\(\int \frac {-4+9 x-2 x^2+(-x^2+2 x^3) \log (2)+(-12+x-x^2+(24 x-2 x^2+2 x^3) \log (2)) \log (-12+x-x^2)}{12-x+x^2} \, dx\) [9027]

Optimal result

Integrand size = 70, antiderivative size = 21 \[ \int \frac {-4+9 x-2 x^2+\left (-x^2+2 x^3\right ) \log (2)+\left (-12+x-x^2+\left (24 x-2 x^2+2 x^3\right ) \log (2)\right ) \log \left (-12+x-x^2\right )}{12-x+x^2} \, dx=\left (4-x+x^2 \log (2)\right ) \log \left (-12+x-x^2\right ) \]

[Out]

ln(-x^2+x-12)*(4-x+x^2*ln(2))

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(162\) vs. \(2(21)=42\).

Time = 0.21 (sec) , antiderivative size = 162, normalized size of antiderivative = 7.71, number of steps used = 15, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {6860, 1642, 648, 632, 210, 642, 2605, 814} \[ \int \frac {-4+9 x-2 x^2+\left (-x^2+2 x^3\right ) \log (2)+\left (-12+x-x^2+\left (24 x-2 x^2+2 x^3\right ) \log (2)\right ) \log \left (-12+x-x^2\right )}{12-x+x^2} \, dx=\frac {1}{2} \sqrt {47} (2-\log (4)) \arctan \left (\frac {1-2 x}{\sqrt {47}}\right )-\sqrt {47} (1-\log (2)) \arctan \left (\frac {1-2 x}{\sqrt {47}}\right )-\frac {\left (2-23 \log ^2(4)-\log (16)\right ) \log \left (x^2-x+12\right )}{4 \log (4)}-\frac {1}{2} x^2 \log (4)+x^2 \log (2)+\frac {(1-x \log (4))^2 \log \left (-x^2+x-12\right )}{2 \log (4)}+\frac {1}{2} (7-23 \log (2)) \log \left (x^2-x+12\right )+\frac {1}{2} x (4-\log (4))-x (2-\log (2)) \]

[In]

Int[(-4 + 9*x - 2*x^2 + (-x^2 + 2*x^3)*Log[2] + (-12 + x - x^2 + (24*x - 2*x^2 + 2*x^3)*Log[2])*Log[-12 + x -
x^2])/(12 - x + x^2),x]

[Out]

-(Sqrt[47]*ArcTan[(1 - 2*x)/Sqrt[47]]*(1 - Log[2])) - x*(2 - Log[2]) + x^2*Log[2] + (Sqrt[47]*ArcTan[(1 - 2*x)
/Sqrt[47]]*(2 - Log[4]))/2 + (x*(4 - Log[4]))/2 - (x^2*Log[4])/2 + ((1 - x*Log[4])^2*Log[-12 + x - x^2])/(2*Lo
g[4]) + ((7 - 23*Log[2])*Log[12 - x + x^2])/2 - ((2 - 23*Log[4]^2 - Log[16])*Log[12 - x + x^2])/(4*Log[4])

Rule 210

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

Rule 632

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 642

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 648

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 814

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 1642

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 2605

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[(d + e*x)^(m +
1)*((a + b*Log[c*RFx^p])^n/(e*(m + 1))), x] - Dist[b*n*(p/(e*(m + 1))), Int[SimplifyIntegrand[(d + e*x)^(m +
1)*(a + b*Log[c*RFx^p])^(n - 1)*(D[RFx, x]/RFx), x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && RationalFunc
tionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 6860

