3.46.76 \(\int \frac {17-37 x+15 x^2+(7-17 x+6 x^2) \log (6)+(-7+17 x-6 x^2) \log (-7+3 x)}{-7+17 x-27 x^2+23 x^3-13 x^4+3 x^5} \, dx\)

Optimal. Leaf size=25 \[ \frac {-2-\log (6)+\log (1+2 (-4+x)+x)}{(-1+x)^2+x} \]

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Rubi [C]  time = 1.22, antiderivative size = 486, normalized size of antiderivative = 19.44, number of steps used = 59, number of rules used = 18, integrand size = 68, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.265, Rules used = {6741, 6742, 740, 800, 634, 618, 204, 628, 822, 1646, 629, 2418, 2395, 36, 31, 2394, 2393, 2391} \begin {gather*} \frac {17 (10-11 x)}{111 \left (x^2-x+1\right )}-\frac {11-x}{3 \left (x^2-x+1\right )}+\frac {5 (10 x+1)}{37 \left (x^2-x+1\right )}-\frac {9}{74} \log \left (x^2-x+1\right )-\frac {\log (6)}{x^2-x+1}+\frac {2 \left (-\sqrt {3}+i\right ) \log (7-3 x)}{3 \sqrt {3}+11 i}+\frac {2 \left (\sqrt {3}+i\right ) \log (7-3 x)}{-3 \sqrt {3}+11 i}-\frac {2 \log (7-3 x)}{11+3 i \sqrt {3}}-\frac {2 \log (7-3 x)}{11-3 i \sqrt {3}}+\frac {9}{37} \log (7-3 x)-\frac {2 \left (\sqrt {3}+i\right ) \log \left (-2 x-i \sqrt {3}+1\right )}{-3 \sqrt {3}+11 i}+\frac {2 \log \left (-2 x-i \sqrt {3}+1\right )}{11+3 i \sqrt {3}}-\frac {2 \left (-\sqrt {3}+i\right ) \log \left (-2 x+i \sqrt {3}+1\right )}{3 \sqrt {3}+11 i}+\frac {2 \log \left (-2 x+i \sqrt {3}+1\right )}{11-3 i \sqrt {3}}+\frac {2 \left (1-i \sqrt {3}\right ) \log (3 x-7)}{3 \left (-2 x-i \sqrt {3}+1\right )}-\frac {2 \log (3 x-7)}{3 \left (-2 x-i \sqrt {3}+1\right )}+\frac {2 \left (1+i \sqrt {3}\right ) \log (3 x-7)}{3 \left (-2 x+i \sqrt {3}+1\right )}-\frac {2 \log (3 x-7)}{3 \left (-2 x+i \sqrt {3}+1\right )}+\frac {11}{37} \sqrt {3} \tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(17 - 37*x + 15*x^2 + (7 - 17*x + 6*x^2)*Log[6] + (-7 + 17*x - 6*x^2)*Log[-7 + 3*x])/(-7 + 17*x - 27*x^2 +
23*x^3 - 13*x^4 + 3*x^5),x]

[Out]

(17*(10 - 11*x))/(111*(1 - x + x^2)) - (11 - x)/(3*(1 - x + x^2)) + (5*(1 + 10*x))/(37*(1 - x + x^2)) + (11*Sq
rt[3]*ArcTan[(1 - 2*x)/Sqrt[3]])/37 - Log[6]/(1 - x + x^2) + (9*Log[7 - 3*x])/37 - (2*Log[7 - 3*x])/(11 - (3*I
)*Sqrt[3]) - (2*Log[7 - 3*x])/(11 + (3*I)*Sqrt[3]) + (2*(I + Sqrt[3])*Log[7 - 3*x])/(11*I - 3*Sqrt[3]) + (2*(I
- Sqrt[3])*Log[7 - 3*x])/(11*I + 3*Sqrt[3]) + (2*Log[1 - I*Sqrt[3] - 2*x])/(11 + (3*I)*Sqrt[3]) - (2*(I + Sqr
t[3])*Log[1 - I*Sqrt[3] - 2*x])/(11*I - 3*Sqrt[3]) + (2*Log[1 + I*Sqrt[3] - 2*x])/(11 - (3*I)*Sqrt[3]) - (2*(I
- Sqrt[3])*Log[1 + I*Sqrt[3] - 2*x])/(11*I + 3*Sqrt[3]) - (2*Log[-7 + 3*x])/(3*(1 - I*Sqrt[3] - 2*x)) + (2*(1
- I*Sqrt[3])*Log[-7 + 3*x])/(3*(1 - I*Sqrt[3] - 2*x)) - (2*Log[-7 + 3*x])/(3*(1 + I*Sqrt[3] - 2*x)) + (2*(1 +
I*Sqrt[3])*Log[-7 + 3*x])/(3*(1 + I*Sqrt[3] - 2*x)) - (9*Log[1 - x + x^2])/74

