\(\int \frac {e^x (-2+64 x+27 x^2-66 x^3-30 x^4+17 x^5+8 x^6)+e^x (-32+2 x+32 x^2-x^3-8 x^4) \log (\frac {-4+2 x^2}{-80+5 x+40 x^2})}{32-2 x-32 x^2+x^3+8 x^4} \, dx\) [3820]

Optimal result

Integrand size = 101, antiderivative size = 28 \[ \int \frac {e^x \left (-2+64 x+27 x^2-66 x^3-30 x^4+17 x^5+8 x^6\right )+e^x \left (-32+2 x+32 x^2-x^3-8 x^4\right ) \log \left (\frac {-4+2 x^2}{-80+5 x+40 x^2}\right )}{32-2 x-32 x^2+x^3+8 x^4} \, dx=e^x \left (x^2-\log \left (\frac {2}{5 \left (8+\frac {x}{-2+x^2}\right )}\right )\right ) \]

[Out]

(x^2-ln(2/(40+5*x/(x^2-2))))*exp(x)

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 28.19 (sec) , antiderivative size = 983, normalized size of antiderivative = 35.11, number of steps used = 191, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.119, Rules used = {6874, 6820, 6857, 2209, 2225, 2301, 2207, 2227, 2634, 2302, 6860, 2300} \[ \int \frac {e^x \left (-2+64 x+27 x^2-66 x^3-30 x^4+17 x^5+8 x^6\right )+e^x \left (-32+2 x+32 x^2-x^3-8 x^4\right ) \log \left (\frac {-4+2 x^2}{-80+5 x+40 x^2}\right )}{32-2 x-32 x^2+x^3+8 x^4} \, dx =\text {Too large to display} \]

[In]

Int[(E^x*(-2 + 64*x + 27*x^2 - 66*x^3 - 30*x^4 + 17*x^5 + 8*x^6) + E^x*(-32 + 2*x + 32*x^2 - x^3 - 8*x^4)*Log[
(-4 + 2*x^2)/(-80 + 5*x + 40*x^2)])/(32 - 2*x - 32*x^2 + x^3 + 8*x^4),x]

[Out]

E^x*x^2 - E^Sqrt[2]*ExpIntegralEi[-Sqrt[2] + x] - (38 - 23*Sqrt[2])*E^Sqrt[2]*ExpIntegralEi[-Sqrt[2] + x] + (3
9 - 23*Sqrt[2])*E^Sqrt[2]*ExpIntegralEi[-Sqrt[2] + x] - ExpIntegralEi[Sqrt[2] + x]/E^Sqrt[2] - ((38 + 23*Sqrt[
2])*ExpIntegralEi[Sqrt[2] + x])/E^Sqrt[2] + ((39 + 23*Sqrt[2])*ExpIntegralEi[Sqrt[2] + x])/E^Sqrt[2] - (512*E^
((-1 + 3*Sqrt[57])/16)*ExpIntegralEi[(1 - 3*Sqrt[57] + 16*x)/16])/(3*Sqrt[57]) - ((2867499 - 82561*Sqrt[57])*E
^((-1 + 3*Sqrt[57])/16)*ExpIntegralEi[(1 - 3*Sqrt[57] + 16*x)/16])/175104 + (17*(43947 - 33281*Sqrt[57])*E^((-
1 + 3*Sqrt[57])/16)*ExpIntegralEi[(1 - 3*Sqrt[57] + 16*x)/16])/175104 + (5*(22059 - 385*Sqrt[57])*E^((-1 + 3*S
qrt[57])/16)*ExpIntegralEi[(1 - 3*Sqrt[57] + 16*x)/16])/3648 - (11*(171 - 257*Sqrt[57])*E^((-1 + 3*Sqrt[57])/1
6)*ExpIntegralEi[(1 - 3*Sqrt[57] + 16*x)/16])/456 + ((1297 - 147*Sqrt[57])*E^((-1 + 3*Sqrt[57])/16)*ExpIntegra
lEi[(1 - 3*Sqrt[57] + 16*x)/16])/32 - (3*(443 - 49*Sqrt[57])*E^((-1 + 3*Sqrt[57])/16)*ExpIntegralEi[(1 - 3*Sqr
t[57] + 16*x)/16])/32 - (3*(171 - Sqrt[57])*E^((-1 + 3*Sqrt[57])/16)*ExpIntegralEi[(1 - 3*Sqrt[57] + 16*x)/16]
)/38 + ((171 + Sqrt[57])*E^((-1 + 3*Sqrt[57])/16)*ExpIntegralEi[(1 - 3*Sqrt[57] + 16*x)/16])/342 + (512*E^((-1
- 3*Sqrt[57])/16)*ExpIntegralEi[(1 + 3*Sqrt[57] + 16*x)/16])/(3*Sqrt[57]) + ((171 - Sqrt[57])*E^((-1 - 3*Sqrt
[57])/16)*ExpIntegralEi[(1 + 3*Sqrt[57] + 16*x)/16])/342 - (3*(171 + Sqrt[57])*E^((-1 - 3*Sqrt[57])/16)*ExpInt
egralEi[(1 + 3*Sqrt[57] + 16*x)/16])/38 - (3*(443 + 49*Sqrt[57])*E^((-1 - 3*Sqrt[57])/16)*ExpIntegralEi[(1 + 3
*Sqrt[57] + 16*x)/16])/32 + ((1297 + 147*Sqrt[57])*E^((-1 - 3*Sqrt[57])/16)*ExpIntegralEi[(1 + 3*Sqrt[57] + 16
*x)/16])/32 - (11*(171 + 257*Sqrt[57])*E^((-1 - 3*Sqrt[57])/16)*ExpIntegralEi[(1 + 3*Sqrt[57] + 16*x)/16])/456
+ (5*(22059 + 385*Sqrt[57])*E^((-1 - 3*Sqrt[57])/16)*ExpIntegralEi[(1 + 3*Sqrt[57] + 16*x)/16])/3648 + (17*(4
3947 + 33281*Sqrt[57])*E^((-1 - 3*Sqrt[57])/16)*ExpIntegralEi[(1 + 3*Sqrt[57] + 16*x)/16])/175104 - ((2867499
+ 82561*Sqrt[57])*E^((-1 - 3*Sqrt[57])/16)*ExpIntegralEi[(1 + 3*Sqrt[57] + 16*x)/16])/175104 - E^x*Log[(2*(2 -
x^2))/(5*(16 - x - 8*x^2))]