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

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(-1+2 x) \left (4-x+x^2 \log (2)\right )}{12-x+x^2}+(-1+x \log (4)) \log \left (-12+x-x^2\right )\right ) \, dx \\ & = \int \frac {(-1+2 x) \left (4-x+x^2 \log (2)\right )}{12-x+x^2} \, dx+\int (-1+x \log (4)) \log \left (-12+x-x^2\right ) \, dx \\ & = \frac {(1-x \log (4))^2 \log \left (-12+x-x^2\right )}{2 \log (4)}-\frac {\int \frac {(1-2 x) (-1+x \log (4))^2}{-12+x-x^2} \, dx}{2 \log (4)}+\int \left (-2+\frac {x (7-23 \log (2))+4 (5-3 \log (2))}{12-x+x^2}+\log (2)+2 x \log (2)\right ) \, dx \\ & = -x (2-\log (2))+x^2 \log (2)+\frac {(1-x \log (4))^2 \log \left (-12+x-x^2\right )}{2 \log (4)}-\frac {\int \left (-((4-\log (4)) \log (4))+2 x \log ^2(4)-\frac {-1+48 \log (4)-12 \log ^2(4)+x \left (2-2 \log (4)-23 \log ^2(4)\right )}{-12+x-x^2}\right ) \, dx}{2 \log (4)}+\int \frac {x (7-23 \log (2))+4 (5-3 \log (2))}{12-x+x^2} \, dx \\ & = -x (2-\log (2))+x^2 \log (2)+\frac {1}{2} x (4-\log (4))-\frac {1}{2} x^2 \log (4)+\frac {(1-x \log (4))^2 \log \left (-12+x-x^2\right )}{2 \log (4)}+\frac {1}{2} (7-23 \log (2)) \int \frac {-1+2 x}{12-x+x^2} \, dx+\frac {1}{2} (47 (1-\log (2))) \int \frac {1}{12-x+x^2} \, dx+\frac {\int \frac {-1+48 \log (4)-12 \log ^2(4)+x \left (2-2 \log (4)-23 \log ^2(4)\right )}{-12+x-x^2} \, dx}{2 \log (4)} \\ & = -x (2-\log (2))+x^2 \log (2)+\frac {1}{2} x (4-\log (4))-\frac {1}{2} x^2 \log (4)+\frac {(1-x \log (4))^2 \log \left (-12+x-x^2\right )}{2 \log (4)}+\frac {1}{2} (7-23 \log (2)) \log \left (12-x+x^2\right )-(47 (1-\log (2))) \text {Subst}\left (\int \frac {1}{-47-x^2} \, dx,x,-1+2 x\right )+\frac {1}{4} (47 (2-\log (4))) \int \frac {1}{-12+x-x^2} \, dx+\frac {\left (-2+23 \log ^2(4)+\log (16)\right ) \int \frac {1-2 x}{-12+x-x^2} \, dx}{4 \log (4)} \\ & = -\sqrt {47} \arctan \left (\frac {1-2 x}{\sqrt {47}}\right ) (1-\log (2))-x (2-\log (2))+x^2 \log (2)+\frac {1}{2} x (4-\log (4))-\frac {1}{2} x^2 \log (4)+\frac {(1-x \log (4))^2 \log \left (-12+x-x^2\right )}{2 \log (4)}+\frac {1}{2} (7-23 \log (2)) \log \left (12-x+x^2\right )-\frac {\left (2-23 \log ^2(4)-\log (16)\right ) \log \left (12-x+x^2\right )}{4 \log (4)}-\frac {1}{2} (47 (2-\log (4))) \text {Subst}\left (\int \frac {1}{-47-x^2} \, dx,x,1-2 x\right ) \\ & = -\sqrt {47} \arctan \left (\frac {1-2 x}{\sqrt {47}}\right ) (1-\log (2))-x (2-\log (2))+x^2 \log (2)+\frac {1}{2} \sqrt {47} \arctan \left (\frac {1-2 x}{\sqrt {47}}\right ) (2-\log (4))+\frac {1}{2} x (4-\log (4))-\frac {1}{2} x^2 \log (4)+\frac {(1-x \log (4))^2 \log \left (-12+x-x^2\right )}{2 \log (4)}+\frac {1}{2} (7-23 \log (2)) \log \left (12-x+x^2\right )-\frac {\left (2-23 \log ^2(4)-\log (16)\right ) \log \left (12-x+x^2\right )}{4 \log (4)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.38 \[ \int \frac {-4+9 x-2 x^2+\left (-x^2+2 x^3\right ) \log (2)+\left (-12+x-x^2+\left (24 x-2 x^2+2 x^3\right ) \log (2)\right ) \log \left (-12+x-x^2\right )}{12-x+x^2} \, dx=x (-1+x \log (2)) \log \left (-12+x-x^2\right )+4 \log \left (12-x+x^2\right ) \]

[In]

Integrate[(-4 + 9*x - 2*x^2 + (-x^2 + 2*x^3)*Log[2] + (-12 + x - x^2 + (24*x - 2*x^2 + 2*x^3)*Log[2])*Log[-12
+ x - x^2])/(12 - x + x^2),x]