Rule 31

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

Rule 36

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

Rule 204

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

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 629

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

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 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 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 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 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2394

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

Rule 2395

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

Rule 2418

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

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 {-17+37 x-15 x^2-\left (7-17 x+6 x^2\right ) \log (6)-\left (-7+17 x-6 x^2\right ) \log (-7+3 x)}{(7-3 x) \left (1-x+x^2\right )^2} \, dx\\ &=\int \left (\frac {17}{(-7+3 x) \left (1-x+x^2\right )^2}-\frac {37 x}{(-7+3 x) \left (1-x+x^2\right )^2}+\frac {15 x^2}{(-7+3 x) \left (1-x+x^2\right )^2}+\frac {(-1+2 x) \log (6)}{\left (1-x+x^2\right )^2}-\frac {(-1+2 x) \log (-7+3 x)}{\left (1-x+x^2\right )^2}\right ) \, dx\\ &=15 \int \frac {x^2}{(-7+3 x) \left (1-x+x^2\right )^2} \, dx+17 \int \frac {1}{(-7+3 x) \left (1-x+x^2\right )^2} \, dx-37 \int \frac {x}{(-7+3 x) \left (1-x+x^2\right )^2} \, dx+\log (6) \int \frac {-1+2 x}{\left (1-x+x^2\right )^2} \, dx-\int \frac {(-1+2 x) \log (-7+3 x)}{\left (1-x+x^2\right )^2} \, dx\\ &=\frac {17 (10-11 x)}{111 \left (1-x+x^2\right )}-\frac {11-x}{3 \left (1-x+x^2\right )}+\frac {5 (1+10 x)}{37 \left (1-x+x^2\right )}-\frac {\log (6)}{1-x+x^2}+\frac {17}{111} \int \frac {104-33 x}{(-7+3 x) \left (1-x+x^2\right )} \, dx-\frac {1}{3} \int \frac {70-3 x}{(-7+3 x) \left (1-x+x^2\right )} \, dx+5 \int \frac {\frac {77}{37}+\frac {30 x}{37}}{(-7+3 x) \left (1-x+x^2\right )} \, dx-\int \left (-\frac {\log (-7+3 x)}{\left (1-x+x^2\right )^2}+\frac {2 x \log (-7+3 x)}{\left (1-x+x^2\right )^2}\right ) \, dx\\ &=\frac {17 (10-11 x)}{111 \left (1-x+x^2\right )}-\frac {11-x}{3 \left (1-x+x^2\right )}+\frac {5 (1+10 x)}{37 \left (1-x+x^2\right )}-\frac {\log (6)}{1-x+x^2}+\frac {17}{111} \int \left (\frac {243}{37 (-7+3 x)}+\frac {-515-81 x}{37 \left (1-x+x^2\right )}\right ) \, dx-\frac {1}{3} \int \left (\frac {567}{37 (-7+3 x)}+\frac {-289-189 x}{37 \left (1-x+x^2\right )}\right ) \, dx-2 \int \frac {x \log (-7+3 x)}{\left (1-x+x^2\right )^2} \, dx+5 \int \left (\frac {1323}{1369 (-7+3 x)}+\frac {-218-441 x}{1369 \left (1-x+x^2\right )}\right ) \, dx+\int \frac {\log (-7+3 x)}{\left (1-x+x^2\right )^2} \, dx\\ &=\frac {17 (10-11 x)}{111 \left (1-x+x^2\right )}-\frac {11-x}{3 \left (1-x+x^2\right )}+\frac {5 (1+10 x)}{37 \left (1-x+x^2\right )}-\frac {\log (6)}{1-x+x^2}+\frac {9}{37} \log (7-3 x)+\frac {5 \int \frac {-218-441 x}{1-x+x^2} \, dx}{1369}+\frac {17 \int \frac {-515-81 x}{1-x+x^2} \, dx}{4107}-\frac {1}{111} \int \frac {-289-189 x}{1-x+x^2} \, dx-2 \int \left (-\frac {2 \left (1+i \sqrt {3}\right ) \log (-7+3 x)}{3 \left (1+i \sqrt {3}-2 x\right )^2}+\frac {2 i \log (-7+3 x)}{3 \sqrt {3} \left (1+i \sqrt {3}-2 x\right )}-\frac {2 \left (1-i \sqrt {3}\right ) \log (-7+3 x)}{3 \left (-1+i \sqrt {3}+2 x\right )^2}+\frac {2 i \log (-7+3 x)}{3 \sqrt {3} \left (-1+i \sqrt {3}+2 x\right )}\right ) \, dx+\int \left (-\frac {4 \log (-7+3 x)}{3 \left (1+i \sqrt {3}-2 x\right )^2}+\frac {4 i \log (-7+3 x)}{3 \sqrt {3} \left (1+i \sqrt {3}-2 x\right )}-\frac {4 \log (-7+3 x)}{3 \left (-1+i \sqrt {3}+2 x\right )^2}+\frac {4 i \log (-7+3 x)}{3 \sqrt {3} \left (-1+i \sqrt {3}+2 x\right )}\right ) \, dx\\ &=\frac {17 (10-11 x)}{111 \left (1-x+x^2\right )}-\frac {11-x}{3 \left (1-x+x^2\right )}+\frac {5 (1+10 x)}{37 \left (1-x+x^2\right )}-\frac {\log (6)}{1-x+x^2}+\frac {9}{37} \log (7-3 x)-\frac {459 \int \frac {-1+2 x}{1-x+x^2} \, dx}{2738}-\frac {2205 \int \frac {-1+2 x}{1-x+x^2} \, dx}{2738}+\frac {63}{74} \int \frac {-1+2 x}{1-x+x^2} \, dx-\frac {4}{3} \int \frac {\log (-7+3 x)}{\left (1+i \sqrt {3}-2 x\right )^2} \, dx-\frac {4}{3} \int \frac {\log (-7+3 x)}{\left (-1+i \sqrt {3}+2 x\right )^2} \, dx-\frac {4385 \int \frac {1}{1-x+x^2} \, dx}{2738}-\frac {18887 \int \frac {1}{1-x+x^2} \, dx}{8214}+\frac {767}{222} \int \frac {1}{1-x+x^2} \, dx+\frac {1}{3} \left (4 \left (1-i \sqrt {3}\right )\right ) \int \frac {\log (-7+3 x)}{\left (-1+i \sqrt {3}+2 x\right )^2} \, dx+\frac {1}{3} \left (4 \left (1+i \sqrt {3}\right )\right ) \int \frac {\log (-7+3 x)}{\left (1+i \sqrt {3}-2 x\right )^2} \, dx\\ &=\frac {17 (10-11 x)}{111 \left (1-x+x^2\right )}-\frac {11-x}{3 \left (1-x+x^2\right )}+\frac {5 (1+10 x)}{37 \left (1-x+x^2\right )}-\frac {\log (6)}{1-x+x^2}+\frac {9}{37} \log (7-3 x)-\frac {2 \log (-7+3 x)}{3 \left (1-i \sqrt {3}-2 x\right )}+\frac {2 \left (1-i \sqrt {3}\right ) \log (-7+3 x)}{3 \left (1-i \sqrt {3}-2 x\right )}-\frac {2 \log (-7+3 x)}{3 \left (1+i \sqrt {3}-2 x\right )}+\frac {2 \left (1+i \sqrt {3}\right ) \log (-7+3 x)}{3 \left (1+i \sqrt {3}-2 x\right )}-\frac {9}{74} \log \left (1-x+x^2\right )+2 \int \frac {1}{\left (1+i \sqrt {3}-2 x\right ) (-7+3 x)} \, dx-2 \int \frac {1}{\left (-1+i \sqrt {3}+2 x\right ) (-7+3 x)} \, dx+\frac {4385 \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 x\right )}{1369}+\frac {18887 \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 x\right )}{4107}-\frac {767}{111} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 x\right )+\left (2 \left (1-i \sqrt {3}\right )\right ) \int \frac {1}{\left (-1+i \sqrt {3}+2 x\right ) (-7+3 x)} \, dx-\left (2 \left (1+i \sqrt {3}\right )\right ) \int \frac {1}{\left (1+i \sqrt {3}-2 x\right ) (-7+3 x)} \, dx\\ &=\frac {17 (10-11 x)}{111 \left (1-x+x^2\right )}-\frac {11-x}{3 \left (1-x+x^2\right )}+\frac {5 (1+10 x)}{37 \left (1-x+x^2\right )}+\frac {11}{37} \sqrt {3} \tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )-\frac {\log (6)}{1-x+x^2}+\frac {9}{37} \log (7-3 x)-\frac {2 \log (-7+3 x)}{3 \left (1-i \sqrt {3}-2 x\right )}+\frac {2 \left (1-i \sqrt {3}\right ) \log (-7+3 x)}{3 \left (1-i \sqrt {3}-2 x\right )}-\frac {2 \log (-7+3 x)}{3 \left (1+i \sqrt {3}-2 x\right )}+\frac {2 \left (1+i \sqrt {3}\right ) \log (-7+3 x)}{3 \left (1+i \sqrt {3}-2 x\right )}-\frac {9}{74} \log \left (1-x+x^2\right )-\frac {4 \int \frac {1}{1+i \sqrt {3}-2 x} \, dx}{11-3 i \sqrt {3}}-\frac {6 \int \frac {1}{-7+3 x} \, dx}{11-3 i \sqrt {3}}+\frac {4 \int \frac {1}{-1+i \sqrt {3}+2 x} \, dx}{11+3 i \sqrt {3}}-\frac {6 \int \frac {1}{-7+3 x} \, dx}{11+3 i \sqrt {3}}-\frac {\left (4 \left (i+\sqrt {3}\right )\right ) \int \frac {1}{-1+i \sqrt {3}+2 x} \, dx}{11 i-3 \sqrt {3}}+\frac {\left (6 \left (i+\sqrt {3}\right )\right ) \int \frac {1}{-7+3 x} \, dx}{11 i-3 \sqrt {3}}+\frac {\left (4 \left (i-\sqrt {3}\right )\right ) \int \frac {1}{1+i \sqrt {3}-2 x} \, dx}{11 i+3 \sqrt {3}}+\frac {\left (6 \left (i-\sqrt {3}\right )\right ) \int \frac {1}{-7+3 x} \, dx}{11 i+3 \sqrt {3}}\\ &=\frac {17 (10-11 x)}{111 \left (1-x+x^2\right )}-\frac {11-x}{3 \left (1-x+x^2\right )}+\frac {5 (1+10 x)}{37 \left (1-x+x^2\right )}+\frac {11}{37} \sqrt {3} \tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )-\frac {\log (6)}{1-x+x^2}+\frac {9}{37} \log (7-3 x)-\frac {2 \log (7-3 x)}{11-3 i \sqrt {3}}-\frac {2 \log (7-3 x)}{11+3 i \sqrt {3}}+\frac {2 \left (i+\sqrt {3}\right ) \log (7-3 x)}{11 i-3 \sqrt {3}}+\frac {2 \left (i-\sqrt {3}\right ) \log (7-3 x)}{11 i+3 \sqrt {3}}+\frac {2 \log \left (1-i \sqrt {3}-2 x\right )}{11+3 i \sqrt {3}}-\frac {2 \left (i+\sqrt {3}\right ) \log \left (1-i \sqrt {3}-2 x\right )}{11 i-3 \sqrt {3}}+\frac {2 \log \left (1+i \sqrt {3}-2 x\right )}{11-3 i \sqrt {3}}-\frac {2 \left (i-\sqrt {3}\right ) \log \left (1+i \sqrt {3}-2 x\right )}{11 i+3 \sqrt {3}}-\frac {2 \log (-7+3 x)}{3 \left (1-i \sqrt {3}-2 x\right )}+\frac {2 \left (1-i \sqrt {3}\right ) \log (-7+3 x)}{3 \left (1-i \sqrt {3}-2 x\right )}-\frac {2 \log (-7+3 x)}{3 \left (1+i \sqrt {3}-2 x\right )}+\frac {2 \left (1+i \sqrt {3}\right ) \log (-7+3 x)}{3 \left (1+i \sqrt {3}-2 x\right )}-\frac {9}{74} \log \left (1-x+x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.38, size = 32, normalized size = 1.28 \begin {gather*} \frac {-222-257 \log (6)+73 \log (36)+111 \log (-7+3 x)}{111 \left (1-x+x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(17 - 37*x + 15*x^2 + (7 - 17*x + 6*x^2)*Log[6] + (-7 + 17*x - 6*x^2)*Log[-7 + 3*x])/(-7 + 17*x - 27
*x^2 + 23*x^3 - 13*x^4 + 3*x^5),x]

