3.28.29 \(\int \frac {-28 x^2-48 x^3-20 x^4+(i \pi +\log (5-e))^4 (-20-10 \log (\frac {4}{x^2}))+(-14 x^2-28 x^3-10 x^4) \log (\frac {4}{x^2})+(i \pi +\log (5-e))^3 (-80 x-40 x \log (\frac {4}{x^2}))+(i \pi +\log (5-e))^2 (-48 x-120 x^2+(-20 x-60 x^2) \log (\frac {4}{x^2}))+(i \pi +\log (5-e)) (-96 x^2-80 x^3+(-48 x^2-40 x^3) \log (\frac {4}{x^2}))}{5 x^4+10 x^5+5 x^6+(20 x^4+20 x^5) (i \pi +\log (5-e))+(10 x^3+30 x^4) (i \pi +\log (5-e))^2+20 x^3 (i \pi +\log (5-e))^3+5 x^2 (i \pi +\log (5-e))^4} \, dx\)

Optimal. Leaf size=36 \[ \frac {\left (2+\frac {4 x}{5 \left (x+(i \pi +x+\log (5-e))^2\right )}\right ) \log \left (\frac {4}{x^2}\right )}{x} \]

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Rubi [B]  time = 15.67, antiderivative size = 2688, normalized size of antiderivative = 74.67, number of steps used = 62, number of rules used = 19, integrand size = 281, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.068, Rules used = {6688, 12, 6742, 614, 618, 206, 638, 722, 740, 800, 634, 628, 822, 2357, 2304, 2314, 31, 2317, 2391}

result too large to display

Antiderivative was successfully verified.

[In]

Int[(-28*x^2 - 48*x^3 - 20*x^4 + (I*Pi + Log[5 - E])^4*(-20 - 10*Log[4/x^2]) + (-14*x^2 - 28*x^3 - 10*x^4)*Log
[4/x^2] + (I*Pi + Log[5 - E])^3*(-80*x - 40*x*Log[4/x^2]) + (I*Pi + Log[5 - E])^2*(-48*x - 120*x^2 + (-20*x -
60*x^2)*Log[4/x^2]) + (I*Pi + Log[5 - E])*(-96*x^2 - 80*x^3 + (-48*x^2 - 40*x^3)*Log[4/x^2]))/(5*x^4 + 10*x^5
+ 5*x^6 + (20*x^4 + 20*x^5)*(I*Pi + Log[5 - E]) + (10*x^3 + 30*x^4)*(I*Pi + Log[5 - E])^2 + 20*x^3*(I*Pi + Log
[5 - E])^3 + 5*x^2*(I*Pi + Log[5 - E])^4),x]

[Out]

