3.30.22 \(\int \frac {-240+380 x-160 x^2+20 x^3+(-7 x^2+6 x^3-x^4) \log ^2(2)+(-60+80 x-20 x^2) \log (3-4 x+x^2)}{(12 x^2-19 x^3+8 x^4-x^5) \log ^2(2)+(3 x^2-4 x^3+x^4) \log ^2(2) \log (3-4 x+x^2)} \, dx\)

Optimal. Leaf size=30 \[ \frac {20}{x \log ^2(2)}+\log (4-x+\log (3-x-(3-x) x)) \]

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Rubi [A]  time = 0.76, antiderivative size = 25, normalized size of antiderivative = 0.83, number of steps used = 5, number of rules used = 4, integrand size = 114, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.035, Rules used = {6741, 12, 6728, 6684} \begin {gather*} \log \left (\log \left (x^2-4 x+3\right )-x+4\right )+\frac {20}{x \log ^2(2)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-240 + 380*x - 160*x^2 + 20*x^3 + (-7*x^2 + 6*x^3 - x^4)*Log[2]^2 + (-60 + 80*x - 20*x^2)*Log[3 - 4*x + x
^2])/((12*x^2 - 19*x^3 + 8*x^4 - x^5)*Log[2]^2 + (3*x^2 - 4*x^3 + x^4)*Log[2]^2*Log[3 - 4*x + x^2]),x]

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rule 6728

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]

Rule 6741

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-240+380 x-160 x^2+20 x^3+\left (-7 x^2+6 x^3-x^4\right ) \log ^2(2)+\left (-60+80 x-20 x^2\right ) \log \left (3-4 x+x^2\right )}{x^2 \left (3-4 x+x^2\right ) \log ^2(2) \left (4-x+\log \left (3-4 x+x^2\right )\right )} \, dx\\ &=\frac {\int \frac {-240+380 x-160 x^2+20 x^3+\left (-7 x^2+6 x^3-x^4\right ) \log ^2(2)+\left (-60+80 x-20 x^2\right ) \log \left (3-4 x+x^2\right )}{x^2 \left (3-4 x+x^2\right ) \left (4-x+\log \left (3-4 x+x^2\right )\right )} \, dx}{\log ^2(2)}\\ &=\frac {\int \left (-\frac {20}{x^2}+\frac {\left (7-6 x+x^2\right ) \log ^2(2)}{(-3+x) (-1+x) \left (-4+x-\log \left (3-4 x+x^2\right )\right )}\right ) \, dx}{\log ^2(2)}\\ &=\frac {20}{x \log ^2(2)}+\int \frac {7-6 x+x^2}{(-3+x) (-1+x) \left (-4+x-\log \left (3-4 x+x^2\right )\right )} \, dx\\ &=\frac {20}{x \log ^2(2)}+\log \left (4-x+\log \left (3-4 x+x^2\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 31, normalized size = 1.03 \begin {gather*} \frac {\frac {20}{x}+\log ^2(2) \log \left (4-x+\log \left (3-4 x+x^2\right )\right )}{\log ^2(2)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-240 + 380*x - 160*x^2 + 20*x^3 + (-7*x^2 + 6*x^3 - x^4)*Log[2]^2 + (-60 + 80*x - 20*x^2)*Log[3 - 4
*x + x^2])/((12*x^2 - 19*x^3 + 8*x^4 - x^5)*Log[2]^2 + (3*x^2 - 4*x^3 + x^4)*Log[2]^2*Log[3 - 4*x + x^2]),x]

[Out]

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

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fricas [A]  time = 0.94, size = 31, normalized size = 1.03 \begin {gather*} \frac {x \log \relax (2)^{2} \log \left (-x + \log \left (x^{2} - 4 \, x + 3\right ) + 4\right ) + 20}{x \log \relax (2)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-20*x^2+80*x-60)*log(x^2-4*x+3)+(-x^4+6*x^3-7*x^2)*log(2)^2+20*x^3-160*x^2+380*x-240)/((x^4-4*x^3+
3*x^2)*log(2)^2*log(x^2-4*x+3)+(-x^5+8*x^4-19*x^3+12*x^2)*log(2)^2),x, algorithm="fricas")

