3.69.55 \(\int \frac {-900+300 x+(462 x-4 x^2) \log (4)-6 x^2 \log ^2(4)+((-462 x+154 x^2) \log (4)+(12 x^2-2 x^3) \log ^2(4)) \log (-3+x)+(-6 x^2+2 x^3) \log ^2(4) \log ^2(-3+x)+(-12+4 x+6 x \log (4)+(-6 x+2 x^2) \log (4) \log (-3+x)) \log (9 x^2)}{-1875 x+625 x^2} \, dx\)

Optimal. Leaf size=28 \[ \left (3+\frac {1}{25} \left (-\log (4) (x-x \log (-3+x))+\log \left (9 x^2\right )\right )\right )^2 \]

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Rubi [A]  time = 0.43, antiderivative size = 27, normalized size of antiderivative = 0.96, number of steps used = 4, number of rules used = 4, integrand size = 127, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {1593, 6688, 12, 6686} \begin {gather*} \frac {1}{625} \left (\log \left (9 x^2\right )+x \log (4) \log (x-3)-x \log (4)+75\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-900 + 300*x + (462*x - 4*x^2)*Log[4] - 6*x^2*Log[4]^2 + ((-462*x + 154*x^2)*Log[4] + (12*x^2 - 2*x^3)*Lo
g[4]^2)*Log[-3 + x] + (-6*x^2 + 2*x^3)*Log[4]^2*Log[-3 + x]^2 + (-12 + 4*x + 6*x*Log[4] + (-6*x + 2*x^2)*Log[4
]*Log[-3 + x])*Log[9*x^2])/(-1875*x + 625*x^2),x]

[Out]

(75 - x*Log[4] + x*Log[4]*Log[-3 + x] + Log[9*x^2])^2/625

Rule 12

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

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

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 {-900+300 x+\left (462 x-4 x^2\right ) \log (4)-6 x^2 \log ^2(4)+\left (\left (-462 x+154 x^2\right ) \log (4)+\left (12 x^2-2 x^3\right ) \log ^2(4)\right ) \log (-3+x)+\left (-6 x^2+2 x^3\right ) \log ^2(4) \log ^2(-3+x)+\left (-12+4 x+6 x \log (4)+\left (-6 x+2 x^2\right ) \log (4) \log (-3+x)\right ) \log \left (9 x^2\right )}{x (-1875+625 x)} \, dx\\ &=\int \frac {2 (6-x (2+\log (64))-(-3+x) x \log (4) \log (-3+x)) \left (75-x \log (4)+x \log (4) \log (-3+x)+\log \left (9 x^2\right )\right )}{625 (3-x) x} \, dx\\ &=\frac {2}{625} \int \frac {(6-x (2+\log (64))-(-3+x) x \log (4) \log (-3+x)) \left (75-x \log (4)+x \log (4) \log (-3+x)+\log \left (9 x^2\right )\right )}{(3-x) x} \, dx\\ &=\frac {1}{625} \left (75-x \log (4)+x \log (4) \log (-3+x)+\log \left (9 x^2\right )\right )^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 27, normalized size = 0.96 \begin {gather*} \frac {1}{625} \left (75-x \log (4)+x \log (4) \log (-3+x)+\log \left (9 x^2\right )\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-900 + 300*x + (462*x - 4*x^2)*Log[4] - 6*x^2*Log[4]^2 + ((-462*x + 154*x^2)*Log[4] + (12*x^2 - 2*x
^3)*Log[4]^2)*Log[-3 + x] + (-6*x^2 + 2*x^3)*Log[4]^2*Log[-3 + x]^2 + (-12 + 4*x + 6*x*Log[4] + (-6*x + 2*x^2)
*Log[4]*Log[-3 + x])*Log[9*x^2])/(-1875*x + 625*x^2),x]

[Out]

