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

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

________________________________________________________________________________________

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} $$\begin{array}{c}\frac{1}{625}{(\log \left(9x^2\right)+x\log (4)\log (x-3)-x\log (4)+75)}^2\end{array}$$

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

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{-900+300x+(462x-4x^2)\log (4)-6x^2{\log }^2(4)+((-462x+154x^2)\log (4)+(12x^2-2x^3){\log }^2(4))\log (-3+x)+(-6x^2+2x^3){\log }^2(4){\log }^2(-3+x)+(-12+4x+6x\log (4)+(-6x+2x^2)\log (4)\log (-3+x))\log \left(9x^2\right)}{x(-1875+625x)}dx\\ & =\int \frac{2(6-x(2+\log (64))-(-3+x)x\log (4)\log (-3+x))(75-x\log (4)+x\log (4)\log (-3+x)+\log \left(9x^2\right))}{625(3-x)x}dx\\ & =\frac{2}{625}\int \frac{(6-x(2+\log (64))-(-3+x)x\log (4)\log (-3+x))(75-x\log (4)+x\log (4)\log (-3+x)+\log \left(9x^2\right))}{(3-x)x}dx\\ & =\frac{1}{625}{(75-x\log (4)+x\log (4)\log (-3+x)+\log \left(9x^2\right))}^2\end{array}\end{array}$$

________________________________________________________________________________________

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

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

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

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

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

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

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

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

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

________________________________________________________________________________________

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

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

________________________________________________________________________________________