Rule 2207

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] &&  !TrueQ[$UseGamma]

Rule 2209

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - c*(f/d)))/d)*ExpInteg
ralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2225

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 2227

Int[(F_)^((c_.)*(v_))*(u_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), u, x], x] /; FreeQ[{F, c
}, x] && PolynomialQ[u, x] && LinearQ[v, x] &&  !TrueQ[$UseGamma]

Rule 2300

Int[(F_)^((g_.)*((d_.) + (e_.)*(x_))^(n_.))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegr
and[F^(g*(d + e*x)^n), 1/(a + b*x + c*x^2), x], x] /; FreeQ[{F, a, b, c, d, e, g, n}, x]

Rule 2301

Int[(F_)^((g_.)*((d_.) + (e_.)*(x_))^(n_.))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[F^(g*(d +
e*x)^n), 1/(a + c*x^2), x], x] /; FreeQ[{F, a, c, d, e, g, n}, x]

Rule 2302

Int[((F_)^((g_.)*((d_.) + (e_.)*(x_))^(n_.))*(u_)^(m_.))/((a_.) + (b_.)*(x_) + (c_)*(x_)^2), x_Symbol] :> Int[
ExpandIntegrand[F^(g*(d + e*x)^n), u^m/(a + b*x + c*x^2), x], x] /; FreeQ[{F, a, b, c, d, e, g, n}, x] && Poly
nomialQ[u, x] && IntegerQ[m]