(75 - x*Log[4] + x*Log[4]*Log[-3 + x] + Log[9*x^2])^2/625

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fricas [B]  time = 0.77, size = 85, normalized size = 3.04 \begin {gather*} \frac {4}{625} \, x^{2} \log \relax (2)^{2} \log \left (x - 3\right )^{2} + \frac {4}{625} \, x^{2} \log \relax (2)^{2} - \frac {12}{25} \, x \log \relax (2) + \frac {2}{625} \, {\left (2 \, x \log \relax (2) \log \left (x - 3\right ) - 2 \, x \log \relax (2) + 75\right )} \log \left (9 \, x^{2}\right ) + \frac {1}{625} \, \log \left (9 \, x^{2}\right )^{2} - \frac {4}{625} \, {\left (2 \, x^{2} \log \relax (2)^{2} - 75 \, x \log \relax (2)\right )} \log \left (x - 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*(2*x^2-6*x)*log(2)*log(x-3)+12*x*log(2)+4*x-12)*log(9*x^2)+4*(2*x^3-6*x^2)*log(2)^2*log(x-3)^2+(
4*(-2*x^3+12*x^2)*log(2)^2+2*(154*x^2-462*x)*log(2))*log(x-3)-24*x^2*log(2)^2+2*(-4*x^2+462*x)*log(2)+300*x-90
0)/(625*x^2-1875*x),x, algorithm="fricas")

[Out]

4/625*x^2*log(2)^2*log(x - 3)^2 + 4/625*x^2*log(2)^2 - 12/25*x*log(2) + 2/625*(2*x*log(2)*log(x - 3) - 2*x*log
(2) + 75)*log(9*x^2) + 1/625*log(9*x^2)^2 - 4/625*(2*x^2*log(2)^2 - 75*x*log(2))*log(x - 3)

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giac [B]  time = 0.21, size = 85, normalized size = 3.04 \begin {gather*} \frac {4}{625} \, x^{2} \log \relax (2)^{2} \log \left (x - 3\right )^{2} + \frac {4}{625} \, x^{2} \log \relax (2)^{2} - \frac {12}{25} \, x \log \relax (2) + \frac {4}{625} \, {\left (x \log \relax (2) \log \left (x - 3\right ) - x \log \relax (2) + \log \relax (x)\right )} \log \left (9 \, x^{2}\right ) - \frac {4}{625} \, {\left (2 \, x^{2} \log \relax (2)^{2} - 75 \, x \log \relax (2)\right )} \log \left (x - 3\right ) - \frac {4}{625} \, \log \relax (x)^{2} + \frac {12}{25} \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*(2*x^2-6*x)*log(2)*log(x-3)+12*x*log(2)+4*x-12)*log(9*x^2)+4*(2*x^3-6*x^2)*log(2)^2*log(x-3)^2+(
4*(-2*x^3+12*x^2)*log(2)^2+2*(154*x^2-462*x)*log(2))*log(x-3)-24*x^2*log(2)^2+2*(-4*x^2+462*x)*log(2)+300*x-90
0)/(625*x^2-1875*x),x, algorithm="giac")

[Out]

4/625*x^2*log(2)^2*log(x - 3)^2 + 4/625*x^2*log(2)^2 - 12/25*x*log(2) + 4/625*(x*log(2)*log(x - 3) - x*log(2)
+ log(x))*log(9*x^2) - 4/625*(2*x^2*log(2)^2 - 75*x*log(2))*log(x - 3) - 4/625*log(x)^2 + 12/25*log(x)

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maple [B]  time = 0.18, size = 127, normalized size = 4.54