[Out]

(-222 - 257*Log[6] + 73*Log[36] + 111*Log[-7 + 3*x])/(111*(1 - x + x^2))

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fricas [A]  time = 0.66, size = 24, normalized size = 0.96 \begin {gather*} -\frac {\log \relax (6) - \log \left (3 \, x - 7\right ) + 2}{x^{2} - x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x^2+17*x-7)*log(3*x-7)+(6*x^2-17*x+7)*log(6)+15*x^2-37*x+17)/(3*x^5-13*x^4+23*x^3-27*x^2+17*x-7
),x, algorithm="fricas")

[Out]

-(log(6) - log(3*x - 7) + 2)/(x^2 - x + 1)

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giac [A]  time = 0.16, size = 34, normalized size = 1.36 \begin {gather*} -\frac {\log \relax (6) + 2}{x^{2} - x + 1} + \frac {\log \left (3 \, x - 7\right )}{x^{2} - x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x^2+17*x-7)*log(3*x-7)+(6*x^2-17*x+7)*log(6)+15*x^2-37*x+17)/(3*x^5-13*x^4+23*x^3-27*x^2+17*x-7
),x, algorithm="giac")

[Out]

-(log(6) + 2)/(x^2 - x + 1) + log(3*x - 7)/(x^2 - x + 1)

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maple [A]  time = 0.08, size = 24, normalized size = 0.96

method	result	size
norman	\(\frac {\ln \left (3 x -7\right )-\ln \relax (6)-2}{x^{2}-x +1}\)	\(24\)
risch	\(\frac {\ln \left (3 x -7\right )}{x^{2}-x +1}-\frac {\ln \relax (3)}{x^{2}-x +1}-\frac {\ln \relax (2)}{x^{2}-x +1}-\frac {2}{x^{2}-x +1}\)	\(59\)
derivativedivides	\(-\frac {9 \ln \relax (6)}{\left (3 x -7\right )^{2}+33 x -40}-\frac {9 \ln \left (3 x -7\right ) \left (3 x -7\right ) \left (4+3 x \right )}{37 \left (\left (3 x -7\right )^{2}+33 x -40\right )}-\frac {18}{\left (3 x -7\right )^{2}+33 x -40}+\frac {9 \ln \left (3 x -7\right )}{37}\)	\(76\)
default	\(-\frac {9 \ln \relax (6)}{\left (3 x -7\right )^{2}+33 x -40}-\frac {9 \ln \left (3 x -7\right ) \left (3 x -7\right ) \left (4+3 x \right )}{37 \left (\left (3 x -7\right )^{2}+33 x -40\right )}-\frac {18}{\left (3 x -7\right )^{2}+33 x -40}+\frac {9 \ln \left (3 x -7\right )}{37}\)	\(76\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-6*x^2+17*x-7)*ln(3*x-7)+(6*x^2-17*x+7)*ln(6)+15*x^2-37*x+17)/(3*x^5-13*x^4+23*x^3-27*x^2+17*x-7),x,meth
od=_RETURNVERBOSE)

