3.23.53 \(\int \frac {4 x^4+\frac {e^{8 x} (-8+8 x)}{x^8}+e^{3 x^2} (-16 x^4-24 x^6)+e^{x^2} (-16 x^4-8 x^6)+e^{4 x^2} (4 x^4+8 x^6)+e^{2 x^2} (24 x^4+24 x^6)+\frac {e^{6 x} (20 x-24 x^2+e^{x^2} (-20 x+24 x^2+8 x^3))}{x^6}+\frac {e^{4 x} (-12 x^2+24 x^3+e^{x^2} (24 x^2-48 x^3-24 x^4)+e^{2 x^2} (-12 x^2+24 x^3+24 x^4))}{x^4}+\frac {e^{2 x} (-4 x^3-8 x^4+e^{2 x^2} (-12 x^3-24 x^4-48 x^5)+e^{3 x^2} (4 x^3+8 x^4+24 x^5)+e^{x^2} (12 x^3+24 x^4+24 x^5))}{x^2}}{x} \, dx\)

Optimal. Leaf size=24 \[ \left (\frac {e^{2 x}}{x^2}-\left (1-e^{x^2}\right ) x\right )^4 \]

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Rubi [B]  time = 1.62, antiderivative size = 163, normalized size of antiderivative = 6.79, number of steps used = 19, number of rules used = 9, integrand size = 292, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {14, 2226, 2212, 2209, 6742, 2197, 2176, 2194, 2288} \begin {gather*} \frac {e^{8 x}}{x^8}-\frac {4 e^{6 x}}{x^5}+x^4+\frac {6 e^{4 x}}{x^2}+\frac {4 e^{3 x^2} \left (e^{2 x} x^2-x^5\right )}{x}+e^{4 x^2} x^4+\frac {6 e^{2 x^2} \left (x^8-2 e^{2 x} x^5+e^{4 x} x^2\right )}{x^4}+\frac {4 e^{x^2} \left (e^{2 x}-x^3\right )^2 \left (e^{2 x} x^2-x^5\right )}{x^7}+2 e^{2 x}-2 e^{2 x} (2 x+1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4*x^4 + (E^(8*x)*(-8 + 8*x))/x^8 + E^(3*x^2)*(-16*x^4 - 24*x^6) + E^x^2*(-16*x^4 - 8*x^6) + E^(4*x^2)*(4*
x^4 + 8*x^6) + E^(2*x^2)*(24*x^4 + 24*x^6) + (E^(6*x)*(20*x - 24*x^2 + E^x^2*(-20*x + 24*x^2 + 8*x^3)))/x^6 +
(E^(4*x)*(-12*x^2 + 24*x^3 + E^x^2*(24*x^2 - 48*x^3 - 24*x^4) + E^(2*x^2)*(-12*x^2 + 24*x^3 + 24*x^4)))/x^4 +
(E^(2*x)*(-4*x^3 - 8*x^4 + E^(2*x^2)*(-12*x^3 - 24*x^4 - 48*x^5) + E^(3*x^2)*(4*x^3 + 8*x^4 + 24*x^5) + E^x^2*
(12*x^3 + 24*x^4 + 24*x^5)))/x^2)/x,x]

[Out]

2*E^(2*x) + E^(8*x)/x^8 - (4*E^(6*x))/x^5 + (6*E^(4*x))/x^2 + x^4 + E^(4*x^2)*x^4 - 2*E^(2*x)*(1 + 2*x) + (4*E
^(3*x^2)*(E^(2*x)*x^2 - x^5))/x + (4*E^x^2*(E^(2*x) - x^3)^2*(E^(2*x)*x^2 - x^5))/x^7 + (6*E^(2*x^2)*(E^(4*x)*
x^2 - 2*E^(2*x)*x^5 + x^8))/x^4

