\(\int \frac {-4608-768 x+120 x^2+18 x^3+(-1536-320 x) \log (4+x)+(-128-32 x) \log ^2(4+x)}{2304 x+1156 x^2+181 x^3+9 x^4+(768 x+288 x^2+24 x^3) \log (4+x)+(64 x+16 x^2) \log ^2(4+x)} \, dx\) [2382]

Optimal result

Integrand size = 94, antiderivative size = 22 \[ \int \frac {-4608-768 x+120 x^2+18 x^3+(-1536-320 x) \log (4+x)+(-128-32 x) \log ^2(4+x)}{2304 x+1156 x^2+181 x^3+9 x^4+\left (768 x+288 x^2+24 x^3\right ) \log (4+x)+\left (64 x+16 x^2\right ) \log ^2(4+x)} \, dx=\log \left (\frac {\left (x+(x-4 (6+x+\log (4+x)))^2\right )^2}{x^2}\right ) \]

[Out]

ln((x+(-3*x-24-4*ln(4+x))^2)^2/x^2)

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.77, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {6873, 6874, 6816} \[ \int \frac {-4608-768 x+120 x^2+18 x^3+(-1536-320 x) \log (4+x)+(-128-32 x) \log ^2(4+x)}{2304 x+1156 x^2+181 x^3+9 x^4+\left (768 x+288 x^2+24 x^3\right ) \log (4+x)+\left (64 x+16 x^2\right ) \log ^2(4+x)} \, dx=2 \log \left (9 x^2+145 x+16 \log ^2(x+4)+24 x \log (x+4)+192 \log (x+4)+576\right )-2 \log (x) \]

[In]

Int[(-4608 - 768*x + 120*x^2 + 18*x^3 + (-1536 - 320*x)*Log[4 + x] + (-128 - 32*x)*Log[4 + x]^2)/(2304*x + 115
6*x^2 + 181*x^3 + 9*x^4 + (768*x + 288*x^2 + 24*x^3)*Log[4 + x] + (64*x + 16*x^2)*Log[4 + x]^2),x]

[Out]

-2*Log[x] + 2*Log[576 + 145*x + 9*x^2 + 192*Log[4 + x] + 24*x*Log[4 + x] + 16*Log[4 + x]^2]

Rule 6816

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

Rule 6873

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

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {-4608-768 x+120 x^2+18 x^3+(-1536-320 x) \log (4+x)+(-128-32 x) \log ^2(4+x)}{x (4+x) \left (576+145 x+9 x^2+192 \log (4+x)+24 x \log (4+x)+16 \log ^2(4+x)\right )} \, dx \\ & = \int \left (-\frac {2}{x}+\frac {2 \left (772+241 x+18 x^2+128 \log (4+x)+24 x \log (4+x)\right )}{(4+x) \left (576+145 x+9 x^2+192 \log (4+x)+24 x \log (4+x)+16 \log ^2(4+x)\right )}\right ) \, dx \\ & = -2 \log (x)+2 \int \frac {772+241 x+18 x^2+128 \log (4+x)+24 x \log (4+x)}{(4+x) \left (576+145 x+9 x^2+192 \log (4+x)+24 x \log (4+x)+16 \log ^2(4+x)\right )} \, dx \\ & = -2 \log (x)+2 \log \left (576+145 x+9 x^2+192 \log (4+x)+24 x \log (4+x)+16 \log ^2(4+x)\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.77 \[ \int \frac {-4608-768 x+120 x^2+18 x^3+(-1536-320 x) \log (4+x)+(-128-32 x) \log ^2(4+x)}{2304 x+1156 x^2+181 x^3+9 x^4+\left (768 x+288 x^2+24 x^3\right ) \log (4+x)+\left (64 x+16 x^2\right ) \log ^2(4+x)} \, dx=2 \left (-\log (x)+\log \left (576+145 x+9 x^2+192 \log (4+x)+24 x \log (4+x)+16 \log ^2(4+x)\right )\right ) \]

[In]

Integrate[(-4608 - 768*x + 120*x^2 + 18*x^3 + (-1536 - 320*x)*Log[4 + x] + (-128 - 32*x)*Log[4 + x]^2)/(2304*x
+ 1156*x^2 + 181*x^3 + 9*x^4 + (768*x + 288*x^2 + 24*x^3)*Log[4 + x] + (64*x + 16*x^2)*Log[4 + x]^2),x]