Rule 2634

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 6820

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

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rule 6860

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {e^x x \left (-2+64 x+27 x^2-66 x^3-30 x^4+17 x^5+8 x^6-32 \log \left (\frac {2 \left (-2+x^2\right )}{5 \left (-16+x+8 x^2\right )}\right )+2 x \log \left (\frac {2 \left (-2+x^2\right )}{5 \left (-16+x+8 x^2\right )}\right )+32 x^2 \log \left (\frac {2 \left (-2+x^2\right )}{5 \left (-16+x+8 x^2\right )}\right )-x^3 \log \left (\frac {2 \left (-2+x^2\right )}{5 \left (-16+x+8 x^2\right )}\right )-8 x^4 \log \left (\frac {2 \left (-2+x^2\right )}{5 \left (-16+x+8 x^2\right )}\right )\right )}{2 \left (-2+x^2\right )}-\frac {e^x (1+8 x) \left (-2+64 x+27 x^2-66 x^3-30 x^4+17 x^5+8 x^6-32 \log \left (\frac {2 \left (-2+x^2\right )}{5 \left (-16+x+8 x^2\right )}\right )+2 x \log \left (\frac {2 \left (-2+x^2\right )}{5 \left (-16+x+8 x^2\right )}\right )+32 x^2 \log \left (\frac {2 \left (-2+x^2\right )}{5 \left (-16+x+8 x^2\right )}\right )-x^3 \log \left (\frac {2 \left (-2+x^2\right )}{5 \left (-16+x+8 x^2\right )}\right )-8 x^4 \log \left (\frac {2 \left (-2+x^2\right )}{5 \left (-16+x+8 x^2\right )}\right )\right )}{2 \left (-16+x+8 x^2\right )}\right ) \, dx \\ & = \frac {1}{2} \int \frac {e^x x \left (-2+64 x+27 x^2-66 x^3-30 x^4+17 x^5+8 x^6-32 \log \left (\frac {2 \left (-2+x^2\right )}{5 \left (-16+x+8 x^2\right )}\right )+2 x \log \left (\frac {2 \left (-2+x^2\right )}{5 \left (-16+x+8 x^2\right )}\right )+32 x^2 \log \left (\frac {2 \left (-2+x^2\right )}{5 \left (-16+x+8 x^2\right )}\right )-x^3 \log \left (\frac {2 \left (-2+x^2\right )}{5 \left (-16+x+8 x^2\right )}\right )-8 x^4 \log \left (\frac {2 \left (-2+x^2\right )}{5 \left (-16+x+8 x^2\right )}\right )\right )}{-2+x^2} \, dx-\frac {1}{2} \int \frac {e^x (1+8 x) \left (-2+64 x+27 x^2-66 x^3-30 x^4+17 x^5+8 x^6-32 \log \left (\frac {2 \left (-2+x^2\right )}{5 \left (-16+x+8 x^2\right )}\right )+2 x \log \left (\frac {2 \left (-2+x^2\right )}{5 \left (-16+x+8 x^2\right )}\right )+32 x^2 \log \left (\frac {2 \left (-2+x^2\right )}{5 \left (-16+x+8 x^2\right )}\right )-x^3 \log \left (\frac {2 \left (-2+x^2\right )}{5 \left (-16+x+8 x^2\right )}\right )-8 x^4 \log \left (\frac {2 \left (-2+x^2\right )}{5 \left (-16+x+8 x^2\right )}\right )\right )}{-16+x+8 x^2} \, dx \\ & = -\left (\frac {1}{2} \int \frac {e^x (1+8 x) \left (2-64 x-27 x^2+66 x^3+30 x^4-17 x^5-8 x^6+\left (32-2 x-32 x^2+x^3+8 x^4\right ) \log \left (\frac {2 \left (-2+x^2\right )}{5 \left (-16+x+8 x^2\right )}\right )\right )}{16-x-8 x^2} \, dx\right )+\frac {1}{2} \int \frac {e^x x \left (2-64 x-27 x^2+66 x^3+30 x^4-17 x^5-8 x^6+\left (32-2 x-32 x^2+x^3+8 x^4\right ) \log \left (\frac {2 \left (-2+x^2\right )}{5 \left (-16+x+8 x^2\right )}\right )\right )}{2-x^2} \, dx \\ & = -\left (\frac {1}{2} \int \left (-\frac {2 e^x (1+8 x)}{-16+x+8 x^2}+\frac {64 e^x x (1+8 x)}{-16+x+8 x^2}+\frac {27 e^x x^2 (1+8 x)}{-16+x+8 x^2}-\frac {66 e^x x^3 (1+8 x)}{-16+x+8 x^2}-\frac {30 e^x x^4 (1+8 x)}{-16+x+8 x^2}+\frac {17 e^x x^5 (1+8 x)}{-16+x+8 x^2}+\frac {8 e^x x^6 (1+8 x)}{-16+x+8 x^2}-e^x (1+8 x) \left (-2+x^2\right ) \log \left (\frac {2 \left (-2+x^2\right )}{5 \left (-16+x+8 x^2\right )}\right )\right ) \, dx\right )+\frac {1}{2} \int \left (-\frac {2 e^x x}{-2+x^2}+\frac {64 e^x x^2}{-2+x^2}+\frac {27 e^x x^3}{-2+x^2}-\frac {66 e^x x^4}{-2+x^2}-\frac {30 e^x x^5}{-2+x^2}+\frac {17 e^x x^6}{-2+x^2}+\frac {8 e^x x^7}{-2+x^2}-e^x x \left (-16+x+8 x^2\right ) \log \left (\frac {2 \left (-2+x^2\right )}{5 \left (-16+x+8 x^2\right )}\right )\right ) \, dx \\ & = \frac {1}{2} \int e^x (1+8 x) \left (-2+x^2\right ) \log \left (\frac {2 \left (-2+x^2\right )}{5 \left (-16+x+8 x^2\right )}\right ) \, dx-\frac {1}{2} \int e^x x \left (-16+x+8 x^2\right ) \log \left (\frac {2 \left (-2+x^2\right )}{5 \left (-16+x+8 x^2\right )}\right ) \, dx+4 \int \frac {e^x x^7}{-2+x^2} \, dx-4 \int \frac {e^x x^6 (1+8 x)}{-16+x+8 x^2} \, dx+\frac {17}{2} \int \frac {e^x x^6}{-2+x^2} \, dx-\frac {17}{2} \int \frac {e^x x^5 (1+8 x)}{-16+x+8 x^2} \, dx+\frac {27}{2} \int \frac {e^x x^3}{-2+x^2} \, dx-\frac {27}{2} \int \frac {e^x x^2 (1+8 x)}{-16+x+8 x^2} \, dx-15 \int \frac {e^x x^5}{-2+x^2} \, dx+15 \int \frac {e^x x^4 (1+8 x)}{-16+x+8 x^2} \, dx+32 \int \frac {e^x x^2}{-2+x^2} \, dx-32 \int \frac {e^x x (1+8 x)}{-16+x+8 x^2} \, dx-33 \int \frac {e^x x^4}{-2+x^2} \, dx+33 \int \frac {e^x x^3 (1+8 x)}{-16+x+8 x^2} \, dx-\int \frac {e^x x}{-2+x^2} \, dx+\int \frac {e^x (1+8 x)}{-16+x+8 x^2} \, dx \\ & = -e^x \log \left (\frac {2 \left (2-x^2\right )}{5 \left (16-x-8 x^2\right )}\right )-\frac {1}{2} \int \frac {e^x \left (2+x^2\right ) \left (-32+30 x-23 x^2+8 x^3\right )}{\left (16-x-8 x^2\right ) \left (2-x^2\right )} \, dx+\frac {1}{2} \int \frac {e^x \left (2+x^2\right ) \left (-30+30 x-23 x^2+8 x^3\right )}{\left (16-x-8 x^2\right ) \left (2-x^2\right )} \, dx+4 \int \left (4 e^x x+2 e^x x^3+e^x x^5+\frac {8 e^x x}{-2+x^2}\right ) \, dx-4 \int \left (-\frac {257 e^x}{256}+\frac {129 e^x x}{32}-\frac {e^x x^2}{4}+2 e^x x^3+e^x x^5-\frac {e^x (4112-16769 x)}{256 \left (-16+x+8 x^2\right )}\right ) \, dx+\frac {17}{2} \int \left (4 e^x+2 e^x x^2+e^x x^4+\frac {8 e^x}{-2+x^2}\right ) \, dx-\frac {17}{2} \int \left (\frac {129 e^x}{32}-\frac {e^x x}{4}+2 e^x x^2+e^x x^4+\frac {e^x (2064-257 x)}{32 \left (-16+x+8 x^2\right )}\right ) \, dx+\frac {27}{2} \int \left (e^x x+\frac {2 e^x x}{-2+x^2}\right ) \, dx-\frac {27}{2} \int \left (e^x x+\frac {16 e^x x}{-16+x+8 x^2}\right ) \, dx-15 \int \left (2 e^x x+e^x x^3+\frac {4 e^x x}{-2+x^2}\right ) \, dx+15 \int \left (-\frac {e^x}{4}+2 e^x x+e^x x^3-\frac {e^x (16-129 x)}{4 \left (-16+x+8 x^2\right )}\right ) \, dx+32 \int \left (e^x+\frac {2 e^x}{-2+x^2}\right ) \, dx-32 \int \left (e^x+\frac {16 e^x}{-16+x+8 x^2}\right ) \, dx-33 \int \left (2 e^x+e^x x^2+\frac {4 e^x}{-2+x^2}\right ) \, dx+33 \int \left (2 e^x+e^x x^2+\frac {2 e^x (16-x)}{-16+x+8 x^2}\right ) \, dx-\int \left (-\frac {e^x}{2 \left (\sqrt {2}-x\right )}+\frac {e^x}{2 \left (\sqrt {2}+x\right )}\right ) \, dx+\int \left (\frac {\left (8+\frac {8}{3 \sqrt {57}}\right ) e^x}{1-3 \sqrt {57}+16 x}+\frac {\left (8-\frac {8}{3 \sqrt {57}}\right ) e^x}{1+3 \sqrt {57}+16 x}\right ) \, dx \\ & = -e^x \log \left (\frac {2 \left (2-x^2\right )}{5 \left (16-x-8 x^2\right )}\right )+\frac {1}{64} \int \frac {e^x (4112-16769 x)}{-16+x+8 x^2} \, dx-\frac {17}{64} \int \frac {e^x (2064-257 x)}{-16+x+8 x^2} \, dx+\frac {1}{2} \int \frac {e^x}{\sqrt {2}-x} \, dx-\frac {1}{2} \int \frac {e^x}{\sqrt {2}+x} \, dx-\frac {1}{2} \int \left (-3 e^x+e^x x-\frac {4 e^x (-46+39 x)}{-2+x^2}+\frac {3 e^x (-496+443 x)}{-16+x+8 x^2}\right ) \, dx+\frac {1}{2} \int \left (-3 e^x+e^x x-\frac {8 e^x (-23+19 x)}{-2+x^2}+\frac {e^x (-1490+1297 x)}{-16+x+8 x^2}\right ) \, dx+\frac {17}{8} \int e^x x \, dx-\frac {15 \int e^x \, dx}{4}-\frac {15}{4} \int \frac {e^x (16-129 x)}{-16+x+8 x^2} \, dx+\frac {257 \int e^x \, dx}{64}+16 \int e^x x \, dx-\frac {129}{8} \int e^x x \, dx+27 \int \frac {e^x x}{-2+x^2} \, dx+32 \int \frac {e^x x}{-2+x^2} \, dx+34 \int e^x \, dx-\frac {2193 \int e^x \, dx}{64}-60 \int \frac {e^x x}{-2+x^2} \, dx+64 \int \frac {e^x}{-2+x^2} \, dx+66 \int \frac {e^x (16-x)}{-16+x+8 x^2} \, dx+68 \int \frac {e^x}{-2+x^2} \, dx-132 \int \frac {e^x}{-2+x^2} \, dx-216 \int \frac {e^x x}{-16+x+8 x^2} \, dx-512 \int \frac {e^x}{-16+x+8 x^2} \, dx+\frac {1}{171} \left (8 \left (171-\sqrt {57}\right )\right ) \int \frac {e^x}{1+3 \sqrt {57}+16 x} \, dx+\frac {1}{171} \left (8 \left (171+\sqrt {57}\right )\right ) \int \frac {e^x}{1-3 \sqrt {57}+16 x} \, dx+\int e^x x^2 \, dx \\ & = 2 e^x x+e^x x^2-\frac {1}{2} e^{\sqrt {2}} \text {Ei}\left (-\sqrt {2}+x\right )-\frac {1}{2} e^{-\sqrt {2}} \text {Ei}\left (\sqrt {2}+x\right )+\frac {1}{342} \left (171+\sqrt {57}\right ) e^{\frac {1}{16} \left (-1+3 \sqrt {57}\right )} \text {Ei}\left (\frac {1}{16} \left (1-3 \sqrt {57}+16 x\right )\right )+\frac {1}{342} \left (171-\sqrt {57}\right ) e^{\frac {1}{16} \left (-1-3 \sqrt {57}\right )} \text {Ei}\left (\frac {1}{16} \left (1+3 \sqrt {57}+16 x\right )\right )-e^x \log \left (\frac {2 \left (2-x^2\right )}{5 \left (16-x-8 x^2\right )}\right )+\frac {1}{64} \int \left (\frac {\left (-16769+\frac {82561}{3 \sqrt {57}}\right ) e^x}{1-3 \sqrt {57}+16 x}+\frac {\left (-16769-\frac {82561}{3 \sqrt {57}}\right ) e^x}{1+3 \sqrt {57}+16 x}\right ) \, dx-\frac {17}{64} \int \left (\frac {\left (-257+\frac {33281}{3 \sqrt {57}}\right ) e^x}{1-3 \sqrt {57}+16 