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2197

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> With[{b = Coefficient[v, x, 1], d = Coefficient[u, x, 0],
e = Coefficient[u, x, 1], f = Coefficient[w, x, 0], g = Coefficient[w, x, 1]}, Simp[(g*u^(m + 1)*F^(c*v))/(b*c
*e*Log[F]), x] /; EqQ[e*g*(m + 1) - b*c*(e*f - d*g)*Log[F], 0]] /; FreeQ[{F, c, m}, x] && LinearQ[{u, v, w}, x
]

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]

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (4 e^{4 x^2} x^3 \left (1+2 x^2\right )+\frac {4 \left (-e^{2 x}+x^3\right )^3 \left (2 e^{2 x}-2 e^{2 x} x+x^3\right )}{x^9}+4 e^{3 x^2} \left (e^{2 x}+2 e^{2 x} x+6 e^{2 x} x^2-4 x^3-6 x^5\right )-\frac {4 e^{x^2} \left (-e^{2 x}+x^3\right )^2 \left (5 e^{2 x}-6 e^{2 x} x-2 e^{2 x} x^2+4 x^3+2 x^5\right )}{x^6}+\frac {12 e^{2 x^2} \left (-e^{4 x}+2 e^{4 x} x+2 e^{4 x} x^2-e^{2 x} x^3-2 e^{2 x} x^4-4 e^{2 x} x^5+2 x^6+2 x^8\right )}{x^3}\right ) \, dx\\ &=4 \int e^{4 x^2} x^3 \left (1+2 x^2\right ) \, dx+4 \int \frac {\left (-e^{2 x}+x^3\right )^3 \left (2 e^{2 x}-2 e^{2 x} x+x^3\right )}{x^9} \, dx+4 \int e^{3 x^2} \left (e^{2 x}+2 e^{2 x} x+6 e^{2 x} x^2-4 x^3-6 x^5\right ) \, dx-4 \int \frac {e^{x^2} \left (-e^{2 x}+x^3\right )^2 \left (5 e^{2 x}-6 e^{2 x} x-2 e^{2 x} x^2+4 x^3+2 x^5\right )}{x^6} \, dx+12 \int \frac {e^{2 x^2} \left (-e^{4 x}+2 e^{4 x} x+2 e^{4 x} x^2-e^{2 x} x^3-2 e^{2 x} x^4-4 e^{2 x} x^5+2 x^6+2 x^8\right )}{x^3} \, dx\\ &=\frac {4 e^{3 x^2} \left (e^{2 x} x^2-x^5\right )}{x}+\frac {4 e^{x^2} \left (e^{2 x}-x^3\right )^2 \left (e^{2 x} x^2-x^5\right )}{x^7}+\frac {6 e^{2 x^2} \left (e^{4 x} x^2-2 e^{2 x} x^5+x^8\right )}{x^4}+4 \int \left (e^{4 x^2} x^3+2 e^{4 x^2} x^5\right ) \, dx+4 \int \left (\frac {2 e^{8 x} (-1+x)}{x^9}+x^3+\frac {3 e^{4 x} (-1+2 x)}{x^3}-e^{2 x} (1+2 x)-\frac {e^{6 x} (-5+6 x)}{x^6}\right ) \, dx\\ &=x^4+\frac {4 e^{3 x^2} \left (e^{2 x} x^2-x^5\right )}{x}+\frac {4 e^{x^2} \left (e^{2 x}-x^3\right )^2 \left (e^{2 x} x^2-x^5\right )}{x^7}+\frac {6 e^{2 x^2} \left (e^{4 