3.71.69 \(\int \frac {-40 x-4 x^3+e (-8+14 x+2 x^2)+e^{x^2} (8-14 x-58 x^2+28 x^3)+(-12 x^2+e (-20+20 x-2 x^2)+e^{x^2} (20-20 x-38 x^2+28 x^3-4 x^4)) \log (x)}{4 x^2-4 x^3+x^4} \, dx\)

Optimal. Leaf size=32 \[ \frac {\left (4+\frac {2 \left (-e+e^{x^2}\right )}{x}\right ) (7+(5-x) \log (x))}{-2+x} \]

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Rubi [B]  time = 3.48, antiderivative size = 124, normalized size of antiderivative = 3.88, number of steps used = 43, number of rules used = 15, integrand size = 106, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.142, Rules used = {1594, 27, 6742, 44, 43, 893, 2314, 31, 2357, 2304, 2220, 2204, 2214, 2210, 2554} \begin {gather*} -\frac {7 e^{x^2}}{2-x}-\frac {7 e^{x^2}}{x}-\frac {3 e^{x^2} \log (x)}{2-x}-\frac {5 e^{x^2} \log (x)}{x}+\frac {7 e}{2-x}-\frac {28}{2-x}+\frac {7 e}{x}+\frac {3 e x \log (x)}{2 (2-x)}-\frac {6 x \log (x)}{2-x}+\frac {5 e \log (x)}{x}+\frac {3}{2} e \log (x)-10 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-40*x - 4*x^3 + E*(-8 + 14*x + 2*x^2) + E^x^2*(8 - 14*x - 58*x^2 + 28*x^3) + (-12*x^2 + E*(-20 + 20*x - 2
*x^2) + E^x^2*(20 - 20*x - 38*x^2 + 28*x^3 - 4*x^4))*Log[x])/(4*x^2 - 4*x^3 + x^4),x]

[Out]

-28/(2 - x) + (7*E)/(2 - x) - (7*E^x^2)/(2 - x) + (7*E)/x - (7*E^x^2)/x - 10*Log[x] + (3*E*Log[x])/2 - (3*E^x^
2*Log[x])/(2 - x) + (5*E*Log[x])/x - (5*E^x^2*Log[x])/x - (6*x*Log[x])/(2 - x) + (3*E*x*Log[x])/(2*(2 - x))

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 893

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rule 1594

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

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 2210

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

Rule 2214

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
+ 1)*F^(a + b*(c + d*x)^n))/(d*(m + 1)), x] - Dist[(b*n*Log[F])/(m + 1), 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[-4, (m + 1)/n, 5] && IntegerQ[n
] && ((GtQ[n, 0] && LtQ[m, -1]) || (GtQ[-n, 0] && LeQ[-n, m + 1]))

Rule 2220

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

Rule 2304

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

Rule 2314

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x_Symbol] :> Simp[(x*(d + e*x^r)^(q
+ 1)*(a + b*Log[c*x^n]))/d, x] - Dist[(b*n)/d, Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q,
r}, x] && EqQ[r*(q + 1) + 1, 0]

Rule 2357

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*x^
n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]