x}+\frac {\left (-257-\frac {33281}{3 \sqrt {57}}\right ) e^x}{1+3 \sqrt {57}+16 x}\right ) \, dx+\frac {1}{2} \int \frac {e^x (-1490+1297 x)}{-16+x+8 x^2} \, dx-\frac {3}{2} \int \frac {e^x (-496+443 x)}{-16+x+8 x^2} \, dx-2 \int e^x x \, dx+2 \int \frac {e^x (-46+39 x)}{-2+x^2} \, dx-\frac {17 \int e^x \, dx}{8}-\frac {15}{4} \int \left (\frac {\left (-129+\frac {385}{3 \sqrt {57}}\right ) e^x}{1-3 \sqrt {57}+16 x}+\frac {\left (-129-\frac {385}{3 \sqrt {57}}\right ) e^x}{1+3 \sqrt {57}+16 x}\right ) \, dx-4 \int \frac {e^x (-23+19 x)}{-2+x^2} \, dx-16 \int e^x \, dx+\frac {129 \int e^x \, dx}{8}+27 \int \left (-\frac {e^x}{2 \left (\sqrt {2}-x\right )}+\frac {e^x}{2 \left (\sqrt {2}+x\right )}\right ) \, dx+32 \int \left (-\frac {e^x}{2 \left (\sqrt {2}-x\right )}+\frac {e^x}{2 \left (\sqrt {2}+x\right )}\right ) \, dx-60 \int \left (-\frac {e^x}{2 \left (\sqrt {2}-x\right )}+\frac {e^x}{2 \left (\sqrt {2}+x\right )}\right ) \, dx+64 \int \left (-\frac {e^x}{2 \sqrt {2} \left (\sqrt {2}-x\right )}-\frac {e^x}{2 \sqrt {2} \left (\sqrt {2}+x\right )}\right ) \, dx+66 \int \left (\frac {\left (-1+\frac {257}{3 \sqrt {57}}\right ) e^x}{1-3 \sqrt {57}+16 x}+\frac {\left (-1-\frac {257}{3 \sqrt {57}}\right ) e^x}{1+3 \sqrt {57}+16 x}\right ) \, dx+68 \int \left (-\frac {e^x}{2 \sqrt {2} \left (\sqrt {2}-x\right )}-\frac {e^x}{2 \sqrt {2} \left (\sqrt {2}+x\right )}\right ) \, dx-132 \int \left (-\frac {e^x}{2 \sqrt {2} \left (\sqrt {2}-x\right )}-\frac {e^x}{2 \sqrt {2} \left (\sqrt {2}+x\right )}\right ) \, dx-216 \int \left (\frac {\left (1-\frac {1}{3 \sqrt {57}}\right ) e^x}{1-3 \sqrt {57}+16 x}+\frac {\left (1+\frac {1}{3 \sqrt {57}}\right ) e^x}{1+3 \sqrt {57}+16 x}\right ) \, dx-512 \int \left (-\frac {16 e^x}{3 \sqrt {57} \left (-1+3 \sqrt {57}-16 x\right )}-\frac {16 e^x}{3 \sqrt {57} \left (1+3 \sqrt {57}+16 x\right )}\right ) \, dx \\ & = -2 e^x+e^x x^2-\frac {1}{2} e^{\sqrt {2}} \text {Ei}\left (-\sqrt {2}+x\right )-\frac {1}{2} e^{-\sqrt {2}} \text {Ei}\left (\sqrt {2}+x\right )+\frac {1}{342} \left (171+\sqrt {57}\right ) e^{\frac {1}{16} \left (-1+3 \sqrt {57}\right )} \text {Ei}\left (\frac {1}{16} \left (1-3 \sqrt {57}+16 x\right )\right )+\frac {1}{342} \left (171-\sqrt {57}\right ) e^{\frac {1}{16} \left (-1-3 \sqrt {57}\right )} \text {Ei}\left (\frac {1}{16} \left (1+3 \sqrt {57}+16 x\right )\right )-e^x \log \left (\frac {2 \left (2-x^2\right )}{5 \left (16-x-8 x^2\right )}\right )+\frac {1}{2} \int \left (\frac {\left (1297-147 \sqrt {57}\right ) e^x}{1-3 \sqrt {57}+16 x}+\frac {\left (1297+147 \sqrt {57}\right ) e^x}{1+3 \sqrt {57}+16 x}\right ) \, dx-\frac {3}{2} \int \left (\frac {\left (443-49 \sqrt {57}\right ) e^x}{1-3 \sqrt {57}+16 x}+\frac {\left (443+49 \sqrt {57}\right ) e^x}{1+3 \sqrt {57}+16 x}\right ) \, dx+2 \int e^x \, dx+2 \int \left (-\frac {\left (78-46 \sqrt {2}\right ) e^x}{4 \left (\sqrt {2}-x\right )}-\frac {\left (-78-46 \sqrt {2}\right ) e^x}{4 \left (\sqrt {2}+x\right )}\right ) \, dx-4 \int \left (-\frac {\left (38-23 \sqrt {2}\right ) e^x}{4 \left (\sqrt {2}-x\right )}-\frac {\left (-38-23 \sqrt {2}\right ) e^x}{4 \left (\sqrt {2}+x\right )}\right ) \, dx-\frac {27}{2} \int \frac {e^x}{\sqrt {2}-x} \, dx+\frac {27}{2} \int \frac {e^x}{\sqrt {2}+x} \, dx-16 \int \frac {e^x}{\sqrt {2}-x} \, dx+16 \int \frac {e^x}{\sqrt {2}+x} \, dx+30 \int \frac {e^x}{\sqrt {2}-x} \, dx-30 \int \frac {e^x}{\sqrt {2}+x} \, dx-\left (16 \sqrt {2}\right ) \int \frac {e^x}{\sqrt {2}-x} \, dx-\left (16 \sqrt {2}\right ) \int \frac {e^x}{\sqrt {2}+x} \, dx-\left (17 \sqrt {2}\right ) \int \frac {e^x}{\sqrt {2}-x} \, dx-\left (17 \sqrt {2}\right ) \int \frac {e^x}{\sqrt {2}+x} \, dx+\left (33 \sqrt {2}\right ) \int \frac {e^x}{\sqrt {2}-x} \, dx+\left (33 \sqrt {2}\right ) \int \frac {e^x}{\sqrt {2}+x} \, dx+\frac {8192 \int \frac {e^x}{-1+3 \sqrt {57}-16 x} \, dx}{3 \sqrt {57}}+\frac {8192 \int \frac {e^x}{1+3 \sqrt {57}+16 x} \, dx}{3 \sqrt {57}}+\frac {\left (17 \left (43947-33281 \sqrt {57}\right )\right ) \int \frac {e^x}{1-3 \sqrt {57}+16 x} \, dx}{10944}+\frac {1}{228} \left (5 \left (22059-385 \sqrt {57}\right )\right ) \int \frac {e^x}{1-3 \sqrt {57}+16 x} \, dx-\frac {1}{57} \left (22 \left (171-257 \sqrt {57}\right )\right ) \int \frac {e^x}{1-3 \sqrt {57}+16 x} \, dx-\frac {1}{19} \left (24 \left (171-\sqrt {57}\right )\right ) \int \frac {e^x}{1-3 \sqrt {57}+16 x} \, dx-\frac {1}{19} \left (24 \left (171+\sqrt {57}\right )\right ) \int \frac {e^x}{1+3 \sqrt {57}+16 x} \, dx-\frac {1}{57} \left (22 \left (171+257 \sqrt {57}\right )\right ) \int \frac {e^x}{1+3 \sqrt {57}+16 x} \, dx+\frac {1}{228} \left (5 \left (22059+385 \sqrt {57}\right )\right ) \int \frac {e^x}{1+3 \sqrt {57}+16 x} \, dx+\frac {\left (17 \left (43947+33281 \sqrt {57}\right )\right ) \int \frac {e^x}{1+3 \sqrt {57}+16 x} \, dx}{10944}+\frac {\left (-2867499+82561 \sqrt {57}\right ) \int \frac {e^x}{1-3 \sqrt {57}+16 x} \, dx}{10944}-\frac {\left (2867499+82561 \sqrt {57}\right ) \int \frac {e^x}{1+3 \sqrt {57}+16 x} \, dx}{10944} \\ & = e^x x^2-e^{\sqrt {2}} \text {Ei}\left (-\sqrt {2}+x\right )-e^{-\sqrt {2}} \text {Ei}\left (\sqrt {2}+x\right )-\frac {512 e^{\frac {1}{16} \left (-1+3 \sqrt {57}\right )} \text {Ei}\left (\frac {1}{16} \left (1-3 \sqrt {57}+16 x\right )\right )}{3 \sqrt {57}}-\frac {\left (2867499-82561 \sqrt {57}\right ) e^{\frac {1}{16} \left (-1+3 \sqrt {57}\right )} \text {Ei}\left (\frac {1}{16} \left (1-3 \sqrt {57}+16 x\right )\right )}{175104}+\frac {17 \left (43947-33281 \sqrt {57}\right ) e^{\frac {1}{16} \left (-1+3 \sqrt {57}\right )} \text {Ei}\left (\frac {1}{16} \left (1-3 \sqrt {57}+16 x\right )\right )}{175104}+\frac {5 \left (22059-385 \sqrt {57}\right ) e^{\frac {1}{16} \left (-1+3 \sqrt {57}\right )} \text {Ei}\left (\frac {1}{16} \left (1-3 \sqrt {57}+16 x\right )\right )}{3648}-\frac {11}{456} \left (171-257 \sqrt {57}\right ) e^{\frac {1}{16} \left (-1+3 \sqrt {57}\right )} \text {Ei}\left (\frac {1}{16} \left (1-3 \sqrt {57}+16 x\right )\right )-\frac {3}{38} \left (171-\sqrt {57}\right ) e^{\frac {1}{16} \left (-1+3 \sqrt {57}\right )} \text {Ei}\left (\frac {1}{16} \left (1-3 \sqrt {57}+16 x\right )\right )+\frac {1}{342} \left (171+\sqrt {57}\right ) e^{\frac {1}{16} \left (-1+3 \sqrt {57}\right )} \text {Ei}\left (\frac {1}{16} \left (1-3 \sqrt {57}+16 x\right )\right )+\frac {512 e^{\frac {1}{16} \left (-1-3 \sqrt {57}\right )} \text {Ei}\left (\frac {1}{16} \left (1+3 \sqrt {57}+16 x\right )\right )}{3 \sqrt {57}}+\frac {1}{342} \left (171-\sqrt {57}\right ) e^{\frac {1}{16} \left (-1-3 \sqrt {57}\right )} \text {Ei}\left (\frac {1}{16} \left (1+3 \sqrt {57}+16 x\right )\right )-\frac {3}{38} \left (171+\sqrt {57}\right ) e^{\frac {1}{16} \left (-1-3 \sqrt {57}\right )} \text {Ei}\left (\frac {1}{16} \left (1+3 \sqrt {57}+16 x\right )\right )-\frac {11}{456} \left (171+257 \sqrt {57}\right ) e^{\frac {1}{16} \left (-1-3 \sqrt {57}\right )} \text {Ei}\left (\frac {1}{16} \left (1+3 \sqrt {57}+16 x\right )\right )+\frac {5 \left (22059+385 \sqrt {57}\right ) e^{\frac {1}{16} \left (-1-3 \sqrt {57}\right )} \text {Ei}\left (\frac {1}{16} \left (1+3 \sqrt {57}+16 x\right )\right )}{3648}+\frac {17 \left (43947+33281 \sqrt {57}\right ) e^{\frac {1}{16} \left (-1-3 \sqrt {57}\right )} \text {Ei}\left (\frac {1}{16} \left (1+3 \sqrt {57}+16 x\right )\right )}{175104}-\frac {\left (2867499+82561 \sqrt {57}\right ) e^{\frac {1}{16} \left (-1-3 \sqrt {57}\right )} \text {Ei}\left (\frac {1}{16} \left (1+3 \sqrt {57}+16 x\right )\right )}{175104}-e^x \log \left (\frac {2 \left (2-x^2\right )}{5 \left (16-x-8 x^2\right )}\right )+\left (-39+23 \sqrt {2}\right ) \int \frac {e^x}{\sqrt {2}-x} \, dx-\left (-38+23 \sqrt {2}\right ) \int \frac {e^x}{\sqrt {2}-x} \, dx-\left (38+23 \sqrt {2}\right ) \int \frac {e^x}{\sqrt {2}+x} \, dx+\left (39+23 \sqrt {2}\right ) \int \frac {e^x}{\sqrt {2}+x} \, dx+\frac {1}{2} \left (1297-147 \sqrt {57}\right ) \int \frac {e^x}{1-3 \sqrt {57}+16 x} \, dx-\frac {1}{2} \left (3 \left (443-49 \sqrt {57}\right )\right ) \int \frac {e^x}{1-3 \sqrt {57}+16 x} \, dx-\frac {1}{2} \left (3 \left (443+49 \sqrt {57}\right )\right ) \int \frac {e^x}{1+3 \sqrt {57}+16 x} \, dx+\frac {1}{2} \left (1297+147 \sqrt {57}\right ) \int \frac {e^x}{1+3 \sqrt {57}+16 x} \, dx \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {e^x \left (-2+64 x+27 x^2-66 x^3-30 x^4+17 x^5+8 x^6\right )+e^x \left (-32+2 x+32 x^2-x^3-8 x^4\right ) \log \left (\frac {-4+2 x^2}{-80+5 x+40 x^2}\right )}{32-2 x-32 x^2+x^3+8 x^4} \, dx=e^x \left (x^2-\log \left (\frac {2 \left (-2+x^2\right )}{5 \left (-16+x+8 x^2\right )}\right )\right ) \]