[Out]

(ln(3*x-7)-ln(6)-2)/(x^2-x+1)

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maxima [B]  time = 0.65, size = 255, normalized size = 10.20 \begin {gather*} -\frac {7}{24642} \, {\left (2222 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {222 \, {\left (11 \, x - 10\right )}}{x^{2} - x + 1} + 243 \, \log \left (x^{2} - x + 1\right ) - 486 \, \log \left (3 \, x - 7\right )\right )} \log \relax (6) - \frac {1}{4107} \, {\left (1754 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \frac {222 \, {\left (10 \, x + 1\right )}}{x^{2} - x + 1} + 1323 \, \log \left (x^{2} - x + 1\right ) - 2646 \, \log \left (3 \, x - 7\right )\right )} \log \relax (6) + \frac {17}{24642} \, {\left (1534 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {222 \, {\left (x - 11\right )}}{x^{2} - x + 1} + 567 \, \log \left (x^{2} - x + 1\right ) - 1134 \, \log \left (3 \, x - 7\right )\right )} \log \relax (6) - \frac {{\left (9 \, x^{2} - 9 \, x - 28\right )} \log \left (3 \, x - 7\right )}{37 \, {\left (x^{2} - x + 1\right )}} - \frac {17 \, {\left (11 \, x - 10\right )}}{111 \, {\left (x^{2} - x + 1\right )}} + \frac {5 \, {\left (10 \, x + 1\right )}}{37 \, {\left (x^{2} - x + 1\right )}} + \frac {x - 11}{3 \, {\left (x^{2} - x + 1\right )}} + \frac {9}{37} \, \log \left (3 \, x - 7\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x^2+17*x-7)*log(3*x-7)+(6*x^2-17*x+7)*log(6)+15*x^2-37*x+17)/(3*x^5-13*x^4+23*x^3-27*x^2+17*x-7
),x, algorithm="maxima")

[Out]

-7/24642*(2222*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) + 222*(11*x - 10)/(x^2 - x + 1) + 243*log(x^2 - x + 1) -
486*log(3*x - 7))*log(6) - 1/4107*(1754*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) - 222*(10*x + 1)/(x^2 - x + 1) +
1323*log(x^2 - x + 1) - 2646*log(3*x - 7))*log(6) + 17/24642*(1534*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) + 22
2*(x - 11)/(x^2 - x + 1) + 567*log(x^2 - x + 1) - 1134*log(3*x - 7))*log(6) - 1/37*(9*x^2 - 9*x - 28)*log(3*x
- 7)/(x^2 - x + 1) - 17/111*(11*x - 10)/(x^2 - x + 1) + 5/37*(10*x + 1)/(x^2 - x + 1) + 1/3*(x - 11)/(x^2 - x
+ 1) + 9/37*log(3*x - 7)

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mupad [B]  time = 0.28, size = 19, normalized size = 0.76 \begin {gather*} \frac {\ln \left (\frac {x}{2}-\frac {7}{6}\right )-2}{x^2-x+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(6)*(6*x^2 - 17*x + 7) - 37*x - log(3*x - 7)*(6*x^2 - 17*x + 7) + 15*x^2 + 17)/(17*x - 27*x^2 + 23*x^3
- 13*x^4 + 3*x^5 - 7),x)

[Out]

(log(x/2 - 7/6) - 2)/(x^2 - x + 1)

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sympy [A]  time = 0.31, size = 26, normalized size = 1.04 \begin {gather*} \frac {\log {\left (3 x - 7 \right )}}{x^{2} - x + 1} + \frac {-2 - \log {\relax (6 )}}{x^{2} - x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x**2+17*x-7)*ln(3*x-7)+(6*x**2-17*x+7)*ln(6)+15*x**2-37*x+17)/(3*x**5-13*x**4+23*x**3-27*x**2+1
7*x-7),x)

[Out]

log(3*x - 7)/(x**2 - x + 1) + (-2 - log(6))/(x**2 - x + 1)

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