[Out]

2*(-Log[x] + Log[576 + 145*x + 9*x^2 + 192*Log[4 + x] + 24*x*Log[4 + x] + 16*Log[4 + x]^2])

Maple [A] (verified)

Time = 0.50 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.59

method	result	size
risch	\(-2 \ln \left (x \right )+2 \ln \left (\ln \left (4+x \right )^{2}+\left (\frac {3 x}{2}+12\right ) \ln \left (4+x \right )+\frac {9 x^{2}}{16}+\frac {145 x}{16}+36\right )\)	\(35\)
parallelrisch	\(-2 \ln \left (x \right )+2 \ln \left (x^{2}+\frac {8 \ln \left (4+x \right ) x}{3}+\frac {16 \ln \left (4+x \right )^{2}}{9}+\frac {145 x}{9}+\frac {64 \ln \left (4+x \right )}{3}+64\right )\)	\(38\)
norman	\(-2 \ln \left (x \right )+2 \ln \left (16 \ln \left (4+x \right )^{2}+24 \ln \left (4+x \right ) x +9 x^{2}+192 \ln \left (4+x \right )+145 x +576\right )\)	\(40\)
derivativedivides	\(2 \ln \left (16 \ln \left (4+x \right )^{2}+24 \left (4+x \right ) \ln \left (4+x \right )+9 \left (4+x \right )^{2}+96 \ln \left (4+x \right )+432+73 x \right )-2 \ln \left (x \right )\)	\(44\)
default	\(2 \ln \left (16 \ln \left (4+x \right )^{2}+24 \left (4+x \right ) \ln \left (4+x \right )+9 \left (4+x \right )^{2}+96 \ln \left (4+x \right )+432+73 x \right )-2 \ln \left (x \right )\)	\(44\)

[In]

int(((-32*x-128)*ln(4+x)^2+(-320*x-1536)*ln(4+x)+18*x^3+120*x^2-768*x-4608)/((16*x^2+64*x)*ln(4+x)^2+(24*x^3+2
88*x^2+768*x)*ln(4+x)+9*x^4+181*x^3+1156*x^2+2304*x),x,method=_RETURNVERBOSE)

[Out]

-2*ln(x)+2*ln(ln(4+x)^2+(3/2*x+12)*ln(4+x)+9/16*x^2+145/16*x+36)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.59 \[ \int \frac {-4608-768 x+120 x^2+18 x^3+(-1536-320 x) \log (4+x)+(-128-32 x) \log ^2(4+x)}{2304 x+1156 x^2+181 x^3+9 x^4+\left (768 x+288 x^2+24 x^3\right ) \log (4+x)+\left (64 x+16 x^2\right ) \log ^2(4+x)} \, dx=2 \, \log \left (9 \, x^{2} + 24 \, {\left (x + 8\right )} \log \left (x + 4\right ) + 16 \, \log \left (x + 4\right )^{2} + 145 \, x + 576\right ) - 2 \, \log \left (x\right ) \]

[In]

integrate(((-32*x-128)*log(4+x)^2+(-320*x-1536)*log(4+x)+18*x^3+120*x^2-768*x-4608)/((16*x^2+64*x)*log(4+x)^2+
(24*x^3+288*x^2+768*x)*log(4+x)+9*x^4+181*x^3+1156*x^2+2304*x),x, algorithm="fricas")

[Out]

2*log(9*x^2 + 24*(x + 8)*log(x + 4) + 16*log(x + 4)^2 + 145*x + 576) - 2*log(x)

Sympy [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.77 \[ \int \frac {-4608-768 x+120 x^2+18 x^3+(-1536-320 x) \log (4+x)+(-128-32 x) \log ^2(4+x)}{2304 x+1156 x^2+181 x^3+9 x^4+\left (768 x+288 x^2+24 x^3\right ) \log (4+x)+\left (64 x+16 x^2\right ) \log ^2(4+x)} \, dx=- 2 \log {\left (x \right )} + 2 \log {\left (\frac {9 x^{2}}{16} + \frac {145 x}{16} + \left (\frac {3 x}{2} + 12\right ) \log {\left (x + 4 \right )} + \log {\left (x + 4 \right )}^{2} + 36 \right )} \]