[In]

Integrate[(E^x*(-2 + 64*x + 27*x^2 - 66*x^3 - 30*x^4 + 17*x^5 + 8*x^6) + E^x*(-32 + 2*x + 32*x^2 - x^3 - 8*x^4
)*Log[(-4 + 2*x^2)/(-80 + 5*x + 40*x^2)])/(32 - 2*x - 32*x^2 + x^3 + 8*x^4),x]

[Out]

E^x*(x^2 - Log[(2*(-2 + x^2))/(5*(-16 + x + 8*x^2))])

Maple [A] (verified)

Time = 4.53 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07

method	result	size
parallelrisch	\({\mathrm e}^{x} x^{2}-{\mathrm e}^{x} \ln \left (\frac {\frac {2 x^{2}}{5}-\frac {4}{5}}{8 x^{2}+x -16}\right )\)	\(30\)
risch	\({\mathrm e}^{x} \ln \left (x^{2}+\frac {1}{8} x -2\right )-\ln \left (x^{2}-2\right ) {\mathrm e}^{x}+\frac {i {\mathrm e}^{x} \pi \,\operatorname {csgn}\left (\frac {i}{x^{2}+\frac {1}{8} x -2}\right ) \operatorname {csgn}\left (i \left (x^{2}-2\right )\right ) \operatorname {csgn}\left (\frac {i \left (x^{2}-2\right )}{x^{2}+\frac {1}{8} x -2}\right )}{2}-\frac {i {\mathrm e}^{x} \pi \,\operatorname {csgn}\left (\frac {i}{x^{2}+\frac {1}{8} x -2}\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}-2\right )}{x^{2}+\frac {1}{8} x -2}\right )}^{2}}{2}-\frac {i {\mathrm e}^{x} \pi \,\operatorname {csgn}\left (i \left (x^{2}-2\right )\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}-2\right )}{x^{2}+\frac {1}{8} x -2}\right )}^{2}}{2}+\frac {i {\mathrm e}^{x} \pi {\operatorname {csgn}\left (\frac {i \left (x^{2}-2\right )}{x^{2}+\frac {1}{8} x -2}\right )}^{3}}{2}+{\mathrm e}^{x} x^{2}+2 \,{\mathrm e}^{x} \ln \left (2\right )+{\mathrm e}^{x} \ln \left (5\right )\)	\(193\)

[In]

int(((-8*x^4-x^3+32*x^2+2*x-32)*exp(x)*ln((2*x^2-4)/(40*x^2+5*x-80))+(8*x^6+17*x^5-30*x^4-66*x^3+27*x^2+64*x-2
)*exp(x))/(8*x^4+x^3-32*x^2-2*x+32),x,method=_RETURNVERBOSE)

