3.27.14 \(\int \frac {3-3 x^2-3 x^3+(4+4 x^2+8 x^3) \log (\frac {x^2 \log (3)}{1-2 x^2-2 x^3+x^4+2 x^5+x^6})+(1-x^2-x^3) \log ^2(\frac {x^2 \log (3)}{1-2 x^2-2 x^3+x^4+2 x^5+x^6})}{-3+3 x^2+3 x^3} \, dx\)

Optimal. Leaf size=27 \[ -x-\frac {1}{3} x \log ^2\left (\frac {\log (3)}{\left (-\frac {1}{x}+x+x^2\right )^2}\right ) \]

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Rubi [A]  time = 22.90, antiderivative size = 32, normalized size of antiderivative = 1.19, number of steps used = 38, number of rules used = 18, integrand size = 119, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.151, Rules used = {6688, 2528, 2523, 12, 6742, 2100, 2081, 2079, 800, 634, 618, 204, 628, 2067, 2065, 705, 31, 1628} \begin {gather*} -\frac {1}{3} x \log ^2\left (\frac {x^2 \log (3)}{\left (-x^3-x^2+1\right )^2}\right )-x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 - 3*x^2 - 3*x^3 + (4 + 4*x^2 + 8*x^3)*Log[(x^2*Log[3])/(1 - 2*x^2 - 2*x^3 + x^4 + 2*x^5 + x^6)] + (1 -
x^2 - x^3)*Log[(x^2*Log[3])/(1 - 2*x^2 - 2*x^3 + x^4 + 2*x^5 + x^6)]^2)/(-3 + 3*x^2 + 3*x^3),x]

[Out]

-x - (x*Log[(x^2*Log[3])/(1 - x^2 - x^3)^2]^2)/3

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 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 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 705

Int[1/(((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[e^2/(c*d^2 - b*d*e + a*e^2
), Int[1/(d + e*x), x], x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[(c*d - b*e - c*e*x)/(a + b*x + c*x^2), 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]

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 1628

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 2065

Int[((a_.) + (b_.)*(x_) + (d_.)*(x_)^3)^(p_), x_Symbol] :> With[{r = Rt[-9*a*d^2 + Sqrt[3]*d*Sqrt[4*b^3*d + 27
*a^2*d^2], 3]}, Dist[1/d^(2*p), Int[Simp[(18^(1/3)*b*d)/(3*r) - r/18^(1/3) + d*x, x]^p*Simp[(b*d)/3 + (12^(1/3
)*b^2*d^2)/(3*r^2) + r^2/(3*12^(1/3)) - d*((2^(1/3)*b*d)/(3^(1/3)*r) - r/18^(1/3))*x + d^2*x^2, x]^p, x], x]]
/; FreeQ[{a, b, d}, x] && NeQ[4*b^3 + 27*a^2*d, 0] && IntegerQ[p]

Rule 2067

Int[(P3_)^(p_), x_Symbol] :> With[{a = Coeff[P3, x, 0], b = Coeff[P3, x, 1], c = Coeff[P3, x, 2], d = Coeff[P3
, x, 3]}, Subst[Int[Simp[(2*c^3 - 9*b*c*d + 27*a*d^2)/(27*d^2) - ((c^2 - 3*b*d)*x)/(3*d) + d*x^3, x]^p, x], x,
x + c/(3*d)] /; NeQ[c, 0]] /; FreeQ[p, x] && PolyQ[P3, x, 3]

Rule 2079

Int[((e_.) + (f_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (d_.)*(x_)^3)^(p_), x_Symbol] :> With[{r = Rt[-9*a*d^2 + S
qrt[3]*d*Sqrt[4*b^3*d + 27*a^2*d^2], 3]}, Dist[1/d^(2*p), Int[(e + f*x)^m*Simp[(18^(1/3)*b*d)/(3*r) - r/18^(1/
3) + d*x, x]^p*Simp[(b*d)/3 + (12^(1/3)*b^2*d^2)/(3*r^2) + r^2/(3*12^(1/3)) - d*((2^(1/3)*b*d)/(3^(1/3)*r) - r
/18^(1/3))*x + d^2*x^2, x]^p, x], x]] /; FreeQ[{a, b, d, e, f, m}, x] && NeQ[4*b^3 + 27*a^2*d, 0] && ILtQ[p, 0
]