x} x^2-2 e^{2 x} x^5+x^8\right )}{x^4}+4 \int e^{4 x^2} x^3 \, dx-4 \int e^{2 x} (1+2 x) \, dx-4 \int \frac {e^{6 x} (-5+6 x)}{x^6} \, dx+8 \int \frac {e^{8 x} (-1+x)}{x^9} \, dx+8 \int e^{4 x^2} x^5 \, dx+12 \int \frac {e^{4 x} (-1+2 x)}{x^3} \, dx\\ &=\frac {e^{8 x}}{x^8}-\frac {4 e^{6 x}}{x^5}+\frac {6 e^{4 x}}{x^2}+\frac {1}{2} e^{4 x^2} x^2+x^4+e^{4 x^2} x^4-2 e^{2 x} (1+2 x)+\frac {4 e^{3 x^2} \left (e^{2 x} x^2-x^5\right )}{x}+\frac {4 e^{x^2} \left (e^{2 x}-x^3\right )^2 \left (e^{2 x} x^2-x^5\right )}{x^7}+\frac {6 e^{2 x^2} \left (e^{4 x} x^2-2 e^{2 x} x^5+x^8\right )}{x^4}+4 \int e^{2 x} \, dx-4 \int e^{4 x^2} x^3 \, dx-\int e^{4 x^2} x \, dx\\ &=2 e^{2 x}-\frac {e^{4 x^2}}{8}+\frac {e^{8 x}}{x^8}-\frac {4 e^{6 x}}{x^5}+\frac {6 e^{4 x}}{x^2}+x^4+e^{4 x^2} x^4-2 e^{2 x} (1+2 x)+\frac {4 e^{3 x^2} \left (e^{2 x} x^2-x^5\right )}{x}+\frac {4 e^{x^2} \left (e^{2 x}-x^3\right )^2 \left (e^{2 x} x^2-x^5\right )}{x^7}+\frac {6 e^{2 x^2} \left (e^{4 x} x^2-2 e^{2 x} x^5+x^8\right )}{x^4}+\int e^{4 x^2} x \, dx\\ &=2 e^{2 x}+\frac {e^{8 x}}{x^8}-\frac {4 e^{6 x}}{x^5}+\frac {6 e^{4 x}}{x^2}+x^4+e^{4 x^2} x^4-2 e^{2 x} (1+2 x)+\frac {4 e^{3 x^2} \left (e^{2 x} x^2-x^5\right )}{x}+\frac {4 e^{x^2} \left (e^{2 x}-x^3\right )^2 \left (e^{2 x} x^2-x^5\right )}{x^7}+\frac {6 e^{2 x^2} \left (e^{4 x} x^2-2 e^{2 x} x^5+x^8\right )}{x^4}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.86, size = 26, normalized size = 1.08 \begin {gather*} \frac {\left (e^{2 x}-x^3+e^{x^2} x^3\right )^4}{x^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4*x^4 + (E^(8*x)*(-8 + 8*x))/x^8 + E^(3*x^2)*(-16*x^4 - 24*x^6) + E^x^2*(-16*x^4 - 8*x^6) + E^(4*x^
2)*(4*x^4 + 8*x^6) + E^(2*x^2)*(24*x^4 + 24*x^6) + (E^(6*x)*(20*x - 24*x^2 + E^x^2*(-20*x + 24*x^2 + 8*x^3)))/
x^6 + (E^(4*x)*(-12*x^2 + 24*x^3 + E^x^2*(24*x^2 - 48*x^3 - 24*x^4) + E^(2*x^2)*(-12*x^2 + 24*x^3 + 24*x^4)))/
x^4 + (E^(2*x)*(-4*x^3 - 8*x^4 + E^(2*x^2)*(-12*x^3 - 24*x^4 - 48*x^5) + E^(3*x^2)*(4*x^3 + 8*x^4 + 24*x^5) +
E^x^2*(12*x^3 + 24*x^4 + 24*x^5)))/x^2)/x,x]

[Out]

