3.73.15 \(\int \frac {10+13 x+7 x^2+e (-10-8 x+2 x^2-2 x^3)+(-2-3 x-2 x^2+e (2+2 x)) \log (4)+(7 x+4 x^2+x^3+e (-2 x-4 x^2)-x \log (4)) \log (2 x)}{200 x-80 x^2-72 x^3+16 x^4+8 x^5+(-80 x+16 x^2+16 x^3) \log (4)+8 x \log ^2(4)} \, dx\)

Optimal. Leaf size=28 \[ \frac {\left (-1+e-\frac {x}{2}\right ) (x+\log (2 x))}{4 \left (-5+x+x^2+\log (4)\right )} \]

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Rubi [C]  time = 3.00, antiderivative size = 1473, normalized size of antiderivative = 52.61, number of steps used = 62, number of rules used = 20, integrand size = 131, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.153, Rules used = {6, 6741, 12, 6742, 1646, 800, 634, 618, 206, 628, 614, 740, 638, 2357, 2316, 2315, 2317, 2391, 2314, 31}

result too large to display

Antiderivative was successfully verified.

[In]

Int[(10 + 13*x + 7*x^2 + E*(-10 - 8*x + 2*x^2 - 2*x^3) + (-2 - 3*x - 2*x^2 + E*(2 + 2*x))*Log[4] + (7*x + 4*x^
2 + x^3 + E*(-2*x - 4*x^2) - x*Log[4])*Log[2*x])/(200*x - 80*x^2 - 72*x^3 + 16*x^4 + 8*x^5 + (-80*x + 16*x^2 +
16*x^3)*Log[4] + 8*x*Log[4]^2),x]

[Out]

-1/4*(E*ArcTanh[(1 + 2*x)/Sqrt[21 - 4*Log[4]]]*(10*Log[4]*(11 - Log[16]) - 2*Log[4]^2*(11 - Log[16]) + 5*(21 -
Log[256])))/((21 - 4*Log[4])^(3/2)*(5 - Log[4])^2) + (19*ArcTanh[(1 + 2*x)/Sqrt[21 - 4*Log[4]]])/(4*(21 - Log
[256])^(3/2)) + (13*(1 + 2*x))/(8*(5 - x - x^2 - Log[4])*(21 - Log[256])) - (Log[4]*(137 - E*(32 - 6*Log[4]) -
47*Log[4] + 4*Log[4]^2 + 2*(1 - E)*x*(11 - Log[16])))/(8*(5 - Log[4])*(5 - x - x^2 - Log[4])*(21 - Log[256]))
+ (7*(10 - x - Log[16]))/(8*(5 - x - x^2 - Log[4])*(21 - Log[256])) + (5*(11 + x - Log[16]))/(4*(5 - Log[4])*
(5 - x - x^2 - Log[4])*(21 - Log[256])) - (E*(x*(105 - 30*Log[4] + 2*Log[4]^2) + Log[4]*(16 - Log[64])))/(4*(5
- Log[4])*(5 - x - x^2 - Log[4])*(21 - Log[256])) - ((1 - E)*ArcTanh[(1 + 2*x)/Sqrt[21 - 4*Log[4]]]*Log[4]*(1
31 - 22*Log[4] - 10*Log[16] + Log[16]^2 - Log[256]))/(4*(21 - 4*Log[4])^(3/2)*(5 - Log[4])^2) + (5*ArcTanh[(1
+ 2*x)/Sqrt[21 - 4*Log[4]]]*(31 - Log[4096]))/(4*(21 - 4*Log[4])^(3/2)*(5 - Log[4])^2) + (5*Log[x])/(4*(5 - Lo
g[4])^2) - (5*E*Log[x])/(4*(5 - Log[4])^2) - ((1 - E)*Log[4]*Log[x])/(4*(5 - Log[4])^2) - ((3 - 4*E)*x*Log[2*x
])/(4*(1 + 2*x - Sqrt[21 - 4*Log[4]])*(21 - Log[256])) - ((3 - 4*E)*x*Log[2*x])/(4*(1 + 2*x + Sqrt[21 - 4*Log[
4]])*(21 - Log[256])) + (x*(6 - E - Log[4])*Log[2*x])/((1 - Sqrt[21 - 4*Log[4]])*(1 + 2*x - Sqrt[21 - 4*Log[4]
])*(21 - Log[256])) + (x*(6 - E - Log[4])*Log[2*x])/((1 + Sqrt[21 - 4*Log[4]])*(1 + 2*x + Sqrt[21 - 4*Log[4]])
*(21 - Log[256])) - ((3 - 4*E)*Log[2*x]*Log[1 + (2*x)/(1 + Sqrt[21 - 4*Log[4]])])/(8*(21 - 4*Log[4])^(3/2)) -
(Log[2*x]*Log[1 + (2*x)/(1 + Sqrt[21 - 4*Log[4]])])/(8*Sqrt[21 - 4*Log[4]]) + ((6 - E - Log[4])*Log[2*x]*Log[1
+ (2*x)/(1 + Sqrt[21 - 4*Log[4]])])/(2*(21 - Log[256])^(3/2)) + ((3 - 4*E)*Log[1 + 2*x - Sqrt[21 - 4*Log[4]]]
)/(8*(21 - Log[256])) - ((6 - E - Log[4])*Log[1 + 2*x - Sqrt[21 - 4*Log[4]]])/(2*(1 - Sqrt[21 - 4*Log[4]])*(21
- Log[256])) + ((3 - 4*E)*Log[-1 + Sqrt[21 - 4*Log[4]]]*Log[-1 - 2*x + Sqrt[21 - 4*Log[4]]])/(8*(21 - 4*Log[4
])^(3/2)) + (Log[-1 + Sqrt[21 - 4*Log[4]]]*Log[-1 - 2*x + Sqrt[21 - 4*Log[4]]])/(8*Sqrt[21 - 4*Log[4]]) - ((6
- E - Log[4])*Log[-1 + Sqrt[21 - 4*Log[4]]]*Log[-1 - 2*x + Sqrt[21 - 4*Log[4]]])/(2*(21 - 4*Log[4])^(3/2)) + (
(3 - 4*E)*Log[1 + 2*x + Sqrt[21 - 4*Log[4]]])/(8*(21 - Log[256])) - ((6 - E - Log[4])*Log[1 + 2*x + Sqrt[21 -
4*Log[4]]])/(2*(1 + Sqrt[21 - 4*Log[4]])*(21 - Log[256])) - (5*Log[5 - x - x^2 - Log[4]])/(8*(5 - Log[4])^2) +
(5*E*Log[5 - x - x^2 - Log[4]])/(8*(5 - Log[4])^2) + ((1 - E)*Log[4]*Log[5 - x - x^2 - Log[4]])/(8*(5 - Log[4
])^2) - ((3 - 4*E)*PolyLog[2, 1 + (2*x)/(1 - Sqrt[21 - 4*Log[4]])])/(8*(21 - 4*Log[4])^(3/2)) - PolyLog[2, 1 +
(2*x)/(1 - Sqrt[21 - 4*Log[4]])]/(8*Sqrt[21 - 4*Log[4]]) + ((6 - E - Log[4])*PolyLog[2, 1 + (2*x)/(1 - Sqrt[2
1 - 4*Log[4]])])/(2*(21 - Log[256])^(3/2)) - ((3 - 4*E)*PolyLog[2, (-2*x)/(1 + Sqrt[21 - 4*Log[4]])])/(8*(21 -
4*Log[4])^(3/2)) - PolyLog[2, (-2*x)/(1 + Sqrt[21 - 4*Log[4]])]/(8*Sqrt[21 - 4*Log[4]]) + ((6 - E - Log[4])*P
olyLog[2, (-2*x)/(1 + Sqrt[21 - 4*Log[4]])])/(2*(21 - Log[256])^(3/2))

