3.37.2 \(\int \frac {1530+1002 x-2902 x^2+1394 x^3-186 x^4+2 x^5+(-1530 x+1038 x^2-182 x^3+2 x^4) \log (x)+(18+12 x-34 x^2+16 x^3-2 x^4+(-18 x+12 x^2-2 x^3) \log (x)) \log (\frac {-4 x+x^2+(-3+x) \log (x)}{-3+x})}{12 x^2-7 x^3+x^4+(9 x-6 x^2+x^3) \log (x)} \, dx\)

Optimal. Leaf size=23 \[ e+\left (85-x+\log \left (x+\frac {x}{3-x}+\log (x)\right )\right )^2 \]

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Rubi [A]  time = 0.38, antiderivative size = 25, normalized size of antiderivative = 1.09, number of steps used = 3, number of rules used = 3, integrand size = 137, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {6688, 12, 6686} \begin {gather*} \left (-x+\log \left (\frac {(4-x) x}{3-x}+\log (x)\right )+85\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1530 + 1002*x - 2902*x^2 + 1394*x^3 - 186*x^4 + 2*x^5 + (-1530*x + 1038*x^2 - 182*x^3 + 2*x^4)*Log[x] + (
18 + 12*x - 34*x^2 + 16*x^3 - 2*x^4 + (-18*x + 12*x^2 - 2*x^3)*Log[x])*Log[(-4*x + x^2 + (-3 + x)*Log[x])/(-3
+ x)])/(12*x^2 - 7*x^3 + x^4 + (9*x - 6*x^2 + x^3)*Log[x]),x]

[Out]

(85 - x + Log[((4 - x)*x)/(3 - x) + Log[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 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

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 \frac {2 \left (9+6 x-17 x^2+8 x^3-x^4-(-3+x)^2 x \log (x)\right ) \left (-85+x-\log \left (\frac {(-4+x) x}{-3+x}+\log (x)\right )\right )}{(3-x) x ((-4+x) x+(-3+x) \log (x))} \, dx\\ &=2 \int \frac {\left (9+6 x-17 x^2+8 x^3-x^4-(-3+x)^2 x \log (x)\right ) \left (-85+x-\log \left (\frac {(-4+x) x}{-3+x}+\log (x)\right )\right )}{(3-x) x ((-4+x) x+(-3+x) \log (x))} \, dx\\ &=\left (85-x+\log \left (\frac {(4-x) x}{3-x}+\log (x)\right )\right )^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 21, normalized size = 0.91 \begin {gather*} \left (-85+x-\log \left (\frac {(-4+x) x}{-3+x}+\log (x)\right )\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1530 + 1002*x - 2902*x^2 + 1394*x^3 - 186*x^4 + 2*x^5 + (-1530*x + 1038*x^2 - 182*x^3 + 2*x^4)*Log[
x] + (18 + 12*x - 34*x^2 + 16*x^3 - 2*x^4 + (-18*x + 12*x^2 - 2*x^3)*Log[x])*Log[(-4*x + x^2 + (-3 + x)*Log[x]
)/(-3 + x)])/(12*x^2 - 7*x^3 + x^4 + (9*x - 6*x^2 + x^3)*Log[x]),x]

[Out]

(-85 + x - Log[((-4 + x)*x)/(-3 + x) + Log[x]])^2

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fricas [B]  time = 0.92, size = 54, normalized size = 2.35 \begin {gather*} x^{2} - 2 \, {\left (x - 85\right )} \log \left (\frac {x^{2} + {\left (x - 3\right )} \log \relax (x) - 4 \, x}{x - 3}\right ) + \log \left (\frac {x^{2} + {\left (x - 3\right )} \log \relax (x) - 4 \, x}{x - 3}\right )^{2} - 170 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x^3+12*x^2-18*x)*log(x)-2*x^4+16*x^3-34*x^2+12*x+18)*log((log(x)*(x-3)+x^2-4*x)/(x-3))+(2*x^4-
182*x^3+1038*x^2-1530*x)*log(x)+2*x^5-186*x^4+1394*x^3-2902*x^2+1002*x+1530)/((x^3-6*x^2+9*x)*log(x)+x^4-7*x^3
+12*x^2),x, algorithm="fricas")