-4/x - (112*ArcTanh[(1 + (2*I)*Pi + 2*x + 2*Log[5 - E])/Sqrt[1 + (4*I)*Pi + 4*Log[5 - E]]])/(5*(1 + (4*I)*Pi +
4*Log[5 - E])^(3/2)) + (16*ArcTanh[(1 + (2*I)*Pi + 2*x + 2*Log[5 - E])/Sqrt[1 + (4*I)*Pi + 4*Log[5 - E]]]*(Pi
- I*Log[5 - E])^2)/(1 + (4*I)*Pi + 4*Log[5 - E])^(3/2) - (32*ArcTanh[(1 + (2*I)*Pi + 2*x + 2*Log[5 - E])/Sqrt
[1 + (4*I)*Pi + 4*Log[5 - E]]]*(Pi - I*Log[5 - E])*(7*I + 10*Pi - (10*I)*Log[5 - E]))/(5*(1 + (4*I)*Pi + 4*Log
[5 - E])^(3/2)) + (96*ArcTanh[(1 + (2*I)*Pi + 2*x + 2*Log[5 - E])/Sqrt[1 + (4*I)*Pi + 4*Log[5 - E]]]*(1 + (2*I
)*Pi + 2*Log[5 - E]))/(5*(1 + (4*I)*Pi + 4*Log[5 - E])^(3/2)) + (8*(1 - Pi^2 + 4*Log[5 - E] + Log[5 - E]^2 + (
2*I)*Pi*(2 + Log[5 - E])))/(x*(1 + (4*I)*Pi + 4*Log[5 - E])) - (16*ArcTanh[(1 + (2*I)*Pi + 2*x + 2*Log[5 - E])
/Sqrt[1 + (4*I)*Pi + 4*Log[5 - E]]]*(1 + 2*Pi^2 + (4*I)*Pi*(1 - Log[5 - E]) + 4*Log[5 - E] - 2*Log[5 - E]^2)*(
2*Pi - I*(1 + 2*Log[5 - E])))/((Pi - I*Log[5 - E])*(1 + (4*I)*Pi + 4*Log[5 - E])^(3/2)) + (28*(1 + (2*I)*Pi +
2*x + 2*Log[5 - E]))/(5*(1 + (4*I)*Pi + 4*Log[5 - E])*(x^2 - (Pi - I*Log[5 - E])^2 + x*(1 + (2*I)*Pi + 2*Log[5
- E]))) + (48*(2*(Pi - I*Log[5 - E])^2 - x*(1 + (2*I)*Pi + 2*Log[5 - E])))/(5*(1 + (4*I)*Pi + 4*Log[5 - E])*(
x^2 - (Pi - I*Log[5 - E])^2 + x*(1 + (2*I)*Pi + 2*Log[5 - E]))) + (4*x*(2*(Pi - I*Log[5 - E])^2 - x*(1 + (2*I)
*Pi + 2*Log[5 - E])))/((1 + (4*I)*Pi + 4*Log[5 - E])*(x^2 - (Pi - I*Log[5 - E])^2 + x*(1 + (2*I)*Pi + 2*Log[5
- E]))) + (16*(Pi - I*Log[5 - E])*(2*(Pi - I*Log[5 - E])^2 + x*(1 + (2*I)*Pi + 2*Log[5 - E]) + (1 + (2*I)*Pi +
2*Log[5 - E])^2))/((I - 4*Pi + (4*I)*Log[5 - E])*(x^2 - (Pi - I*Log[5 - E])^2 + x*(1 + (2*I)*Pi + 2*Log[5 - E
]))) + (4*(Pi - I*Log[5 - E])^2*(2*(Pi - I*Log[5 - E])^2 + x*(1 + (2*I)*Pi + 2*Log[5 - E]) + (1 + (2*I)*Pi + 2
*Log[5 - E])^2))/(x*(1 + (4*I)*Pi + 4*Log[5 - E])*(x^2 - (Pi - I*Log[5 - E])^2 + x*(1 + (2*I)*Pi + 2*Log[5 - E
]))) - (24*(4*(Pi - I*Log[5 - E])^2 + 5*(Pi - I*Log[5 - E])^2*(1 + (2*I)*Pi + 2*Log[5 - E]) + 2*(1 + (2*I)*Pi
+ 2*Log[5 - E])^2 + 2*x*(1 + (2*I)*Pi + 5*(Pi - I*Log[5 - E])^2 + 2*Log[5 - E])))/(5*(1 + (4*I)*Pi + 4*Log[5 -
E])*(x^2 - (Pi - I*Log[5 - E])^2 + x*(1 + (2*I)*Pi + 2*Log[5 - E]))) - (8*ArcTanh[(1 + (2*I)*Pi + 2*x + 2*Log
[5 - E])/Sqrt[1 + (4*I)*Pi + 4*Log[5 - E]]]*(1 - 2*Pi^4 - (8*I)*Pi^3*(1 - Log[5 - E]) + 8*Log[5 - E] + 18*Log[
5 - E]^2 + 8*Log[5 - E]^3 - 2*Log[5 - E]^4 - 6*Pi^2*(3 + 4*Log[5 - E] - 2*Log[5 - E]^2) + (4*I)*Pi*(2 + 9*Log[