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

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 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 1646

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = Polynomi
alQuotient[(d + e*x)^m*Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2,
x], x, 0], g = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2, x], x, 1]}, Simp[((b*f - 2*a*g + (2
*c*f - b*g)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(d
+ e*x)^m*(a + b*x + c*x^2)^(p + 1)*ExpandToSum[((p + 1)*(b^2 - 4*a*c)*Q)/(d + e*x)^m - ((2*p + 3)*(2*c*f - b*
g))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && LtQ[p, -1] && ILtQ[m, 0]

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 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rule 2316

Int[((a_.) + Log[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[((a + b*Log[-((c*d)/e)])*Log[d + e*
x])/e, x] + Dist[b, Int[Log[-((e*x)/d)]/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e}, x] && GtQ[-((c*d)/e), 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 6741

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

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 {10+13 x+7 x^2+e \left (-10-8 x+2 x^2-2 x^3\right )+\left (-2-3 x-2 x^2+e (2+2 x)\right ) \log (4)+\left (7 x+4 x^2+x^3+e \left (-2 x-4 x^2\right )-x \log (4)\right ) \log (2 x)}{-80 x^2-72 x^3+16 x^4+8 x^5+\left (-80 x+16 x^2+16 x^3\right ) \log (4)+x \left (200+8 \log ^2(4)\right )} \, dx\\ &=\int \frac {10+13 x+7 x^2+e \left (-10-8 x+2 x^2-2 x^3\right )+\left (-2-3 x-2 x^2+e (2+2 x)\right ) \log (4)+\left (7 x+4 x^2+x^3+e \left (-2 x-4 x^2\right )-x \log (4)\right ) \log (2 x)}{8 x \left (5-x-x^2-\log (4)\right )^2} \, dx\\ &=\frac {1}{8} \int \frac {10+13 x+7 x^2+e \left (-10-8 x+2 x^2-2 x^3\right )+\left (-2-3 x-2 x^2+e (2+2 x)\right ) \log (4)+\left (7 x+4 x^2+x^3+e \left (-2 x-4 x^2\right )-x \log (4)\right ) \log (2 x)}{x \left (5-x-x^2-\log (4)\right )^2} \, dx\\ &=\frac {1}{8} \int \left (\frac {\left (-2 (1-e)-(3-2 e) x-2 x^2\right ) \log (4)}{x \left (5-x-x^2-\log (4)\right )^2}+\frac {13}{\left (-5+x+x^2+\log (4)\right )^2}+\frac {10}{x \left (-5+x+x^2+\log (4)\right )^2}+\frac {7 x}{\left (-5+x+x^2+\log (4)\right )^2}-\frac {2 e \left (5+4 x-x^2+x^3\right )}{x \left (-5+x+x^2+\log (4)\right )^2}+\frac {\left (7-2 e+4 (1-e) x+x^2-\log (4)\right ) \log (2 x)}{\left (5-x-x^2-\log (4)\right )^2}\right ) \, dx\\ &=\frac {1}{8} \int \frac {\left (7-2 e+4 (1-e) x+x^2-\log (4)\right ) \log (2 x)}{\left (5-x-x^2-\log (4)\right )^2} \, dx+\frac {7}{8} \int \frac {x}{\left (-5+x+x^2+\log (4)\right )^2} \, dx+\frac {5}{4} \int \frac {1}{x \left (-5+x+x^2+\log (4)\right )^2} \, dx+\frac {13}{8} \int \frac {1}{\left (-5+x+x^2+\log (4)\right )^2} \, dx-\frac {1}{4} e \int \frac {5+4 x-x^2+x^3}{x \left (-5+x+x^2+\log (4)\right )^2} \, dx+\frac {1}{8} \log (4) \int \frac {-2 (1-e)-(3-2 e) x-2 x^2}{x \left (5-x-x^2-\log (4)\right )^2} \, dx\\ &=\frac {13 (1+2 x)}{8 \left (5-x-x^2-\log (4)\right ) (21-\log (256))}-\frac {\log (4) \left (137-e (32-6 \log (4))-47 \log (4)+4 \log ^2(4)+2 (1-e) x (11-\log (16))\right )}{8 (5-\log (4)) \left (5-x-x^2-\log (4)\right ) (21-\log (256))}+\frac {7 (10-x-\log (16))}{8 \left (5-x-x^2-\log (4)\right ) (21-\log (256))}+\frac {5 (11+x-\log (16))}{4 (5-\log (4)) \left (5-x-x^2-\log (4)\right ) (21-\log (256))}-\frac {e \left (x \left (105-30 \log (4)+2 \log ^2(4)\right )+\log (4) (16-\log (64))\right )}{4 (5-\log (4)) \left (5-x-x^2-\log (4)\right ) (21-\log (256))}+\frac {1}{8} \int \left (\frac {((3-4 e) x+2 (6-e-\log (4))) \log (2 x)}{\left (5-x-x^2-\log (4)\right )^2}+\frac {\log (2 x)}{-5+x+x^2+\log (4)}\right ) \, dx+\frac {7 \int \frac {1}{-5+x+x^2+\log (4)} \, dx}{8 (21-\log (256))}-\frac {13 \int \frac {1}{-5+x+x^2+\log (4)} \, dx}{4 (21-\log (256))}+\frac {e \int \frac {\frac {x \log (4) (11-\log (16))}{5-\log (4)}+\frac {5 (21-\log (256))}{5-\log (4)}}{x \left (-5+x+x^2+\log (4)\right )} \, dx}{4 (21-\log (256))}+\frac {5 \int \frac {-21-x+\log (256)}{x \left (-5+x+x^2+\log (4)\right )} \, dx}{4 (5-\log (4)) (21-\log (256))}-\frac {\log (4) \int \frac {\frac {2 (1-e) x (11-\log (16))}{5-\log (4)}+\frac {2 (1-e) (21-\log (256))}{5-\log (4)}}{x \left (5-x-x^2-\log (4)\right )} \, dx}{8 (21-\log (256))}\\ &=\frac {13 (1+2 x)}{8 \left (5-x-x^2-\log (4)\right ) (21-\log (256))}-\frac {\log (4) \left (137-e (32-6 \log (4))-47 \log (4)+4 \log ^2(4)+2 (1-e) x (11-\log (16))\right )}{8 (5-\log (4)) \left (5-x-x^2-\log (4)\right ) (21-\log (256))}+\frac {7 (10-x-\log (16))}{8 \left (5-x-x^2-\log (4)\right ) (21-\log (256))}+\frac {5 (11+x-\log (16))}{4 (5-\log (4)) \left (5-x-x^2-\log (4)\right ) (21-\log (256))}-\frac {e \left (x \left (105-30 \log (4)+2 \log ^2(4)\right )+\log (4) (16-\log (64))\right )}{4 (5-\log (4)) \left (5-x-x^2-\log (4)\right ) (21-\log (256))}+\frac {1}{8} \int \frac {((3-4 e) x+2 (6-e-\log (4))) \log (2 x)}{\left (5-x-x^2-\log (4)\right )^2} \, dx+\frac {1}{8} \int \frac {\log (2 x)}{-5+x+x^2+\log (4)} \, dx-\frac {7 \operatorname {Subst}\left (\int \frac {1}{21-x^2-4 \log (4)} \, dx,x,1+2 x\right )}{4 (21-\log (256))}+\frac {13 \operatorname {Subst}\left (\int \frac {1}{21-x^2-4 \log (4)} \, dx,x,1+2 x\right )}{2 (21-\log (256))}+\frac {e \int \left (\frac {-5 \log (4) (11-\log (16))+\log ^2(4) (11-\log (16))-5 (21-\log (256))-5 x (21-\log (256))}{(5-\log (4))^2 \left (5-x-x^2-\log (4)\right )}+\frac {5 (-21+\log (256))}{x (-5+\log (4))^2}\right ) \, dx}{4 (21-\log (256))}+\frac {5 \int \left (\frac {-21+\log (256)}{x (-5+\log (4))}+\frac {26+x (21-\log (256))-\log (1024)}{(5-\log (4)) \left (5-x-x^2-\log (4)\right )}\right ) \, dx}{4 (5-\log (4)) (21-\log (256))}-\frac {\log (4) \int \left (\frac {2 (1-e) (76-\log (4) (11-\log (16))-5 \log (16)+x (21-\log (256))-\log (256))}{(5-\log (4))^2 \left (5-x-x^2-\log (4)\right )}+\frac {2 (-1+e) (-21+\log (256))}{x (-5+\log (4))^2}\right ) \, dx}{8 (21-\log (256))}\\ &=\frac {19 \tanh ^{-1}\left (\frac {1+2 x}{\sqrt {21-4 \log (4)}}\right )}{4 (21-\log (256))^{3/2}}+\frac {13 (1+2 x)}{8 \left (5-x-x^2-\log (4)\right ) (21-\log (256))}-\frac {\log (4) \left (137-e (32-6 \log (4))-47 \log (4)+4 \log ^2(4)+2 (1-e) x (11-\log (16))\right )}{8 (5-\log (4)) \left (5-x-x^2-\log (4)\right ) (21-\log (256))}+\frac {7 (10-x-\log (16))}{8 \left (5-x-x^2-\log (4)\right ) (21-\log (256))}+\frac {5 (11+x-\log (16))}{4 (5-\log (4)) \left (5-x-x^2-\log (4)\right ) (21-\log (256))}-\frac {e \left (x \left (105-30 \log (4)+2 \log ^2(4)\right )+\log (4) (16-\log (64))\right )}{4 (5-\log (4)) \left (5-x-x^2-\log (4)\right ) (21-\log (256))}+\frac {5 \log (x)}{4 (5-\log (4))^2}-\frac {5 e \log (x)}{4 (5-\log (4))^2}-\frac {(1-e) \log (4) \log (x)}{4 (5-\log (4))^2}+\frac {1}{8} \int \left (-\frac {2 \log (2 x)}{\left (-1-2 x+\sqrt {21-4 \log (4)}\right ) \sqrt {21-4 \log (4)}}-\frac {2 \log (2 x)}{\left (1+2 x+\sqrt {21-4 \log (4)}\right ) \sqrt {21-4 \log (4)}}\right ) \, dx+\frac {1}{8} \int \left (-\frac {(-3+4 e) x \log (2 x)}{\left (-5+x+x^2+\log (4)\right )^2}-\frac {2 (-6+e+\log (4)) \log (2 x)}{\left (-5+x+x^2+\log (4)\right )^2}\right ) \, dx+\frac {5 \int \frac {26+x (21-\log (256))-\log (1024)}{5-x-x^2-\log (4)} \, dx}{4 (5-\log (4))^2 (21-\log (256))}+\frac {e \int \frac {-5 \log (4) (11-\log (16))+\log ^2(4) (11-\log (16))-5 (21-\log (256))-5 x (21-\log (256))}{5-x-x^2-\log (4)} \, dx}{4 (5-\log (4))^2 (21-\log (256))}-\frac {((1-e) \log (4)) \int \frac {76-\log (4) (11-\log (16))-5 \log (16)+x (21-\log (256))-\log (256)}{5-x-x^2-\log (4)} \, dx}{4 (5-\log (4))^2 (21-\log (256))}\\ &=\frac {19 \tanh ^{-1}\left (\frac {1+2 x}{\sqrt {21-4 \log (4)}}\right )}{4 (21-\log (256))^{3/2}}+\frac {13 (1+2 x)}{8 \left (5-x-x^2-\log (4)\right ) (21-\log (256))}-\frac {\log (4) \left (137-e (32-6 \log (4))-47 \log (4)+4 \log ^2(4)+2 (1-e) x (11-\log (16))\right )}{8 (5-\log (4)) \left (5-x-x^2-\log (4)\right ) (21-\log (256))}+\frac {7 (10-x-\log (16))}{8 \left (5-x-x^2-\log (4)\right ) (21-\log (256))}+\frac {5 (11+x-\log (16))}{4 (5-\log (4)) \left (5-x-x^2-\log (4)\right ) (21-\log (256))}-\frac {e \left (x \left (105-30 \log (4)+2 \log ^2(4)\right )+\log (4) (16-\log (64))\right )}{4 (5-\log (4)) \left (5-x-x^2-\log (4)\right ) (21-\log (256))}+\frac {5 \log (x)}{4 (5-\log (4))^2}-\frac {5 e \log (x)}{4 (5-\log (4))^2}-\frac {(1-e) \log (4) \log (x)}{4 (5-\log (4))^2}+\frac {1}{8} (3-4 e) \int \frac {x \log (2 x)}{\left (-5+x+x^2+\log (4)\right )^2} \, dx-\frac {\int \frac {\log (2 x)}{-1-2 x+\sqrt {21-4 \log (4)}} \, dx}{4 \sqrt {21-4 \log (4)}}-\frac {\int \frac {\log (2 x)}{1+2 x+\sqrt {21-4 \log (4)}} \, dx}{4 \sqrt {21-4 \log (4)}}-\frac {5 \int \frac {-1-2 x}{5-x-x^2-\log (4)} \, dx}{8 (5-\log (4))^2}+\frac {(5 e) \int \frac {-1-2 x}{5-x-x^2-\log (4)} \, dx}{8 (5-\log (4))^2}+\frac {1}{4} (6-e-\log (4)) \int \frac {\log (2 x)}{\left (-5+x+x^2+\log (4)\right )^2} \, dx+\frac {((1-e) \log (4)) \int \frac {-1-2 x}{5-x-x^2-\log (4)} \, dx}{8 (5-\log (4))^2}-\frac {\left (e \left (10 \log (4) (11-\log (16))-2 \log ^2(4) (11-\log (16))+5 (21-\log (256))\right )\right ) \int \frac {1}{5-x-x^2-\log (4)} \, dx}{8 (5-\log (4))^2 (21-\log (256))}-\frac {\left ((1-e) \log (4) \left (131-22 \log (4)-10 \log (16)+\log ^2(16)-\log (256)\right )\right ) \int \frac {1}{5-x-x^2-\log (4)} \, dx}{8 (5-\log (4))^2 (21-\log (256))}+\frac {(5 (31-\log (4096))) \int \frac {1}{5-x-x^2-\log (4)} \, dx}{8 (5-\log (4))^2 (21-\log (256))}\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 38, normalized size = 1.36 \begin {gather*} \frac {-5+(-1+2 e) x+\log (4)+(-2+2 e-x) \log (2 x)}{8 \left (-5+x+x^2+\log (4)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(10 + 13*x + 7*x^2 + E*(-10 - 8*x + 2*x^2 - 2*x^3) + (-2 - 3*x - 2*x^2 + E*(2 + 2*x))*Log[4] + (7*x
+ 4*x^2 + x^3 + E*(-2*x - 4*x^2) - x*Log[4])*Log[2*x])/(200*x - 80*x^2 - 72*x^3 + 16*x^4 + 8*x^5 + (-80*x + 16
*x^2 + 16*x^3)*Log[4] + 8*x*Log[4]^2),x]