(E^(2*x) - x^3 + E^x^2*x^3)^4/x^8

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fricas [B]  time = 0.97, size = 156, normalized size = 6.50 \begin {gather*} x^{4} e^{\left (4 \, x^{2}\right )} - 4 \, x^{4} e^{\left (3 \, x^{2}\right )} + 6 \, x^{4} e^{\left (2 \, x^{2}\right )} - 4 \, x^{4} e^{\left (x^{2}\right )} + x^{4} + 4 \, {\left (x e^{\left (x^{2}\right )} - x\right )} e^{\left (6 \, x - 6 \, \log \relax (x)\right )} + 6 \, {\left (x^{2} e^{\left (2 \, x^{2}\right )} - 2 \, x^{2} e^{\left (x^{2}\right )} + x^{2}\right )} e^{\left (4 \, x - 4 \, \log \relax (x)\right )} + 4 \, {\left (x^{3} e^{\left (3 \, x^{2}\right )} - 3 \, x^{3} e^{\left (2 \, x^{2}\right )} + 3 \, x^{3} e^{\left (x^{2}\right )} - x^{3}\right )} e^{\left (2 \, x - 2 \, \log \relax (x)\right )} + e^{\left (8 \, x - 8 \, \log \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x-8)*exp(x-log(x))^8+((8*x^3+24*x^2-20*x)*exp(x^2)-24*x^2+20*x)*exp(x-log(x))^6+((24*x^4+24*x^3-
12*x^2)*exp(x^2)^2+(-24*x^4-48*x^3+24*x^2)*exp(x^2)+24*x^3-12*x^2)*exp(x-log(x))^4+((24*x^5+8*x^4+4*x^3)*exp(x
^2)^3+(-48*x^5-24*x^4-12*x^3)*exp(x^2)^2+(24*x^5+24*x^4+12*x^3)*exp(x^2)-8*x^4-4*x^3)*exp(x-log(x))^2+(8*x^6+4
*x^4)*exp(x^2)^4+(-24*x^6-16*x^4)*exp(x^2)^3+(24*x^6+24*x^4)*exp(x^2)^2+(-8*x^6-16*x^4)*exp(x^2)+4*x^4)/x,x, a
lgorithm="fricas")

[Out]

x^4*e^(4*x^2) - 4*x^4*e^(3*x^2) + 6*x^4*e^(2*x^2) - 4*x^4*e^(x^2) + x^4 + 4*(x*e^(x^2) - x)*e^(6*x - 6*log(x))
+ 6*(x^2*e^(2*x^2) - 2*x^2*e^(x^2) + x^2)*e^(4*x - 4*log(x)) + 4*(x^3*e^(3*x^2) - 3*x^3*e^(2*x^2) + 3*x^3*e^(
x^2) - x^3)*e^(2*x - 2*log(x)) + e^(8*x - 8*log(x))

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giac [B]  time = 0.22, size = 164, normalized size = 6.83 \begin {gather*} \frac {x^{12} e^{\left (4 \, x^{2}\right )} - 4 \, x^{12} e^{\left (3 \, x^{2}\right )} + 6 \, x^{12} e^{\left (2 \, x^{2}\right )} - 4 \, x^{12} e^{\left (x^{2}\right )} + x^{12} + 4 \, x^{9} e^{\left (3 \, x^{2} + 2 \, x\right )} - 12 \, x^{9} e^{\left (2 \, x^{2} + 2 \, x\right )} + 12 \, x^{9} e^{\left (x^{2} + 2 \, x\right )} - 4 \, x^{9} e^{\left (2 \, x\right )} + 6 \, x^{6} e^{\left (2 \, x^{2} + 4 \, x\right )} - 12 \, x^{6} e^{\left (x^{2} + 4 \, x\right )} + 6 \, x^{6} e^{\left (4 \, x\right )} + 4 \, x^{3} e^{\left (x^{2} + 6 \, x\right )} - 4 \, x^{3} e^{\left (6 \, x\right )} + e^{\left (8 \, x\right )}}{x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x-8)*exp(x-log(x))^8+((8*x^3+24*x^2-20*x)*exp(x^2)-24*x^2+20*x)*exp(x-log(x))^6+((24*x^4+24*x^3-
12*x^2)*exp(x^2)^2+(-24*x^4-48*x^3+24*x^2)*exp(x^2)+24*x^3-12*x^2)*exp(x-log(x))^4+((24*x^5+8*x^4+4*x^3)*exp(x
^2)^3+(-48*x^5-24*x^4-12*x^3)*exp(x^2)^2+(24*x^5+24*x^4+12*x^3)*exp(x^2)-8*x^4-4*x^3)*exp(x-log(x))^2+(8*x^6+4
*x^4)*exp(x^2)^4+(-24*x^6-16*x^4)*exp(x^2)^3+(24*x^6+24*x^4)*exp(x^2)^2+(-8*x^6-16*x^4)*exp(x^2)+4*x^4)/x,x, a
lgorithm="giac")

