\(\int \frac {8+16 x+96 x^2+128 x^3+2 \log (x^2)}{4 x^2+16 x^3+80 x^4+192 x^5+384 x^6+512 x^7+256 x^8+(4 x^2+8 x^3+32 x^4+32 x^5) \log (x^2)+x^2 \log ^2(x^2)} \, dx\) [6906]

Optimal result

Integrand size = 96, antiderivative size = 27 \[ \int \frac {8+16 x+96 x^2+128 x^3+2 \log \left (x^2\right )}{4 x^2+16 x^3+80 x^4+192 x^5+384 x^6+512 x^7+256 x^8+\left (4 x^2+8 x^3+32 x^4+32 x^5\right ) \log \left (x^2\right )+x^2 \log ^2\left (x^2\right )} \, dx=\frac {2}{-2 x-x^2 (2+4 x)^2-x \log \left (x^2\right )} \]

[Out]

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

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {6820, 12, 6819} \[ \int \frac {8+16 x+96 x^2+128 x^3+2 \log \left (x^2\right )}{4 x^2+16 x^3+80 x^4+192 x^5+384 x^6+512 x^7+256 x^8+\left (4 x^2+8 x^3+32 x^4+32 x^5\right ) \log \left (x^2\right )+x^2 \log ^2\left (x^2\right )} \, dx=-\frac {2}{x \left (16 x^3+16 x^2+\log \left (x^2\right )+4 x+2\right )} \]

[In]

Int[(8 + 16*x + 96*x^2 + 128*x^3 + 2*Log[x^2])/(4*x^2 + 16*x^3 + 80*x^4 + 192*x^5 + 384*x^6 + 512*x^7 + 256*x^
8 + (4*x^2 + 8*x^3 + 32*x^4 + 32*x^5)*Log[x^2] + x^2*Log[x^2]^2),x]

[Out]

-2/(x*(2 + 4*x + 16*x^2 + 16*x^3 + 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 6819

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

Rule 6820

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {2 \left (4+8 x+48 x^2+64 x^3+\log \left (x^2\right )\right )}{x^2 \left (2+4 x+16 x^2+16 x^3+\log \left (x^2\right )\right )^2} \, dx \\ & = 2 \int \frac {4+8 x+48 x^2+64 x^3+\log \left (x^2\right )}{x^2 \left (2+4 x+16 x^2+16 x^3+\log \left (x^2\right )\right )^2} \, dx \\ & = -\frac {2}{x \left (2+4 x+16 x^2+16 x^3+\log \left (x^2\right )\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {8+16 x+96 x^2+128 x^3+2 \log \left (x^2\right )}{4 x^2+16 x^3+80 x^4+192 x^5+384 x^6+512 x^7+256 x^8+\left (4 x^2+8 x^3+32 x^4+32 x^5\right ) \log \left (x^2\right )+x^2 \log ^2\left (x^2\right )} \, dx=-\frac {2}{x \left (2+4 x+16 x^2+16 x^3+\log \left (x^2\right )\right )} \]

[In]

Integrate[(8 + 16*x + 96*x^2 + 128*x^3 + 2*Log[x^2])/(4*x^2 + 16*x^3 + 80*x^4 + 192*x^5 + 384*x^6 + 512*x^7 +
256*x^8 + (4*x^2 + 8*x^3 + 32*x^4 + 32*x^5)*Log[x^2] + x^2*Log[x^2]^2),x]

[Out]

-2/(x*(2 + 4*x + 16*x^2 + 16*x^3 + Log[x^2]))

Maple [A] (verified)

Time = 0.86 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00

method	result	size
risch	\(-\frac {2}{x \left (16 x^{3}+16 x^{2}+\ln \left (x^{2}\right )+4 x +2\right )}\)	\(27\)
parallelrisch	\(-\frac {2}{x \left (16 x^{3}+16 x^{2}+\ln \left (x^{2}\right )+4 x +2\right )}\)	\(27\)

[In]

int((2*ln(x^2)+128*x^3+96*x^2+16*x+8)/(x^2*ln(x^2)^2+(32*x^5+32*x^4+8*x^3+4*x^2)*ln(x^2)+256*x^8+512*x^7+384*x
^6+192*x^5+80*x^4+16*x^3+4*x^2),x,method=_RETURNVERBOSE)

[Out]

-2/x/(16*x^3+16*x^2+ln(x^2)+4*x+2)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {8+16 x+96 x^2+128 x^3+2 \log \left (x^2\right )}{4 x^2+16 x^3+80 x^4+192 x^5+384 x^6+512 x^7+256 x^8+\left (4 x^2+8 x^3+32 x^4+32 x^5\right ) \log \left (x^2\right )+x^2 \log ^2\left (x^2\right )} \, dx=-\frac {2}{16 \, x^{4} + 16 \, x^{3} + 4 \, x^{2} + x \log \left (x^{2}\right ) + 2 \, x} \]