[Out]

(x*log(2)^2*log(-x + log(x^2 - 4*x + 3) + 4) + 20)/(x*log(2)^2)

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giac [A]  time = 0.22, size = 25, normalized size = 0.83 \begin {gather*} \frac {20}{x \log \relax (2)^{2}} + \log \left (x - \log \left (x^{2} - 4 \, x + 3\right ) - 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-20*x^2+80*x-60)*log(x^2-4*x+3)+(-x^4+6*x^3-7*x^2)*log(2)^2+20*x^3-160*x^2+380*x-240)/((x^4-4*x^3+
3*x^2)*log(2)^2*log(x^2-4*x+3)+(-x^5+8*x^4-19*x^3+12*x^2)*log(2)^2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.18, size = 26, normalized size = 0.87

method	result	size
default	\(\frac {20}{\ln \relax (2)^{2} x}+\ln \left (x -\ln \left (x^{2}-4 x +3\right )-4\right )\)	\(26\)
norman	\(\frac {20}{\ln \relax (2)^{2} x}+\ln \left (x -\ln \left (x^{2}-4 x +3\right )-4\right )\)	\(26\)
risch	\(\frac {20}{\ln \relax (2)^{2} x}+\ln \left (\ln \left (x^{2}-4 x +3\right )-x +4\right )\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-20*x^2+80*x-60)*ln(x^2-4*x+3)+(-x^4+6*x^3-7*x^2)*ln(2)^2+20*x^3-160*x^2+380*x-240)/((x^4-4*x^3+3*x^2)*l
n(2)^2*ln(x^2-4*x+3)+(-x^5+8*x^4-19*x^3+12*x^2)*ln(2)^2),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.94, size = 24, normalized size = 0.80 \begin {gather*} \frac {20}{x \log \relax (2)^{2}} + \log \left (-x + \log \left (x - 1\right ) + \log \left (x - 3\right ) + 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-20*x^2+80*x-60)*log(x^2-4*x+3)+(-x^4+6*x^3-7*x^2)*log(2)^2+20*x^3-160*x^2+380*x-240)/((x^4-4*x^3+
3*x^2)*log(2)^2*log(x^2-4*x+3)+(-x^5+8*x^4-19*x^3+12*x^2)*log(2)^2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.89, size = 25, normalized size = 0.83 \begin {gather*} \ln \left (\ln \left (x^2-4\,x+3\right )-x+4\right )+\frac {20}{x\,{\ln \relax (2)}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x^2 - 4*x + 3)*(20*x^2 - 80*x + 60) - 380*x + log(2)^2*(7*x^2 - 6*x^3 + x^4) + 160*x^2 - 20*x^3 + 24
0)/(log(2)^2*(12*x^2 - 19*x^3 + 8*x^4 - x^5) + log(2)^2*log(x^2 - 4*x + 3)*(3*x^2 - 4*x^3 + x^4)),x)

[Out]

log(log(x^2 - 4*x + 3) - x + 4) + 20/(x*log(2)^2)

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sympy [A]  time = 0.28, size = 22, normalized size = 0.73 \begin {gather*} \log {\left (- x + \log {\left (x^{2} - 4 x + 3 \right )} + 4 \right )} + \frac {20}{x \log {\relax (2 )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-20*x**2+80*x-60)*ln(x**2-4*x+3)+(-x**4+6*x**3-7*x**2)*ln(2)**2+20*x**3-160*x**2+380*x-240)/((x**4
-4*x**3+3*x**2)*ln(2)**2*ln(x**2-4*x+3)+(-x**5+8*x**4-19*x**3+12*x**2)*ln(2)**2),x)

[Out]

log(-x + log(x**2 - 4*x + 3) + 4) + 20/(x*log(2)**2)

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