[Out]

exp(x)*x^2-exp(x)*ln(2/5*(x^2-2)/(8*x^2+x-16))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {e^x \left (-2+64 x+27 x^2-66 x^3-30 x^4+17 x^5+8 x^6\right )+e^x \left (-32+2 x+32 x^2-x^3-8 x^4\right ) \log \left (\frac {-4+2 x^2}{-80+5 x+40 x^2}\right )}{32-2 x-32 x^2+x^3+8 x^4} \, dx=x^{2} e^{x} - e^{x} \log \left (\frac {2 \, {\left (x^{2} - 2\right )}}{5 \, {\left (8 \, x^{2} + x - 16\right )}}\right ) \]

[In]

integrate(((-8*x^4-x^3+32*x^2+2*x-32)*exp(x)*log((2*x^2-4)/(40*x^2+5*x-80))+(8*x^6+17*x^5-30*x^4-66*x^3+27*x^2
+64*x-2)*exp(x))/(8*x^4+x^3-32*x^2-2*x+32),x, algorithm="fricas")

[Out]

x^2*e^x - e^x*log(2/5*(x^2 - 2)/(8*x^2 + x - 16))

Sympy [A] (verification not implemented)

Time = 0.45 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {e^x \left (-2+64 x+27 x^2-66 x^3-30 x^4+17 x^5+8 x^6\right )+e^x \left (-32+2 x+32 x^2-x^3-8 x^4\right ) \log \left (\frac {-4+2 x^2}{-80+5 x+40 x^2}\right )}{32-2 x-32 x^2+x^3+8 x^4} \, dx=\left (x^{2} - \log {\left (\frac {2 x^{2} - 4}{40 x^{2} + 5 x - 80} \right )}\right ) e^{x} \]

[In]

integrate(((-8*x**4-x**3+32*x**2+2*x-32)*exp(x)*ln((2*x**2-4)/(40*x**2+5*x-80))+(8*x**6+17*x**5-30*x**4-66*x**
3+27*x**2+64*x-2)*exp(x))/(8*x**4+x**3-32*x**2-2*x+32),x)

[Out]

(x**2 - log((2*x**2 - 4)/(40*x**2 + 5*x - 80)))*exp(x)

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.29 \[ \int \frac {e^x \left (-2+64 x+27 x^2-66 x^3-30 x^4+17 x^5+8 x^6\right )+e^x \left (-32+2 x+32 x^2-x^3-8 x^4\right ) \log \left (\frac {-4+2 x^2}{-80+5 x+40 x^2}\right )}{32-2 x-32 x^2+x^3+8 x^4} \, dx={\left (x^{2} + \log \left (5\right ) - \log \left (2\right )\right )} e^{x} + e^{x} \log \left (8 \, x^{2} + x - 16\right ) - e^{x} \log \left (x^{2} - 2\right ) \]

[In]

integrate(((-8*x^4-x^3+32*x^2+2*x-32)*exp(x)*log((2*x^2-4)/(40*x^2+5*x-80))+(8*x^6+17*x^5-30*x^4-66*x^3+27*x^2
+64*x-2)*exp(x))/(8*x^4+x^3-32*x^2-2*x+32),x, algorithm="maxima")

[Out]

(x^2 + log(5) - log(2))*e^x + e^x*log(8*x^2 + x - 16) - e^x*log(x^2 - 2)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {e^x \left (-2+64 x+27 x^2-66 x^3-30 x^4+17 x^5+8 x^6\right )+e^x \left (-32+2 x+32 x^2-x^3-8 x^4\right ) \log \left (\frac {-4+2 x^2}{-80+5 x+40 x^2}\right )}{32-2 x-32 x^2+x^3+8 x^4} \, dx=x^{2} e^{x} - e^{x} \log \left (\frac {2 \, {\left (x^{2} - 2\right )}}{5 \, {\left (8 \, x^{2} + x - 16\right )}}\right ) \]

[In]

integrate(((-8*x^4-x^3+32*x^2+2*x-32)*exp(x)*log((2*x^2-4)/(40*x^2+5*x-80))+(8*x^6+17*x^5-30*x^4-66*x^3+27*x^2
+64*x-2)*exp(x))/(8*x^4+x^3-32*x^2-2*x+32),x, algorithm="giac")

[Out]

x^2*e^x - e^x*log(2/5*(x^2 - 2)/(8*x^2 + x - 16))

Mupad [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \frac {e^x \left (-2+64 x+27 x^2-66 x^3-30 x^4+17 x^5+8 x^6\right )+e^x \left (-32+2 x+32 x^2-x^3-8 x^4\right ) \log \left (\frac {-4+2 x^2}{-80+5 x+40 x^2}\right )}{32-2 x-32 x^2+x^3+8 x^4} \, dx=-{\mathrm {e}}^x\,\left (\ln \left (\frac {2\,x^2-4}{40\,x^2+5\,x-80}\right )-x^2\right ) \]

[In]

int((exp(x)*(64*x + 27*x^2 - 66*x^3 - 30*x^4 + 17*x^5 + 8*x^6 - 2) - log((2*x^2 - 4)/(5*x + 40*x^2 - 80))*exp(
x)*(x^3 - 32*x^2 - 2*x + 8*x^4 + 32))/(x^3 - 32*x^2 - 2*x + 8*x^4 + 32),x)

[Out]

-exp(x)*(log((2*x^2 - 4)/(5*x + 40*x^2 - 80)) - x^2)