2*x+12)*exp(2
))*exp(exp(1)-x^2)+36*exp(2)*exp(4)^2+(12*x+72)*exp(2)*exp(4)+(x^2+12*x+36)*exp(2)),x, algorithm="fricas")

[Out]

5*x/((x + 6)*e^2 + 6*e^6 + e^(-x^2 + e + 2))

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giac [B]  time = 0.34, size = 501, normalized size = 20.04 \begin {gather*} -\frac {5 \, {\left (12 \, x^{3} e^{\left (x^{2} + 6\right )} + 12 \, x^{3} e^{\left (x^{2} + 2\right )} + 2 \, x^{3} e^{\left (e + 2\right )} + 144 \, x^{2} e^{\left (x^{2} + 10\right )} + 288 \, x^{2} e^{\left (x^{2} + 6\right )} + 144 \, x^{2} e^{\left (x^{2} + 2\right )} + 2 \, x^{2} e^{\left (-x^{2} + 2 \, e + 2\right )} + 36 \, x^{2} e^{\left (e + 6\right )} + 36 \, x^{2} e^{\left (e + 2\right )} + 432 \, x e^{\left (x^{2} + 14\right )} + 1296 \, x e^{\left (x^{2} + 10\right )} + 1302 \, x e^{\left (x^{2} + 6\right )} + 438 \, x e^{\left (x^{2} + 2\right )} + 12 \, x e^{\left (-x^{2} + 2 \, e + 6\right )} + 12 \, x e^{\left (-x^{2} + 2 \, e + 2\right )} + 144 \, x e^{\left (e + 10\right )} + 288 \, x e^{\left (e + 6\right )} + 145 \, x e^{\left (e + 2\right )} + 36 \, e^{\left (x^{2} + 10\right )} + 72 \, e^{\left (x^{2} + 6\right )} + 36 \, e^{\left (x^{2} + 2\right )} + e^{\left (-x^{2} + 2 \, e + 2\right )} + 12 \, e^{\left (e + 6\right )} + 12 \, e^{\left (e + 2\right )}\right )}}{2 \, x^{4} e^{\left (x^{2} + 4\right )} + 36 \, x^{3} e^{\left (x^{2} + 8\right )} + 36 \, x^{3} e^{\left (x^{2} + 4\right )} + 4 \, x^{3} e^{\left (e + 4\right )} + 216 \, x^{2} e^{\left (x^{2} + 12\right )} + 432 \, x^{2} e^{\left (x^{2} + 8\right )} + 217 \, x^{2} e^{\left (x^{2} + 4\right )} + 2 \, x^{2} e^{\left (-x^{2} + 2 \, e + 4\right )} + 48 \, x^{2} e^{\left (e + 8\right )} + 48 \, x^{2} e^{\left (e + 4\right )} + 432 \, x e^{\left (x^{2} + 16\right )} + 1296 \, x e^{\left (x^{2} + 12\right )} + 1308 \, x e^{\left (x^{2} + 8\right )} + 444 \, x e^{\left (x^{2} + 4\right )} + 12 \, x e^{\left (-x^{2} + 2 \, e + 8\right )} + 12 \, x e^{\left (-x^{2} + 2 \, e + 4\right )} + 144 \, x e^{\left (e + 12\right )} + 288 \, x e^{\left (e + 8\right )} + 146 \, x e^{\left (e + 4\right )} + 36 \, e^{\left (x^{2} + 12\right )} + 72 \, e^{\left (x^{2} + 8\right )} + 36 \, e^{\left (x^{2} + 4\right )} + e^{\left (-x^{2} + 2 \, e + 4\right )} + 12 \, e^{\left (e + 8\right )} + 12 \, e^{\left (e + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x^2+5)*exp(exp(1)-x^2)+30*exp(4)+30)/(exp(2)*exp(exp(1)-x^2)^2+(12*exp(2)*exp(4)+(2*x+12)*exp(2
))*exp(exp(1)-x^2)+36*exp(2)*exp(4)^2+(12*x+72)*exp(2)*exp(4)+(x^2+12*x+36)*exp(2)),x, algorithm="giac")

[Out]