5 - E] + 6*Log[5 - E]^2 - 2*Log[5 - E]^3)))/((Pi - I*Log[5 - E])^2*(1 + (4*I)*Pi + 4*Log[5 - E])^(3/2)) + (48*
ArcTanh[(1 + (2*I)*Pi + 2*x + 2*Log[5 - E])/Sqrt[1 + (4*I)*Pi + 4*Log[5 - E]]]*(1 + 10*Pi^4 + (4*I)*Pi^3*(1 -
10*Log[5 - E]) + 6*Log[5 - E] + 6*Log[5 - E]^2 - 4*Log[5 - E]^3 + 10*Log[5 - E]^4 - 6*Pi^2*(1 - 2*Log[5 - E] +
10*Log[5 - E]^2) + (2*I)*Pi*(3 + 6*Log[5 - E] - 6*Log[5 - E]^2 + 20*Log[5 - E]^3)))/(5*(Pi - I*Log[5 - E])^2*
(1 + (4*I)*Pi + 4*Log[5 - E])^(3/2)) - (16*(Pi - I*Log[5 - E])*(x*(7 - (10*I)*Pi - 10*Log[5 - E]) + 2*(5*(Pi -
I*Log[5 - E])^2 + 3*(1 + (2*I)*Pi + 2*Log[5 - E]))))/(5*(I - 4*Pi + (4*I)*Log[5 - E])*(x^2 - (Pi - I*Log[5 -
E])^2 + x*(1 + (2*I)*Pi + 2*Log[5 - E]))) + (2*Log[4/x^2])/x + (16*x*Log[4/x^2])/(5*(1 + (4*I)*Pi + 4*Log[5 -
E])*(1 + (2*I)*Pi + 2*x + 2*Log[5 - E] - Sqrt[1 + (4*I)*Pi + 4*Log[5 - E]])) - (16*x*(1 + (2*I)*Pi + 2*Log[5 -
E])*Log[4/x^2])/(5*(1 + (4*I)*Pi + 4*Log[5 - E])*(1 + (2*I)*Pi + 2*Log[5 - E] - Sqrt[1 + (4*I)*Pi + 4*Log[5 -
E]])*(1 + (2*I)*Pi + 2*x + 2*Log[5 - E] - Sqrt[1 + (4*I)*Pi + 4*Log[5 - E]])) + (16*x*Log[4/x^2])/(5*(1 + (4*
I)*Pi + 4*Log[5 - E])*(1 + (2*I)*Pi + 2*x + 2*Log[5 - E] + Sqrt[1 + (4*I)*Pi + 4*Log[5 - E]])) - (16*x*(I - 2*
Pi + (2*I)*Log[5 - E])*Log[4/x^2])/(5*(I - 4*Pi + (4*I)*Log[5 - E])*(1 + (2*I)*Pi + 2*Log[5 - E] + Sqrt[1 + (4
*I)*Pi + 4*Log[5 - E]])*(1 + (2*I)*Pi + 2*x + 2*Log[5 - E] + Sqrt[1 + (4*I)*Pi + 4*Log[5 - E]])) + (48*Log[x])
/(5*(Pi - I*Log[5 - E])^2) - (16*Log[x])/(I*Pi + Log[5 - E]) - (8*(1 + (2*I)*Pi + 2*Log[5 - E])*Log[x])/(Pi -
I*Log[5 - E])^2 - (24*Log[x^2 - (Pi - I*Log[5 - E])^2 + x*(1 + (2*I)*Pi + 2*Log[5 - E])])/(5*(Pi - I*Log[5 - E
])^2) + (8*Log[x^2 - (Pi - I*Log[5 - E])^2 + x*(1 + (2*I)*Pi + 2*Log[5 - E])])/(I*Pi + Log[5 - E]) + (4*(1 + (
2*I)*Pi + 2*Log[5 - E])*Log[x^2 - (Pi - I*Log[5 - E])^2 + x*(1 + (2*I)*Pi + 2*Log[5 - E])])/(Pi - I*Log[5 - E]
)^2 + (16*Log[1 + (2*I)*Pi + 2*x + 2*Log[5 - E] - Sqrt[1 + (4*I)*Pi + 4*Log[5 - E]]])/(5*(1 + (4*I)*Pi + 4*Log
[5 - E])) - (16*(1 + (2*I)*Pi + 2*Log[5 - E])*Log[1 + (2*I)*Pi + 2*x + 2*Log[5 - E] - Sqrt[1 + (4*I)*Pi + 4*Lo
g[5 - E]]])/(5*(1 + (4*I)*Pi + 4*Log[5 - E])*(1 + (2*I)*Pi + 2*Log[5 - E] - Sqrt[1 + (4*I)*Pi + 4*Log[5 - E]])
) + (16*Log[1 + (2*I)*Pi + 2*x + 2*Log[5 - E] + Sqrt[1 + (4*I)*Pi + 4*Log[5 - E]]])/(5*(1 + (4*I)*Pi + 4*Log[5
- E])) - (16*(I - 2*Pi + (2*I)*Log[5 - E])*Log[1 + (2*I)*Pi + 2*x + 2*Log[5 - E] + Sqrt[1 + (4*I)*Pi + 4*Log[
5 - E]]])/(5*(I - 4*Pi + (4*I)*Log[5 - E])*(1 + (2*I)*Pi + 2*Log[5 - E] + Sqrt[1 + (4*I)*Pi + 4*Log[5 - E]]))