Rule 2554

Int[Log[u_]*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Dist[Log[u], w, x] - Int[SimplifyIntegrand[(w*D[u, x]
)/u, x], x] /; InverseFunctionFreeQ[w, x]] /; InverseFunctionFreeQ[u, 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 \frac {-40 x-4 x^3+e \left (-8+14 x+2 x^2\right )+e^{x^2} \left (8-14 x-58 x^2+28 x^3\right )+\left (-12 x^2+e \left (-20+20 x-2 x^2\right )+e^{x^2} \left (20-20 x-38 x^2+28 x^3-4 x^4\right )\right ) \log (x)}{x^2 \left (4-4 x+x^2\right )} \, dx\\ &=\int \frac {-40 x-4 x^3+e \left (-8+14 x+2 x^2\right )+e^{x^2} \left (8-14 x-58 x^2+28 x^3\right )+\left (-12 x^2+e \left (-20+20 x-2 x^2\right )+e^{x^2} \left (20-20 x-38 x^2+28 x^3-4 x^4\right )\right ) \log (x)}{(-2+x)^2 x^2} \, dx\\ &=\int \left (-\frac {40}{(-2+x)^2 x}-\frac {4 x}{(-2+x)^2}+\frac {2 e \left (-4+7 x+x^2\right )}{(-2+x)^2 x^2}-\frac {12 \log (x)}{(-2+x)^2}-\frac {2 e \left (10-10 x+x^2\right ) \log (x)}{(-2+x)^2 x^2}-\frac {2 e^{x^2} \left (-4+7 x+29 x^2-14 x^3-10 \log (x)+10 x \log (x)+19 x^2 \log (x)-14 x^3 \log (x)+2 x^4 \log (x)\right )}{(-2+x)^2 x^2}\right ) \, dx\\ &=-\left (2 \int \frac {e^{x^2} \left (-4+7 x+29 x^2-14 x^3-10 \log (x)+10 x \log (x)+19 x^2 \log (x)-14 x^3 \log (x)+2 x^4 \log (x)\right )}{(-2+x)^2 x^2} \, dx\right )-4 \int \frac {x}{(-2+x)^2} \, dx-12 \int \frac {\log (x)}{(-2+x)^2} \, dx-40 \int \frac {1}{(-2+x)^2 x} \, dx+(2 e) \int \frac {-4+7 x+x^2}{(-2+x)^2 x^2} \, dx-(2 e) \int \frac {\left (10-10 x+x^2\right ) \log (x)}{(-2+x)^2 x^2} \, dx\\ &=-\frac {6 x \log (x)}{2-x}-2 \int \left (\frac {29 e^{x^2}}{(-2+x)^2}-\frac {4 e^{x^2}}{(-2+x)^2 x^2}+\frac {7 e^{x^2}}{(-2+x)^2 x}-\frac {14 e^{x^2} x}{(-2+x)^2}+\frac {e^{x^2} \left (-10+10 x+19 x^2-14 x^3+2 x^4\right ) \log (x)}{(-2+x)^2 x^2}\right ) \, dx-4 \int \left (\frac {2}{(-2+x)^2}+\frac {1}{-2+x}\right ) \, dx-6 \int \frac {1}{-2+x} \, dx-40 \int \left (\frac {1}{2 (-2+x)^2}-\frac {1}{4 (-2+x)}+\frac {1}{4 x}\right ) \, dx+(2 e) \int \left (\frac {7}{2 (-2+x)^2}-\frac {3}{4 (-2+x)}-\frac {1}{x^2}+\frac {3}{4 x}\right ) \, dx-(2 e) \int \left (-\frac {3 \log (x)}{2 (-2+x)^2}+\frac {5 \log (x)}{2 x^2}\right ) \, dx\\ &=-\frac {28}{2-x}+\frac {7 e}{2-x}+\frac {2 e}{x}-\frac {3}{2} e \log (2-x)-10 \log (x)+\frac {3}{2} e \log (x)-\frac {6 x \log (x)}{2-x}-2 \int \frac {e^{x^2} \left (-10+10 x+19 x^2-14 x^3+2 x^4\right ) \log (x)}{(-2+x)^2 x^2} \, dx+8 \int \frac {e^{x^2}}{(-2+x)^2 x^2} \, dx-14 \int \frac {e^{x^2}}{(-2+x)^2 x} \, dx+28 \int \frac {e^{x^2} x}{(-2+x)^2} \, dx-58 \int \frac {e^{x^2}}{(-2+x)^2} \, dx+(3 e) \int \frac {\log (x)}{(-2+x)^2} \, dx-(5 e) \int \frac {\log (x)}{x^2} \, dx\\ &=-\frac {28}{2-x}+\frac {7 e}{2-x}-\frac {58 e^{x^2}}{2-x}+\frac {7 e}{x}-\frac {3}{2} e \log (2-x)-10 \log (x)+\frac {3}{2} e \log (x)-\frac {3 e^{x^2} \log (x)}{2-x}+\frac {5 e \log (x)}{x}-\frac {5 e^{x^2} \log (x)}{x}-\frac {6 x \log (x)}{2-x}+\frac {3 e x \log (x)}{2 (2-x)}+2 \int \frac {e^{x^2} (5-x)}{(2-x) x^2} \, dx+8 \int \left (\frac {e^{x^2}}{4 (-2+x)^2}-\frac {e^{x^2}}{4 (-2+x)}+\frac {e^{x^2}}{4 x^2}+\frac {e^{x^2}}{4 x}\right ) \, dx-14 \int \left (\frac {e^{x^2}}{2 (-2+x)^2}-\frac {e^{x^2}}{4 (-2+x)}+\frac {e^{x^2}}{4 x}\right ) \, dx+28 \int \left (\frac {2 e^{x^2}}{(-2+x)^2}+\frac {e^{x^2}}{-2+x}\right ) \, dx-116 \int e^{x^2} \, dx-232 \int \frac {e^{x^2}}{-2+x} \, dx+\frac {1}{2} (3 e) \int \frac {1}{-2+x} \, dx\\ &=-\frac {28}{2-x}+\frac {7 e}{2-x}-\frac {58 e^{x^2}}{2-x}+\frac {7 e}{x}-58 \sqrt {\pi } \text {erfi}(x)-10 \log (x)+\frac {3}{2} e \log (x)-\frac {3 e^{x^2} \log (x)}{2-x}+\frac {5 e \log (x)}{x}-\frac {5 e^{x^2} \log (x)}{x}-\frac {6 x \log (x)}{2-x}+\frac {3 e x \log (x)}{2 (2-x)}+2 \int \left (-\frac {3 e^{x^2}}{4 (-2+x)}+\frac {5 e^{x^2}}{2 x^2}+\frac {3 e^{x^2}}{4 x}\right ) \, dx+2 \int \frac {e^{x^2}}{(-2+x)^2} \, dx-2 \int \frac {e^{x^2}}{-2+x} \, dx+2 \int \frac {e^{x^2}}{x^2} \, dx+2 \int \frac {e^{x^2}}{x} \, dx+\frac {7}{2} \int \frac {e^{x^2}}{-2+x} \, dx-\frac {7}{2} \int \frac {e^{x^2}}{x} \, dx-7 \int \frac {e^{x^2}}{(-2+x)^2} \, dx+28 \int \frac {e^{x^2}}{-2+x} \, dx+56 \int \frac {e^{x^2}}{(-2+x)^2} \, dx-232 \int \frac {e^{x^2}}{-2+x} \, dx\\ &=-\frac {28}{2-x}+\frac {7 e}{2-x}-\frac {7 e^{x^2}}{2-x}+\frac {7 e}{x}-\frac {2 e^{x^2}}{x}-58 \sqrt {\pi } \text {erfi}(x)-\frac {3 \text {Ei}\left (x^2\right )}{4}-10 \log (x)+\frac {3}{2} e \log (x)-\frac {3 e^{x^2} \log (x)}{2-x}+\frac {5 e \log (x)}{x}-\frac {5 e^{x^2} \log (x)}{x}-\frac {6 x \log (x)}{2-x}+\frac {3 e x \log (x)}{2 (2-x)}-\frac {3}{2} \int \frac {e^{x^2}}{-2+x} \, dx+\frac {3}{2} \int \frac {e^{x^2}}{x} \, dx-2 \int \frac {e^{x^2}}{-2+x} \, dx+\frac {7}{2} \int \frac {e^{x^2}}{-2+x} \, dx+2 \left (4 \int e^{x^2} \, dx\right )+5 \int \frac {e^{x^2}}{x^2} \, dx+8 \int \frac {e^{x^2}}{-2+x} \, dx-14 \int e^{x^2} \, dx+112 \int e^{x^2} \, dx+224 \int \frac {e^{x^2}}{-2+x} \, dx-232 \int \frac {e^{x^2}}{-2+x} \, dx\\ &=-\frac {28}{2-x}+\frac {7 e}{2-x}-\frac {7 e^{x^2}}{2-x}+\frac {7 e}{x}-\frac {7 e^{x^2}}{x}-5 \sqrt {\pi } \text {erfi}(x)-10 \log (x)+\frac {3}{2} e \log (x)-\frac {3 e^{x^2} \log (x)}{2-x}+\frac {5 e \log (x)}{x}-\frac {5 e^{x^2} \log (x)}{x}-\frac {6 x \log (x)}{2-x}+\frac {3 e x \log (x)}{2 (2-x)}-\frac {3}{2} \int \frac {e^{x^2}}{-2+x} \, dx-2 \int \frac {e^{x^2}}{-2+x} \, dx+\frac {7}{2} \int \frac {e^{x^2}}{-2+x} \, dx+8 \int \frac {e^{x^2}}{-2+x} \, dx+10 \int e^{x^2} \, dx+224 \int \frac {e^{x^2}}{-2+x} \, dx-232 \int \frac {e^{x^2}}{-2+x} \, dx\\ &=-\frac {28}{2-x}+\frac {7 e}{2-x}-\frac {7 e^{x^2}}{2-x}+\frac {7 e}{x}-\frac {7 e^{x^2}}{x}-10 \log (x)+\frac {3}{2} e \log (x)-\frac {3 e^{x^2} \log (x)}{2-x}+\frac {5 e \log (x)}{x}-\frac {5 e^{x^2} \log (x)}{x}-\frac {6 x \log (x)}{2-x}+\frac {3 e x \log (x)}{2 (2-x)}-\frac {3}{2} \int \frac {e^{x^2}}{-2+x} \, dx-2 \int \frac {e^{x^2}}{-2+x} \, dx+\frac {7}{2} \int \frac {e^{x^2}}{-2+x} \, dx+8 \int \frac {e^{x^2}}{-2+x} \, dx+224 \int \frac {e^{x^2}}{-2+x} \, dx-232 \int \frac {e^{x^2}}{-2+x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.10, size = 30, normalized size = 0.94 \begin {gather*} -\frac {2 \left (-e+e^{x^2}+2 x\right ) (-7+(-5+x) \log (x))}{(-2+x) x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-40*x - 4*x^3 + E*(-8 + 14*x + 2*x^2) + E^x^2*(8 - 14*x - 58*x^2 + 28*x^3) + (-12*x^2 + E*(-20 + 20
*x - 2*x^2) + E^x^2*(20 - 20*x - 38*x^2 + 28*x^3 - 4*x^4))*Log[x])/(4*x^2 - 4*x^3 + x^4),x]

