\(\int \frac {400-160 x-72 x^3+16 x^4+24 x^5-8 x^6}{-100 x+80 x^2+24 x^3-36 x^4+4 x^5+4 x^6-x^7+x^3 \log ^2(4)} \, dx\) [2332]

Optimal result

Integrand size = 70, antiderivative size = 27 \[ \int \frac {400-160 x-72 x^3+16 x^4+24 x^5-8 x^6}{-100 x+80 x^2+24 x^3-36 x^4+4 x^5+4 x^6-x^7+x^3 \log ^2(4)} \, dx=\log \left (\left (-\left (4-\frac {10}{x}+2 x-x^2\right )^2+\log ^2(4)\right )^2\right ) \]

[Out]

ln((4*ln(2)^2-(2*x-10/x-x^2+4)^2)^2)

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.74, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {6, 2099, 1601} \[ \int \frac {400-160 x-72 x^3+16 x^4+24 x^5-8 x^6}{-100 x+80 x^2+24 x^3-36 x^4+4 x^5+4 x^6-x^7+x^3 \log ^2(4)} \, dx=2 \log \left (x^3-2 x^2-x (4-\log (4))+10\right )+2 \log \left (x^3-2 x^2-x (4+\log (4))+10\right )-4 \log (x) \]

[In]

Int[(400 - 160*x - 72*x^3 + 16*x^4 + 24*x^5 - 8*x^6)/(-100*x + 80*x^2 + 24*x^3 - 36*x^4 + 4*x^5 + 4*x^6 - x^7
+ x^3*Log[4]^2),x]

[Out]

-4*Log[x] + 2*Log[10 - 2*x^2 + x^3 - x*(4 - Log[4])] + 2*Log[10 - 2*x^2 + x^3 - x*(4 + Log[4])]

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

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 \frac {400-160 x-72 x^3+16 x^4+24 x^5-8 x^6}{-100 x+80 x^2-36 x^4+4 x^5+4 x^6-x^7+x^3 \left (24+\log ^2(4)\right )} \, dx \\ & = \int \left (-\frac {4}{x}+\frac {2 \left (-4-4 x+3 x^2-\log (4)\right )}{10-2 x^2+x^3-x (4+\log (4))}+\frac {-8-8 x+6 x^2+\log (16)}{10-2 x^2+x^3-x (4-\log (4))}\right ) \, dx \\ & = -4 \log (x)+2 \int \frac {-4-4 x+3 x^2-\log (4)}{10-2 x^2+x^3-x (4+\log (4))} \, dx+\int \frac {-8-8 x+6 x^2+\log (16)}{10-2 x^2+x^3-x (4-\log (4))} \, dx \\ & = -4 \log (x)+2 \log \left (10-2 x^2+x^3-x (4-\log (4))\right )+2 \log \left (10-2 x^2+x^3-x (4+\log (4))\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.89 \[ \int \frac {400-160 x-72 x^3+16 x^4+24 x^5-8 x^6}{-100 x+80 x^2+24 x^3-36 x^4+4 x^5+4 x^6-x^7+x^3 \log ^2(4)} \, dx=8 \left (-\frac {\log (x)}{2}+\frac {1}{4} \log \left (100-80 x-24 x^2+36 x^3-4 x^4-4 x^5+x^6-x^2 \log ^2(4)\right )\right ) \]

[In]

Integrate[(400 - 160*x - 72*x^3 + 16*x^4 + 24*x^5 - 8*x^6)/(-100*x + 80*x^2 + 24*x^3 - 36*x^4 + 4*x^5 + 4*x^6
- x^7 + x^3*Log[4]^2),x]

[Out]

8*(-1/2*Log[x] + Log[100 - 80*x - 24*x^2 + 36*x^3 - 4*x^4 - 4*x^5 + x^6 - x^2*Log[4]^2]/4)

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.63

method	result	size
risch	\(-4 \ln \left (x \right )+2 \ln \left (100+x^{6}-4 x^{5}-4 x^{4}+36 x^{3}+\left (-4 \ln \left (2\right )^{2}-24\right ) x^{2}-80 x \right )\)	\(44\)
default	\(2 \ln \left (x^{3}+2 x \ln \left (2\right )-2 x^{2}-4 x +10\right )-4 \ln \left (x \right )+2 \ln \left (x^{3}-2 x \ln \left (2\right )-2 x^{2}-4 x +10\right )\)	\(48\)
parallelrisch	\(2 \ln \left (x^{3}+2 x \ln \left (2\right )-2 x^{2}-4 x +10\right )-4 \ln \left (x \right )+2 \ln \left (x^{3}-2 x \ln \left (2\right )-2 x^{2}-4 x +10\right )\)	\(48\)
norman	\(-4 \ln \left (x \right )+2 \ln \left (-x^{3}+2 x \ln \left (2\right )+2 x^{2}+4 x -10\right )+2 \ln \left (x^{3}+2 x \ln \left (2\right )-2 x^{2}-4 x +10\right )\)	\(50\)

[In]

int((-8*x^6+24*x^5+16*x^4-72*x^3-160*x+400)/(4*x^3*ln(2)^2-x^7+4*x^6+4*x^5-36*x^4+24*x^3+80*x^2-100*x),x,metho
d=_RETURNVERBOSE)

[Out]