[In]

integrate((2*log(x^2)+128*x^3+96*x^2+16*x+8)/(x^2*log(x^2)^2+(32*x^5+32*x^4+8*x^3+4*x^2)*log(x^2)+256*x^8+512*
x^7+384*x^6+192*x^5+80*x^4+16*x^3+4*x^2),x, algorithm="fricas")

[Out]

-2/(16*x^4 + 16*x^3 + 4*x^2 + x*log(x^2) + 2*x)

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {8+16 x+96 x^2+128 x^3+2 \log \left (x^2\right )}{4 x^2+16 x^3+80 x^4+192 x^5+384 x^6+512 x^7+256 x^8+\left (4 x^2+8 x^3+32 x^4+32 x^5\right ) \log \left (x^2\right )+x^2 \log ^2\left (x^2\right )} \, dx=- \frac {2}{16 x^{4} + 16 x^{3} + 4 x^{2} + x \log {\left (x^{2} \right )} + 2 x} \]

[In]

integrate((2*ln(x**2)+128*x**3+96*x**2+16*x+8)/(x**2*ln(x**2)**2+(32*x**5+32*x**4+8*x**3+4*x**2)*ln(x**2)+256*
x**8+512*x**7+384*x**6+192*x**5+80*x**4+16*x**3+4*x**2),x)

[Out]

-2/(16*x**4 + 16*x**3 + 4*x**2 + x*log(x**2) + 2*x)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {8+16 x+96 x^2+128 x^3+2 \log \left (x^2\right )}{4 x^2+16 x^3+80 x^4+192 x^5+384 x^6+512 x^7+256 x^8+\left (4 x^2+8 x^3+32 x^4+32 x^5\right ) \log \left (x^2\right )+x^2 \log ^2\left (x^2\right )} \, dx=-\frac {1}{8 \, x^{4} + 8 \, x^{3} + 2 \, x^{2} + x \log \left (x\right ) + x} \]

[In]

integrate((2*log(x^2)+128*x^3+96*x^2+16*x+8)/(x^2*log(x^2)^2+(32*x^5+32*x^4+8*x^3+4*x^2)*log(x^2)+256*x^8+512*
x^7+384*x^6+192*x^5+80*x^4+16*x^3+4*x^2),x, algorithm="maxima")

[Out]

-1/(8*x^4 + 8*x^3 + 2*x^2 + x*log(x) + x)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {8+16 x+96 x^2+128 x^3+2 \log \left (x^2\right )}{4 x^2+16 x^3+80 x^4+192 x^5+384 x^6+512 x^7+256 x^8+\left (4 x^2+8 x^3+32 x^4+32 x^5\right ) \log \left (x^2\right )+x^2 \log ^2\left (x^2\right )} \, dx=-\frac {2}{16 \, x^{4} + 16 \, x^{3} + 4 \, x^{2} + x \log \left (x^{2}\right ) + 2 \, x} \]

[In]

integrate((2*log(x^2)+128*x^3+96*x^2+16*x+8)/(x^2*log(x^2)^2+(32*x^5+32*x^4+8*x^3+4*x^2)*log(x^2)+256*x^8+512*
x^7+384*x^6+192*x^5+80*x^4+16*x^3+4*x^2),x, algorithm="giac")

[Out]

-2/(16*x^4 + 16*x^3 + 4*x^2 + x*log(x^2) + 2*x)

Mupad [F(-1)]

Timed out. \[ \int \frac {8+16 x+96 x^2+128 x^3+2 \log \left (x^2\right )}{4 x^2+16 x^3+80 x^4+192 x^5+384 x^6+512 x^7+256 x^8+\left (4 x^2+8 x^3+32 x^4+32 x^5\right ) \log \left (x^2\right )+x^2 \log ^2\left (x^2\right )} \, dx=\int \frac {16\,x+2\,\ln \left (x^2\right )+96\,x^2+128\,x^3+8}{4\,x^2+16\,x^3+80\,x^4+192\,x^5+384\,x^6+512\,x^7+256\,x^8+x^2\,{\ln \left (x^2\right )}^2+\ln \left (x^2\right )\,\left (32\,x^5+32\,x^4+8\,x^3+4\,x^2\right )} \,d x \]

[In]

int((16*x + 2*log(x^2) + 96*x^2 + 128*x^3 + 8)/(4*x^2 + 16*x^3 + 80*x^4 + 192*x^5 + 384*x^6 + 512*x^7 + 256*x^
8 + x^2*log(x^2)^2 + log(x^2)*(4*x^2 + 8*x^3 + 32*x^4 + 32*x^5)),x)

[Out]

int((16*x + 2*log(x^2) + 96*x^2 + 128*x^3 + 8)/(4*x^2 + 16*x^3 + 80*x^4 + 192*x^5 + 384*x^6 + 512*x^7 + 256*x^
8 + x^2*log(x^2)^2 + log(x^2)*(4*x^2 + 8*x^3 + 32*x^4 + 32*x^5)), x)