\(\int \frac {-32-16 x+3 x^3+4 x^6-x^7}{16 x-3 x^3+x^7} \, dx\) [9313]

Optimal result

Integrand size = 35, antiderivative size = 21 \[ \int \frac {-32-16 x+3 x^3+4 x^6-x^7}{16 x-3 x^3+x^7} \, dx=-8+\log \left (e^{-x} \left (3-\frac {16}{x^2}-x^4\right )\right ) \]

[Out]

ln((3-16/x^2-x^4)/exp(x))-8

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {1608, 6874, 1601} \[ \int \frac {-32-16 x+3 x^3+4 x^6-x^7}{16 x-3 x^3+x^7} \, dx=\log \left (x^6-3 x^2+16\right )-x-2 \log (x) \]

[In]

Int[(-32 - 16*x + 3*x^3 + 4*x^6 - x^7)/(16*x - 3*x^3 + x^7),x]

[Out]

-x - 2*Log[x] + Log[16 - 3*x^2 + x^6]

Rule 1601

Int[(Pp_)/(Qq_), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[Coeff[Pp, x, p]*(Log[RemoveConte
nt[Qq, x]]/(q*Coeff[Qq, x, q])), x] /; EqQ[p, q - 1] && EqQ[Pp, Simplify[(Coeff[Pp, x, p]/(q*Coeff[Qq, x, q]))
*D[Qq, x]]]] /; PolyQ[Pp, x] && PolyQ[Qq, x]

Rule 1608

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

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {-32-16 x+3 x^3+4 x^6-x^7}{x \left (16-3 x^2+x^6\right )} \, dx \\ & = \int \left (-1-\frac {2}{x}+\frac {6 x \left (-1+x^4\right )}{16-3 x^2+x^6}\right ) \, dx \\ & = -x-2 \log (x)+6 \int \frac {x \left (-1+x^4\right )}{16-3 x^2+x^6} \, dx \\ & = -x-2 \log (x)+\log \left (16-3 x^2+x^6\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {-32-16 x+3 x^3+4 x^6-x^7}{16 x-3 x^3+x^7} \, dx=-x-2 \log (x)+\log \left (16-3 x^2+x^6\right ) \]

[In]

Integrate[(-32 - 16*x + 3*x^3 + 4*x^6 - x^7)/(16*x - 3*x^3 + x^7),x]

[Out]

-x - 2*Log[x] + Log[16 - 3*x^2 + x^6]

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95

method	result	size
default	\(-x +\ln \left (x^{6}-3 x^{2}+16\right )-2 \ln \left (x \right )\)	\(20\)
norman	\(-x +\ln \left (x^{6}-3 x^{2}+16\right )-2 \ln \left (x \right )\)	\(20\)
risch	\(-x +\ln \left (x^{6}-3 x^{2}+16\right )-2 \ln \left (x \right )\)	\(20\)
parallelrisch	\(-x +\ln \left (x^{6}-3 x^{2}+16\right )-2 \ln \left (x \right )\)	\(20\)

[In]

int((-x^7+4*x^6+3*x^3-16*x-32)/(x^7-3*x^3+16*x),x,method=_RETURNVERBOSE)

[Out]

-x+ln(x^6-3*x^2+16)-2*ln(x)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {-32-16 x+3 x^3+4 x^6-x^7}{16 x-3 x^3+x^7} \, dx=-x + \log \left (x^{6} - 3 \, x^{2} + 16\right ) - 2 \, \log \left (x\right ) \]

[In]

integrate((-x^7+4*x^6+3*x^3-16*x-32)/(x^7-3*x^3+16*x),x, algorithm="fricas")

[Out]

-x + log(x^6 - 3*x^2 + 16) - 2*log(x)

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {-32-16 x+3 x^3+4 x^6-x^7}{16 x-3 x^3+x^7} \, dx=- x - 2 \log {\left (x \right )} + \log {\left (x^{6} - 3 x^{2} + 16 \right )} \]

[In]

integrate((-x**7+4*x**6+3*x**3-16*x-32)/(x**7-3*x**3+16*x),x)

[Out]

-x - 2*log(x) + log(x**6 - 3*x**2 + 16)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {-32-16 x+3 x^3+4 x^6-x^7}{16 x-3 x^3+x^7} \, dx=-x + \log \left (x^{6} - 3 \, x^{2} + 16\right ) - 2 \, \log \left (x\right ) \]

[In]

integrate((-x^7+4*x^6+3*x^3-16*x-32)/(x^7-3*x^3+16*x),x, algorithm="maxima")

[Out]

-x + log(x^6 - 3*x^2 + 16) - 2*log(x)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {-32-16 x+3 x^3+4 x^6-x^7}{16 x-3 x^3+x^7} \, dx=-x + \log \left (x^{6} - 3 \, x^{2} + 16\right ) - 2 \, \log \left ({\left | x \right |}\right ) \]

[In]

integrate((-x^7+4*x^6+3*x^3-16*x-32)/(x^7-3*x^3+16*x),x, algorithm="giac")

[Out]

-x + log(x^6 - 3*x^2 + 16) - 2*log(abs(x))

Mupad [B] (verification not implemented)

Time = 13.20 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {-32-16 x+3 x^3+4 x^6-x^7}{16 x-3 x^3+x^7} \, dx=\ln \left (x^6-3\,x^2+16\right )-x-2\,\ln \left (x\right ) \]

[In]

int(-(16*x - 3*x^3 - 4*x^6 + x^7 + 32)/(16*x - 3*x^3 + x^7),x)

[Out]

log(x^6 - 3*x^2 + 16) - x - 2*log(x)