method	result	size
default	\(\frac {4 x^{2} \ln \relax (2)^{2} \ln \left (x -3\right )^{2}}{625}-\frac {8 \ln \left (x -3\right ) \ln \relax (2)^{2} x^{2}}{625}+\frac {4 x^{2} \ln \relax (2)^{2}}{625}-\frac {108 \ln \relax (2)^{2}}{625}+\frac {8 \ln \relax (2) \ln \left (x -3\right ) x \ln \relax (3)}{625}+\frac {4 \ln \relax (2) \ln \left (x -3\right ) x \ln \left (x^{2}\right )}{625}-\frac {8 x \ln \relax (2) \ln \relax (3)}{625}-\frac {4 x \ln \relax (2) \ln \left (x^{2}\right )}{625}+\frac {12 \ln \relax (2) \ln \left (x -3\right ) x}{25}+\frac {24 \ln \relax (2) \ln \relax (3)}{625}-\frac {12 x \ln \relax (2)}{25}+\frac {924 \ln \relax (2)}{625}+\frac {12 \ln \relax (x )}{25}+\frac {8 \ln \relax (3) \ln \relax (x )}{625}+\frac {\ln \left (x^{2}\right )^{2}}{625}\)	\(127\)
risch	\(\frac {4 x^{2} \ln \relax (2)^{2} \ln \left (x -3\right )^{2}}{625}+\left (\frac {8 x \ln \relax (2) \ln \relax (x )}{625}-\frac {2 i \pi \ln \relax (2) x \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )}{625}+\frac {4 i \pi \ln \relax (2) x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}}{625}-\frac {2 i \pi \ln \relax (2) x \mathrm {csgn}\left (i x^{2}\right )^{3}}{625}-\frac {8 x^{2} \ln \relax (2)^{2}}{625}+\frac {8 x \ln \relax (2) \ln \relax (3)}{625}+\frac {12 x \ln \relax (2)}{25}\right ) \ln \left (x -3\right )+\frac {4 \ln \relax (x )^{2}}{625}-\frac {8 x \ln \relax (2) \ln \relax (x )}{625}-\frac {2 i \pi \ln \relax (x ) \mathrm {csgn}\left (i x^{2}\right )^{3}}{625}+\frac {2 i \pi \ln \relax (2) x \mathrm {csgn}\left (i x^{2}\right )^{3}}{625}-\frac {4 i \pi \ln \relax (2) x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}}{625}+\frac {4 i \pi \ln \relax (x ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}}{625}-\frac {2 i \pi \ln \relax (x ) \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )}{625}+\frac {2 i \pi \ln \relax (2) x \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )}{625}+\frac {4 x^{2} \ln \relax (2)^{2}}{625}-\frac {8 x \ln \relax (2) \ln \relax (3)}{625}-\frac {12 x \ln \relax (2)}{25}+\frac {8 \ln \relax (3) \ln \relax (x )}{625}+\frac {12 \ln \relax (x )}{25}\)	\(266\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*(2*x^2-6*x)*ln(2)*ln(x-3)+12*x*ln(2)+4*x-12)*ln(9*x^2)+4*(2*x^3-6*x^2)*ln(2)^2*ln(x-3)^2+(4*(-2*x^3+12
*x^2)*ln(2)^2+2*(154*x^2-462*x)*ln(2))*ln(x-3)-24*x^2*ln(2)^2+2*(-4*x^2+462*x)*ln(2)+300*x-900)/(625*x^2-1875*
x),x,method=_RETURNVERBOSE)

[Out]

4/625*x^2*ln(2)^2*ln(x-3)^2-8/625*ln(x-3)*ln(2)^2*x^2+4/625*x^2*ln(2)^2-108/625*ln(2)^2+8/625*ln(2)*ln(x-3)*x*
ln(3)+4/625*ln(2)*ln(x-3)*x*ln(x^2)-8/625*x*ln(2)*ln(3)-4/625*x*ln(2)*ln(x^2)+12/25*ln(2)*ln(x-3)*x+24/625*ln(
2)*ln(3)-12/25*x*ln(2)+924/625*ln(2)+12/25*ln(x)+8/625*ln(3)*ln(x)+1/625*ln(x^2)^2

