\(\int \frac {64-48 x-84 x^2+23 x^3+42 x^4-24 x^5-8 x^6}{64 x-112 x^2-36 x^3+143 x^4-8 x^5-60 x^6+4 x^7+8 x^8} \, dx\) [2273]

Optimal result

Integrand size = 72, antiderivative size = 27 \[ \int \frac {64-48 x-84 x^2+23 x^3+42 x^4-24 x^5-8 x^6}{64 x-112 x^2-36 x^3+143 x^4-8 x^5-60 x^6+4 x^7+8 x^8} \, dx=4-\log \left (-1+\frac {1}{x}+\frac {3 x^2}{\left (4-x-2 x^2\right )^2}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.44, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {2099, 642, 1601} \[ \int \frac {64-48 x-84 x^2+23 x^3+42 x^4-24 x^5-8 x^6}{64 x-112 x^2-36 x^3+143 x^4-8 x^5-60 x^6+4 x^7+8 x^8} \, dx=2 \log \left (-2 x^2-x+4\right )-\log \left (-4 x^5+22 x^3-7 x^2-24 x+16\right )+\log (x) \]

[In]

Int[(64 - 48*x - 84*x^2 + 23*x^3 + 42*x^4 - 24*x^5 - 8*x^6)/(64*x - 112*x^2 - 36*x^3 + 143*x^4 - 8*x^5 - 60*x^
6 + 4*x^7 + 8*x^8),x]

[Out]

Log[x] + 2*Log[4 - x - 2*x^2] - Log[16 - 24*x - 7*x^2 + 22*x^3 - 4*x^5]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

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 {1}{x}+\frac {2 (1+4 x)}{-4+x+2 x^2}-\frac {2 \left (12+7 x-33 x^2+10 x^4\right )}{-16+24 x+7 x^2-22 x^3+4 x^5}\right ) \, dx \\ & = \log (x)+2 \int \frac {1+4 x}{-4+x+2 x^2} \, dx-2 \int \frac {12+7 x-33 x^2+10 x^4}{-16+24 x+7 x^2-22 x^3+4 x^5} \, dx \\ & = \log (x)+2 \log \left (4-x-2 x^2\right )-\log \left (16-24 x-7 x^2+22 x^3-4 x^5\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.44 \[ \int \frac {64-48 x-84 x^2+23 x^3+42 x^4-24 x^5-8 x^6}{64 x-112 x^2-36 x^3+143 x^4-8 x^5-60 x^6+4 x^7+8 x^8} \, dx=\log (x)+2 \log \left (4-x-2 x^2\right )-\log \left (16-24 x-7 x^2+22 x^3-4 x^5\right ) \]

[In]

Integrate[(64 - 48*x - 84*x^2 + 23*x^3 + 42*x^4 - 24*x^5 - 8*x^6)/(64*x - 112*x^2 - 36*x^3 + 143*x^4 - 8*x^5 -
60*x^6 + 4*x^7 + 8*x^8),x]

[Out]

Log[x] + 2*Log[4 - x - 2*x^2] - Log[16 - 24*x - 7*x^2 + 22*x^3 - 4*x^5]

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.33

method	result	size
parallelrisch	\(\ln \left (x \right )+2 \ln \left (x^{2}+\frac {1}{2} x -2\right )-\ln \left (x^{5}-\frac {11}{2} x^{3}+\frac {7}{4} x^{2}+6 x -4\right )\)	\(36\)
default	\(\ln \left (x \right )-\ln \left (4 x^{5}-22 x^{3}+7 x^{2}+24 x -16\right )+2 \ln \left (2 x^{2}+x -4\right )\)	\(38\)
norman	\(\ln \left (x \right )-\ln \left (4 x^{5}-22 x^{3}+7 x^{2}+24 x -16\right )+2 \ln \left (2 x^{2}+x -4\right )\)	\(38\)
risch	\(\ln \left (x \right )-\ln \left (4 x^{5}-22 x^{3}+7 x^{2}+24 x -16\right )+2 \ln \left (2 x^{2}+x -4\right )\)	\(38\)

[In]

int((-8*x^6-24*x^5+42*x^4+23*x^3-84*x^2-48*x+64)/(8*x^8+4*x^7-60*x^6-8*x^5+143*x^4-36*x^3-112*x^2+64*x),x,meth
od=_RETURNVERBOSE)