-5*(12*x^3*e^(x^2 + 6) + 12*x^3*e^(x^2 + 2) + 2*x^3*e^(e + 2) + 144*x^2*e^(x^2 + 10) + 288*x^2*e^(x^2 + 6) + 1
44*x^2*e^(x^2 + 2) + 2*x^2*e^(-x^2 + 2*e + 2) + 36*x^2*e^(e + 6) + 36*x^2*e^(e + 2) + 432*x*e^(x^2 + 14) + 129
6*x*e^(x^2 + 10) + 1302*x*e^(x^2 + 6) + 438*x*e^(x^2 + 2) + 12*x*e^(-x^2 + 2*e + 6) + 12*x*e^(-x^2 + 2*e + 2)
+ 144*x*e^(e + 10) + 288*x*e^(e + 6) + 145*x*e^(e + 2) + 36*e^(x^2 + 10) + 72*e^(x^2 + 6) + 36*e^(x^2 + 2) + e
^(-x^2 + 2*e + 2) + 12*e^(e + 6) + 12*e^(e + 2))/(2*x^4*e^(x^2 + 4) + 36*x^3*e^(x^2 + 8) + 36*x^3*e^(x^2 + 4)
+ 4*x^3*e^(e + 4) + 216*x^2*e^(x^2 + 12) + 432*x^2*e^(x^2 + 8) + 217*x^2*e^(x^2 + 4) + 2*x^2*e^(-x^2 + 2*e + 4
) + 48*x^2*e^(e + 8) + 48*x^2*e^(e + 4) + 432*x*e^(x^2 + 16) + 1296*x*e^(x^2 + 12) + 1308*x*e^(x^2 + 8) + 444*
x*e^(x^2 + 4) + 12*x*e^(-x^2 + 2*e + 8) + 12*x*e^(-x^2 + 2*e + 4) + 144*x*e^(e + 12) + 288*x*e^(e + 8) + 146*x
*e^(e + 4) + 36*e^(x^2 + 12) + 72*e^(x^2 + 8) + 36*e^(x^2 + 4) + e^(-x^2 + 2*e + 4) + 12*e^(e + 8) + 12*e^(e +
4))

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maple [A]  time = 0.18, size = 24, normalized size = 0.96

method	result	size
risch	\(\frac {5 x \,{\mathrm e}^{-2}}{{\mathrm e}^{{\mathrm e}-x^{2}}+6+x +6 \,{\mathrm e}^{4}}\)	\(24\)
norman	\(\frac {5 x \,{\mathrm e}^{-2}}{{\mathrm e}^{{\mathrm e}-x^{2}}+6+x +6 \,{\mathrm e}^{4}}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((10*x^2+5)*exp(exp(1)-x^2)+30*exp(4)+30)/(exp(2)*exp(exp(1)-x^2)^2+(12*exp(2)*exp(4)+(2*x+12)*exp(2))*exp
(exp(1)-x^2)+36*exp(2)*exp(4)^2+(12*x+72)*exp(2)*exp(4)+(x^2+12*x+36)*exp(2)),x,method=_RETURNVERBOSE)

[Out]

5*x*exp(-2)/(exp(exp(1)-x^2)+6+x+6*exp(4))

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maxima [A]  time = 0.49, size = 42, normalized size = 1.68 \begin {gather*} -\frac {5 \, {\left (6 \, {\left (e^{4} + 1\right )} e^{\left (x^{2}\right )} + e^{e}\right )}}{{\left (x e^{2} + 6 \, e^{6} + 6 \, e^{2}\right )} e^{\left (x^{2}\right )} + e^{\left (e + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x^2+5)*exp(exp(1)-x^2)+30*exp(4)+30)/(exp(2)*exp(exp(1)-x^2)^2+(12*exp(2)*exp(4)+(2*x+12)*exp(2
))*exp(exp(1)-x^2)+36*exp(2)*exp(4)^2+(12*x+72)*exp(2)*exp(4)+(x^2+12*x+36)*exp(2)),x, algorithm="maxima")

[Out]

-5*(6*(e^4 + 1)*e^(x^2) + e^e)/((x*e^2 + 6*e^6 + 6*e^2)*e^(x^2) + e^(e + 2))

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mupad [B]  time = 2.77, size = 40, normalized size = 1.60 \begin {gather*} \frac {x\,\left (30\,{\mathrm {e}}^4+30\right )}{6\,\left ({\mathrm {e}}^4+1\right )\,\left ({\mathrm {e}}^{-x^2+\mathrm {e}+2}+6\,{\mathrm {e}}^2+6\,{\mathrm {e}}^6+x\,{\mathrm {e}}^2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((30*exp(4) + exp(exp(1) - x^2)*(10*x^2 + 5) + 30)/(36*exp(10) + exp(2)*exp(2*exp(1) - 2*x^2) + exp(2)*(12*
x + x^2 + 36) + exp(exp(1) - x^2)*(12*exp(6) + exp(2)*(2*x + 12)) + exp(6)*(12*x + 72)),x)

[Out]

(x*(30*exp(4) + 30))/(6*(exp(4) + 1)*(exp(exp(1) - x^2 + 2) + 6*exp(2) + 6*exp(6) + x*exp(2)))

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sympy [A]  time = 0.15, size = 29, normalized size = 1.16 \begin {gather*} \frac {5 x}{x e^{2} + e^{2} e^{e - x^{2}} + 6 e^{2} + 6 e^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x**2+5)*exp(exp(1)-x**2)+30*exp(4)+30)/(exp(2)*exp(exp(1)-x**2)**2+(12*exp(2)*exp(4)+(2*x+12)*e
xp(2))*exp(exp(1)-x**2)+36*exp(2)*exp(4)**2+(12*x+72)*exp(2)*exp(4)+(x**2+12*x+36)*exp(2)),x)

[Out]

5*x/(x*exp(2) + exp(2)*exp(E - x**2) + 6*exp(2) + 6*exp(6))

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