\(\int \frac {-48-40 x^2+96 x^4-40 x^6+14 x^8}{24 x+x^2-20 x^3+16 x^5-4 x^7+x^9} \, dx\) [59]

Optimal result

Integrand size = 50, antiderivative size = 26 \[ \int \frac {-48-40 x^2+96 x^4-40 x^6+14 x^8}{24 x+x^2-20 x^3+16 x^5-4 x^7+x^9} \, dx=\log \left (\frac {\left (-12+x+4 x^2+\left (5+\left (-1+x^2\right )^2\right )^2\right )^2}{x^2}\right ) \]

[Out]

ln((4*x^2-12+x+(5+(x^2-1)^2)^2)^2/x^2)

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2099, 1601} \[ \int \frac {-48-40 x^2+96 x^4-40 x^6+14 x^8}{24 x+x^2-20 x^3+16 x^5-4 x^7+x^9} \, dx=2 \log \left (x^8-4 x^6+16 x^4-20 x^2+x+24\right )-2 \log (x) \]

[In]

Int[(-48 - 40*x^2 + 96*x^4 - 40*x^6 + 14*x^8)/(24*x + x^2 - 20*x^3 + 16*x^5 - 4*x^7 + x^9),x]

[Out]

-2*Log[x] + 2*Log[24 + x - 20*x^2 + 16*x^4 - 4*x^6 + x^8]

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 2099

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {2}{x}+\frac {2 \left (1-40 x+64 x^3-24 x^5+8 x^7\right )}{24+x-20 x^2+16 x^4-4 x^6+x^8}\right ) \, dx \\ & = -2 \log (x)+2 \int \frac {1-40 x+64 x^3-24 x^5+8 x^7}{24+x-20 x^2+16 x^4-4 x^6+x^8} \, dx \\ & = -2 \log (x)+2 \log \left (24+x-20 x^2+16 x^4-4 x^6+x^8\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {-48-40 x^2+96 x^4-40 x^6+14 x^8}{24 x+x^2-20 x^3+16 x^5-4 x^7+x^9} \, dx=2 \left (-\log (x)+\log \left (24+x-20 x^2+16 x^4-4 x^6+x^8\right )\right ) \]

[In]

Integrate[(-48 - 40*x^2 + 96*x^4 - 40*x^6 + 14*x^8)/(24*x + x^2 - 20*x^3 + 16*x^5 - 4*x^7 + x^9),x]

[Out]

2*(-Log[x] + Log[24 + x - 20*x^2 + 16*x^4 - 4*x^6 + x^8])

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15

method	result	size
default	\(-2 \ln \left (x \right )+2 \ln \left (x^{8}-4 x^{6}+16 x^{4}-20 x^{2}+x +24\right )\)	\(30\)
norman	\(-2 \ln \left (x \right )+2 \ln \left (x^{8}-4 x^{6}+16 x^{4}-20 x^{2}+x +24\right )\)	\(30\)
risch	\(-2 \ln \left (x \right )+2 \ln \left (x^{8}-4 x^{6}+16 x^{4}-20 x^{2}+x +24\right )\)	\(30\)
parallelrisch	\(-2 \ln \left (x \right )+2 \ln \left (x^{8}-4 x^{6}+16 x^{4}-20 x^{2}+x +24\right )\)	\(30\)

[In]

int((14*x^8-40*x^6+96*x^4-40*x^2-48)/(x^9-4*x^7+16*x^5-20*x^3+x^2+24*x),x,method=_RETURNVERBOSE)

[Out]

-2*ln(x)+2*ln(x^8-4*x^6+16*x^4-20*x^2+x+24)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {-48-40 x^2+96 x^4-40 x^6+14 x^8}{24 x+x^2-20 x^3+16 x^5-4 x^7+x^9} \, dx=2 \, \log \left (x^{8} - 4 \, x^{6} + 16 \, x^{4} - 20 \, x^{2} + x + 24\right ) - 2 \, \log \left (x\right ) \]

[In]

integrate((14*x^8-40*x^6+96*x^4-40*x^2-48)/(x^9-4*x^7+16*x^5-20*x^3+x^2+24*x),x, algorithm="fricas")

[Out]

2*log(x^8 - 4*x^6 + 16*x^4 - 20*x^2 + x + 24) - 2*log(x)

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {-48-40 x^2+96 x^4-40 x^6+14 x^8}{24 x+x^2-20 x^3+16 x^5-4 x^7+x^9} \, dx=- 2 \log {\left (x \right )} + 2 \log {\left (x^{8} - 4 x^{6} + 16 x^{4} - 20 x^{2} + x + 24 \right )} \]

[In]

integrate((14*x**8-40*x**6+96*x**4-40*x**2-48)/(x**9-4*x**7+16*x**5-20*x**3+x**2+24*x),x)

[Out]

-2*log(x) + 2*log(x**8 - 4*x**6 + 16*x**4 - 20*x**2 + x + 24)

Maxima [A] (verification not implemented)

none

Time = 0.17 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {-48-40 x^2+96 x^4-40 x^6+14 x^8}{24 x+x^2-20 x^3+16 x^5-4 x^7+x^9} \, dx=2 \, \log \left (x^{8} - 4 \, x^{6} + 16 \, x^{4} - 20 \, x^{2} + x + 24\right ) - 2 \, \log \left (x\right ) \]

[In]

integrate((14*x^8-40*x^6+96*x^4-40*x^2-48)/(x^9-4*x^7+16*x^5-20*x^3+x^2+24*x),x, algorithm="maxima")

[Out]

2*log(x^8 - 4*x^6 + 16*x^4 - 20*x^2 + x + 24) - 2*log(x)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15 \[ \int \frac {-48-40 x^2+96 x^4-40 x^6+14 x^8}{24 x+x^2-20 x^3+16 x^5-4 x^7+x^9} \, dx=2 \, \log \left (x^{8} - 4 \, x^{6} + 16 \, x^{4} - 20 \, x^{2} + x + 24\right ) - 2 \, \log \left ({\left | x \right |}\right ) \]

[In]

integrate((14*x^8-40*x^6+96*x^4-40*x^2-48)/(x^9-4*x^7+16*x^5-20*x^3+x^2+24*x),x, algorithm="giac")

[Out]

2*log(x^8 - 4*x^6 + 16*x^4 - 20*x^2 + x + 24) - 2*log(abs(x))

Mupad [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {-48-40 x^2+96 x^4-40 x^6+14 x^8}{24 x+x^2-20 x^3+16 x^5-4 x^7+x^9} \, dx=2\,\ln \left (x^8-4\,x^6+16\,x^4-20\,x^2+x+24\right )-2\,\ln \left (x\right ) \]

[In]

int(-(40*x^2 - 96*x^4 + 40*x^6 - 14*x^8 + 48)/(24*x + x^2 - 20*x^3 + 16*x^5 - 4*x^7 + x^9),x)

[Out]

2*log(x - 20*x^2 + 16*x^4 - 4*x^6 + x^8 + 24) - 2*log(x)