Rule 2081

Int[(P3_)^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> With[{a = Coeff[P3, x, 0], b = Coeff[P3, x, 1], c = C
oeff[P3, x, 2], d = Coeff[P3, x, 3]}, Subst[Int[((3*d*e - c*f)/(3*d) + f*x)^m*Simp[(2*c^3 - 9*b*c*d + 27*a*d^2
)/(27*d^2) - ((c^2 - 3*b*d)*x)/(3*d) + d*x^3, x]^p, x], x, x + c/(3*d)] /; NeQ[c, 0]] /; FreeQ[{e, f, m, p}, x
] && PolyQ[P3, x, 3]

Rule 2100

Int[(Pm_)/(Qn_), x_Symbol] :> With[{m = Expon[Pm, x], n = Expon[Qn, x]}, Simp[(Coeff[Pm, x, m]*Log[Qn])/(n*Coe
ff[Qn, x, n]), x] + Dist[1/(n*Coeff[Qn, x, n]), Int[ExpandToSum[n*Coeff[Qn, x, n]*Pm - Coeff[Pm, x, m]*D[Qn, x
], x]/Qn, x], x] /; EqQ[m, n - 1]] /; PolyQ[Pm, x] && PolyQ[Qn, x]

Rule 2523

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*Log[c*RFx^p])^n, x] - Dist[b*n*p
, Int[SimplifyIntegrand[(x*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, p}, x] &
& RationalFunctionQ[RFx, x] && IGtQ[n, 0]