[Out]

x^2 - 2*(x - 85)*log((x^2 + (x - 3)*log(x) - 4*x)/(x - 3)) + log((x^2 + (x - 3)*log(x) - 4*x)/(x - 3))^2 - 170
*x

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, {\left (x^{5} - 93 \, x^{4} + 697 \, x^{3} - 1451 \, x^{2} + {\left (x^{4} - 91 \, x^{3} + 519 \, x^{2} - 765 \, x\right )} \log \relax (x) - {\left (x^{4} - 8 \, x^{3} + 17 \, x^{2} + {\left (x^{3} - 6 \, x^{2} + 9 \, x\right )} \log \relax (x) - 6 \, x - 9\right )} \log \left (\frac {x^{2} + {\left (x - 3\right )} \log \relax (x) - 4 \, x}{x - 3}\right ) + 501 \, x + 765\right )}}{x^{4} - 7 \, x^{3} + 12 \, x^{2} + {\left (x^{3} - 6 \, x^{2} + 9 \, x\right )} \log \relax (x)}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x^3+12*x^2-18*x)*log(x)-2*x^4+16*x^3-34*x^2+12*x+18)*log((log(x)*(x-3)+x^2-4*x)/(x-3))+(2*x^4-
182*x^3+1038*x^2-1530*x)*log(x)+2*x^5-186*x^4+1394*x^3-2902*x^2+1002*x+1530)/((x^3-6*x^2+9*x)*log(x)+x^4-7*x^3
+12*x^2),x, algorithm="giac")

[Out]

integrate(2*(x^5 - 93*x^4 + 697*x^3 - 1451*x^2 + (x^4 - 91*x^3 + 519*x^2 - 765*x)*log(x) - (x^4 - 8*x^3 + 17*x
^2 + (x^3 - 6*x^2 + 9*x)*log(x) - 6*x - 9)*log((x^2 + (x - 3)*log(x) - 4*x)/(x - 3)) + 501*x + 765)/(x^4 - 7*x
^3 + 12*x^2 + (x^3 - 6*x^2 + 9*x)*log(x)), x)

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maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (\left (-2 x^{3}+12 x^{2}-18 x \right ) \ln \relax (x )-2 x^{4}+16 x^{3}-34 x^{2}+12 x +18\right ) \ln \left (\frac {\ln \relax (x ) \left (x -3\right )+x^{2}-4 x}{x -3}\right )+\left (2 x^{4}-182 x^{3}+1038 x^{2}-1530 x \right ) \ln \relax (x )+2 x^{5}-186 x^{4}+1394 x^{3}-2902 x^{2}+1002 x +1530}{\left (x^{3}-6 x^{2}+9 x \right ) \ln \relax (x )+x^{4}-7 x^{3}+12 x^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-2*x^3+12*x^2-18*x)*ln(x)-2*x^4+16*x^3-34*x^2+12*x+18)*ln((ln(x)*(x-3)+x^2-4*x)/(x-3))+(2*x^4-182*x^3+1
038*x^2-1530*x)*ln(x)+2*x^5-186*x^4+1394*x^3-2902*x^2+1002*x+1530)/((x^3-6*x^2+9*x)*ln(x)+x^4-7*x^3+12*x^2),x)

[Out]

int((((-2*x^3+12*x^2-18*x)*ln(x)-2*x^4+16*x^3-34*x^2+12*x+18)*ln((ln(x)*(x-3)+x^2-4*x)/(x-3))+(2*x^4-182*x^3+1
038*x^2-1530*x)*ln(x)+2*x^5-186*x^4+1394*x^3-2902*x^2+1002*x+1530)/((x^3-6*x^2+9*x)*ln(x)+x^4-7*x^3+12*x^2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} 2 \, \int \frac {x^{5} - 93 \, x^{4} + 697 \, x^{3} - 1451 \, x^{2} + {\left (x^{4} - 91 \, x^{3} + 519 \, x^{2} - 765 \, x\right )} \log \relax (x) - {\left (x^{4} - 8 \, x^{3} + 17 \, x^{2} + {\left (x^{3} - 6 \, x^{2} + 9 \, x\right )} \log \relax (x) - 6 \, x - 9\right )} \log \left (\frac {x^{2} + {\left (x - 3\right )} \log \relax (x) - 4 \, x}{x - 3}\right ) + 501 \, x + 765}{x^{4} - 7 \, x^{3} + 12 \, x^{2} + {\left (x^{3} - 6 \, x^{2} + 9 \, x\right )} \log \relax (x)}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x^3+12*x^2-18*x)*log(x)-2*x^4+16*x^3-34*x^2+12*x+18)*log((log(x)*(x-3)+x^2-4*x)/(x-3))+(2*x^4-
182*x^3+1038*x^2-1530*x)*log(x)+2*x^5-186*x^4+1394*x^3-2902*x^2+1002*x+1530)/((x^3-6*x^2+9*x)*log(x)+x^4-7*x^3
+12*x^2),x, algorithm="maxima")