[Out]

(-5 + (-1 + 2*E)*x + Log[4] + (-2 + 2*E - x)*Log[2*x])/(8*(-5 + x + x^2 + Log[4]))

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fricas [A]  time = 0.68, size = 41, normalized size = 1.46 \begin {gather*} \frac {2 \, x e - {\left (x - 2 \, e + 2\right )} \log \left (2 \, x\right ) - x + 2 \, \log \relax (2) - 5}{8 \, {\left (x^{2} + x + 2 \, \log \relax (2) - 5\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x*log(2)+(-4*x^2-2*x)*exp(1)+x^3+4*x^2+7*x)*log(2*x)+2*((2*x+2)*exp(1)-2*x^2-3*x-2)*log(2)+(-2*
x^3+2*x^2-8*x-10)*exp(1)+7*x^2+13*x+10)/(32*x*log(2)^2+2*(16*x^3+16*x^2-80*x)*log(2)+8*x^5+16*x^4-72*x^3-80*x^
2+200*x),x, algorithm="fricas")

[Out]

1/8*(2*x*e - (x - 2*e + 2)*log(2*x) - x + 2*log(2) - 5)/(x^2 + x + 2*log(2) - 5)

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giac [A]  time = 0.16, size = 50, normalized size = 1.79 \begin {gather*} \frac {2 \, x e - x \log \relax (2) + 2 \, e \log \relax (2) - x \log \relax (x) + 2 \, e \log \relax (x) - x - 2 \, \log \relax (x) - 5}{8 \, {\left (x^{2} + x + 2 \, \log \relax (2) - 5\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x*log(2)+(-4*x^2-2*x)*exp(1)+x^3+4*x^2+7*x)*log(2*x)+2*((2*x+2)*exp(1)-2*x^2-3*x-2)*log(2)+(-2*
x^3+2*x^2-8*x-10)*exp(1)+7*x^2+13*x+10)/(32*x*log(2)^2+2*(16*x^3+16*x^2-80*x)*log(2)+8*x^5+16*x^4-72*x^3-80*x^
2+200*x),x, algorithm="giac")

[Out]

1/8*(2*x*e - x*log(2) + 2*e*log(2) - x*log(x) + 2*e*log(x) - x - 2*log(x) - 5)/(x^2 + x + 2*log(2) - 5)

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maple [A]  time = 0.20, size = 46, normalized size = 1.64