Rule 2528

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

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 \left (-1+\frac {4 \left (1+x^2+2 x^3\right ) \log \left (\frac {x^2 \log (3)}{\left (-1+x^2+x^3\right )^2}\right )}{3 \left (-1+x^2+x^3\right )}-\frac {1}{3} \log ^2\left (\frac {x^2 \log (3)}{\left (-1+x^2+x^3\right )^2}\right )\right ) \, dx\\ &=-x-\frac {1}{3} \int \log ^2\left (\frac {x^2 \log (3)}{\left (-1+x^2+x^3\right )^2}\right ) \, dx+\frac {4}{3} \int \frac {\left (1+x^2+2 x^3\right ) \log \left (\frac {x^2 \log (3)}{\left (-1+x^2+x^3\right )^2}\right )}{-1+x^2+x^3} \, dx\\ &=-x-\frac {1}{3} x \log ^2\left (\frac {x^2 \log (3)}{\left (1-x^2-x^3\right )^2}\right )+\frac {2}{3} \int \frac {2 \left (1+x^2+2 x^3\right ) \log \left (\frac {x^2 \log (3)}{\left (-1+x^2+x^3\right )^2}\right )}{1-x^2-x^3} \, dx+\frac {4}{3} \int \left (2 \log \left (\frac {x^2 \log (3)}{\left (-1+x^2+x^3\right )^2}\right )+\frac {\left (3-x^2\right ) \log \left (\frac {x^2 \log (3)}{\left (-1+x^2+x^3\right )^2}\right )}{-1+x^2+x^3}\right ) \, dx\\ &=-x-\frac {1}{3} x \log ^2\left (\frac {x^2 \log (3)}{\left (1-x^2-x^3\right )^2}\right )+\frac {4}{3} \int \frac {\left (3-x^2\right ) \log \left (\frac {x^2 \log (3)}{\left (-1+x^2+x^3\right )^2}\right )}{-1+x^2+x^3} \, dx+\frac {4}{3} \int \frac {\left (1+x^2+2 x^3\right ) \log \left (\frac {x^2 \log (3)}{\left (-1+x^2+x^3\right )^2}\right )}{1-x^2-x^3} \, dx+\frac {8}{3} \int \log \left (\frac {x^2 \log (3)}{\left (-1+x^2+x^3\right )^2}\right ) \, dx\\ &=-x+\frac {8}{3} x \log \left (\frac {x^2 \log (3)}{\left (1-x^2-x^3\right )^2}\right )-\frac {1}{3} x \log ^2\left (\frac {x^2 \log (3)}{\left (1-x^2-x^3\right )^2}\right )+\frac {4}{3} \int \left (-2 \log \left (\frac {x^2 \log (3)}{\left (-1+x^2+x^3\right )^2}\right )+\frac {\left (3-x^2\right ) \log \left (\frac {x^2 \log (3)}{\left (-1+x^2+x^3\right )^2}\right )}{1-x^2-x^3}\right ) \, dx+\frac {4}{3} \int \left (\frac {3 \log \left (\frac {x^2 \log (3)}{\left (-1+x^2+x^3\right )^2}\right )}{-1+x^2+x^3}-\frac {x^2 \log \left (\frac {x^2 \log (3)}{\left (-1+x^2+x^3\right )^2}\right )}{-1+x^2+x^3}\right ) \, dx-\frac {8}{3} \int \frac {2 \left (1+x^2+2 x^3\right )}{1-x^2-x^3} \, dx\\ &=-x+\frac {8}{3} x \log \left (\frac {x^2 \log (3)}{\left (1-x^2-x^3\right )^2}\right )-\frac {1}{3} x \log ^2\left (\frac {x^2 \log (3)}{\left (1-x^2-x^3\right )^2}\right )+\frac {4}{3} \int \frac {\left (3-x^2\right ) \log \left (\frac {x^2 \log (3)}{\left (-1+x^2+x^3\right )^2}\right )}{1-x^2-x^3} \, dx-\frac {4}{3} \int \frac {x^2 \log \left (\frac {x^2 \log (3)}{\left (-1+x^2+x^3\right )^2}\right )}{-1+x^2+x^3} \, dx-\frac {8}{3} \int \log \left (\frac {x^2 \log (3)}{\left (-1+x^2+x^3\right )^2}\right ) \, dx+4 \int \frac {\log \left (\frac {x^2 \log (3)}{\left (-1+x^2+x^3\right )^2}\right )}{-1+x^2+x^3} \, dx-\frac {16}{3} \int \frac {1+x^2+2 x^3}{1-x^2-x^3} \, dx\\ &=-x-\frac {1}{3} x \log ^2\left (\frac {x^2 \log (3)}{\left (1-x^2-x^3\right )^2}\right )-\frac {4}{3} \int \frac {x^2 \log \left (\frac {x^2 \log (3)}{\left (-1+x^2+x^3\right )^2}\right )}{-1+x^2+x^3} \, dx+\frac {4}{3} \int \left (-\frac {3 \log \left (\frac {x^2 \log (3)}{\left (-1+x^2+x^3\right )^2}\right )}{-1+x^2+x^3}+\frac {x^2 \log \left (\frac {x^2 \log (3)}{\left (-1+x^2+x^3\right )^2}\right )}{-1+x^2+x^3}\right ) \, dx+\frac {8}{3} \int \frac {2 \left (1+x^2+2 x^3\right )}{1-x^2-x^3} \, dx+4 \int \frac {\log \left (\frac {x^2 \log (3)}{\left (-1+x^2+x^3\right )^2}\right )}{-1+x^2+x^3} \, dx-\frac {16}{3} \int \left (-2+\frac {3-x^2}{1-x^2-x^3}\right ) \, dx\\ &=\frac {29 x}{3}-\frac {1}{3} x \log ^2\left (\frac {x^2 \log (3)}{\left (1-x^2-x^3\right )^2}\right )-\frac {16}{3} \int \frac {3-x^2}{1-x^2-x^3} \, dx+\frac {16}{3} \int \frac {1+x^2+2 x^3}{1-x^2-x^3} \, dx\\ &=\frac {29 x}{3}-\frac {16}{9} \log \left (1-x^2-x^3\right )-\frac {1}{3} x \log ^2\left (\frac {x^2 \log (3)}{\left (1-x^2-x^3\right )^2}\right )+\frac {16}{9} \int \frac {-9-2 x}{1-x^2-x^3} \, dx+\frac {16}{3} \int \left (-2+\frac {3-x^2}{1-x^2-x^3}\right ) \, dx\\ &=-x-\frac {16}{9} \log \left (1-x^2-x^3\right )-\frac {1}{3} x \log ^2\left (\frac {x^2 \log (3)}{\left (1-x^2-x^3\right )^2}\right )+\frac {16}{9} \operatorname {Subst}\left (\int \frac {-\frac {25}{3}-2 x}{\frac {25}{27}+\frac {x}{3}-x^3} \, dx,x,\frac {1}{3}+x\right )+\frac {16}{3} \int \frac {3-x^2}{1-x^2-x^3} \, dx\\ &=-x-\frac {1}{3} x \log ^2\left (\frac {x^2 \log (3)}{\left (1-x^2-x^3\right )^2}\right )-\frac {16}{9} \int \frac {-9-2 x}{1-x^2-x^3} \, dx+\frac {16}{9} \operatorname {Subst}\left (\int \frac {-\frac {25}{3}-2 x}{\left (\frac {\frac {2}{\sqrt [3]{25+3 \sqrt {69}}}+\sqrt [3]{50+6 \sqrt {69}}}{3\ 2^{2/3}}-x\right ) \left (\frac {1}{18} \left (-2+2 \left (\frac {2}{25+3 \sqrt {69}}\right )^{2/3}+\sqrt [3]{2} \left (25+3 \sqrt {69}\right )^{2/3}\right )+\frac {1}{3} \left (\sqrt [3]{\frac {2}{25+3 \sqrt {69}}}+\sqrt [3]{\frac {1}{2} \left (25+3 \sqrt {69}\right )}\right ) x+x^2\right )} \, dx,x,\frac {1}{3}+x\right )\\ &=-x-\frac {1}{3} x \log ^2\left (\frac {x^2 \log (3)}{\left (1-x^2-x^3\right )^2}\right )-\frac {16}{9} \operatorname {Subst}\left (\int \frac {-\frac {25}{3}-2 x}{\frac {25}{27}+\frac {x}{3}-x^3} \, dx,x,\frac {1}{3}+x\right )+\frac {16}{9} \operatorname {Subst}\left (\int \left (\frac {36 \left (-2 \sqrt [3]{2}-25 \sqrt [3]{25+3 \sqrt {69}}-\left (50+6 \sqrt {69}\right )^{2/3}\right )}{\left (6+6 \left (\frac {2}{25+3 \sqrt {69}}\right )^{2/3}+2 \sqrt [3]{2} \left (25+3 \sqrt {69}\right )^{2/3}+2^{2/3} \sqrt [3]{623+75 \sqrt {69}}\right ) \left (2 \sqrt [3]{2}+\left (2 \left (25+3 \sqrt {69}\right )\right )^{2/3}-6 \sqrt [3]{25+3 \sqrt {69}} x\right )}+\frac {36 \left (-2 \left (25+3 \sqrt {69}\right )^{2/3}-\left (25-3 \sqrt {69}\right ) \sqrt [3]{50+6 \sqrt {69}}-2^{2/3} \left (623+75 \sqrt {69}\right )-3 \left (25 \left (25+3 \sqrt {69}\right )^{2/3}+2^{2/3} \left (25+3 \sqrt {69}\right )+2 \sqrt [3]{50+6 \sqrt {69}}\right ) x\right )}{\left (6+6 \left (\frac {2}{25+3 \sqrt {69}}\right )^{2/3}+2 \sqrt [3]{2} \left (25+3 \sqrt {69}\right )^{2/3}+2^{2/3} \sqrt [3]{623+75 \sqrt {69}}\right ) \left (2\ 2^{2/3}-2 \left (25+3 \sqrt {69}\right )^{2/3}+\sqrt [3]{2} \left (25+3 \sqrt {69}\right )^{4/3}+3\ 2^{2/3} \left (25+3 \sqrt {69}+2^{2/3} \sqrt [3]{25+3 \sqrt {69}}\right ) x+18 \left (25+3 \sqrt {69}\right )^{2/3} x^2\right )}\right ) \, dx,x,\frac {1}{3}+x\right )\\ &=-x+\frac {32 \left (25+2 \sqrt [3]{\frac {2}{25+3 \sqrt {69}}}+2^{2/3} \sqrt [3]{25+3 \sqrt {69}}\right ) \log \left (2 \sqrt [3]{2}+\left (50+6 \sqrt {69}\right )^{2/3}-2 \sqrt [3]{25+3 \sqrt {69}} (1+3 x)\right )}{3 \left (6+6 \left (\frac {2}{25+3 \sqrt {69}}\right )^{2/3}+2 \sqrt [3]{2} \left (25+3 \sqrt {69}\right )^{2/3}+2^{2/3} \sqrt [3]{623+75 \sqrt {69}}\right )}-\frac {1}{3} x \log ^2\left (\frac {x^2 \log (3)}{\left (1-x^2-x^3\right )^2}\right )-\frac {16}{9} \operatorname {Subst}\left (\int \frac {-\frac {25}{3}-2 x}{\left (\frac {\frac {2}{\sqrt [3]{25+3 \sqrt {69}}}+\sqrt [3]{50+6 \sqrt {69}}}{3\ 2^{2/3}}-x\right ) \left (\frac {1}{18} \left (-2+2 \left (\frac {2}{25+3 \sqrt {69}}\right )^{2/3}+\sqrt [3]{2} \left (25+3 \sqrt {69}\right )^{2/3}\right )+\frac {1}{3} \left (\sqrt [3]{\frac {2}{25+3 \sqrt {69}}}+\sqrt [3]{\frac {1}{2} \left (25+3 \sqrt {69}\right )}\right ) x+x^2\right )} \, dx,x,\frac {1}{3}+x\right )+\frac {64 \operatorname {Subst}\left (\int \frac {-2 \left (25+3 \sqrt {69}\right )^{2/3}-\left (25-3 \sqrt {69}\right ) \sqrt [3]{50+6 \sqrt {69}}-2^{2/3} \left (623+75 \sqrt {69}\right )-3 \left (25 \left (25+3 \sqrt {69}\right )^{2/3}+2^{2/3} \left (25+3 \sqrt {69}\right )+2 \sqrt [3]{50+6 \sqrt {69}}\right ) x}{2\ 2^{2/3}-2 \left (25+3 \sqrt {69}\right )^{2/3}+\sqrt [3]{2} \left (25+3 \sqrt {69}\right )^{4/3}+3\ 2^{2/3} \left (25+3 \sqrt {69}+2^{2/3} \sqrt [3]{25+3 \sqrt {69}}\right ) x+18 \left (25+3 \sqrt {69}\right )^{2/3} x^2} \, dx,x,\frac {1}{3}+x\right )}{6+6 \left (\frac {2}{25+3 \sqrt {69}}\right )^{2/3}+2 \sqrt [3]{2} \left (25+3 \sqrt {69}\right )^{2/3}+2^{2/3} \sqrt [3]{623+75 \sqrt {69}}}\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [C]  time = 2.67, size = 223, normalized size = 8.26 \begin {gather*} \frac {x \left (3+\log ^2\left (\frac {x^2 \log (3)}{\left (-1+x^2+x^3\right )^2}\right )\right ) \left (3+2 \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-1\&,1\right ] \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-1\&,2\right ]^2+2 \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-1\&,1\right ]^2 \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-1\&,3\right ]+2 \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-1\&,2\right ] \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-1\&,3\right ]^2\right )}{3 \left (\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-1\&,1\right ]-\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-1\&,2\right ]\right ) \left (\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-1\&,1\right ]-\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-1\&,3\right ]\right ) \left (\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-1\&,2\right ]-\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-1\&,3\right ]\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 - 3*x^2 - 3*x^3 + (4 + 4*x^2 + 8*x^3)*Log[(x^2*Log[3])/(1 - 2*x^2 - 2*x^3 + x^4 + 2*x^5 + x^6)] +
(1 - x^2 - x^3)*Log[(x^2*Log[3])/(1 - 2*x^2 - 2*x^3 + x^4 + 2*x^5 + x^6)]^2)/(-3 + 3*x^2 + 3*x^3),x]