[Out]

(-2*(-E + E^x^2 + 2*x)*(-7 + (-5 + x)*Log[x]))/((-2 + x)*x)

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fricas [A]  time = 0.53, size = 52, normalized size = 1.62 \begin {gather*} -\frac {2 \, {\left ({\left (2 \, x^{2} - {\left (x - 5\right )} e + {\left (x - 5\right )} e^{\left (x^{2}\right )} - 10 \, x\right )} \log \relax (x) - 14 \, x + 7 \, e - 7 \, e^{\left (x^{2}\right )}\right )}}{x^{2} - 2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*x^4+28*x^3-38*x^2-20*x+20)*exp(x^2)+(-2*x^2+20*x-20)*exp(1)-12*x^2)*log(x)+(28*x^3-58*x^2-14*x
+8)*exp(x^2)+(2*x^2+14*x-8)*exp(1)-4*x^3-40*x)/(x^4-4*x^3+4*x^2),x, algorithm="fricas")

[Out]

-2*((2*x^2 - (x - 5)*e + (x - 5)*e^(x^2) - 10*x)*log(x) - 14*x + 7*e - 7*e^(x^2))/(x^2 - 2*x)

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giac [B]  time = 0.22, size = 66, normalized size = 2.06 \begin {gather*} -\frac {2 \, {\left (2 \, x^{2} \log \relax (x) - x e \log \relax (x) + x e^{\left (x^{2}\right )} \log \relax (x) - 10 \, x \log \relax (x) + 5 \, e \log \relax (x) - 5 \, e^{\left (x^{2}\right )} \log \relax (x) - 14 \, x + 7 \, e - 7 \, e^{\left (x^{2}\right )}\right )}}{x^{2} - 2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*x^4+28*x^3-38*x^2-20*x+20)*exp(x^2)+(-2*x^2+20*x-20)*exp(1)-12*x^2)*log(x)+(28*x^3-58*x^2-14*x
+8)*exp(x^2)+(2*x^2+14*x-8)*exp(1)-4*x^3-40*x)/(x^4-4*x^3+4*x^2),x, algorithm="giac")

[Out]

-2*(2*x^2*log(x) - x*e*log(x) + x*e^(x^2)*log(x) - 10*x*log(x) + 5*e*log(x) - 5*e^(x^2)*log(x) - 14*x + 7*e -
7*e^(x^2))/(x^2 - 2*x)

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maple [A]  time = 0.15, size = 64, normalized size = 2.00

method	result	size
norman	\(\frac {28 x -4 x^{2} \ln \relax (x )+\left (2 \,{\mathrm e}+20\right ) x \ln \relax (x )-10 \,{\mathrm e} \ln \relax (x )+10 \,{\mathrm e}^{x^{2}} \ln \relax (x )-2 x \,{\mathrm e}^{x^{2}} \ln \relax (x )-14 \,{\mathrm e}+14 \,{\mathrm e}^{x^{2}}}{\left (x -2\right ) x}\)	\(64\)
risch	\(\frac {2 \left (x \,{\mathrm e}-{\mathrm e}^{x^{2}} x -5 \,{\mathrm e}+6 x +5 \,{\mathrm e}^{x^{2}}\right ) \ln \relax (x )}{\left (x -2\right ) x}-\frac {2 \left (2 x^{2} \ln \relax (x )-4 x \ln \relax (x )+7 \,{\mathrm e}-14 x -7 \,{\mathrm e}^{x^{2}}\right )}{\left (x -2\right ) x}\)	\(75\)
default	\(\frac {10 \,{\mathrm e}^{x^{2}} \ln \relax (x )-2 x \,{\mathrm e}^{x^{2}} \ln \relax (x )+14 \,{\mathrm e}^{x^{2}}}{x \left (x -2\right )}+\frac {3 \,{\mathrm e} \ln \relax (x )}{2}-10 \ln \relax (x )+\frac {7 \,{\mathrm e}}{x}-\frac {7 \,{\mathrm e}}{x -2}+\frac {28}{x -2}+\frac {5 \,{\mathrm e} \ln \relax (x )}{x}-\frac {3 \,{\mathrm e} \ln \relax (x ) x}{2 \left (x -2\right )}+\frac {6 \ln \relax (x ) x}{x -2}\)	\(99\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-4*x^4+28*x^3-38*x^2-20*x+20)*exp(x^2)+(-2*x^2+20*x-20)*exp(1)-12*x^2)*ln(x)+(28*x^3-58*x^2-14*x+8)*exp
(x^2)+(2*x^2+14*x-8)*exp(1)-4*x^3-40*x)/(x^4-4*x^3+4*x^2),x,method=_RETURNVERBOSE)

[Out]

(28*x-4*x^2*ln(x)+(2*exp(1)+20)*x*ln(x)-10*exp(1)*ln(x)+10*exp(x^2)*ln(x)-2*x*exp(x^2)*ln(x)-14*exp(1)+14*exp(
x^2))/(x-2)/x