[Out]

x*(-1 + x*Log[2])*Log[-12 + x - x^2] + 4*Log[12 - x + x^2]

Maple [A] (verified)

Time = 4.17 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.57

method	result	size
risch	\(\left (x^{2} \ln \left (2\right )-x \right ) \ln \left (-x^{2}+x -12\right )+4 \ln \left (x^{2}-x +12\right )\)	\(33\)
default	\(\ln \left (2\right ) \ln \left (-x^{2}+x -12\right ) x^{2}+4 \ln \left (x^{2}-x +12\right )-\ln \left (-x^{2}+x -12\right ) x\)	\(40\)
norman	\(4 \ln \left (-x^{2}+x -12\right )+\ln \left (2\right ) \ln \left (-x^{2}+x -12\right ) x^{2}-\ln \left (-x^{2}+x -12\right ) x\)	\(40\)
parallelrisch	\(4 \ln \left (-x^{2}+x -12\right )+\ln \left (2\right ) \ln \left (-x^{2}+x -12\right ) x^{2}-\ln \left (-x^{2}+x -12\right ) x\)	\(40\)
parts	\(2 \ln \left (2\right ) \left (\frac {\ln \left (-x^{2}+x -12\right ) x^{2}}{2}-\frac {x}{2}-\frac {x^{2}}{2}+\frac {23 \ln \left (x^{2}-x +12\right )}{4}+\frac {\sqrt {47}\, \arctan \left (\frac {\left (-1+2 x \right ) \sqrt {47}}{47}\right )}{2}\right )-\ln \left (-x^{2}+x -12\right ) x +\frac {\ln \left (x^{2}-x +12\right )}{2}-\sqrt {47}\, \arctan \left (\frac {\left (-1+2 x \right ) \sqrt {47}}{47}\right )+x^{2} \ln \left (2\right )+x \ln \left (2\right )+\frac {\left (7-23 \ln \left (2\right )\right ) \ln \left (x^{2}-x +12\right )}{2}+\frac {2 \left (\frac {47}{2}-\frac {47 \ln \left (2\right )}{2}\right ) \sqrt {47}\, \arctan \left (\frac {\left (-1+2 x \right ) \sqrt {47}}{47}\right )}{47}\)	\(144\)

[In]

int((((2*x^3-2*x^2+24*x)*ln(2)-x^2+x-12)*ln(-x^2+x-12)+(2*x^3-x^2)*ln(2)-2*x^2+9*x-4)/(x^2-x+12),x,method=_RET
URNVERBOSE)

[Out]

(x^2*ln(2)-x)*ln(-x^2+x-12)+4*ln(x^2-x+12)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {-4+9 x-2 x^2+\left (-x^2+2 x^3\right ) \log (2)+\left (-12+x-x^2+\left (24 x-2 x^2+2 x^3\right ) \log (2)\right ) \log \left (-12+x-x^2\right )}{12-x+x^2} \, dx={\left (x^{2} \log \left (2\right ) - x + 4\right )} \log \left (-x^{2} + x - 12\right ) \]

[In]

integrate((((2*x^3-2*x^2+24*x)*log(2)-x^2+x-12)*log(-x^2+x-12)+(2*x^3-x^2)*log(2)-2*x^2+9*x-4)/(x^2-x+12),x, a
lgorithm="fricas")

[Out]

(x^2*log(2) - x + 4)*log(-x^2 + x - 12)

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.24 \[ \int \frac {-4+9 x-2 x^2+\left (-x^2+2 x^3\right ) \log (2)+\left (-12+x-x^2+\left (24 x-2 x^2+2 x^3\right ) \log (2)\right ) \log \left (-12+x-x^2\right )}{12-x+x^2} \, dx=\left (x^{2} \log {\left (2 \right )} - x\right ) \log {\left (- x^{2} + x - 12 \right )} + 4 \log {\left (x^{2} - x + 12 \right )} \]

[In]

integrate((((2*x**3-2*x**2+24*x)*ln(2)-x**2+x-12)*ln(-x**2+x-12)+(2*x**3-x**2)*ln(2)-2*x**2+9*x-4)/(x**2-x+12)
,x)

[Out]

(x**2*log(2) - x)*log(-x**2 + x - 12) + 4*log(x**2 - x + 12)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 165 vs. \(2 (21) = 42\).

