3.9.29 \(\int \frac {-906+1328 x-438 x^2+16 x^3+(144-150 x+6 x^2) \log (-24+x)+(-906+422 x-16 x^2+(144-6 x) \log (-24+x)) \log (-11+8 x+3 \log (-24+x))}{264-203 x+8 x^2+(-72+3 x) \log (-24+x)} \, dx\)

Optimal. Leaf size=24 \[ (-1+x-\log (-2-x+3 (-3+3 x+\log (-24+x))))^2 \]

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Rubi [A]  time = 0.22, antiderivative size = 19, normalized size of antiderivative = 0.79, number of steps used = 3, number of rules used = 3, integrand size = 86, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.035, Rules used = {6688, 12, 6686} \begin {gather*} (-x+\log (8 x+3 \log (x-24)-11)+1)^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-906 + 1328*x - 438*x^2 + 16*x^3 + (144 - 150*x + 6*x^2)*Log[-24 + x] + (-906 + 422*x - 16*x^2 + (144 - 6
*x)*Log[-24 + x])*Log[-11 + 8*x + 3*Log[-24 + x]])/(264 - 203*x + 8*x^2 + (-72 + 3*x)*Log[-24 + x]),x]

[Out]

(1 - x + Log[-11 + 8*x + 3*Log[-24 + 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 (453-211 x+8 x^2+3 (-24+x) \log (-24+x)\right ) (-1+x-\log (-11+8 x+3 \log (-24+x)))}{(24-x) (11-8 x-3 \log (-24+x))} \, dx\\ &=2 \int \frac {\left (453-211 x+8 x^2+3 (-24+x) \log (-24+x)\right ) (-1+x-\log (-11+8 x+3 \log (-24+x)))}{(24-x) (11-8 x-3 \log (-24+x))} \, dx\\ &=(1-x+\log (-11+8 x+3 \log (-24+x)))^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 19, normalized size = 0.79 \begin {gather*} (-1+x-\log (-11+8 x+3 \log (-24+x)))^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-906 + 1328*x - 438*x^2 + 16*x^3 + (144 - 150*x + 6*x^2)*Log[-24 + x] + (-906 + 422*x - 16*x^2 + (1
44 - 6*x)*Log[-24 + x])*Log[-11 + 8*x + 3*Log[-24 + x]])/(264 - 203*x + 8*x^2 + (-72 + 3*x)*Log[-24 + x]),x]

[Out]

(-1 + x - Log[-11 + 8*x + 3*Log[-24 + x]])^2

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fricas [A]  time = 0.58, size = 38, normalized size = 1.58 \begin {gather*} x^{2} - 2 \, {\left (x - 1\right )} \log \left (8 \, x + 3 \, \log \left (x - 24\right ) - 11\right ) + \log \left (8 \, x + 3 \, \log \left (x - 24\right ) - 11\right )^{2} - 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-6*x+144)*log(x-24)-16*x^2+422*x-906)*log(3*log(x-24)+8*x-11)+(6*x^2-150*x+144)*log(x-24)+16*x^3-
438*x^2+1328*x-906)/((3*x-72)*log(x-24)+8*x^2-203*x+264),x, algorithm="fricas")

[Out]

x^2 - 2*(x - 1)*log(8*x + 3*log(x - 24) - 11) + log(8*x + 3*log(x - 24) - 11)^2 - 2*x

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giac [B]  time = 0.50, size = 50, normalized size = 2.08 \begin {gather*} x^{2} - 2 \, x \log \left (8 \, x + 3 \, \log \left (x - 24\right ) - 11\right ) + \log \left (8 \, x + 3 \, \log \left (x - 24\right ) - 11\right )^{2} - 2 \, x + 2 \, \log \left (8 \, x + 3 \, \log \left (x - 24\right ) - 11\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-6*x+144)*log(x-24)-16*x^2+422*x-906)*log(3*log(x-24)+8*x-11)+(6*x^2-150*x+144)*log(x-24)+16*x^3-
438*x^2+1328*x-906)/((3*x-72)*log(x-24)+8*x^2-203*x+264),x, algorithm="giac")