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maxima [B]  time = 0.44, size = 142, normalized size = 4.44 \begin {gather*} 2 \, {\left (\frac {2 \, {\left (x - 1\right )}}{x^{2} - 2 \, x} + \log \left (x - 2\right ) - \log \relax (x)\right )} e - \frac {7}{2} \, {\left (\frac {2}{x - 2} + \log \left (x - 2\right ) - \log \relax (x)\right )} e + \frac {3}{2} \, {\left (e - 4\right )} \log \left (x - 2\right ) + \frac {10 \, x e - 4 \, {\left ({\left (x - 5\right )} \log \relax (x) - 7\right )} e^{\left (x^{2}\right )} - {\left (3 \, x^{2} {\left (e - 4\right )} - 10 \, x e + 20 \, e\right )} \log \relax (x) - 20 \, e}{2 \, {\left (x^{2} - 2 \, x\right )}} - \frac {2 \, e}{x - 2} + \frac {28}{x - 2} + 6 \, \log \left (x - 2\right ) - 10 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*x^4+28*x^3-38*x^2-20*x+20)*exp(x^2)+(-2*x^2+20*x-20)*exp(1)-12*x^2)*log(x)+(28*x^3-58*x^2-14*x
+8)*exp(x^2)+(2*x^2+14*x-8)*exp(1)-4*x^3-40*x)/(x^4-4*x^3+4*x^2),x, algorithm="maxima")

[Out]

2*(2*(x - 1)/(x^2 - 2*x) + log(x - 2) - log(x))*e - 7/2*(2/(x - 2) + log(x - 2) - log(x))*e + 3/2*(e - 4)*log(
x - 2) + 1/2*(10*x*e - 4*((x - 5)*log(x) - 7)*e^(x^2) - (3*x^2*(e - 4) - 10*x*e + 20*e)*log(x) - 20*e)/(x^2 -
2*x) - 2*e/(x - 2) + 28/(x - 2) + 6*log(x - 2) - 10*log(x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {40\,x-\mathrm {e}\,\left (2\,x^2+14\,x-8\right )+{\mathrm {e}}^{x^2}\,\left (-28\,x^3+58\,x^2+14\,x-8\right )+\ln \relax (x)\,\left (\mathrm {e}\,\left (2\,x^2-20\,x+20\right )+{\mathrm {e}}^{x^2}\,\left (4\,x^4-28\,x^3+38\,x^2+20\,x-20\right )+12\,x^2\right )+4\,x^3}{x^4-4\,x^3+4\,x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(40*x - exp(1)*(14*x + 2*x^2 - 8) + exp(x^2)*(14*x + 58*x^2 - 28*x^3 - 8) + log(x)*(exp(1)*(2*x^2 - 20*x
+ 20) + exp(x^2)*(20*x + 38*x^2 - 28*x^3 + 4*x^4 - 20) + 12*x^2) + 4*x^3)/(4*x^2 - 4*x^3 + x^4),x)

[Out]

int(-(40*x - exp(1)*(14*x + 2*x^2 - 8) + exp(x^2)*(14*x + 58*x^2 - 28*x^3 - 8) + log(x)*(exp(1)*(2*x^2 - 20*x
+ 20) + exp(x^2)*(20*x + 38*x^2 - 28*x^3 + 4*x^4 - 20) + 12*x^2) + 4*x^3)/(4*x^2 - 4*x^3 + x^4), x)

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sympy [B]  time = 0.74, size = 70, normalized size = 2.19 \begin {gather*} - \frac {- 28 x + 14 e}{x^{2} - 2 x} - 4 \log {\relax (x )} + \frac {\left (2 e x + 12 x - 10 e\right ) \log {\relax (x )}}{x^{2} - 2 x} + \frac {\left (- 2 x \log {\relax (x )} + 10 \log {\relax (x )} + 14\right ) e^{x^{2}}}{x^{2} - 2 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*x**4+28*x**3-38*x**2-20*x+20)*exp(x**2)+(-2*x**2+20*x-20)*exp(1)-12*x**2)*ln(x)+(28*x**3-58*x*
*2-14*x+8)*exp(x**2)+(2*x**2+14*x-8)*exp(1)-4*x**3-40*x)/(x**4-4*x**3+4*x**2),x)

[Out]

-(-28*x + 14*E)/(x**2 - 2*x) - 4*log(x) + (2*E*x + 12*x - 10*E)*log(x)/(x**2 - 2*x) + (-2*x*log(x) + 10*log(x)
+ 14)*exp(x**2)/(x**2 - 2*x)

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