[Out]

(x^12*e^(4*x^2) - 4*x^12*e^(3*x^2) + 6*x^12*e^(2*x^2) - 4*x^12*e^(x^2) + x^12 + 4*x^9*e^(3*x^2 + 2*x) - 12*x^9
*e^(2*x^2 + 2*x) + 12*x^9*e^(x^2 + 2*x) - 4*x^9*e^(2*x) + 6*x^6*e^(2*x^2 + 4*x) - 12*x^6*e^(x^2 + 4*x) + 6*x^6
*e^(4*x) + 4*x^3*e^(x^2 + 6*x) - 4*x^3*e^(6*x) + e^(8*x))/x^8

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maple [B]  time = 0.20, size = 152, normalized size = 6.33

method	result	size
risch	\(x^{4} {\mathrm e}^{4 x^{2}}+\frac {{\mathrm e}^{8 x}}{x^{8}}-4 x^{4} {\mathrm e}^{3 x^{2}}+6 x^{4} {\mathrm e}^{2 x^{2}}-4 x^{4} {\mathrm e}^{x^{2}}+x^{4}+\frac {\left (4 \,{\mathrm e}^{x^{2}} x -4 x \right ) {\mathrm e}^{6 x}}{x^{6}}+\frac {\left (6 x^{2} {\mathrm e}^{2 x^{2}}-12 x^{2} {\mathrm e}^{x^{2}}+6 x^{2}\right ) {\mathrm e}^{4 x}}{x^{4}}+\frac {\left (4 x^{3} {\mathrm e}^{3 x^{2}}-12 x^{3} {\mathrm e}^{2 x^{2}}+12 x^{3} {\mathrm e}^{x^{2}}-4 x^{3}\right ) {\mathrm e}^{2 x}}{x^{2}}\)	\(152\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((8*x-8)*exp(x-ln(x))^8+((8*x^3+24*x^2-20*x)*exp(x^2)-24*x^2+20*x)*exp(x-ln(x))^6+((24*x^4+24*x^3-12*x^2)*
exp(x^2)^2+(-24*x^4-48*x^3+24*x^2)*exp(x^2)+24*x^3-12*x^2)*exp(x-ln(x))^4+((24*x^5+8*x^4+4*x^3)*exp(x^2)^3+(-4
8*x^5-24*x^4-12*x^3)*exp(x^2)^2+(24*x^5+24*x^4+12*x^3)*exp(x^2)-8*x^4-4*x^3)*exp(x-ln(x))^2+(8*x^6+4*x^4)*exp(
x^2)^4+(-24*x^6-16*x^4)*exp(x^2)^3+(24*x^6+24*x^4)*exp(x^2)^2+(-8*x^6-16*x^4)*exp(x^2)+4*x^4)/x,x,method=_RETU
RNVERBOSE)

[Out]

x^4*exp(4*x^2)+1/x^8*exp(8*x)-4*x^4*exp(3*x^2)+6*x^4*exp(2*x^2)-4*x^4*exp(x^2)+x^4+(4*exp(x^2)*x-4*x)/x^6*exp(
6*x)+(6*x^2*exp(2*x^2)-12*x^2*exp(x^2)+6*x^2)/x^4*exp(4*x)+(4*x^3*exp(3*x^2)-12*x^3*exp(2*x^2)+12*x^3*exp(x^2)
-4*x^3)/x^2*exp(2*x)