[Out]

ln(x)+2*ln(x^2+1/2*x-2)-ln(x^5-11/2*x^3+7/4*x^2+6*x-4)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.37 \[ \int \frac {64-48 x-84 x^2+23 x^3+42 x^4-24 x^5-8 x^6}{64 x-112 x^2-36 x^3+143 x^4-8 x^5-60 x^6+4 x^7+8 x^8} \, dx=-\log \left (4 \, x^{5} - 22 \, x^{3} + 7 \, x^{2} + 24 \, x - 16\right ) + 2 \, \log \left (2 \, x^{2} + x - 4\right ) + \log \left (x\right ) \]

[In]

integrate((-8*x^6-24*x^5+42*x^4+23*x^3-84*x^2-48*x+64)/(8*x^8+4*x^7-60*x^6-8*x^5+143*x^4-36*x^3-112*x^2+64*x),
x, algorithm="fricas")

[Out]

-log(4*x^5 - 22*x^3 + 7*x^2 + 24*x - 16) + 2*log(2*x^2 + x - 4) + log(x)

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.33 \[ \int \frac {64-48 x-84 x^2+23 x^3+42 x^4-24 x^5-8 x^6}{64 x-112 x^2-36 x^3+143 x^4-8 x^5-60 x^6+4 x^7+8 x^8} \, dx=\log {\left (x \right )} + 2 \log {\left (2 x^{2} + x - 4 \right )} - \log {\left (4 x^{5} - 22 x^{3} + 7 x^{2} + 24 x - 16 \right )} \]

[In]

integrate((-8*x**6-24*x**5+42*x**4+23*x**3-84*x**2-48*x+64)/(8*x**8+4*x**7-60*x**6-8*x**5+143*x**4-36*x**3-112
*x**2+64*x),x)

[Out]

log(x) + 2*log(2*x**2 + x - 4) - log(4*x**5 - 22*x**3 + 7*x**2 + 24*x - 16)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.37 \[ \int \frac {64-48 x-84 x^2+23 x^3+42 x^4-24 x^5-8 x^6}{64 x-112 x^2-36 x^3+143 x^4-8 x^5-60 x^6+4 x^7+8 x^8} \, dx=-\log \left (4 \, x^{5} - 22 \, x^{3} + 7 \, x^{2} + 24 \, x - 16\right ) + 2 \, \log \left (2 \, x^{2} + x - 4\right ) + \log \left (x\right ) \]

[In]

integrate((-8*x^6-24*x^5+42*x^4+23*x^3-84*x^2-48*x+64)/(8*x^8+4*x^7-60*x^6-8*x^5+143*x^4-36*x^3-112*x^2+64*x),
x, algorithm="maxima")

[Out]

-log(4*x^5 - 22*x^3 + 7*x^2 + 24*x - 16) + 2*log(2*x^2 + x - 4) + log(x)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.48 \[ \int \frac {64-48 x-84 x^2+23 x^3+42 x^4-24 x^5-8 x^6}{64 x-112 x^2-36 x^3+143 x^4-8 x^5-60 x^6+4 x^7+8 x^8} \, dx=-\log \left ({\left | 4 \, x^{5} - 22 \, x^{3} + 7 \, x^{2} + 24 \, x - 16 \right |}\right ) + 2 \, \log \left ({\left | 2 \, x^{2} + x - 4 \right |}\right ) + \log \left ({\left | x \right |}\right ) \]

[In]

integrate((-8*x^6-24*x^5+42*x^4+23*x^3-84*x^2-48*x+64)/(8*x^8+4*x^7-60*x^6-8*x^5+143*x^4-36*x^3-112*x^2+64*x),
x, algorithm="giac")

[Out]

-log(abs(4*x^5 - 22*x^3 + 7*x^2 + 24*x - 16)) + 2*log(abs(2*x^2 + x - 4)) + log(abs(x))

Mupad [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.30 \[ \int \frac {64-48 x-84 x^2+23 x^3+42 x^4-24 x^5-8 x^6}{64 x-112 x^2-36 x^3+143 x^4-8 x^5-60 x^6+4 x^7+8 x^8} \, dx=2\,\ln \left (2\,x^2+x-4\right )+\ln \left (x\right )-\ln \left (x^5-\frac {11\,x^3}{2}+\frac {7\,x^2}{4}+6\,x-4\right ) \]

[In]

int(-(48*x + 84*x^2 - 23*x^3 - 42*x^4 + 24*x^5 + 8*x^6 - 64)/(64*x - 112*x^2 - 36*x^3 + 143*x^4 - 8*x^5 - 60*x
^6 + 4*x^7 + 8*x^8),x)

[Out]

2*log(x + 2*x^2 - 4) + log(x) - log(6*x + (7*x^2)/4 - (11*x^3)/2 + x^5 - 4)