method	result	size
norman	\(\frac {\left (\frac {{\mathrm e}}{4}-\frac {1}{8}\right ) x +\left (\frac {{\mathrm e}}{4}-\frac {1}{4}\right ) \ln \left (2 x \right )-\frac {x \ln \left (2 x \right )}{8}-\frac {5}{8}+\frac {\ln \relax (2)}{4}}{x^{2}+2 \ln \relax (2)+x -5}\)	\(46\)
risch	\(\frac {\left (-x +2 \,{\mathrm e}-2\right ) \ln \left (2 x \right )}{8 x^{2}+16 \ln \relax (2)+8 x -40}+\frac {2 x \,{\mathrm e}+2 \ln \relax (2)-x -5}{8 x^{2}+16 \ln \relax (2)+8 x -40}\)	\(57\)
derivativedivides	\(\frac {\ln \relax (2) {\mathrm e} x}{4 \left (2 \ln \relax (2)-5\right ) \left (\frac {x^{2}}{2}+\ln \relax (2)+\frac {x}{2}-\frac {5}{2}\right )}-\frac {x \ln \relax (2)}{8 \left (2 \ln \relax (2)-5\right ) \left (\frac {x^{2}}{2}+\ln \relax (2)+\frac {x}{2}-\frac {5}{2}\right )}-\frac {5 \,{\mathrm e} x}{8 \left (2 \ln \relax (2)-5\right ) \left (\frac {x^{2}}{2}+\ln \relax (2)+\frac {x}{2}-\frac {5}{2}\right )}+\frac {5 x}{16 \left (2 \ln \relax (2)-5\right ) \left (\frac {x^{2}}{2}+\ln \relax (2)+\frac {x}{2}-\frac {5}{2}\right )}+\frac {\ln \relax (2)^{2}}{4 \left (2 \ln \relax (2)-5\right ) \left (\frac {x^{2}}{2}+\ln \relax (2)+\frac {x}{2}-\frac {5}{2}\right )}-\frac {5 \ln \relax (2)}{4 \left (2 \ln \relax (2)-5\right ) \left (\frac {x^{2}}{2}+\ln \relax (2)+\frac {x}{2}-\frac {5}{2}\right )}+\frac {25}{16 \left (2 \ln \relax (2)-5\right ) \left (\frac {x^{2}}{2}+\ln \relax (2)+\frac {x}{2}-\frac {5}{2}\right )}+\frac {\ln \left (2 x \right ) {\mathrm e}}{8 \ln \relax (2)-20}-\frac {\ln \left (2 x \right )}{2 \left (4 \ln \relax (2)-10\right )}+\frac {\ln \left (2 x \right ) \left (\ln \left (\frac {-1+\sqrt {-8 \ln \relax (2)+21}-2 x}{-1+\sqrt {-8 \ln \relax (2)+21}}\right )-\ln \left (\frac {1+\sqrt {-8 \ln \relax (2)+21}+2 x}{1+\sqrt {-8 \ln \relax (2)+21}}\right )\right )}{8 \sqrt {-8 \ln \relax (2)+21}}-\frac {\ln \left (2 x \right ) \left (8 \ln \relax (2) \ln \left (\frac {-1+\sqrt {-8 \ln \relax (2)+21}-2 x}{-1+\sqrt {-8 \ln \relax (2)+21}}\right ) x^{2}-8 \ln \relax (2) \ln \left (\frac {1+\sqrt {-8 \ln \relax (2)+21}+2 x}{1+\sqrt {-8 \ln \relax (2)+21}}\right ) x^{2}+8 \,{\mathrm e} \sqrt {-8 \ln \relax (2)+21}\, x^{2}+16 \ln \relax (2)^{2} \ln \left (\frac {-1+\sqrt {-8 \ln \relax (2)+21}-2 x}{-1+\sqrt {-8 \ln \relax (2)+21}}\right )-16 \ln \relax (2)^{2} \ln \left (\frac {1+\sqrt {-8 \ln \relax (2)+21}+2 x}{1+\sqrt {-8 \ln \relax (2)+21}}\right )+8 \ln \relax (2) \ln \left (\frac {-1+\sqrt {-8 \ln \relax (2)+21}-2 x}{-1+\sqrt {-8 \ln \relax (2)+21}}\right ) x -8 \ln \relax (2) \ln \left (\frac {1+\sqrt {-8 \ln \relax (2)+21}+2 x}{1+\sqrt {-8 \ln \relax (2)+21}}\right ) x +8 \ln \relax (2) \sqrt {-8 \ln \relax (2)+21}\, x +8 \,{\mathrm e} \sqrt {-8 \ln \relax (2)+21}\, x -20 \ln \left (\frac {-1+\sqrt {-8 \ln \relax (2)+21}-2 x}{-1+\sqrt {-8 \ln \relax (2)+21}}\right ) x^{2}+20 \ln \left (\frac {1+\sqrt {-8 \ln \relax (2)+21}+2 x}{1+\sqrt {-8 \ln \relax (2)+21}}\right ) x^{2}-8 \sqrt {-8 \ln \relax (2)+21}\, x^{2}-80 \ln \relax (2) \ln \left (\frac {-1+\sqrt {-8 \ln \relax (2)+21}-2 x}{-1+\sqrt {-8 \ln \relax (2)+21}}\right )+80 \ln \relax (2) \ln \left (\frac {1+\sqrt {-8 \ln \relax (2)+21}+2 x}{1+\sqrt {-8 \ln \relax (2)+21}}\right )-20 \ln \left (\frac {-1+\sqrt {-8 \ln \relax (2)+21}-2 x}{-1+\sqrt {-8 \ln \relax (2)+21}}\right ) x +20 \ln \left (\frac {1+\sqrt {-8 \ln \relax (2)+21}+2 x}{1+\sqrt {-8 \ln \relax (2)+21}}\right ) x -28 x \sqrt {-8 \ln \relax (2)+21}+100 \ln \left (\frac {-1+\sqrt {-8 \ln \relax (2)+21}-2 x}{-1+\sqrt {-8 \ln \relax (2)+21}}\right )-100 \ln \left (\frac {1+\sqrt {-8 \ln \relax (2)+21}+2 x}{1+\sqrt {-8 \ln \relax (2)+21}}\right )\right )}{8 \left (4 x^{2}+8 \ln \relax (2)+4 x -20\right ) \left (2 \ln \relax (2)-5\right ) \sqrt {-8 \ln \relax (2)+21}}\)	\(835\)