[Out]

(x*(3 + Log[(x^2*Log[3])/(-1 + x^2 + x^3)^2]^2)*(3 + 2*Root[-1 + #1^2 + #1^3 & , 1, 0]*Root[-1 + #1^2 + #1^3 &
, 2, 0]^2 + 2*Root[-1 + #1^2 + #1^3 & , 1, 0]^2*Root[-1 + #1^2 + #1^3 & , 3, 0] + 2*Root[-1 + #1^2 + #1^3 & ,
2, 0]*Root[-1 + #1^2 + #1^3 & , 3, 0]^2))/(3*(Root[-1 + #1^2 + #1^3 & , 1, 0] - Root[-1 + #1^2 + #1^3 & , 2,
0])*(Root[-1 + #1^2 + #1^3 & , 1, 0] - Root[-1 + #1^2 + #1^3 & , 3, 0])*(Root[-1 + #1^2 + #1^3 & , 2, 0] - Roo
t[-1 + #1^2 + #1^3 & , 3, 0]))

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fricas [A]  time = 0.52, size = 41, normalized size = 1.52 \begin {gather*} -\frac {1}{3} \, x \log \left (\frac {x^{2} \log \relax (3)}{x^{6} + 2 \, x^{5} + x^{4} - 2 \, x^{3} - 2 \, x^{2} + 1}\right )^{2} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^3-x^2+1)*log(x^2*log(3)/(x^6+2*x^5+x^4-2*x^3-2*x^2+1))^2+(8*x^3+4*x^2+4)*log(x^2*log(3)/(x^6+2*
x^5+x^4-2*x^3-2*x^2+1))-3*x^3-3*x^2+3)/(3*x^3+3*x^2-3),x, algorithm="fricas")

[Out]

-1/3*x*log(x^2*log(3)/(x^6 + 2*x^5 + x^4 - 2*x^3 - 2*x^2 + 1))^2 - x

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giac [B]  time = 1.12, size = 79, normalized size = 2.93 \begin {gather*} -\frac {1}{3} \, x \log \left (x^{6} + 2 \, x^{5} + x^{4} - 2 \, x^{3} - 2 \, x^{2} + 1\right )^{2} + \frac {2}{3} \, x \log \left (x^{6} + 2 \, x^{5} + x^{4} - 2 \, x^{3} - 2 \, x^{2} + 1\right ) \log \left (x^{2} \log \relax (3)\right ) - \frac {1}{3} \, x \log \left (x^{2} \log \relax (3)\right )^{2} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^3-x^2+1)*log(x^2*log(3)/(x^6+2*x^5+x^4-2*x^3-2*x^2+1))^2+(8*x^3+4*x^2+4)*log(x^2*log(3)/(x^6+2*
x^5+x^4-2*x^3-2*x^2+1))-3*x^3-3*x^2+3)/(3*x^3+3*x^2-3),x, algorithm="giac")

[Out]

-1/3*x*log(x^6 + 2*x^5 + x^4 - 2*x^3 - 2*x^2 + 1)^2 + 2/3*x*log(x^6 + 2*x^5 + x^4 - 2*x^3 - 2*x^2 + 1)*log(x^2
*log(3)) - 1/3*x*log(x^2*log(3))^2 - x

