Integrand size = 83, antiderivative size = 26 \[ \int \frac {16 x^2-8 x^3+x^4+e^{4/x} \left (-64+32 x-4 x^2\right )+\left (16 x^5-4 x^6\right ) \log \left (x^2\right )+\left (16 x^5-3 x^6\right ) \log ^2\left (x^2\right )}{16 x^2-8 x^3+x^4} \, dx=e^{4/x}+x+\frac {x^4 \log ^2\left (x^2\right )}{4-x} \]
Time = 0.08 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int \frac {16 x^2-8 x^3+x^4+e^{4/x} \left (-64+32 x-4 x^2\right )+\left (16 x^5-4 x^6\right ) \log \left (x^2\right )+\left (16 x^5-3 x^6\right ) \log ^2\left (x^2\right )}{16 x^2-8 x^3+x^4} \, dx=e^{4/x}+x-\frac {x^4 \log ^2\left (x^2\right )}{-4+x} \]
Integrate[(16*x^2 - 8*x^3 + x^4 + E^(4/x)*(-64 + 32*x - 4*x^2) + (16*x^5 - 4*x^6)*Log[x^2] + (16*x^5 - 3*x^6)*Log[x^2]^2)/(16*x^2 - 8*x^3 + x^4),x]
Leaf count is larger than twice the leaf count of optimal. \(56\) vs. \(2(26)=52\).
Time = 1.27 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.15, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.060, Rules used = {2026, 7277, 27, 7293, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {x^4-8 x^3+16 x^2+e^{4/x} \left (-4 x^2+32 x-64\right )+\left (16 x^5-3 x^6\right ) \log ^2\left (x^2\right )+\left (16 x^5-4 x^6\right ) \log \left (x^2\right )}{x^4-8 x^3+16 x^2} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {x^4-8 x^3+16 x^2+e^{4/x} \left (-4 x^2+32 x-64\right )+\left (16 x^5-3 x^6\right ) \log ^2\left (x^2\right )+\left (16 x^5-4 x^6\right ) \log \left (x^2\right )}{x^2 \left (x^2-8 x+16\right )}dx\) |
\(\Big \downarrow \) 7277 |
\(\displaystyle 4 \int \frac {x^4-8 x^3+16 x^2+\left (16 x^5-3 x^6\right ) \log ^2\left (x^2\right )-4 e^{4/x} \left (x^2-8 x+16\right )+4 \left (4 x^5-x^6\right ) \log \left (x^2\right )}{4 (4-x)^2 x^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {x^4-8 x^3+16 x^2-4 e^{4/x} \left (x^2-8 x+16\right )+\left (16 x^5-3 x^6\right ) \log ^2\left (x^2\right )+4 \left (4 x^5-x^6\right ) \log \left (x^2\right )}{(4-x)^2 x^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {x^2}{(x-4)^2}-\frac {4 e^{4/x}}{x^2}-\frac {(3 x-16) x^3 \log ^2\left (x^2\right )}{(x-4)^2}-\frac {4 x^3 \log \left (x^2\right )}{x-4}-\frac {8 x}{(x-4)^2}+\frac {16}{(x-4)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -4 x^2 \log ^2\left (x^2\right )+\frac {64 x \log ^2\left (x^2\right )}{4-x}-16 x \log ^2\left (x^2\right )+x^3 \left (-\log ^2\left (x^2\right )\right )+x+e^{4/x}\) |
Int[(16*x^2 - 8*x^3 + x^4 + E^(4/x)*(-64 + 32*x - 4*x^2) + (16*x^5 - 4*x^6 )*Log[x^2] + (16*x^5 - 3*x^6)*Log[x^2]^2)/(16*x^2 - 8*x^3 + x^4),x]
E^(4/x) + x - 16*x*Log[x^2]^2 + (64*x*Log[x^2]^2)/(4 - x) - 4*x^2*Log[x^2] ^2 - x^3*Log[x^2]^2
3.25.72.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Int[(u_)*((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_.), x_Symbol] :> Simp[1/(4^p*c^p) Int[u*(b + 2*c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n} , x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p] && !AlgebraicFu nctionQ[u, x]
Time = 0.61 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.65
method | result | size |
parallelrisch | \(\frac {-4 x^{4} \ln \left (x^{2}\right )^{2}-64+4 x^{2}+4 x \,{\mathrm e}^{\frac {4}{x}}-16 \,{\mathrm e}^{\frac {4}{x}}}{4 x -16}\) | \(43\) |
risch | \(-\frac {4 x^{4} \ln \left (x \right )^{2}}{x -4}+\frac {2 i \pi \,\operatorname {csgn}\left (i x^{2}\right ) \left (x^{4} \operatorname {csgn}\left (i x \right )^{2}-2 x^{4} \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )+x^{4} \operatorname {csgn}\left (i x^{2}\right )^{2}-64 x \operatorname {csgn}\left (i x \right )^{2}+128 x \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )-64 x \operatorname {csgn}\left (i x^{2}\right )^{2}+256 \operatorname {csgn}\left (i x \right )^{2}-512 \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )+256 \operatorname {csgn}\left (i x^{2}\right )^{2}\right ) \ln \left (x \right )}{x -4}+\frac {512 i \ln \left (x \right ) \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right ) x -1024 i \ln \left (x \right ) \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2} x +512 i \ln \left (x \right ) \pi \operatorname {csgn}\left (i x^{2}\right )^{3} x -2048 i \ln \left (x \right ) \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )+4096 i \ln \left (x \right ) \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+256 \operatorname {csgn}\left (i x^{2}\right )^{6} \pi ^{2}-16 x +4 x \,{\mathrm e}^{\frac {4}{x}}-16 \,{\mathrm e}^{\frac {4}{x}}+4 x^{2}-1024 \operatorname {csgn}\left (i x^{2}\right )^{5} \operatorname {csgn}\left (i x \right ) \pi ^{2}+1536 \operatorname {csgn}\left (i x^{2}\right )^{4} \operatorname {csgn}\left (i x \right )^{2} \pi ^{2}-1024 \operatorname {csgn}\left (i x^{2}\right )^{3} \operatorname {csgn}\left (i x \right )^{3} \pi ^{2}+256 \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right )^{4} \pi ^{2}+\pi ^{2} x^{4} \operatorname {csgn}\left (i x^{2}\right )^{6}-64 \pi ^{2} x \operatorname {csgn}\left (i x^{2}\right )^{6}-2048 i \ln \left (x \right ) \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+6 \pi ^{2} x^{4} \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )^{4}-4 \pi ^{2} x^{4} \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{5}+\pi ^{2} x^{4} \operatorname {csgn}\left (i x \right )^{4} \operatorname {csgn}\left (i x^{2}\right )^{2}-4 \pi ^{2} x^{4} \operatorname {csgn}\left (i x \right )^{3} \operatorname {csgn}\left (i x^{2}\right )^{3}-64 \pi ^{2} x \operatorname {csgn}\left (i x \right )^{4} \operatorname {csgn}\left (i x^{2}\right )^{2}+256 \pi ^{2} x \operatorname {csgn}\left (i x \right )^{3} \operatorname {csgn}\left (i x^{2}\right )^{3}-384 \pi ^{2} x \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )^{4}+256 \pi ^{2} x \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{5}}{4 x -16}\) | \(599\) |
int(((-3*x^6+16*x^5)*ln(x^2)^2+(-4*x^6+16*x^5)*ln(x^2)+(-4*x^2+32*x-64)*ex p(4/x)+x^4-8*x^3+16*x^2)/(x^4-8*x^3+16*x^2),x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.42 \[ \int \frac {16 x^2-8 x^3+x^4+e^{4/x} \left (-64+32 x-4 x^2\right )+\left (16 x^5-4 x^6\right ) \log \left (x^2\right )+\left (16 x^5-3 x^6\right ) \log ^2\left (x^2\right )}{16 x^2-8 x^3+x^4} \, dx=-\frac {x^{4} \log \left (x^{2}\right )^{2} - x^{2} - {\left (x - 4\right )} e^{\frac {4}{x}} + 4 \, x}{x - 4} \]
integrate(((-3*x^6+16*x^5)*log(x^2)^2+(-4*x^6+16*x^5)*log(x^2)+(-4*x^2+32* x-64)*exp(4/x)+x^4-8*x^3+16*x^2)/(x^4-8*x^3+16*x^2),x, algorithm=\
Time = 0.21 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.73 \[ \int \frac {16 x^2-8 x^3+x^4+e^{4/x} \left (-64+32 x-4 x^2\right )+\left (16 x^5-4 x^6\right ) \log \left (x^2\right )+\left (16 x^5-3 x^6\right ) \log ^2\left (x^2\right )}{16 x^2-8 x^3+x^4} \, dx=- \frac {x^{4} \log {\left (x^{2} \right )}^{2}}{x - 4} + x + e^{\frac {4}{x}} \]
integrate(((-3*x**6+16*x**5)*ln(x**2)**2+(-4*x**6+16*x**5)*ln(x**2)+(-4*x* *2+32*x-64)*exp(4/x)+x**4-8*x**3+16*x**2)/(x**4-8*x**3+16*x**2),x)
Time = 0.23 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15 \[ \int \frac {16 x^2-8 x^3+x^4+e^{4/x} \left (-64+32 x-4 x^2\right )+\left (16 x^5-4 x^6\right ) \log \left (x^2\right )+\left (16 x^5-3 x^6\right ) \log ^2\left (x^2\right )}{16 x^2-8 x^3+x^4} \, dx=x - \frac {4 \, x^{4} \log \left (x\right )^{2} - {\left (x - 4\right )} e^{\frac {4}{x}}}{x - 4} \]
integrate(((-3*x^6+16*x^5)*log(x^2)^2+(-4*x^6+16*x^5)*log(x^2)+(-4*x^2+32* x-64)*exp(4/x)+x^4-8*x^3+16*x^2)/(x^4-8*x^3+16*x^2),x, algorithm=\
Time = 0.27 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.38 \[ \int \frac {16 x^2-8 x^3+x^4+e^{4/x} \left (-64+32 x-4 x^2\right )+\left (16 x^5-4 x^6\right ) \log \left (x^2\right )+\left (16 x^5-3 x^6\right ) \log ^2\left (x^2\right )}{16 x^2-8 x^3+x^4} \, dx=-{\left (x^{3} + 4 \, x^{2} + 16 \, x + \frac {256}{x - 4} + 64\right )} \log \left (x^{2}\right )^{2} + x + e^{\frac {4}{x}} \]
integrate(((-3*x^6+16*x^5)*log(x^2)^2+(-4*x^6+16*x^5)*log(x^2)+(-4*x^2+32* x-64)*exp(4/x)+x^4-8*x^3+16*x^2)/(x^4-8*x^3+16*x^2),x, algorithm=\
Time = 11.38 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {16 x^2-8 x^3+x^4+e^{4/x} \left (-64+32 x-4 x^2\right )+\left (16 x^5-4 x^6\right ) \log \left (x^2\right )+\left (16 x^5-3 x^6\right ) \log ^2\left (x^2\right )}{16 x^2-8 x^3+x^4} \, dx=x+{\mathrm {e}}^{4/x}-\frac {x^4\,{\ln \left (x^2\right )}^2}{x-4} \]