3.27.86 \(\int \frac {180+72 x+4 e^{8+2 x^2} x+e^8 (168+72 x)+e^4 (348+144 x)+e^{x^2} (e^4 (-12-60 x-24 x^2)+e^8 (-12-56 x-24 x^2))}{9+18 e^4+9 e^8} \, dx\)

Optimal. Leaf size=25 \[ \left (5-\frac {1+e^{x^2}}{3 \left (1+\frac {1}{e^4}\right )}+2 x\right )^2 \]

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Rubi [B]  time = 0.11, antiderivative size = 129, normalized size of antiderivative = 5.16, number of steps used = 9, number of rules used = 5, integrand size = 85, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {12, 2209, 2226, 2204, 2212} \begin {gather*} \frac {4 x^2}{\left (1+e^4\right )^2}-\frac {4 e^{x^2+4} x}{3 \left (1+e^4\right )}-\frac {2 \left (15+14 e^4\right ) e^{x^2+4}}{9 \left (1+e^4\right )^2}+\frac {e^{2 x^2+8}}{9 \left (1+e^4\right )^2}+\frac {20 x}{\left (1+e^4\right )^2}+\frac {4 e^8 (3 x+7)^2}{9 \left (1+e^4\right )^2}+\frac {e^4 (12 x+29)^2}{18 \left (1+e^4\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(180 + 72*x + 4*E^(8 + 2*x^2)*x + E^8*(168 + 72*x) + E^4*(348 + 144*x) + E^x^2*(E^4*(-12 - 60*x - 24*x^2)
+ E^8*(-12 - 56*x - 24*x^2)))/(9 + 18*E^4 + 9*E^8),x]

[Out]

E^(8 + 2*x^2)/(9*(1 + E^4)^2) - (2*E^(4 + x^2)*(15 + 14*E^4))/(9*(1 + E^4)^2) + (20*x)/(1 + E^4)^2 - (4*E^(4 +
x^2)*x)/(3*(1 + E^4)) + (4*x^2)/(1 + E^4)^2 + (4*E^8*(7 + 3*x)^2)/(9*(1 + E^4)^2) + (E^4*(29 + 12*x)^2)/(18*(
1 + E^4)^2)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2204

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2
]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rule 2212

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
- n + 1)*F^(a + b*(c + d*x)^n))/(b*d*n*Log[F]), x] - Dist[(m - n + 1)/(b*n*Log[F]), Int[(c + d*x)^(m - n)*F^(
a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[(2*(m + 1))/n] && LtQ[0, (m + 1)/n, 5] &&
IntegerQ[n] && (LtQ[0, n, m + 1] || LtQ[m, n, 0])