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maple [A]  time = 0.29, size = 42, normalized size = 1.56

method	result	size
norman	\(-x -\frac {x \ln \left (\frac {x^{2} \ln \relax (3)}{x^{6}+2 x^{5}+x^{4}-2 x^{3}-2 x^{2}+1}\right )^{2}}{3}\)	\(42\)
risch	\(-x -\frac {x \ln \left (\frac {x^{2} \ln \relax (3)}{x^{6}+2 x^{5}+x^{4}-2 x^{3}-2 x^{2}+1}\right )^{2}}{3}\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^3-x^2+1)*ln(x^2*ln(3)/(x^6+2*x^5+x^4-2*x^3-2*x^2+1))^2+(8*x^3+4*x^2+4)*ln(x^2*ln(3)/(x^6+2*x^5+x^4-2*
x^3-2*x^2+1))-3*x^3-3*x^2+3)/(3*x^3+3*x^2-3),x,method=_RETURNVERBOSE)

[Out]

-x-1/3*x*ln(x^2*ln(3)/(x^6+2*x^5+x^4-2*x^3-2*x^2+1))^2

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maxima [B]  time = 0.43, size = 62, normalized size = 2.30 \begin {gather*} -\frac {4}{3} \, x \log \left (x^{3} + x^{2} - 1\right )^{2} - \frac {4}{3} \, x \log \relax (x)^{2} - \frac {4}{3} \, x \log \relax (x) \log \left (\log \relax (3)\right ) - \frac {1}{3} \, {\left (\log \left (\log \relax (3)\right )^{2} + 3\right )} x + \frac {4}{3} \, {\left (2 \, x \log \relax (x) + x \log \left (\log \relax (3)\right )\right )} \log \left (x^{3} + x^{2} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^3-x^2+1)*log(x^2*log(3)/(x^6+2*x^5+x^4-2*x^3-2*x^2+1))^2+(8*x^3+4*x^2+4)*log(x^2*log(3)/(x^6+2*
x^5+x^4-2*x^3-2*x^2+1))-3*x^3-3*x^2+3)/(3*x^3+3*x^2-3),x, algorithm="maxima")

[Out]

-4/3*x*log(x^3 + x^2 - 1)^2 - 4/3*x*log(x)^2 - 4/3*x*log(x)*log(log(3)) - 1/3*(log(log(3))^2 + 3)*x + 4/3*(2*x
*log(x) + x*log(log(3)))*log(x^3 + x^2 - 1)

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mupad [B]  time = 1.73, size = 39, normalized size = 1.44 \begin {gather*} -\frac {x\,\left ({\ln \left (\frac {x^2\,\ln \relax (3)}{x^6+2\,x^5+x^4-2\,x^3-2\,x^2+1}\right )}^2+3\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(3*x^2 - log((x^2*log(3))/(x^4 - 2*x^3 - 2*x^2 + 2*x^5 + x^6 + 1))*(4*x^2 + 8*x^3 + 4) + 3*x^3 + log((x^2
*log(3))/(x^4 - 2*x^3 - 2*x^2 + 2*x^5 + x^6 + 1))^2*(x^2 + x^3 - 1) - 3)/(3*x^2 + 3*x^3 - 3),x)

[Out]

-(x*(log((x^2*log(3))/(x^4 - 2*x^3 - 2*x^2 + 2*x^5 + x^6 + 1))^2 + 3))/3

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sympy [A]  time = 0.22, size = 39, normalized size = 1.44 \begin {gather*} - \frac {x \log {\left (\frac {x^{2} \log {\relax (3 )}}{x^{6} + 2 x^{5} + x^{4} - 2 x^{3} - 2 x^{2} + 1} \right )}^{2}}{3} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**3-x**2+1)*ln(x**2*ln(3)/(x**6+2*x**5+x**4-2*x**3-2*x**2+1))**2+(8*x**3+4*x**2+4)*ln(x**2*ln(3)
/(x**6+2*x**5+x**4-2*x**3-2*x**2+1))-3*x**3-3*x**2+3)/(3*x**3+3*x**2-3),x)

[Out]

-x*log(x**2*log(3)/(x**6 + 2*x**5 + x**4 - 2*x**3 - 2*x**2 + 1))**2/3 - x

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