[Out]

2*integrate((x^5 - 93*x^4 + 697*x^3 - 1451*x^2 + (x^4 - 91*x^3 + 519*x^2 - 765*x)*log(x) - (x^4 - 8*x^3 + 17*x
^2 + (x^3 - 6*x^2 + 9*x)*log(x) - 6*x - 9)*log((x^2 + (x - 3)*log(x) - 4*x)/(x - 3)) + 501*x + 765)/(x^4 - 7*x
^3 + 12*x^2 + (x^3 - 6*x^2 + 9*x)*log(x)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {1002\,x-\ln \relax (x)\,\left (-2\,x^4+182\,x^3-1038\,x^2+1530\,x\right )+\ln \left (\frac {\ln \relax (x)\,\left (x-3\right )-4\,x+x^2}{x-3}\right )\,\left (12\,x-34\,x^2+16\,x^3-2\,x^4-\ln \relax (x)\,\left (2\,x^3-12\,x^2+18\,x\right )+18\right )-2902\,x^2+1394\,x^3-186\,x^4+2\,x^5+1530}{\ln \relax (x)\,\left (x^3-6\,x^2+9\,x\right )+12\,x^2-7\,x^3+x^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1002*x - log(x)*(1530*x - 1038*x^2 + 182*x^3 - 2*x^4) + log((log(x)*(x - 3) - 4*x + x^2)/(x - 3))*(12*x -
34*x^2 + 16*x^3 - 2*x^4 - log(x)*(18*x - 12*x^2 + 2*x^3) + 18) - 2902*x^2 + 1394*x^3 - 186*x^4 + 2*x^5 + 1530
)/(log(x)*(9*x - 6*x^2 + x^3) + 12*x^2 - 7*x^3 + x^4),x)

[Out]

int((1002*x - log(x)*(1530*x - 1038*x^2 + 182*x^3 - 2*x^4) + log((log(x)*(x - 3) - 4*x + x^2)/(x - 3))*(12*x -
34*x^2 + 16*x^3 - 2*x^4 - log(x)*(18*x - 12*x^2 + 2*x^3) + 18) - 2902*x^2 + 1394*x^3 - 186*x^4 + 2*x^5 + 1530
)/(log(x)*(9*x - 6*x^2 + x^3) + 12*x^2 - 7*x^3 + x^4), x)

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sympy [B]  time = 0.82, size = 65, normalized size = 2.83 \begin {gather*} x^{2} - 2 x \log {\left (\frac {x^{2} - 4 x + \left (x - 3\right ) \log {\relax (x )}}{x - 3} \right )} - 170 x + \log {\left (\frac {x^{2} - 4 x + \left (x - 3\right ) \log {\relax (x )}}{x - 3} \right )}^{2} + 170 \log {\left (\log {\relax (x )} + \frac {x^{2} - 4 x}{x - 3} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x**3+12*x**2-18*x)*ln(x)-2*x**4+16*x**3-34*x**2+12*x+18)*ln((ln(x)*(x-3)+x**2-4*x)/(x-3))+(2*x
**4-182*x**3+1038*x**2-1530*x)*ln(x)+2*x**5-186*x**4+1394*x**3-2902*x**2+1002*x+1530)/((x**3-6*x**2+9*x)*ln(x)
+x**4-7*x**3+12*x**2),x)

[Out]

x**2 - 2*x*log((x**2 - 4*x + (x - 3)*log(x))/(x - 3)) - 170*x + log((x**2 - 4*x + (x - 3)*log(x))/(x - 3))**2
+ 170*log(log(x) + (x**2 - 4*x)/(x - 3))

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