3.85.62 \(\int \frac {256+16 x^4+32 x^5+x^9+x^{10}+(256+80 x^4+32 x^5+x^{10}) \log (x)}{(256 x+16 x^5+32 x^6+x^{10}+x^{11}) \log (x)} \, dx\)

Optimal. Leaf size=18 \[ \log \left (x+\frac {x}{\frac {16}{x^4}+x}\right )+\log (\log (x)) \]

________________________________________________________________________________________

Rubi [B]  time = 0.69, antiderivative size = 37, normalized size of antiderivative = 2.06, number of steps used = 9, number of rules used = 6, integrand size = 63, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {6741, 6742, 1587, 260, 2302, 29} \begin {gather*} -\log \left (x^5+16\right )+\log \left (x^4-x^3+2 x^2-4 x+8\right )+\log (x)+\log (x+2)+\log (\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(256 + 16*x^4 + 32*x^5 + x^9 + x^10 + (256 + 80*x^4 + 32*x^5 + x^10)*Log[x])/((256*x + 16*x^5 + 32*x^6 + x
^10 + x^11)*Log[x]),x]

[Out]

Log[x] + Log[2 + x] + Log[8 - 4*x + 2*x^2 - x^3 + x^4] - Log[16 + x^5] + Log[Log[x]]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 1587

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]*D[Qq, x])/(q*Coeff[Q
q, x, q])]]] /; PolyQ[Pp, x] && PolyQ[Qq, x]

Rule 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 6741

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

Rule 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {256+16 x^4+32 x^5+x^9+x^{10}+\left (256+80 x^4+32 x^5+x^{10}\right ) \log (x)}{x \left (256+16 x^4+32 x^5+x^9+x^{10}\right ) \log (x)} \, dx\\ &=\int \left (\frac {256+80 x^4+32 x^5+x^{10}}{x \left (256+16 x^4+32 x^5+x^9+x^{10}\right )}+\frac {1}{x \log (x)}\right ) \, dx\\ &=\int \frac {256+80 x^4+32 x^5+x^{10}}{x \left (256+16 x^4+32 x^5+x^9+x^{10}\right )} \, dx+\int \frac {1}{x \log (x)} \, dx\\ &=\int \left (\frac {1}{x}+\frac {1}{2+x}+\frac {-4+4 x-3 x^2+4 x^3}{8-4 x+2 x^2-x^3+x^4}-\frac {5 x^4}{16+x^5}\right ) \, dx+\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log (x)\right )\\ &=\log (x)+\log (2+x)+\log (\log (x))-5 \int \frac {x^4}{16+x^5} \, dx+\int \frac {-4+4 x-3 x^2+4 x^3}{8-4 x+2 x^2-x^3+x^4} \, dx\\ &=\log (x)+\log (2+x)+\log \left (8-4 x+2 x^2-x^3+x^4\right )-\log \left (16+x^5\right )+\log (\log (x))\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.16, size = 23, normalized size = 1.28 \begin {gather*} \log (x)-\log \left (16+x^5\right )+\log \left (16+x^4+x^5\right )+\log (\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(256 + 16*x^4 + 32*x^5 + x^9 + x^10 + (256 + 80*x^4 + 32*x^5 + x^10)*Log[x])/((256*x + 16*x^5 + 32*x
^6 + x^10 + x^11)*Log[x]),x]

[Out]

Log[x] - Log[16 + x^5] + Log[16 + x^4 + x^5] + Log[Log[x]]

________________________________________________________________________________________

fricas [A]  time = 0.86, size = 23, normalized size = 1.28 \begin {gather*} \log \left (x^{6} + x^{5} + 16 \, x\right ) - \log \left (x^{5} + 16\right ) + \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^10+32*x^5+80*x^4+256)*log(x)+x^10+x^9+32*x^5+16*x^4+256)/(x^11+x^10+32*x^6+16*x^5+256*x)/log(x),
x, algorithm="fricas")

