3.97.36 \(\int \frac {-4+4 x^2+(4-3 x+4 x^2) \log (\frac {-4+3 x-4 x^2}{2 x})}{4-3 x+4 x^2} \, dx\)

Optimal. Leaf size=20 \[ x \log \left (2 \left (1-x-\frac {4+x}{4 x}\right )\right ) \]

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Rubi [A]  time = 0.15, antiderivative size = 15, normalized size of antiderivative = 0.75, number of steps used = 16, number of rules used = 8, integrand size = 49, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.163, Rules used = {6688, 1657, 634, 618, 204, 628, 2523, 12} \begin {gather*} x \log \left (-2 x-\frac {2}{x}+\frac {3}{2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x*Log[3/2 - 2/x - 2*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 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 1657

Int[(Pq_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x + c*x^2)^p, x
], x] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

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 6688

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {4 \left (-1+x^2\right )}{4-3 x+4 x^2}+\log \left (\frac {3}{2}-\frac {2}{x}-2 x\right )\right ) \, dx\\ &=4 \int \frac {-1+x^2}{4-3 x+4 x^2} \, dx+\int \log \left (\frac {3}{2}-\frac {2}{x}-2 x\right ) \, dx\\ &=x \log \left (\frac {3}{2}-\frac {2}{x}-2 x\right )+4 \int \left (\frac {1}{4}-\frac {8-3 x}{4 \left (4-3 x+4 x^2\right )}\right ) \, dx-\int \frac {4 \left (-1+x^2\right )}{4-3 x+4 x^2} \, dx\\ &=x+x \log \left (\frac {3}{2}-\frac {2}{x}-2 x\right )-4 \int \frac {-1+x^2}{4-3 x+4 x^2} \, dx-\int \frac {8-3 x}{4-3 x+4 x^2} \, dx\\ &=x+x \log \left (\frac {3}{2}-\frac {2}{x}-2 x\right )+\frac {3}{8} \int \frac {-3+8 x}{4-3 x+4 x^2} \, dx-4 \int \left (\frac {1}{4}-\frac {8-3 x}{4 \left (4-3 x+4 x^2\right )}\right ) \, dx-\frac {55}{8} \int \frac {1}{4-3 x+4 x^2} \, dx\\ &=x \log \left (\frac {3}{2}-\frac {2}{x}-2 x\right )+\frac {3}{8} \log \left (4-3 x+4 x^2\right )+\frac {55}{4} \operatorname {Subst}\left (\int \frac {1}{-55-x^2} \, dx,x,-3+8 x\right )+\int \frac {8-3 x}{4-3 x+4 x^2} \, dx\\ &=\frac {1}{4} \sqrt {55} \tan ^{-1}\left (\frac {3-8 x}{\sqrt {55}}\right )+x \log \left (\frac {3}{2}-\frac {2}{x}-2 x\right )+\frac {3}{8} \log \left (4-3 x+4 x^2\right )-\frac {3}{8} \int \frac {-3+8 x}{4-3 x+4 x^2} \, dx+\frac {55}{8} \int \frac {1}{4-3 x+4 x^2} \, dx\\ &=\frac {1}{4} \sqrt {55} \tan ^{-1}\left (\frac {3-8 x}{\sqrt {55}}\right )+x \log \left (\frac {3}{2}-\frac {2}{x}-2 x\right )-\frac {55}{4} \operatorname {Subst}\left (\int \frac {1}{-55-x^2} \, dx,x,-3+8 x\right )\\ &=x \log \left (\frac {3}{2}-\frac {2}{x}-2 x\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 15, normalized size = 0.75 \begin {gather*} x \log \left (\frac {3}{2}-\frac {2}{x}-2 x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x*Log[3/2 - 2/x - 2*x]

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fricas [A]  time = 0.58, size = 18, normalized size = 0.90 \begin {gather*} x \log \left (-\frac {4 \, x^{2} - 3 \, x + 4}{2 \, x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*log(-1/2*(4*x^2 - 3*x + 4)/x)

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giac [A]  time = 0.21, size = 18, normalized size = 0.90 \begin {gather*} x \log \left (-\frac {4 \, x^{2} - 3 \, x + 4}{2 \, x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*log(-1/2*(4*x^2 - 3*x + 4)/x)

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maple [A]  time = 0.30, size = 19, normalized size = 0.95

method	result	size
norman	\(x \ln \left (\frac {-4 x^{2}+3 x -4}{2 x}\right )\)	\(19\)
risch	\(x \ln \left (\frac {-4 x^{2}+3 x -4}{2 x}\right )\)	\(19\)
default	\(-x \ln \relax (2)+x \ln \left (\frac {-4 x^{2}+3 x -4}{x}\right )\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.69, size = 24, normalized size = 1.20 \begin {gather*} -x \log \relax (2) + x \log \left (-4 \, x^{2} + 3 \, x - 4\right ) - x \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 5.58, size = 18, normalized size = 0.90 \begin {gather*} x\,\ln \left (-\frac {2\,x^2-\frac {3\,x}{2}+2}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*log(-(2*x^2 - (3*x)/2 + 2)/x)

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sympy [A]  time = 0.15, size = 15, normalized size = 0.75 \begin {gather*} x \log {\left (\frac {- 2 x^{2} + \frac {3 x}{2} - 2}{x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*log((-2*x**2 + 3*x/2 - 2)/x)

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