[Out]

x^2 - 2*x*log(8*x + 3*log(x - 24) - 11) + log(8*x + 3*log(x - 24) - 11)^2 - 2*x + 2*log(8*x + 3*log(x - 24) -
11)

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maple [B]  time = 0.05, size = 49, normalized size = 2.04

method	result	size
risch	\(\ln \left (3 \ln \left (x -24\right )+8 x -11\right )^{2}-2 \ln \left (3 \ln \left (x -24\right )+8 x -11\right ) x +x^{2}-2 x +2 \ln \left (\ln \left (x -24\right )+\frac {8 x}{3}-\frac {11}{3}\right )\)	\(49\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-6*x+144)*ln(x-24)-16*x^2+422*x-906)*ln(3*ln(x-24)+8*x-11)+(6*x^2-150*x+144)*ln(x-24)+16*x^3-438*x^2+13
28*x-906)/((3*x-72)*ln(x-24)+8*x^2-203*x+264),x,method=_RETURNVERBOSE)

[Out]

ln(3*ln(x-24)+8*x-11)^2-2*ln(3*ln(x-24)+8*x-11)*x+x^2-2*x+2*ln(ln(x-24)+8/3*x-11/3)

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maxima [A]  time = 0.52, size = 38, normalized size = 1.58 \begin {gather*} x^{2} - 2 \, {\left (x - 1\right )} \log \left (8 \, x + 3 \, \log \left (x - 24\right ) - 11\right ) + \log \left (8 \, x + 3 \, \log \left (x - 24\right ) - 11\right )^{2} - 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-6*x+144)*log(x-24)-16*x^2+422*x-906)*log(3*log(x-24)+8*x-11)+(6*x^2-150*x+144)*log(x-24)+16*x^3-
438*x^2+1328*x-906)/((3*x-72)*log(x-24)+8*x^2-203*x+264),x, algorithm="maxima")

[Out]

x^2 - 2*(x - 1)*log(8*x + 3*log(x - 24) - 11) + log(8*x + 3*log(x - 24) - 11)^2 - 2*x

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mupad [B]  time = 1.44, size = 48, normalized size = 2.00 \begin {gather*} 2\,\ln \left (\frac {8\,x}{3}+\ln \left (x-24\right )-\frac {11}{3}\right )-2\,x-2\,x\,\ln \left (8\,x+3\,\ln \left (x-24\right )-11\right )+{\ln \left (8\,x+3\,\ln \left (x-24\right )-11\right )}^2+x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1328*x + log(x - 24)*(6*x^2 - 150*x + 144) - 438*x^2 + 16*x^3 - log(8*x + 3*log(x - 24) - 11)*(16*x^2 - 4
22*x + log(x - 24)*(6*x - 144) + 906) - 906)/(8*x^2 - 203*x + log(x - 24)*(3*x - 72) + 264),x)

[Out]

2*log((8*x)/3 + log(x - 24) - 11/3) - 2*x - 2*x*log(8*x + 3*log(x - 24) - 11) + log(8*x + 3*log(x - 24) - 11)^
2 + x^2

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sympy [B]  time = 0.55, size = 54, normalized size = 2.25 \begin {gather*} x^{2} - 2 x \log {\left (8 x + 3 \log {\left (x - 24 \right )} - 11 \right )} - 2 x + 2 \log {\left (\frac {8 x}{3} + \log {\left (x - 24 \right )} - \frac {11}{3} \right )} + \log {\left (8 x + 3 \log {\left (x - 24 \right )} - 11 \right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-6*x+144)*ln(x-24)-16*x**2+422*x-906)*ln(3*ln(x-24)+8*x-11)+(6*x**2-150*x+144)*ln(x-24)+16*x**3-4
38*x**2+1328*x-906)/((3*x-72)*ln(x-24)+8*x**2-203*x+264),x)

[Out]

x**2 - 2*x*log(8*x + 3*log(x - 24) - 11) - 2*x + 2*log(8*x/3 + log(x - 24) - 11/3) + log(8*x + 3*log(x - 24) -
11)**2

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