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maxima [B]  time = 0.49, size = 319, normalized size = 11.39 \begin {gather*} -\frac {4}{625} \, {\left (x^{2} + 6 \, x + 18 \, \log \left (x - 3\right )\right )} \log \relax (2)^{2} \log \left (x - 3\right ) + \frac {48}{625} \, {\left (x + 3 \, \log \left (x - 3\right )\right )} \log \relax (2)^{2} \log \left (x - 3\right ) - \frac {8}{625} \, x {\left (\log \relax (3) - 2\right )} \log \relax (2) + \frac {2}{625} \, {\left ({\left (2 \, \log \left (x - 3\right )^{2} - 2 \, \log \left (x - 3\right ) + 1\right )} {\left (x - 3\right )}^{2} + 12 \, \log \left (x - 3\right )^{3} + 24 \, {\left (\log \left (x - 3\right )^{2} - 2 \, \log \left (x - 3\right ) + 2\right )} {\left (x - 3\right )}\right )} \log \relax (2)^{2} - \frac {24}{625} \, {\left (\log \left (x - 3\right )^{3} + {\left (\log \left (x - 3\right )^{2} - 2 \, \log \left (x - 3\right ) + 2\right )} {\left (x - 3\right )}\right )} \log \relax (2)^{2} + \frac {2}{625} \, {\left (x^{2} + 18 \, \log \left (x - 3\right )^{2} + 18 \, x + 54 \, \log \left (x - 3\right )\right )} \log \relax (2)^{2} - \frac {24}{625} \, {\left (3 \, \log \left (x - 3\right )^{2} + 2 \, x + 6 \, \log \left (x - 3\right )\right )} \log \relax (2)^{2} - \frac {24}{625} \, {\left (x + 3 \, \log \left (x - 3\right )\right )} \log \relax (2)^{2} + \frac {308}{625} \, {\left (x + 3 \, \log \left (x - 3\right )\right )} \log \relax (2) \log \left (x - 3\right ) - \frac {462}{625} \, \log \relax (2) \log \left (x - 3\right )^{2} - \frac {154}{625} \, {\left (3 \, \log \left (x - 3\right )^{2} + 2 \, x + 6 \, \log \left (x - 3\right )\right )} \log \relax (2) - \frac {8}{625} \, {\left (x + 3 \, \log \left (x - 3\right )\right )} \log \relax (2) + \frac {8}{625} \, {\left (x {\left (\log \relax (3) - 1\right )} \log \relax (2) + x \log \relax (2) \log \relax (x) + 3 \, \log \relax (2)\right )} \log \left (x - 3\right ) + \frac {924}{625} \, \log \relax (2) \log \left (x - 3\right ) - \frac {8}{625} \, {\left (x \log \relax (2) - \log \relax (3)\right )} \log \relax (x) + \frac {4}{625} \, \log \relax (x)^{2} + \frac {12}{25} \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*(2*x^2-6*x)*log(2)*log(x-3)+12*x*log(2)+4*x-12)*log(9*x^2)+4*(2*x^3-6*x^2)*log(2)^2*log(x-3)^2+(
4*(-2*x^3+12*x^2)*log(2)^2+2*(154*x^2-462*x)*log(2))*log(x-3)-24*x^2*log(2)^2+2*(-4*x^2+462*x)*log(2)+300*x-90
0)/(625*x^2-1875*x),x, algorithm="maxima")

[Out]

-4/625*(x^2 + 6*x + 18*log(x - 3))*log(2)^2*log(x - 3) + 48/625*(x + 3*log(x - 3))*log(2)^2*log(x - 3) - 8/625
*x*(log(3) - 2)*log(2) + 2/625*((2*log(x - 3)^2 - 2*log(x - 3) + 1)*(x - 3)^2 + 12*log(x - 3)^3 + 24*(log(x -
3)^2 - 2*log(x - 3) + 2)*(x - 3))*log(2)^2 - 24/625*(log(x - 3)^3 + (log(x - 3)^2 - 2*log(x - 3) + 2)*(x - 3))
*log(2)^2 + 2/625*(x^2 + 18*log(x - 3)^2 + 18*x + 54*log(x - 3))*log(2)^2 - 24/625*(3*log(x - 3)^2 + 2*x + 6*l
og(x - 3))*log(2)^2 - 24/625*(x + 3*log(x - 3))*log(2)^2 + 308/625*(x + 3*log(x - 3))*log(2)*log(x - 3) - 462/
625*log(2)*log(x - 3)^2 - 154/625*(3*log(x - 3)^2 + 2*x + 6*log(x - 3))*log(2) - 8/625*(x + 3*log(x - 3))*log(
2) + 8/625*(x*(log(3) - 1)*log(2) + x*log(2)*log(x) + 3*log(2))*log(x - 3) + 924/625*log(2)*log(x - 3) - 8/625
*(x*log(2) - log(3))*log(x) + 4/625*log(x)^2 + 12/25*log(x)