Rule 2226

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*(u_), x_Symbol] :> Int[ExpandLinearProduct[F^(a + b*(c + d*
x)^n), u, c, d, x], x] /; FreeQ[{F, a, b, c, d, n}, x] && PolynomialQ[u, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \left (180+72 x+4 e^{8+2 x^2} x+e^8 (168+72 x)+e^4 (348+144 x)+e^{x^2} \left (e^4 \left (-12-60 x-24 x^2\right )+e^8 \left (-12-56 x-24 x^2\right )\right )\right ) \, dx}{9 \left (1+e^4\right )^2}\\ &=\frac {20 x}{\left (1+e^4\right )^2}+\frac {4 x^2}{\left (1+e^4\right )^2}+\frac {4 e^8 (7+3 x)^2}{9 \left (1+e^4\right )^2}+\frac {e^4 (29+12 x)^2}{18 \left (1+e^4\right )^2}+\frac {\int e^{x^2} \left (e^4 \left (-12-60 x-24 x^2\right )+e^8 \left (-12-56 x-24 x^2\right )\right ) \, dx}{9 \left (1+e^4\right )^2}+\frac {4 \int e^{8+2 x^2} x \, dx}{9 \left (1+e^4\right )^2}\\ &=\frac {e^{8+2 x^2}}{9 \left (1+e^4\right )^2}+\frac {20 x}{\left (1+e^4\right )^2}+\frac {4 x^2}{\left (1+e^4\right )^2}+\frac {4 e^8 (7+3 x)^2}{9 \left (1+e^4\right )^2}+\frac {e^4 (29+12 x)^2}{18 \left (1+e^4\right )^2}+\frac {\int \left (-12 e^{x^2} \left (e^4+e^8\right )-4 e^{4+x^2} \left (15+14 e^4\right ) x-24 e^{x^2} \left (e^4+e^8\right ) x^2\right ) \, dx}{9 \left (1+e^4\right )^2}\\ &=\frac {e^{8+2 x^2}}{9 \left (1+e^4\right )^2}+\frac {20 x}{\left (1+e^4\right )^2}+\frac {4 x^2}{\left (1+e^4\right )^2}+\frac {4 e^8 (7+3 x)^2}{9 \left (1+e^4\right )^2}+\frac {e^4 (29+12 x)^2}{18 \left (1+e^4\right )^2}-\frac {\left (4 e^4\right ) \int e^{x^2} \, dx}{3 \left (1+e^4\right )}-\frac {\left (8 e^4\right ) \int e^{x^2} x^2 \, dx}{3 \left (1+e^4\right )}-\frac {\left (4 \left (15+14 e^4\right )\right ) \int e^{4+x^2} x \, dx}{9 \left (1+e^4\right )^2}\\ &=\frac {e^{8+2 x^2}}{9 \left (1+e^4\right )^2}-\frac {2 e^{4+x^2} \left (15+14 e^4\right )}{9 \left (1+e^4\right )^2}+\frac {20 x}{\left (1+e^4\right )^2}-\frac {4 e^{4+x^2} x}{3 \left (1+e^4\right )}+\frac {4 x^2}{\left (1+e^4\right )^2}+\frac {4 e^8 (7+3 x)^2}{9 \left (1+e^4\right )^2}+\frac {e^4 (29+12 x)^2}{18 \left (1+e^4\right )^2}-\frac {2 e^4 \sqrt {\pi } \text {erfi}(x)}{3 \left (1+e^4\right )}+\frac {\left (4 e^4\right ) \int e^{x^2} \, dx}{3 \left (1+e^4\right )}\\ &=\frac {e^{8+2 x^2}}{9 \left (1+e^4\right )^2}-\frac {2 e^{4+x^2} \left (15+14 e^4\right )}{9 \left (1+e^4\right )^2}+\frac {20 x}{\left (1+e^4\right )^2}-\frac {4 e^{4+x^2} x}{3 \left (1+e^4\right )}+\frac {4 x^2}{\left (1+e^4\right )^2}+\frac {4 e^8 (7+3 x)^2}{9 \left (1+e^4\right )^2}+\frac {e^4 (29+12 x)^2}{18 \left (1+e^4\right )^2}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.06, size = 101, normalized size = 4.04 \begin {gather*} \frac {4 \left (-\frac {15}{2} e^{4+x^2}-7 e^{8+x^2}+\frac {1}{4} e^{8+2 x^2}+45 x+87 e^4 x+42 e^8 x-3 e^{4+x^2} x-3 e^{8+x^2} x+9 x^2+18 e^4 x^2+9 e^8 x^2\right )}{9 \left (1+e^4\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(180 + 72*x + 4*E^(8 + 2*x^2)*x + E^8*(168 + 72*x) + E^4*(348 + 144*x) + E^x^2*(E^4*(-12 - 60*x - 24
*x^2) + E^8*(-12 - 56*x - 24*x^2)))/(9 + 18*E^4 + 9*E^8),x]

[Out]

(4*((-15*E^(4 + x^2))/2 - 7*E^(8 + x^2) + E^(8 + 2*x^2)/4 + 45*x + 87*E^4*x + 42*E^8*x - 3*E^(4 + x^2)*x - 3*E
^(8 + x^2)*x + 9*x^2 + 18*E^4*x^2 + 9*E^8*x^2))/(9*(1 + E^4)^2)

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fricas [B]  time = 0.64, size = 80, normalized size = 3.20 \begin {gather*} \frac {36 \, x^{2} + 12 \, {\left (3 \, x^{2} + 14 \, x\right )} e^{8} + 12 \, {\left (6 \, x^{2} + 29 \, x\right )} e^{4} - 2 \, {\left (2 \, {\left (3 \, x + 7\right )} e^{8} + 3 \, {\left (2 \, x + 5\right )} e^{4}\right )} e^{\left (x^{2}\right )} + 180 \, x + e^{\left (2 \, x^{2} + 8\right )}}{9 \, {\left (e^{8} + 2 \, e^{4} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x*exp(4)^2*exp(x^2)^2+((-24*x^2-56*x-12)*exp(4)^2+(-24*x^2-60*x-12)*exp(4))*exp(x^2)+(72*x+168)*e
xp(4)^2+(144*x+348)*exp(4)+72*x+180)/(9*exp(4)^2+18*exp(4)+9),x, algorithm="fricas")

[Out]

1/9*(36*x^2 + 12*(3*x^2 + 14*x)*e^8 + 12*(6*x^2 + 29*x)*e^4 - 2*(2*(3*x + 7)*e^8 + 3*(2*x + 5)*e^4)*e^(x^2) +
180*x + e^(2*x^2 + 8))/(e^8 + 2*e^4 + 1)