default	\(\frac {\ln \relax (2) {\mathrm e} x}{4 \left (2 \ln \relax (2)-5\right ) \left (\frac {x^{2}}{2}+\ln \relax (2)+\frac {x}{2}-\frac {5}{2}\right )}-\frac {x \ln \relax (2)}{8 \left (2 \ln \relax (2)-5\right ) \left (\frac {x^{2}}{2}+\ln \relax (2)+\frac {x}{2}-\frac {5}{2}\right )}-\frac {5 \,{\mathrm e} x}{8 \left (2 \ln \relax (2)-5\right ) \left (\frac {x^{2}}{2}+\ln \relax (2)+\frac {x}{2}-\frac {5}{2}\right )}+\frac {5 x}{16 \left (2 \ln \relax (2)-5\right ) \left (\frac {x^{2}}{2}+\ln \relax (2)+\frac {x}{2}-\frac {5}{2}\right )}+\frac {\ln \relax (2)^{2}}{4 \left (2 \ln \relax (2)-5\right ) \left (\frac {x^{2}}{2}+\ln \relax (2)+\frac {x}{2}-\frac {5}{2}\right )}-\frac {5 \ln \relax (2)}{4 \left (2 \ln \relax (2)-5\right ) \left (\frac {x^{2}}{2}+\ln \relax (2)+\frac {x}{2}-\frac {5}{2}\right )}+\frac {25}{16 \left (2 \ln \relax (2)-5\right ) \left (\frac {x^{2}}{2}+\ln \relax (2)+\frac {x}{2}-\frac {5}{2}\right )}+\frac {\ln \left (2 x \right ) {\mathrm e}}{8 \ln \relax (2)-20}-\frac {\ln \left (2 x \right )}{2 \left (4 \ln \relax (2)-10\right )}+\frac {\ln \left (2 x \right ) \left (\ln \left (\frac {-1+\sqrt {-8 \ln \relax (2)+21}-2 x}{-1+\sqrt {-8 \ln \relax (2)+21}}\right )-\ln \left (\frac {1+\sqrt {-8 \ln \relax (2)+21}+2 x}{1+\sqrt {-8 \ln \relax (2)+21}}\right )\right )}{8 \sqrt {-8 \ln \relax (2)+21}}-\frac {\ln \left (2 x \right ) \left (8 \ln \relax (2) \ln \left (\frac {-1+\sqrt {-8 \ln \relax (2)+21}-2 x}{-1+\sqrt {-8 \ln \relax (2)+21}}\right ) x^{2}-8 \ln \relax (2) \ln \left (\frac {1+\sqrt {-8 \ln \relax (2)+21}+2 x}{1+\sqrt {-8 \ln \relax (2)+21}}\right ) x^{2}+8 \,{\mathrm e} \sqrt {-8 \ln \relax (2)+21}\, x^{2}+16 \ln \relax (2)^{2} \ln \left (\frac {-1+\sqrt {-8 \ln \relax (2)+21}-2 x}{-1+\sqrt {-8 \ln \relax (2)+21}}\right )-16 \ln \relax (2)^{2} \ln \left (\frac {1+\sqrt {-8 \ln \relax (2)+21}+2 x}{1+\sqrt {-8 \ln \relax (2)+21}}\right )+8 \ln \relax (2) \ln \left (\frac {-1+\sqrt {-8 \ln \relax (2)+21}-2 x}{-1+\sqrt {-8 \ln \relax (2)+21}}\right ) x -8 \ln \relax (2) \ln \left (\frac {1+\sqrt {-8 \ln \relax (2)+21}+2 x}{1+\sqrt {-8 \ln \relax (2)+21}}\right ) x +8 \ln \relax (2) \sqrt {-8 \ln \relax (2)+21}\, x +8 \,{\mathrm e} \sqrt {-8 \ln \relax (2)+21}\, x -20 \ln \left (\frac {-1+\sqrt {-8 \ln \relax (2)+21}-2 x}{-1+\sqrt {-8 \ln \relax (2)+21}}\right ) x^{2}+20 \ln \left (\frac {1+\sqrt {-8 \ln \relax (2)+21}+2 x}{1+\sqrt {-8 \ln \relax (2)+21}}\right ) x^{2}-8 \sqrt {-8 \ln \relax (2)+21}\, x^{2}-80 \ln \relax (2) \ln \left (\frac {-1+\sqrt {-8 \ln \relax (2)+21}-2 x}{-1+\sqrt {-8 \ln \relax (2)+21}}\right )+80 \ln \relax (2) \ln \left (\frac {1+\sqrt {-8 \ln \relax (2)+21}+2 x}{1+\sqrt {-8 \ln \relax (2)+21}}\right )-20 \ln \left (\frac {-1+\sqrt {-8 \ln \relax (2)+21}-2 x}{-1+\sqrt {-8 \ln \relax (2)+21}}\right ) x +20 \ln \left (\frac {1+\sqrt {-8 \ln \relax (2)+21}+2 x}{1+\sqrt {-8 \ln \relax (2)+21}}\right ) x -28 x \sqrt {-8 \ln \relax (2)+21}+100 \ln \left (\frac {-1+\sqrt {-8 \ln \relax (2)+21}-2 x}{-1+\sqrt {-8 \ln \relax (2)+21}}\right )-100 \ln \left (\frac {1+\sqrt {-8 \ln \relax (2)+21}+2 x}{1+\sqrt {-8 \ln \relax (2)+21}}\right )\right )}{8 \left (4 x^{2}+8 \ln \relax (2)+4 x -20\right ) \left (2 \ln \relax (2)-5\right ) \sqrt {-8 \ln \relax (2)+21}}\)	\(835\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x*ln(2)+(-4*x^2-2*x)*exp(1)+x^3+4*x^2+7*x)*ln(2*x)+2*((2*x+2)*exp(1)-2*x^2-3*x-2)*ln(2)+(-2*x^3+2*x^2
-8*x-10)*exp(1)+7*x^2+13*x+10)/(32*x*ln(2)^2+2*(16*x^3+16*x^2-80*x)*ln(2)+8*x^5+16*x^4-72*x^3-80*x^2+200*x),x,
method=_RETURNVERBOSE)