[In]

integrate(((-32*x-128)*ln(4+x)**2+(-320*x-1536)*ln(4+x)+18*x**3+120*x**2-768*x-4608)/((16*x**2+64*x)*ln(4+x)**
2+(24*x**3+288*x**2+768*x)*ln(4+x)+9*x**4+181*x**3+1156*x**2+2304*x),x)

[Out]

-2*log(x) + 2*log(9*x**2/16 + 145*x/16 + (3*x/2 + 12)*log(x + 4) + log(x + 4)**2 + 36)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.50 \[ \int \frac {-4608-768 x+120 x^2+18 x^3+(-1536-320 x) \log (4+x)+(-128-32 x) \log ^2(4+x)}{2304 x+1156 x^2+181 x^3+9 x^4+\left (768 x+288 x^2+24 x^3\right ) \log (4+x)+\left (64 x+16 x^2\right ) \log ^2(4+x)} \, dx=2 \, \log \left (\frac {9}{16} \, x^{2} + \frac {3}{2} \, {\left (x + 8\right )} \log \left (x + 4\right ) + \log \left (x + 4\right )^{2} + \frac {145}{16} \, x + 36\right ) - 2 \, \log \left (x\right ) \]

[In]

integrate(((-32*x-128)*log(4+x)^2+(-320*x-1536)*log(4+x)+18*x^3+120*x^2-768*x-4608)/((16*x^2+64*x)*log(4+x)^2+
(24*x^3+288*x^2+768*x)*log(4+x)+9*x^4+181*x^3+1156*x^2+2304*x),x, algorithm="maxima")

[Out]

2*log(9/16*x^2 + 3/2*(x + 8)*log(x + 4) + log(x + 4)^2 + 145/16*x + 36) - 2*log(x)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.77 \[ \int \frac {-4608-768 x+120 x^2+18 x^3+(-1536-320 x) \log (4+x)+(-128-32 x) \log ^2(4+x)}{2304 x+1156 x^2+181 x^3+9 x^4+\left (768 x+288 x^2+24 x^3\right ) \log (4+x)+\left (64 x+16 x^2\right ) \log ^2(4+x)} \, dx=2 \, \log \left (9 \, x^{2} + 24 \, x \log \left (x + 4\right ) + 16 \, \log \left (x + 4\right )^{2} + 145 \, x + 192 \, \log \left (x + 4\right ) + 576\right ) - 2 \, \log \left (x\right ) \]

[In]

integrate(((-32*x-128)*log(4+x)^2+(-320*x-1536)*log(4+x)+18*x^3+120*x^2-768*x-4608)/((16*x^2+64*x)*log(4+x)^2+
(24*x^3+288*x^2+768*x)*log(4+x)+9*x^4+181*x^3+1156*x^2+2304*x),x, algorithm="giac")

[Out]

2*log(9*x^2 + 24*x*log(x + 4) + 16*log(x + 4)^2 + 145*x + 192*log(x + 4) + 576) - 2*log(x)

Mupad [B] (verification not implemented)

Time = 11.05 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.68 \[ \int \frac {-4608-768 x+120 x^2+18 x^3+(-1536-320 x) \log (4+x)+(-128-32 x) \log ^2(4+x)}{2304 x+1156 x^2+181 x^3+9 x^4+\left (768 x+288 x^2+24 x^3\right ) \log (4+x)+\left (64 x+16 x^2\right ) \log ^2(4+x)} \, dx=2\,\ln \left (\frac {9\,x^2}{16}+\frac {3\,x\,\ln \left (x+4\right )}{2}+\frac {145\,x}{16}+{\ln \left (x+4\right )}^2+12\,\ln \left (x+4\right )+36\right )-2\,\ln \left (x\right ) \]

[In]

int(-(768*x + log(x + 4)^2*(32*x + 128) - 120*x^2 - 18*x^3 + log(x + 4)*(320*x + 1536) + 4608)/(2304*x + log(x
+ 4)*(768*x + 288*x^2 + 24*x^3) + log(x + 4)^2*(64*x + 16*x^2) + 1156*x^2 + 181*x^3 + 9*x^4),x)

[Out]

2*log((145*x)/16 + 12*log(x + 4) + (3*x*log(x + 4))/2 + log(x + 4)^2 + (9*x^2)/16 + 36) - 2*log(x)