Rule 12

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

Rule 31

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

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 614

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^(p + 1))/((p +
1)*(b^2 - 4*a*c)), x] - Dist[(2*c*(2*p + 3))/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[4*p]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 638

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b*d - 2*a*e + (2*c*d -
b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] - Dist[((2*p + 3)*(2*c*d - b*e))/((p + 1)*(b^2
- 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^
2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2]

Rule 722

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m - 1)*(
d*b - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] - Dist[(2*(2*p + 3)*(c*d
^2 - b*d*e + a*e^2))/((p + 1)*(b^2 - 4*a*c)), Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ
[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m +
2*p + 2, 0] && LtQ[p, -1]

Rule 740

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e
+ a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m
+ 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x
, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b
*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 800

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[((d + e*x)^m*(f + g*x))/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

Rule 822

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x
)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b,
c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] ||
IntegerQ[p] || IntegersQ[2*m, 2*p])

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 2317

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(Log[1 + (e*x)/d]*(a +
b*Log[c*x^n])^p)/e, x] - Dist[(b*n*p)/e, Int[(Log[1 + (e*x)/d]*(a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[
{a, b, c, d, e, n}, x] && IGtQ[p, 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 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, 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 {-28 x^2-48 x^3-20 x^4-2 x^2 \left (7+14 x+5 x^2\right ) \log \left (\frac {4}{x^2}\right )-10 (\pi -i \log (5-e))^4 \left (2+\log \left (\frac {4}{x^2}\right )\right )-8 x^2 (6+5 x) (i \pi +\log (5-e)) \left (2+\log \left (\frac {4}{x^2}\right )\right )-40 x (i \pi +\log (5-e))^3 \left (2+\log \left (\frac {4}{x^2}\right )\right )-4 x (i \pi +\log (5-e))^2 \left (6 (2+5 x)+5 (1+3 x) \log \left (\frac {4}{x^2}\right )\right )}{5 x^2 \left (x^2-(\pi -i \log (5-e))^2+x (1+2 i \pi +2 \log (5-e))\right )^2} \, dx\\ &=\frac {1}{5} \int \frac {-28 x^2-48 x^3-20 x^4-2 x^2 \left (7+14 x+5 x^2\right ) \log \left (\frac {4}{x^2}\right )-10 (\pi -i \log (5-e))^4 \left (2+\log \left (\frac {4}{x^2}\right )\right )-8 x^2 (6+5 x) (i \pi +\log (5-e)) \left (2+\log \left (\frac {4}{x^2}\right )\right )-40 x (i \pi +\log (5-e))^3 \left (2+\log \left (\frac {4}{x^2}\right )\right )-4 x (i \pi +\log (5-e))^2 \left (6 (2+5 x)+5 (1+3 x) \log \left (\frac {4}{x^2}\right )\right )}{x^2 \left (x^2-(\pi -i \log (5-e))^2+x (1+2 i \pi +2 \log (5-e))\right )^2} \, dx\\ &=\frac {1}{5} \int \left (-\frac {28}{\left (x^2-(\pi -i \log (5-e))^2+x (1+2 i \pi +2 \log (5-e))\right )^2}-\frac {48 x}{\left (x^2-(\pi -i \log (5-e))^2+x (1+2 i \pi +2 \log (5-e))\right )^2}-\frac {20 x^2}{\left (x^2-(\pi -i \log (5-e))^2+x (1+2 i \pi +2 \log (5-e))\right )^2}+\frac {80 (-i \pi -\log (5-e))^3}{x \left (x^2-(\pi -i \log (5-e))^2+x (1+2 i \pi +2 \log (5-e))\right )^2}-\frac {16 i (6+5 x) (\pi -i \log (5-e))}{\left (x^2-(\pi -i \log (5-e))^2+x (1+2 i \pi +2 \log (5-e))\right )^2}+\frac {24 (2+5 x) (\pi -i \log (5-e))^2}{x \left (x^2-(\pi -i \log (5-e))^2+x (1+2 i \pi +2 \log (5-e))\right )^2}-\frac {20 (\pi -i \log (5-e))^4}{x^2 \left (x^2-(\pi -i \log (5-e))^2+x (1+2 i \pi +2 \log (5-e))\right )^2}+\frac {2 \left (-5 x^4+10 x (1+i (2 \pi -2 i \log (5-e))) (\pi -i \log (5-e))^2-5 (\pi -i \log (5-e))^4-2 x^3 (7+10 i \pi +10 \log (5-e))-x^2 \left (7-30 \pi ^2+24 \log (5-e)+30 \log ^2(5-e)+12 i \pi (2+5 \log (5-e))\right )\right ) \log \left (\frac {4}{x^2}\right )}{x^2 \left (x^2-(\pi -i \log (5-e))^2+x (1+2 i \pi +2 \log (5-e))\right )^2}\right ) \, dx\\ &=\frac {2}{5} \int \frac {\left (-5 x^4+10 x (1+i (2 \pi -2 i \log (5-e))) (\pi -i \log (5-e))^2-5 (\pi -i \log (5-e))^4-2 x^3 (7+10 i \pi +10 \log (5-e))-x^2 \left (7-30 \pi ^2+24 \log (5-e)+30 \log ^2(5-e)+12 i \pi (2+5 \log (5-e))\right )\right ) \log \left (\frac {4}{x^2}\right )}{x^2 \left (x^2-(\pi -i \log (5-e))^2+x (1+2 i \pi +2 \log (5-e))\right )^2} \, dx-4 \int \frac {x^2}{\left (x^2-(\pi -i \log (5-e))^2+x (1+2 i \pi +2 \log (5-e))\right )^2} \, dx-\frac {28}{5} \int \frac {1}{\left (x^2-(\pi -i \log (5-e))^2+x (1+2 i \pi +2 \log (5-e))\right )^2} \, dx-\frac {48}{5} \int \frac {x}{\left (x^2-(\pi -i \log (5-e))^2+x (1+2 i \pi +2 \log (5-e))\right )^2} \, dx+\frac {1}{5} \left (24 (\pi -i \log (5-e))^2\right ) \int \frac {2+5 x}{x \left (x^2-(\pi -i \log (5-e))^2+x (1+2 i \pi +2 \log (5-e))\right )^2} \, dx-\left (4 (\pi -i \log (5-e))^4\right ) \int \frac {1}{x^2 \left (x^2-(\pi -i \log (5-e))^2+x (1+2 i \pi +2 \log (5-e))\right )^2} \, dx-\frac {1}{5} (16 (i \pi +\log (5-e))) \int \frac {6+5 x}{\left (x^2-(\pi -i \log (5-e))^2+x (1+2 i \pi +2 \log (5-e))\right )^2} \, dx-\left (16 (i \pi +\log (5-e))^3\right ) \int \frac {1}{x \left (x^2-(\pi -i \log (5-e))^2+x (1+2 i \pi +2 \log (5-e))\right )^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.25, size = 101, normalized size = 2.81 \begin {gather*} \frac {2 \left (-5 \pi ^2+5 x^2+5 \log ^2(5-e)+10 i \pi (x+\log (5-e))+x (7+10 \log (5-e))\right ) \log \left (\frac {4}{x^2}\right )}{5 x \left (-\pi ^2+x+x^2+2 x \log (5-e)+\log ^2(5-e)+2 i \pi (x+\log (5-e))\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-28*x^2 - 48*x^3 - 20*x^4 + (I*Pi + Log[5 - E])^4*(-20 - 10*Log[4/x^2]) + (-14*x^2 - 28*x^3 - 10*x^
4)*Log[4/x^2] + (I*Pi + Log[5 - E])^3*(-80*x - 40*x*Log[4/x^2]) + (I*Pi + Log[5 - E])^2*(-48*x - 120*x^2 + (-2
0*x - 60*x^2)*Log[4/x^2]) + (I*Pi + Log[5 - E])*(-96*x^2 - 80*x^3 + (-48*x^2 - 40*x^3)*Log[4/x^2]))/(5*x^4 + 1
0*x^5 + 5*x^6 + (20*x^4 + 20*x^5)*(I*Pi + Log[5 - E]) + (10*x^3 + 30*x^4)*(I*Pi + Log[5 - E])^2 + 20*x^3*(I*Pi
+ Log[5 - E])^3 + 5*x^2*(I*Pi + Log[5 - E])^4),x]

[Out]

(2*(-5*Pi^2 + 5*x^2 + 5*Log[5 - E]^2 + (10*I)*Pi*(x + Log[5 - E]) + x*(7 + 10*Log[5 - E]))*Log[4/x^2])/(5*x*(-
Pi^2 + x + x^2 + 2*x*Log[5 - E] + Log[5 - E]^2 + (2*I)*Pi*(x + Log[5 - E])))