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giac [B]  time = 0.17, size = 81, normalized size = 3.24 \begin {gather*} \frac {36 \, x^{2} + 12 \, {\left (3 \, x^{2} + 14 \, x\right )} e^{8} + 12 \, {\left (6 \, x^{2} + 29 \, x\right )} e^{4} - 4 \, {\left (3 \, x + 7\right )} e^{\left (x^{2} + 8\right )} - 6 \, {\left (2 \, x + 5\right )} e^{\left (x^{2} + 4\right )} + 180 \, x + e^{\left (2 \, x^{2} + 8\right )}}{9 \, {\left (e^{8} + 2 \, e^{4} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x*exp(4)^2*exp(x^2)^2+((-24*x^2-56*x-12)*exp(4)^2+(-24*x^2-60*x-12)*exp(4))*exp(x^2)+(72*x+168)*e
xp(4)^2+(144*x+348)*exp(4)+72*x+180)/(9*exp(4)^2+18*exp(4)+9),x, algorithm="giac")

[Out]

1/9*(36*x^2 + 12*(3*x^2 + 14*x)*e^8 + 12*(6*x^2 + 29*x)*e^4 - 4*(3*x + 7)*e^(x^2 + 8) - 6*(2*x + 5)*e^(x^2 + 4
) + 180*x + e^(2*x^2 + 8))/(e^8 + 2*e^4 + 1)

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maple [B]  time = 0.05, size = 74, normalized size = 2.96

method	result	size
norman	\(\frac {\left (4 \,{\mathrm e}^{4}+4\right ) x^{2}+\left (\frac {56 \,{\mathrm e}^{4}}{3}+20\right ) x -\frac {4 x \,{\mathrm e}^{4} {\mathrm e}^{x^{2}}}{3}+\frac {{\mathrm e}^{8} {\mathrm e}^{2 x^{2}}}{9 \,{\mathrm e}^{4}+9}-\frac {2 \left (14 \,{\mathrm e}^{4}+15\right ) {\mathrm e}^{4} {\mathrm e}^{x^{2}}}{9 \left ({\mathrm e}^{4}+1\right )}}{{\mathrm e}^{4}+1}\)	\(74\)
default	\(\frac {180 x +{\mathrm e}^{8} \left (36 x^{2}+168 x \right )+{\mathrm e}^{4} \left (72 x^{2}+348 x \right )-6 \,{\mathrm e}^{4} \sqrt {\pi }\, \erfi \relax (x )-6 \,{\mathrm e}^{8} \sqrt {\pi }\, \erfi \relax (x )-30 \,{\mathrm e}^{4} {\mathrm e}^{x^{2}}-28 \,{\mathrm e}^{8} {\mathrm e}^{x^{2}}-24 \,{\mathrm e}^{4} \left (\frac {{\mathrm e}^{x^{2}} x}{2}-\frac {\sqrt {\pi }\, \erfi \relax (x )}{4}\right )-24 \,{\mathrm e}^{8} \left (\frac {{\mathrm e}^{x^{2}} x}{2}-\frac {\sqrt {\pi }\, \erfi \relax (x )}{4}\right )+36 x^{2}+{\mathrm e}^{8} {\mathrm e}^{2 x^{2}}}{9 \,{\mathrm e}^{8}+18 \,{\mathrm e}^{4}+9}\)	\(139\)
risch	\(\frac {36 x^{2} {\mathrm e}^{8}}{9 \,{\mathrm e}^{8}+18 \,{\mathrm e}^{4}+9}+\frac {{\mathrm e}^{2 x^{2}+8}}{9 \,{\mathrm e}^{8}+18 \,{\mathrm e}^{4}+9}+\frac {168 x \,{\mathrm e}^{8}}{9 \,{\mathrm e}^{8}+18 \,{\mathrm e}^{4}+9}+\frac {72 x^{2} {\mathrm e}^{4}}{9 \,{\mathrm e}^{8}+18 \,{\mathrm e}^{4}+9}+\frac {348 x \,{\mathrm e}^{4}}{9 \,{\mathrm e}^{8}+18 \,{\mathrm e}^{4}+9}+\frac {36 x^{2}}{9 \,{\mathrm e}^{8}+18 \,{\mathrm e}^{4}+9}+\frac {180 x}{9 \,{\mathrm e}^{8}+18 \,{\mathrm e}^{4}+9}+\frac {\left (-12 x \,{\mathrm e}^{8}-28 \,{\mathrm e}^{8}-12 x \,{\mathrm e}^{4}-30 \,{\mathrm e}^{4}\right ) {\mathrm e}^{x^{2}}}{9 \,{\mathrm e}^{8}+18 \,{\mathrm e}^{4}+9}\)	\(163\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x*exp(4)^2*exp(x^2)^2+((-24*x^2-56*x-12)*exp(4)^2+(-24*x^2-60*x-12)*exp(4))*exp(x^2)+(72*x+168)*exp(4)^
2+(144*x+348)*exp(4)+72*x+180)/(9*exp(4)^2+18*exp(4)+9),x,method=_RETURNVERBOSE)