Time = 0.51 (sec) , antiderivative size = 165, normalized size of antiderivative = 7.86 \[ \int \frac {-4+9 x-2 x^2+\left (-x^2+2 x^3\right ) \log (2)+\left (-12+x-x^2+\left (24 x-2 x^2+2 x^3\right ) \log (2)\right ) \log \left (-12+x-x^2\right )}{12-x+x^2} \, dx=-x^{2} \log \left (2\right ) + \sqrt {47} {\left (\log \left (2\right ) - 1\right )} \arctan \left (\frac {1}{47} \, \sqrt {47} {\left (2 \, x - 1\right )}\right ) - x {\left (\log \left (2\right ) - 2\right )} + \frac {1}{47} \, {\left (47 \, x^{2} - 70 \, \sqrt {47} \arctan \left (\frac {1}{47} \, \sqrt {47} {\left (2 \, x - 1\right )}\right ) + 94 \, x - 517 \, \log \left (x^{2} - x + 12\right )\right )} \log \left (2\right ) + \frac {1}{94} \, {\left (46 \, \sqrt {47} \arctan \left (\frac {1}{47} \, \sqrt {47} {\left (2 \, x - 1\right )}\right ) - 94 \, x - 47 \, \log \left (x^{2} - x + 12\right )\right )} \log \left (2\right ) + \frac {1}{2} \, {\left (2 \, x^{2} \log \left (2\right ) - 2 \, x + 23 \, \log \left (2\right ) + 1\right )} \log \left (-x^{2} + x - 12\right ) + \sqrt {47} \arctan \left (\frac {1}{47} \, \sqrt {47} {\left (2 \, x - 1\right )}\right ) - 2 \, x + \frac {7}{2} \, \log \left (x^{2} - x + 12\right ) \]

[In]

integrate((((2*x^3-2*x^2+24*x)*log(2)-x^2+x-12)*log(-x^2+x-12)+(2*x^3-x^2)*log(2)-2*x^2+9*x-4)/(x^2-x+12),x, a
lgorithm="maxima")

[Out]

-x^2*log(2) + sqrt(47)*(log(2) - 1)*arctan(1/47*sqrt(47)*(2*x - 1)) - x*(log(2) - 2) + 1/47*(47*x^2 - 70*sqrt(
47)*arctan(1/47*sqrt(47)*(2*x - 1)) + 94*x - 517*log(x^2 - x + 12))*log(2) + 1/94*(46*sqrt(47)*arctan(1/47*sqr
t(47)*(2*x - 1)) - 94*x - 47*log(x^2 - x + 12))*log(2) + 1/2*(2*x^2*log(2) - 2*x + 23*log(2) + 1)*log(-x^2 + x
- 12) + sqrt(47)*arctan(1/47*sqrt(47)*(2*x - 1)) - 2*x + 7/2*log(x^2 - x + 12)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.52 \[ \int \frac {-4+9 x-2 x^2+\left (-x^2+2 x^3\right ) \log (2)+\left (-12+x-x^2+\left (24 x-2 x^2+2 x^3\right ) \log (2)\right ) \log \left (-12+x-x^2\right )}{12-x+x^2} \, dx={\left (x^{2} \log \left (2\right ) - x\right )} \log \left (-x^{2} + x - 12\right ) + 4 \, \log \left (x^{2} - x + 12\right ) \]

[In]

integrate((((2*x^3-2*x^2+24*x)*log(2)-x^2+x-12)*log(-x^2+x-12)+(2*x^3-x^2)*log(2)-2*x^2+9*x-4)/(x^2-x+12),x, a
lgorithm="giac")

[Out]

(x^2*log(2) - x)*log(-x^2 + x - 12) + 4*log(x^2 - x + 12)

Mupad [B] (verification not implemented)

Time = 0.37 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {-4+9 x-2 x^2+\left (-x^2+2 x^3\right ) \log (2)+\left (-12+x-x^2+\left (24 x-2 x^2+2 x^3\right ) \log (2)\right ) \log \left (-12+x-x^2\right )}{12-x+x^2} \, dx=\ln \left (-x^2+x-12\right )\,\left (\ln \left (2\right )\,x^2-x+4\right ) \]

[In]

int(-(log(2)*(x^2 - 2*x^3) - 9*x - log(x - x^2 - 12)*(x + log(2)*(24*x - 2*x^2 + 2*x^3) - x^2 - 12) + 2*x^2 +
4)/(x^2 - x + 12),x)

[Out]

log(x - x^2 - 12)*(x^2*log(2) - x + 4)