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maxima [C]  time = 0.81, size = 385, normalized size = 16.04 \begin {gather*} x^{4} - 6 i \, \sqrt {\pi } \operatorname {erf}\left (i \, x + i\right ) e^{\left (-1\right )} - 12 \, {\left (\frac {{\left (x + 1\right )}^{3} \Gamma \left (\frac {3}{2}, -{\left (x + 1\right )}^{2}\right )}{\left (-{\left (x + 1\right )}^{2}\right )^{\frac {3}{2}}} - \frac {\sqrt {\pi } {\left (x + 1\right )} {\left (\operatorname {erf}\left (\sqrt {-{\left (x + 1\right )}^{2}}\right ) - 1\right )}}{\sqrt {-{\left (x + 1\right )}^{2}}} + 2 \, e^{\left ({\left (x + 1\right )}^{2}\right )}\right )} e^{\left (-1\right )} - 12 \, {\left (\frac {\sqrt {\pi } {\left (x + 1\right )} {\left (\operatorname {erf}\left (\sqrt {-{\left (x + 1\right )}^{2}}\right ) - 1\right )}}{\sqrt {-{\left (x + 1\right )}^{2}}} - e^{\left ({\left (x + 1\right )}^{2}\right )}\right )} e^{\left (-1\right )} + \frac {1}{8} \, {\left (8 \, x^{4} - 4 \, x^{2} + 1\right )} e^{\left (4 \, x^{2}\right )} + \frac {1}{8} \, {\left (4 \, x^{2} - 1\right )} e^{\left (4 \, x^{2}\right )} - \frac {4}{9} \, {\left (9 \, x^{4} - 6 \, x^{2} + 2\right )} e^{\left (3 \, x^{2}\right )} - \frac {8}{9} \, {\left (3 \, x^{2} - 1\right )} e^{\left (3 \, x^{2}\right )} + 3 \, {\left (2 \, x^{4} - 2 \, x^{2} + 1\right )} e^{\left (2 \, x^{2}\right )} + 3 \, {\left (2 \, x^{2} - 1\right )} e^{\left (2 \, x^{2}\right )} - 4 \, {\left (x^{4} - 2 \, x^{2} + 2\right )} e^{\left (x^{2}\right )} - 8 \, {\left (x^{2} - 1\right )} e^{\left (x^{2}\right )} - 2 \, {\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} + \frac {2 \, {\left (2 \, x^{6} e^{\left (3 \, x^{2} + 2 \, x\right )} - 3 \, {\left (2 \, x^{6} e^{\left (2 \, x\right )} - x^{3} e^{\left (4 \, x\right )}\right )} e^{\left (2 \, x^{2}\right )} - 2 \, {\left (3 \, x^{3} e^{\left (4 \, x\right )} - e^{\left (6 \, x\right )}\right )} e^{\left (x^{2}\right )}\right )}}{x^{5}} - 2 \, e^{\left (2 \, x\right )} + 96 \, \Gamma \left (-1, -4 \, x\right ) + 192 \, \Gamma \left (-2, -4 \, x\right ) + 31104 \, \Gamma \left (-4, -6 \, x\right ) + 155520 \, \Gamma \left (-5, -6 \, x\right ) + 16777216 \, \Gamma \left (-7, -8 \, x\right ) + 134217728 \, \Gamma \left (-8, -8 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x-8)*exp(x-log(x))^8+((8*x^3+24*x^2-20*x)*exp(x^2)-24*x^2+20*x)*exp(x-log(x))^6+((24*x^4+24*x^3-
12*x^2)*exp(x^2)^2+(-24*x^4-48*x^3+24*x^2)*exp(x^2)+24*x^3-12*x^2)*exp(x-log(x))^4+((24*x^5+8*x^4+4*x^3)*exp(x
^2)^3+(-48*x^5-24*x^4-12*x^3)*exp(x^2)^2+(24*x^5+24*x^4+12*x^3)*exp(x^2)-8*x^4-4*x^3)*exp(x-log(x))^2+(8*x^6+4
*x^4)*exp(x^2)^4+(-24*x^6-16*x^4)*exp(x^2)^3+(24*x^6+24*x^4)*exp(x^2)^2+(-8*x^6-16*x^4)*exp(x^2)+4*x^4)/x,x, a
lgorithm="maxima")