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fricas [B]  time = 0.75, size = 76, normalized size = 2.11 \begin {gather*} \frac {2 \, {\left (10 \, x \log \left (\frac {4}{x^{2}}\right ) \log \left (e - 5\right ) + 5 \, \log \left (\frac {4}{x^{2}}\right ) \log \left (e - 5\right )^{2} + {\left (5 \, x^{2} + 7 \, x\right )} \log \left (\frac {4}{x^{2}}\right )\right )}}{5 \, {\left (x^{3} + 2 \, x^{2} \log \left (e - 5\right ) + x \log \left (e - 5\right )^{2} + x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*log(4/x^2)-20)*log(exp(1)-5)^4+(-40*x*log(4/x^2)-80*x)*log(exp(1)-5)^3+((-60*x^2-20*x)*log(4/x
^2)-120*x^2-48*x)*log(exp(1)-5)^2+((-40*x^3-48*x^2)*log(4/x^2)-80*x^3-96*x^2)*log(exp(1)-5)+(-10*x^4-28*x^3-14
*x^2)*log(4/x^2)-20*x^4-48*x^3-28*x^2)/(5*x^2*log(exp(1)-5)^4+20*x^3*log(exp(1)-5)^3+(30*x^4+10*x^3)*log(exp(1
)-5)^2+(20*x^5+20*x^4)*log(exp(1)-5)+5*x^6+10*x^5+5*x^4),x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.48, size = 105, normalized size = 2.92 \begin {gather*} \frac {2 \, {\left (10 \, x^{2} \log \relax (2) - 5 \, x^{2} \log \left (x^{2}\right ) + 20 \, x \log \relax (2) \log \left (e - 5\right ) - 10 \, x \log \left (x^{2}\right ) \log \left (e - 5\right ) + 10 \, \log \relax (2) \log \left (e - 5\right )^{2} - 5 \, \log \left (x^{2}\right ) \log \left (e - 5\right )^{2} + 14 \, x \log \relax (2) - 7 \, x \log \left (x^{2}\right )\right )}}{5 \, {\left (x^{3} + 2 \, x^{2} \log \left (e - 5\right ) + x \log \left (e - 5\right )^{2} + x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*log(4/x^2)-20)*log(exp(1)-5)^4+(-40*x*log(4/x^2)-80*x)*log(exp(1)-5)^3+((-60*x^2-20*x)*log(4/x
^2)-120*x^2-48*x)*log(exp(1)-5)^2+((-40*x^3-48*x^2)*log(4/x^2)-80*x^3-96*x^2)*log(exp(1)-5)+(-10*x^4-28*x^3-14
*x^2)*log(4/x^2)-20*x^4-48*x^3-28*x^2)/(5*x^2*log(exp(1)-5)^4+20*x^3*log(exp(1)-5)^3+(30*x^4+10*x^3)*log(exp(1
)-5)^2+(20*x^5+20*x^4)*log(exp(1)-5)+5*x^6+10*x^5+5*x^4),x, algorithm="giac")

[Out]

2/5*(10*x^2*log(2) - 5*x^2*log(x^2) + 20*x*log(2)*log(e - 5) - 10*x*log(x^2)*log(e - 5) + 10*log(2)*log(e - 5)
^2 - 5*log(x^2)*log(e - 5)^2 + 14*x*log(2) - 7*x*log(x^2))/(x^3 + 2*x^2*log(e - 5) + x*log(e - 5)^2 + x^2)

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maple [B]  time = 1.77, size = 60, normalized size = 1.67