-4*ln(x)+2*ln(100+x^6-4*x^5-4*x^4+36*x^3+(-4*ln(2)^2-24)*x^2-80*x)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.67 \[ \int \frac {400-160 x-72 x^3+16 x^4+24 x^5-8 x^6}{-100 x+80 x^2+24 x^3-36 x^4+4 x^5+4 x^6-x^7+x^3 \log ^2(4)} \, dx=2 \, \log \left (x^{6} - 4 \, x^{5} - 4 \, x^{4} - 4 \, x^{2} \log \left (2\right )^{2} + 36 \, x^{3} - 24 \, x^{2} - 80 \, x + 100\right ) - 4 \, \log \left (x\right ) \]

[In]

integrate((-8*x^6+24*x^5+16*x^4-72*x^3-160*x+400)/(4*x^3*log(2)^2-x^7+4*x^6+4*x^5-36*x^4+24*x^3+80*x^2-100*x),
x, algorithm="fricas")

[Out]

2*log(x^6 - 4*x^5 - 4*x^4 - 4*x^2*log(2)^2 + 36*x^3 - 24*x^2 - 80*x + 100) - 4*log(x)

Sympy [A] (verification not implemented)

Time = 2.25 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.63 \[ \int \frac {400-160 x-72 x^3+16 x^4+24 x^5-8 x^6}{-100 x+80 x^2+24 x^3-36 x^4+4 x^5+4 x^6-x^7+x^3 \log ^2(4)} \, dx=- 4 \log {\left (x \right )} + 2 \log {\left (x^{6} - 4 x^{5} - 4 x^{4} + 36 x^{3} + x^{2} \left (-24 - 4 \log {\left (2 \right )}^{2}\right ) - 80 x + 100 \right )} \]

[In]

integrate((-8*x**6+24*x**5+16*x**4-72*x**3-160*x+400)/(4*x**3*ln(2)**2-x**7+4*x**6+4*x**5-36*x**4+24*x**3+80*x
**2-100*x),x)

[Out]

-4*log(x) + 2*log(x**6 - 4*x**5 - 4*x**4 + 36*x**3 + x**2*(-24 - 4*log(2)**2) - 80*x + 100)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.67 \[ \int \frac {400-160 x-72 x^3+16 x^4+24 x^5-8 x^6}{-100 x+80 x^2+24 x^3-36 x^4+4 x^5+4 x^6-x^7+x^3 \log ^2(4)} \, dx=2 \, \log \left (x^{3} - 2 \, x^{2} - 2 \, x {\left (\log \left (2\right ) + 2\right )} + 10\right ) + 2 \, \log \left (x^{3} - 2 \, x^{2} + 2 \, x {\left (\log \left (2\right ) - 2\right )} + 10\right ) - 4 \, \log \left (x\right ) \]

[In]

integrate((-8*x^6+24*x^5+16*x^4-72*x^3-160*x+400)/(4*x^3*log(2)^2-x^7+4*x^6+4*x^5-36*x^4+24*x^3+80*x^2-100*x),
x, algorithm="maxima")

[Out]

2*log(x^3 - 2*x^2 - 2*x*(log(2) + 2) + 10) + 2*log(x^3 - 2*x^2 + 2*x*(log(2) - 2) + 10) - 4*log(x)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.85 \[ \int \frac {400-160 x-72 x^3+16 x^4+24 x^5-8 x^6}{-100 x+80 x^2+24 x^3-36 x^4+4 x^5+4 x^6-x^7+x^3 \log ^2(4)} \, dx=2 \, \log \left ({\left | x^{3} - 2 \, x^{2} + 2 \, x \log \left (2\right ) - 4 \, x + 10 \right |}\right ) + 2 \, \log \left ({\left | x^{3} - 2 \, x^{2} - 2 \, x \log \left (2\right ) - 4 \, x + 10 \right |}\right ) - 4 \, \log \left ({\left | x \right |}\right ) \]

[In]

integrate((-8*x^6+24*x^5+16*x^4-72*x^3-160*x+400)/(4*x^3*log(2)^2-x^7+4*x^6+4*x^5-36*x^4+24*x^3+80*x^2-100*x),
x, algorithm="giac")

[Out]

2*log(abs(x^3 - 2*x^2 + 2*x*log(2) - 4*x + 10)) + 2*log(abs(x^3 - 2*x^2 - 2*x*log(2) - 4*x + 10)) - 4*log(abs(
x))

Mupad [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.67 \[ \int \frac {400-160 x-72 x^3+16 x^4+24 x^5-8 x^6}{-100 x+80 x^2+24 x^3-36 x^4+4 x^5+4 x^6-x^7+x^3 \log ^2(4)} \, dx=2\,\ln \left (36\,x^3-4\,x^2\,{\ln \left (2\right )}^2-24\,x^2-80\,x-4\,x^4-4\,x^5+x^6+100\right )-4\,\ln \left (x\right ) \]

[In]

int(-(160*x + 72*x^3 - 16*x^4 - 24*x^5 + 8*x^6 - 400)/(4*x^3*log(2)^2 - 100*x + 80*x^2 + 24*x^3 - 36*x^4 + 4*x
^5 + 4*x^6 - x^7),x)

[Out]

2*log(36*x^3 - 4*x^2*log(2)^2 - 24*x^2 - 80*x - 4*x^4 - 4*x^5 + x^6 + 100) - 4*log(x)