[Out]

x^4 - 6*I*sqrt(pi)*erf(I*x + I)*e^(-1) - 12*((x + 1)^3*gamma(3/2, -(x + 1)^2)/(-(x + 1)^2)^(3/2) - sqrt(pi)*(x
+ 1)*(erf(sqrt(-(x + 1)^2)) - 1)/sqrt(-(x + 1)^2) + 2*e^((x + 1)^2))*e^(-1) - 12*(sqrt(pi)*(x + 1)*(erf(sqrt(
-(x + 1)^2)) - 1)/sqrt(-(x + 1)^2) - e^((x + 1)^2))*e^(-1) + 1/8*(8*x^4 - 4*x^2 + 1)*e^(4*x^2) + 1/8*(4*x^2 -
1)*e^(4*x^2) - 4/9*(9*x^4 - 6*x^2 + 2)*e^(3*x^2) - 8/9*(3*x^2 - 1)*e^(3*x^2) + 3*(2*x^4 - 2*x^2 + 1)*e^(2*x^2)
+ 3*(2*x^2 - 1)*e^(2*x^2) - 4*(x^4 - 2*x^2 + 2)*e^(x^2) - 8*(x^2 - 1)*e^(x^2) - 2*(2*x - 1)*e^(2*x) + 2*(2*x^
6*e^(3*x^2 + 2*x) - 3*(2*x^6*e^(2*x) - x^3*e^(4*x))*e^(2*x^2) - 2*(3*x^3*e^(4*x) - e^(6*x))*e^(x^2))/x^5 - 2*e
^(2*x) + 96*gamma(-1, -4*x) + 192*gamma(-2, -4*x) + 31104*gamma(-4, -6*x) + 155520*gamma(-5, -6*x) + 16777216*
gamma(-7, -8*x) + 134217728*gamma(-8, -8*x)

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mupad [B]  time = 1.76, size = 156, normalized size = 6.50 \begin {gather*} \frac {6\,{\mathrm {e}}^{2\,x^2+4\,x}}{x^2}-4\,x\,{\mathrm {e}}^{2\,x}+12\,x\,{\mathrm {e}}^{x^2+2\,x}+\frac {6\,{\mathrm {e}}^{4\,x}}{x^2}-\frac {4\,{\mathrm {e}}^{6\,x}}{x^5}+\frac {{\mathrm {e}}^{8\,x}}{x^8}-4\,x^4\,{\mathrm {e}}^{x^2}-12\,x\,{\mathrm {e}}^{2\,x^2+2\,x}+4\,x\,{\mathrm {e}}^{3\,x^2+2\,x}-\frac {12\,{\mathrm {e}}^{x^2+4\,x}}{x^2}+\frac {4\,{\mathrm {e}}^{x^2+6\,x}}{x^5}+6\,x^4\,{\mathrm {e}}^{2\,x^2}-4\,x^4\,{\mathrm {e}}^{3\,x^2}+x^4\,{\mathrm {e}}^{4\,x^2}+x^4 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(8*x - 8*log(x))*(8*x - 8) - exp(x^2)*(16*x^4 + 8*x^6) - exp(2*x - 2*log(x))*(4*x^3 - exp(x^2)*(12*x^3
+ 24*x^4 + 24*x^5) + 8*x^4 - exp(3*x^2)*(4*x^3 + 8*x^4 + 24*x^5) + exp(2*x^2)*(12*x^3 + 24*x^4 + 48*x^5)) - e
xp(4*x - 4*log(x))*(exp(x^2)*(48*x^3 - 24*x^2 + 24*x^4) + 12*x^2 - 24*x^3 - exp(2*x^2)*(24*x^3 - 12*x^2 + 24*x
^4)) + exp(4*x^2)*(4*x^4 + 8*x^6) - exp(3*x^2)*(16*x^4 + 24*x^6) + exp(2*x^2)*(24*x^4 + 24*x^6) + 4*x^4 + exp(
6*x - 6*log(x))*(20*x + exp(x^2)*(24*x^2 - 20*x + 8*x^3) - 24*x^2))/x,x)