[Out]

log(x^6 + x^5 + 16*x) - log(x^5 + 16) + log(log(x))

________________________________________________________________________________________

giac [A]  time = 0.20, size = 23, normalized size = 1.28 \begin {gather*} \log \left (x^{5} + x^{4} + 16\right ) - \log \left (x^{5} + 16\right ) + \log \relax (x) + \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^10+32*x^5+80*x^4+256)*log(x)+x^10+x^9+32*x^5+16*x^4+256)/(x^11+x^10+32*x^6+16*x^5+256*x)/log(x),
x, algorithm="giac")

[Out]

log(x^5 + x^4 + 16) - log(x^5 + 16) + log(x) + log(log(x))

________________________________________________________________________________________

maple [A]  time = 0.09, size = 24, normalized size = 1.33

method	result	size
risch	\(-\ln \left (x^{5}+16\right )+\ln \left (x^{6}+x^{5}+16 x \right )+\ln \left (\ln \relax (x )\right )\)	\(24\)
default	\(\ln \left (\ln \relax (x )\right )+\ln \relax (x )+\ln \left (x^{4}-x^{3}+2 x^{2}-4 x +8\right )-\ln \left (x^{5}+16\right )+\ln \left (2+x \right )\)	\(38\)
norman	\(\ln \left (\ln \relax (x )\right )+\ln \relax (x )+\ln \left (x^{4}-x^{3}+2 x^{2}-4 x +8\right )-\ln \left (x^{5}+16\right )+\ln \left (2+x \right )\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^10+32*x^5+80*x^4+256)*ln(x)+x^10+x^9+32*x^5+16*x^4+256)/(x^11+x^10+32*x^6+16*x^5+256*x)/ln(x),x,method
=_RETURNVERBOSE)

[Out]

-ln(x^5+16)+ln(x^6+x^5+16*x)+ln(ln(x))

________________________________________________________________________________________

maxima [B]  time = 0.39, size = 37, normalized size = 2.06 \begin {gather*} -\log \left (x^{5} + 16\right ) + \log \left (x^{4} - x^{3} + 2 \, x^{2} - 4 \, x + 8\right ) + \log \left (x + 2\right ) + \log \relax (x) + \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^10+32*x^5+80*x^4+256)*log(x)+x^10+x^9+32*x^5+16*x^4+256)/(x^11+x^10+32*x^6+16*x^5+256*x)/log(x),
x, algorithm="maxima")

[Out]

-log(x^5 + 16) + log(x^4 - x^3 + 2*x^2 - 4*x + 8) + log(x + 2) + log(x) + log(log(x))

________________________________________________________________________________________

mupad [B]  time = 5.47, size = 23, normalized size = 1.28 \begin {gather*} \ln \left (\ln \relax (x)\right )+\ln \left (x\,\left (x^5+x^4+16\right )\right )-\ln \left (x^5+16\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((16*x^4 + 32*x^5 + x^9 + x^10 + log(x)*(80*x^4 + 32*x^5 + x^10 + 256) + 256)/(log(x)*(256*x + 16*x^5 + 32*
x^6 + x^10 + x^11)),x)

[Out]

log(log(x)) + log(x*(x^4 + x^5 + 16)) - log(x^5 + 16)

________________________________________________________________________________________

sympy [A]  time = 0.21, size = 22, normalized size = 1.22 \begin {gather*} - \log {\left (x^{5} + 16 \right )} + \log {\left (x^{6} + x^{5} + 16 x \right )} + \log {\left (\log {\relax (x )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**10+32*x**5+80*x**4+256)*ln(x)+x**10+x**9+32*x**5+16*x**4+256)/(x**11+x**10+32*x**6+16*x**5+256*
x)/ln(x),x)

[Out]

-log(x**5 + 16) + log(x**6 + x**5 + 16*x) + log(log(x))

________________________________________________________________________________________