method	result	size
risch	\(\frac {2 \left (5 \ln \left ({\mathrm e}-5\right )^{2}+10 x \ln \left ({\mathrm e}-5\right )+5 x^{2}+7 x \right ) \ln \left (\frac {4}{x^{2}}\right )}{5 \left (\ln \left ({\mathrm e}-5\right )^{2}+2 x \ln \left ({\mathrm e}-5\right )+x^{2}+x \right ) x}\)	\(60\)
norman	\(\frac {2 x^{2} \ln \left (\frac {4}{x^{2}}\right )+\left (\frac {14}{5}+4 \ln \left ({\mathrm e}-5\right )\right ) x \ln \left (\frac {4}{x^{2}}\right )+2 \ln \left ({\mathrm e}-5\right )^{2} \ln \left (\frac {4}{x^{2}}\right )}{x \left (\ln \left ({\mathrm e}-5\right )^{2}+2 x \ln \left ({\mathrm e}-5\right )+x^{2}+x \right )}\)	\(71\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-10*ln(4/x^2)-20)*ln(exp(1)-5)^4+(-40*x*ln(4/x^2)-80*x)*ln(exp(1)-5)^3+((-60*x^2-20*x)*ln(4/x^2)-120*x^2
-48*x)*ln(exp(1)-5)^2+((-40*x^3-48*x^2)*ln(4/x^2)-80*x^3-96*x^2)*ln(exp(1)-5)+(-10*x^4-28*x^3-14*x^2)*ln(4/x^2
)-20*x^4-48*x^3-28*x^2)/(5*x^2*ln(exp(1)-5)^4+20*x^3*ln(exp(1)-5)^3+(30*x^4+10*x^3)*ln(exp(1)-5)^2+(20*x^5+20*
x^4)*ln(exp(1)-5)+5*x^6+10*x^5+5*x^4),x,method=_RETURNVERBOSE)

[Out]

2/5*(5*ln(exp(1)-5)^2+10*x*ln(exp(1)-5)+5*x^2+7*x)/(ln(exp(1)-5)^2+2*x*ln(exp(1)-5)+x^2+x)/x*ln(4/x^2)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*log(4/x^2)-20)*log(exp(1)-5)^4+(-40*x*log(4/x^2)-80*x)*log(exp(1)-5)^3+((-60*x^2-20*x)*log(4/x
^2)-120*x^2-48*x)*log(exp(1)-5)^2+((-40*x^3-48*x^2)*log(4/x^2)-80*x^3-96*x^2)*log(exp(1)-5)+(-10*x^4-28*x^3-14
*x^2)*log(4/x^2)-20*x^4-48*x^3-28*x^2)/(5*x^2*log(exp(1)-5)^4+20*x^3*log(exp(1)-5)^3+(30*x^4+10*x^3)*log(exp(1
)-5)^2+(20*x^5+20*x^4)*log(exp(1)-5)+5*x^6+10*x^5+5*x^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*log(%e-5)+1>0)', see `assume
?` for more

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mupad [B]  time = 2.95, size = 49, normalized size = 1.36 \begin {gather*} \frac {2\,\ln \left (\frac {4}{x^2}\right )}{x}+\frac {4\,\ln \left (\frac {4}{x^2}\right )}{5\,\left (x^2+\left (2\,\ln \left (\mathrm {e}-5\right )+1\right )\,x+{\ln \left (\mathrm {e}-5\right )}^2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(4/x^2)*(14*x^2 + 28*x^3 + 10*x^4) + log(exp(1) - 5)*(log(4/x^2)*(48*x^2 + 40*x^3) + 96*x^2 + 80*x^3)
+ log(exp(1) - 5)^4*(10*log(4/x^2) + 20) + log(exp(1) - 5)^3*(80*x + 40*x*log(4/x^2)) + log(exp(1) - 5)^2*(48
*x + log(4/x^2)*(20*x + 60*x^2) + 120*x^2) + 28*x^2 + 48*x^3 + 20*x^4)/(log(exp(1) - 5)*(20*x^4 + 20*x^5) + lo
g(exp(1) - 5)^2*(10*x^3 + 30*x^4) + 5*x^2*log(exp(1) - 5)^4 + 20*x^3*log(exp(1) - 5)^3 + 5*x^4 + 10*x^5 + 5*x^
6),x)

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*ln(4/x**2)-20)*ln(exp(1)-5)**4+(-40*x*ln(4/x**2)-80*x)*ln(exp(1)-5)**3+((-60*x**2-20*x)*ln(4/x
**2)-120*x**2-48*x)*ln(exp(1)-5)**2+((-40*x**3-48*x**2)*ln(4/x**2)-80*x**3-96*x**2)*ln(exp(1)-5)+(-10*x**4-28*
x**3-14*x**2)*ln(4/x**2)-20*x**4-48*x**3-28*x**2)/(5*x**2*ln(exp(1)-5)**4+20*x**3*ln(exp(1)-5)**3+(30*x**4+10*
x**3)*ln(exp(1)-5)**2+(20*x**5+20*x**4)*ln(exp(1)-5)+5*x**6+10*x**5+5*x**4),x)

[Out]

Timed out

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