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mupad [B]  time = 4.59, size = 88, normalized size = 3.14 \begin {gather*} \frac {12\,\ln \relax (x)}{25}+\frac {4\,x^2\,{\ln \relax (2)}^2}{625}+\frac {{\ln \left (9\,x^2\right )}^2}{625}-\ln \left (9\,x^2\right )\,\left (\frac {4\,x\,\ln \relax (2)}{625}-\frac {4\,x\,\ln \left (x-3\right )\,\ln \relax (2)}{625}\right )-\frac {12\,x\,\ln \relax (2)}{25}-\ln \left (x-3\right )\,\left (\frac {8\,x^2\,{\ln \relax (2)}^2}{625}-\frac {12\,x\,\ln \relax (2)}{25}\right )+\frac {4\,x^2\,{\ln \left (x-3\right )}^2\,{\ln \relax (2)}^2}{625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((24*x^2*log(2)^2 - 300*x + log(x - 3)*(2*log(2)*(462*x - 154*x^2) - 4*log(2)^2*(12*x^2 - 2*x^3)) - 2*log(2
)*(462*x - 4*x^2) - log(9*x^2)*(4*x + 12*x*log(2) - 2*log(x - 3)*log(2)*(6*x - 2*x^2) - 12) + 4*log(x - 3)^2*l
og(2)^2*(6*x^2 - 2*x^3) + 900)/(1875*x - 625*x^2),x)

[Out]

(12*log(x))/25 + (4*x^2*log(2)^2)/625 + log(9*x^2)^2/625 - log(9*x^2)*((4*x*log(2))/625 - (4*x*log(x - 3)*log(
2))/625) - (12*x*log(2))/25 - log(x - 3)*((8*x^2*log(2)^2)/625 - (12*x*log(2))/25) + (4*x^2*log(x - 3)^2*log(2
)^2)/625

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sympy [B]  time = 0.79, size = 109, normalized size = 3.89 \begin {gather*} \frac {4 x^{2} \log {\relax (2 )}^{2} \log {\left (x - 3 \right )}^{2}}{625} + \frac {4 x^{2} \log {\relax (2 )}^{2}}{625} - \frac {12 x \log {\relax (2 )}}{25} + \left (- \frac {8 x^{2} \log {\relax (2 )}^{2}}{625} + \frac {12 x \log {\relax (2 )}}{25}\right ) \log {\left (x - 3 \right )} + \left (\frac {4 x \log {\relax (2 )} \log {\left (x - 3 \right )}}{625} - \frac {4 x \log {\relax (2 )}}{625}\right ) \log {\left (9 x^{2} \right )} + \frac {12 \log {\relax (x )}}{25} + \frac {\log {\left (9 x^{2} \right )}^{2}}{625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*(2*x**2-6*x)*ln(2)*ln(x-3)+12*x*ln(2)+4*x-12)*ln(9*x**2)+4*(2*x**3-6*x**2)*ln(2)**2*ln(x-3)**2+(
4*(-2*x**3+12*x**2)*ln(2)**2+2*(154*x**2-462*x)*ln(2))*ln(x-3)-24*x**2*ln(2)**2+2*(-4*x**2+462*x)*ln(2)+300*x-
900)/(625*x**2-1875*x),x)

[Out]

4*x**2*log(2)**2*log(x - 3)**2/625 + 4*x**2*log(2)**2/625 - 12*x*log(2)/25 + (-8*x**2*log(2)**2/625 + 12*x*log
(2)/25)*log(x - 3) + (4*x*log(2)*log(x - 3)/625 - 4*x*log(2)/625)*log(9*x**2) + 12*log(x)/25 + log(9*x**2)**2/
625

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