[Out]

(6*exp(4*x + 2*x^2))/x^2 - 4*x*exp(2*x) + 12*x*exp(2*x + x^2) + (6*exp(4*x))/x^2 - (4*exp(6*x))/x^5 + exp(8*x)
/x^8 - 4*x^4*exp(x^2) - 12*x*exp(2*x + 2*x^2) + 4*x*exp(2*x + 3*x^2) - (12*exp(4*x + x^2))/x^2 + (4*exp(6*x +
x^2))/x^5 + 6*x^4*exp(2*x^2) - 4*x^4*exp(3*x^2) + x^4*exp(4*x^2) + x^4

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sympy [B]  time = 0.60, size = 153, normalized size = 6.38 \begin {gather*} x^{4} + \frac {x^{11} e^{4 x^{2}} + \left (- 4 x^{11} + 4 x^{8} e^{2 x}\right ) e^{3 x^{2}} + \left (6 x^{11} - 12 x^{8} e^{2 x} + 6 x^{5} e^{4 x}\right ) e^{2 x^{2}} + \left (- 4 x^{11} + 12 x^{8} e^{2 x} - 12 x^{5} e^{4 x} + 4 x^{2} e^{6 x}\right ) e^{x^{2}}}{x^{7}} + \frac {- 4 x^{16} e^{2 x} + 6 x^{13} e^{4 x} - 4 x^{10} e^{6 x} + x^{7} e^{8 x}}{x^{15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x-8)*exp(x-ln(x))**8+((8*x**3+24*x**2-20*x)*exp(x**2)-24*x**2+20*x)*exp(x-ln(x))**6+((24*x**4+24
*x**3-12*x**2)*exp(x**2)**2+(-24*x**4-48*x**3+24*x**2)*exp(x**2)+24*x**3-12*x**2)*exp(x-ln(x))**4+((24*x**5+8*
x**4+4*x**3)*exp(x**2)**3+(-48*x**5-24*x**4-12*x**3)*exp(x**2)**2+(24*x**5+24*x**4+12*x**3)*exp(x**2)-8*x**4-4
*x**3)*exp(x-ln(x))**2+(8*x**6+4*x**4)*exp(x**2)**4+(-24*x**6-16*x**4)*exp(x**2)**3+(24*x**6+24*x**4)*exp(x**2
)**2+(-8*x**6-16*x**4)*exp(x**2)+4*x**4)/x,x)

[Out]

x**4 + (x**11*exp(4*x**2) + (-4*x**11 + 4*x**8*exp(2*x))*exp(3*x**2) + (6*x**11 - 12*x**8*exp(2*x) + 6*x**5*ex
p(4*x))*exp(2*x**2) + (-4*x**11 + 12*x**8*exp(2*x) - 12*x**5*exp(4*x) + 4*x**2*exp(6*x))*exp(x**2))/x**7 + (-4
*x**16*exp(2*x) + 6*x**13*exp(4*x) - 4*x**10*exp(6*x) + x**7*exp(8*x))/x**15

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