[Out]

((4*exp(4)+4)*x^2+(56/3*exp(4)+20)*x-4/3*x*exp(4)*exp(x^2)+1/9*exp(4)^2/(exp(4)+1)*exp(x^2)^2-2/9*(14*exp(4)+1
5)*exp(4)/(exp(4)+1)*exp(x^2))/(exp(4)+1)

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maxima [B]  time = 0.40, size = 78, normalized size = 3.12 \begin {gather*} \frac {36 \, x^{2} + 12 \, {\left (3 \, x^{2} + 14 \, x\right )} e^{8} + 12 \, {\left (6 \, x^{2} + 29 \, x\right )} e^{4} - 2 \, {\left (6 \, x {\left (e^{8} + e^{4}\right )} + 14 \, e^{8} + 15 \, e^{4}\right )} e^{\left (x^{2}\right )} + 180 \, x + e^{\left (2 \, x^{2} + 8\right )}}{9 \, {\left (e^{8} + 2 \, e^{4} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x*exp(4)^2*exp(x^2)^2+((-24*x^2-56*x-12)*exp(4)^2+(-24*x^2-60*x-12)*exp(4))*exp(x^2)+(72*x+168)*e
xp(4)^2+(144*x+348)*exp(4)+72*x+180)/(9*exp(4)^2+18*exp(4)+9),x, algorithm="maxima")

[Out]

1/9*(36*x^2 + 12*(3*x^2 + 14*x)*e^8 + 12*(6*x^2 + 29*x)*e^4 - 2*(6*x*(e^8 + e^4) + 14*e^8 + 15*e^4)*e^(x^2) +
180*x + e^(2*x^2 + 8))/(e^8 + 2*e^4 + 1)

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mupad [B]  time = 1.58, size = 73, normalized size = 2.92 \begin {gather*} \frac {{\mathrm {e}}^{2\,x^2+8}+36\,x^2\,{\left ({\mathrm {e}}^4+1\right )}^2-{\mathrm {e}}^{x^2}\,\left (30\,{\mathrm {e}}^4+28\,{\mathrm {e}}^8\right )+x\,\left (348\,{\mathrm {e}}^4+168\,{\mathrm {e}}^8+180\right )-12\,x\,{\mathrm {e}}^{x^2+4}\,\left ({\mathrm {e}}^4+1\right )}{18\,{\mathrm {e}}^4+9\,{\mathrm {e}}^8+9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((72*x - exp(x^2)*(exp(4)*(60*x + 24*x^2 + 12) + exp(8)*(56*x + 24*x^2 + 12)) + exp(8)*(72*x + 168) + exp(4
)*(144*x + 348) + 4*x*exp(8)*exp(2*x^2) + 180)/(18*exp(4) + 9*exp(8) + 9),x)

[Out]

(exp(2*x^2 + 8) + 36*x^2*(exp(4) + 1)^2 - exp(x^2)*(30*exp(4) + 28*exp(8)) + x*(348*exp(4) + 168*exp(8) + 180)
- 12*x*exp(x^2 + 4)*(exp(4) + 1))/(18*exp(4) + 9*exp(8) + 9)

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sympy [B]  time = 0.25, size = 117, normalized size = 4.68 \begin {gather*} 4 x^{2} + \frac {x \left (60 + 56 e^{4}\right )}{3 + 3 e^{4}} + \frac {\left (- 108 x e^{16} - 324 x e^{12} - 324 x e^{8} - 108 x e^{4} - 252 e^{16} - 774 e^{12} - 792 e^{8} - 270 e^{4}\right ) e^{x^{2}} + \left (9 e^{8} + 18 e^{12} + 9 e^{16}\right ) e^{2 x^{2}}}{81 + 324 e^{4} + 486 e^{8} + 324 e^{12} + 81 e^{16}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x*exp(4)**2*exp(x**2)**2+((-24*x**2-56*x-12)*exp(4)**2+(-24*x**2-60*x-12)*exp(4))*exp(x**2)+(72*x
+168)*exp(4)**2+(144*x+348)*exp(4)+72*x+180)/(9*exp(4)**2+18*exp(4)+9),x)

[Out]

4*x**2 + x*(60 + 56*exp(4))/(3 + 3*exp(4)) + ((-108*x*exp(16) - 324*x*exp(12) - 324*x*exp(8) - 108*x*exp(4) -
252*exp(16) - 774*exp(12) - 792*exp(8) - 270*exp(4))*exp(x**2) + (9*exp(8) + 18*exp(12) + 9*exp(16))*exp(2*x**
2))/(81 + 324*exp(4) + 486*exp(8) + 324*exp(12) + 81*exp(16))

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