[Out]

((1/4*exp(1)-1/8)*x+(1/4*exp(1)-1/4)*ln(2*x)-1/8*x*ln(2*x)-5/8+1/4*ln(2))/(x^2+2*ln(2)+x-5)

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maxima [B]  time = 0.54, size = 1589, normalized size = 56.75 result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x*log(2)+(-4*x^2-2*x)*exp(1)+x^3+4*x^2+7*x)*log(2*x)+2*((2*x+2)*exp(1)-2*x^2-3*x-2)*log(2)+(-2*
x^3+2*x^2-8*x-10)*exp(1)+7*x^2+13*x+10)/(32*x*log(2)^2+2*(16*x^3+16*x^2-80*x)*log(2)+8*x^5+16*x^4-72*x^3-80*x^
2+200*x),x, algorithm="maxima")

[Out]

-1/4*((12*log(2) - 31)*log((2*x - sqrt(-8*log(2) + 21) + 1)/(2*x + sqrt(-8*log(2) + 21) + 1))/((32*log(2)^3 -
244*log(2)^2 + 620*log(2) - 525)*sqrt(-8*log(2) + 21)) + 2*(x - 4*log(2) + 11)/((16*log(2)^2 - 82*log(2) + 105
)*x^2 + 32*log(2)^3 + (16*log(2)^2 - 82*log(2) + 105)*x - 244*log(2)^2 + 620*log(2) - 525) + log(x^2 + x + 2*l
og(2) - 5)/(4*log(2)^2 - 20*log(2) + 25) - 2*log(x)/(4*log(2)^2 - 20*log(2) + 25))*e*log(2) + 1/2*((2*x + 1)/(
x^2*(8*log(2) - 21) + x*(8*log(2) - 21) + 16*log(2)^2 - 82*log(2) + 105) + 2*log((2*x - sqrt(-8*log(2) + 21) +
1)/(2*x + sqrt(-8*log(2) + 21) + 1))/((8*log(2) - 21)*sqrt(-8*log(2) + 21)))*e*log(2) + 5/8*((12*log(2) - 31)
*log((2*x - sqrt(-8*log(2) + 21) + 1)/(2*x + sqrt(-8*log(2) + 21) + 1))/((32*log(2)^3 - 244*log(2)^2 + 620*log
(2) - 525)*sqrt(-8*log(2) + 21)) + 2*(x - 4*log(2) + 11)/((16*log(2)^2 - 82*log(2) + 105)*x^2 + 32*log(2)^3 +
(16*log(2)^2 - 82*log(2) + 105)*x - 244*log(2)^2 + 620*log(2) - 525) + log(x^2 + x + 2*log(2) - 5)/(4*log(2)^2
- 20*log(2) + 25) - 2*log(x)/(4*log(2)^2 - 20*log(2) + 25))*e - 1/4*(2*(2*log(2) - 5)*log((2*x - sqrt(-8*log(
2) + 21) + 1)/(2*x + sqrt(-8*log(2) + 21) + 1))/((8*log(2) - 21)*sqrt(-8*log(2) + 21)) - (x*(4*log(2) - 11) -
2*log(2) + 5)/(x^2*(8*log(2) - 21) + x*(8*log(2) - 21) + 16*log(2)^2 - 82*log(2) + 105))*e - ((2*x + 1)/(x^2*(
8*log(2) - 21) + x*(8*log(2) - 21) + 16*log(2)^2 - 82*log(2) + 105) + 2*log((2*x - sqrt(-8*log(2) + 21) + 1)/(
2*x + sqrt(-8*log(2) + 21) + 1))/((8*log(2) - 21)*sqrt(-8*log(2) + 21)))*e - 1/4*((x + 4*log(2) - 10)/(x^2*(8*
log(2) - 21) + x*(8*log(2) - 21) + 16*log(2)^2 - 82*log(2) + 105) + log((2*x - sqrt(-8*log(2) + 21) + 1)/(2*x
+ sqrt(-8*log(2) + 21) + 1))/((8*log(2) - 21)*sqrt(-8*log(2) + 21)))*e + 1/4*((12*log(2) - 31)*log((2*x - sqrt
(-8*log(2) + 21) + 1)/(2*x + sqrt(-8*log(2) + 21) + 1))/((32*log(2)^3 - 244*log(2)^2 + 620*log(2) - 525)*sqrt(
-8*log(2) + 21)) + 2*(x - 4*log(2) + 11)/((16*log(2)^2 - 82*log(2) + 105)*x^2 + 32*log(2)^3 + (16*log(2)^2 - 8
2*log(2) + 105)*x - 244*log(2)^2 + 620*log(2) - 525) + log(x^2 + x + 2*log(2) - 5)/(4*log(2)^2 - 20*log(2) + 2
5) - 2*log(x)/(4*log(2)^2 - 20*log(2) + 25))*log(2) - 3/4*((2*x + 1)/(x^2*(8*log(2) - 21) + x*(8*log(2) - 21)
+ 16*log(2)^2 - 82*log(2) + 105) + 2*log((2*x - sqrt(-8*log(2) + 21) + 1)/(2*x + sqrt(-8*log(2) + 21) + 1))/((
8*log(2) - 21)*sqrt(-8*log(2) + 21)))*log(2) + 1/2*((x + 4*log(2) - 10)/(x^2*(8*log(2) - 21) + x*(8*log(2) - 2
1) + 16*log(2)^2 - 82*log(2) + 105) + log((2*x - sqrt(-8*log(2) + 21) + 1)/(2*x + sqrt(-8*log(2) + 21) + 1))/(
(8*log(2) - 21)*sqrt(-8*log(2) + 21)))*log(2) + 1/8*(e - 1)*log(x^2 + x + 2*log(2) - 5)/(2*log(2) - 5) - 5/8*(
12*log(2) - 31)*log((2*x - sqrt(-8*log(2) + 21) + 1)/(2*x + sqrt(-8*log(2) + 21) + 1))/((32*log(2)^3 - 244*log
(2)^2 + 620*log(2) - 525)*sqrt(-8*log(2) + 21)) + 1/8*(e + 2*log(2) - 6)*log((2*x - sqrt(-8*log(2) + 21) + 1)/
(2*x + sqrt(-8*log(2) + 21) + 1))/((2*log(2) - 5)*sqrt(-8*log(2) + 21)) - 1/8*((2*log(2)^2 - 5*log(2))*x - 2*(
2*log(2)^2 - 5*log(2))*e + 4*log(2)^2 + (2*x^2*(e - 1) + x*(2*e + 2*log(2) - 7))*log(x) - 10*log(2))/(x^2*(2*l
og(2) - 5) + x*(2*log(2) - 5) + 4*log(2)^2 - 20*log(2) + 25) + 13/8*(2*x + 1)/(x^2*(8*log(2) - 21) + x*(8*log(
2) - 21) + 16*log(2)^2 - 82*log(2) + 105) - 7/8*(x + 4*log(2) - 10)/(x^2*(8*log(2) - 21) + x*(8*log(2) - 21) +
16*log(2)^2 - 82*log(2) + 105) - 5/4*(x - 4*log(2) + 11)/((16*log(2)^2 - 82*log(2) + 105)*x^2 + 32*log(2)^3 +
(16*log(2)^2 - 82*log(2) + 105)*x - 244*log(2)^2 + 620*log(2) - 525) - 5/8*log(x^2 + x + 2*log(2) - 5)/(4*log
(2)^2 - 20*log(2) + 25) + 5/4*log(x)/(4*log(2)^2 - 20*log(2) + 25) + 19/8*log((2*x - sqrt(-8*log(2) + 21) + 1)
/(2*x + sqrt(-8*log(2) + 21) + 1))/((8*log(2) - 21)*sqrt(-8*log(2) + 21))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {13\,x-2\,\ln \relax (2)\,\left (3\,x+2\,x^2-\mathrm {e}\,\left (2\,x+2\right )+2\right )+\ln \left (2\,x\right )\,\left (7\,x-\mathrm {e}\,\left (4\,x^2+2\,x\right )-2\,x\,\ln \relax (2)+4\,x^2+x^3\right )-\mathrm {e}\,\left (2\,x^3-2\,x^2+8\,x+10\right )+7\,x^2+10}{200\,x+2\,\ln \relax (2)\,\left (16\,x^3+16\,x^2-80\,x\right )+32\,x\,{\ln \relax (2)}^2-80\,x^2-72\,x^3+16\,x^4+8\,x^5} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((13*x - 2*log(2)*(3*x + 2*x^2 - exp(1)*(2*x + 2) + 2) + log(2*x)*(7*x - exp(1)*(2*x + 4*x^2) - 2*x*log(2)
+ 4*x^2 + x^3) - exp(1)*(8*x - 2*x^2 + 2*x^3 + 10) + 7*x^2 + 10)/(200*x + 2*log(2)*(16*x^2 - 80*x + 16*x^3) +
32*x*log(2)^2 - 80*x^2 - 72*x^3 + 16*x^4 + 8*x^5),x)

[Out]

int((13*x - 2*log(2)*(3*x + 2*x^2 - exp(1)*(2*x + 2) + 2) + log(2*x)*(7*x - exp(1)*(2*x + 4*x^2) - 2*x*log(2)
+ 4*x^2 + x^3) - exp(1)*(8*x - 2*x^2 + 2*x^3 + 10) + 7*x^2 + 10)/(200*x + 2*log(2)*(16*x^2 - 80*x + 16*x^3) +
32*x*log(2)^2 - 80*x^2 - 72*x^3 + 16*x^4 + 8*x^5), x)

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sympy [A]  time = 2.35, size = 58, normalized size = 2.07 \begin {gather*} \frac {\left (- x - 2 + 2 e\right ) \log {\left (2 x \right )}}{8 x^{2} + 8 x - 40 + 16 \log {\relax (2 )}} + \frac {x \left (-1 + 2 e\right ) - 5 + 2 \log {\relax (2 )}}{8 x^{2} + 8 x - 40 + 16 \log {\relax (2 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x*ln(2)+(-4*x**2-2*x)*exp(1)+x**3+4*x**2+7*x)*ln(2*x)+2*((2*x+2)*exp(1)-2*x**2-3*x-2)*ln(2)+(-2
*x**3+2*x**2-8*x-10)*exp(1)+7*x**2+13*x+10)/(32*x*ln(2)**2+2*(16*x**3+16*x**2-80*x)*ln(2)+8*x**5+16*x**4-72*x*
*3-80*x**2+200*x),x)

[Out]

(-x - 2 + 2*E)*log(2*x)/(8*x**2 + 8*x - 40 + 16*log(2)) + (x*(-1 + 2*E) - 5 + 2*log(2))/(8*x**2 + 8*x - 40 + 1
6*log(2))

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