3.21.84 \(\int \frac {256-10000 x^5+(-256+10000 x^5) \log (x)+(-256-40000 x^5) \log (x) \log (\frac {5 x}{\log (x)})}{(256-20000 x^5+390625 x^{10}) \log (x)} \, dx\)

Optimal. Leaf size=24 \[ \frac {\log \left (\frac {5 x}{\log (x)}\right )}{-\frac {1}{x}+\frac {625 x^4}{16}} \]

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Rubi [A]  time = 1.62, antiderivative size = 20, normalized size of antiderivative = 0.83, number of steps used = 21, number of rules used = 12, integrand size = 54, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {28, 6742, 6725, 202, 634, 618, 204, 628, 31, 383, 2555, 12} \begin {gather*} -\frac {16 x \log \left (\frac {5 x}{\log (x)}\right )}{16-625 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(256 - 10000*x^5 + (-256 + 10000*x^5)*Log[x] + (-256 - 40000*x^5)*Log[x]*Log[(5*x)/Log[x]])/((256 - 20000*
x^5 + 390625*x^10)*Log[x]),x]

[Out]

(-16*x*Log[(5*x)/Log[x]])/(16 - 625*x^5)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 31

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

Rule 202

Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[-(a/b), n]], s = Denominator[Rt[-(a/b
), n]], k, u}, Simp[u = Int[(r + s*Cos[((2*k - 1)*Pi)/n]*x)/(r^2 + 2*r*s*Cos[((2*k - 1)*Pi)/n]*x + s^2*x^2), x
]; (r*Int[1/(r - s*x), x])/(a*n) + Dist[(2*r)/(a*n), Sum[u, {k, 1, (n - 1)/2}], x], x]] /; FreeQ[{a, b}, x] &&
IGtQ[(n - 3)/2, 0] && NegQ[a/b]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 383

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*x*(a + b*x^n)^(p + 1))/a, x]
/; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[a*d - b*c*(n*(p + 1) + 1), 0]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

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 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 2555

Int[Log[u_]*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Dist[Log[u], w, x] - Int[SimplifyIntegrand[w*Simplify
[D[u, x]/u], x], x] /; InverseFunctionFreeQ[w, x]] /; ProductQ[u]

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rule 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=390625 \int \frac {256-10000 x^5+\left (-256+10000 x^5\right ) \log (x)+\left (-256-40000 x^5\right ) \log (x) \log \left (\frac {5 x}{\log (x)}\right )}{\left (-10000+390625 x^5\right )^2 \log (x)} \, dx\\ &=390625 \int \left (\frac {16 (-1+\log (x))}{390625 \left (-16+625 x^5\right ) \log (x)}-\frac {64 \left (4+625 x^5\right ) \log \left (\frac {5 x}{\log (x)}\right )}{390625 \left (-16+625 x^5\right )^2}\right ) \, dx\\ &=16 \int \frac {-1+\log (x)}{\left (-16+625 x^5\right ) \log (x)} \, dx-64 \int \frac {\left (4+625 x^5\right ) \log \left (\frac {5 x}{\log (x)}\right )}{\left (-16+625 x^5\right )^2} \, dx\\ &=-\frac {16 x \log \left (\frac {5 x}{\log (x)}\right )}{16-625 x^5}+16 \int \left (\frac {1}{-16+625 x^5}-\frac {1}{\left (-16+625 x^5\right ) \log (x)}\right ) \, dx+64 \int \frac {-1+\log (x)}{4 \left (16-625 x^5\right ) \log (x)} \, dx\\ &=-\frac {16 x \log \left (\frac {5 x}{\log (x)}\right )}{16-625 x^5}+16 \int \frac {1}{-16+625 x^5} \, dx-16 \int \frac {1}{\left (-16+625 x^5\right ) \log (x)} \, dx+16 \int \frac {-1+\log (x)}{\left (16-625 x^5\right ) \log (x)} \, dx\\ &=-\frac {16 x \log \left (\frac {5 x}{\log (x)}\right )}{16-625 x^5}+16 \int \left (\frac {1}{16-625 x^5}+\frac {1}{\left (-16+625 x^5\right ) \log (x)}\right ) \, dx-16 \int \frac {1}{\left (-16+625 x^5\right ) \log (x)} \, dx-\frac {1}{5} 2^{4/5} \int \frac {1}{2^{4/5}-5^{4/5} x} \, dx-\frac {1}{5} \left (2\ 2^{4/5}\right ) \int \frac {2^{4/5}+\frac {1}{4} 5^{4/5} \left (1-\sqrt {5}\right ) x}{2\ 2^{3/5}+\frac {5^{4/5} \left (1-\sqrt {5}\right ) x}{\sqrt [5]{2}}+5\ 5^{3/5} x^2} \, dx-\frac {1}{5} \left (2\ 2^{4/5}\right ) \int \frac {2^{4/5}+\frac {1}{4} 5^{4/5} \left (1+\sqrt {5}\right ) x}{2\ 2^{3/5}+\frac {5^{4/5} \left (1+\sqrt {5}\right ) x}{\sqrt [5]{2}}+5\ 5^{3/5} x^2} \, dx\\ &=\frac {1}{5} \left (\frac {2}{5}\right )^{4/5} \log \left (2^{4/5}-5^{4/5} x\right )-\frac {16 x \log \left (\frac {5 x}{\log (x)}\right )}{16-625 x^5}+16 \int \frac {1}{16-625 x^5} \, dx-\frac {\left (1-\sqrt {5}\right ) \int \frac {\frac {5^{4/5} \left (1-\sqrt {5}\right )}{\sqrt [5]{2}}+10\ 5^{3/5} x}{2\ 2^{3/5}+\frac {5^{4/5} \left (1-\sqrt {5}\right ) x}{\sqrt [5]{2}}+5\ 5^{3/5} x^2} \, dx}{10 \sqrt [5]{2} 5^{4/5}}-\frac {\left (5-\sqrt {5}\right ) \int \frac {1}{2\ 2^{3/5}+\frac {5^{4/5} \left (1+\sqrt {5}\right ) x}{\sqrt [5]{2}}+5\ 5^{3/5} x^2} \, dx}{5\ 2^{2/5}}-\frac {\left (1+\sqrt {5}\right ) \int \frac {\frac {5^{4/5} \left (1+\sqrt {5}\right )}{\sqrt [5]{2}}+10\ 5^{3/5} x}{2\ 2^{3/5}+\frac {5^{4/5} \left (1+\sqrt {5}\right ) x}{\sqrt [5]{2}}+5\ 5^{3/5} x^2} \, dx}{10 \sqrt [5]{2} 5^{4/5}}-\frac {\left (5+\sqrt {5}\right ) \int \frac {1}{2\ 2^{3/5}+\frac {5^{4/5} \left (1-\sqrt {5}\right ) x}{\sqrt [5]{2}}+5\ 5^{3/5} x^2} \, dx}{5\ 2^{2/5}}\\ &=\frac {1}{5} \left (\frac {2}{5}\right )^{4/5} \log \left (2^{4/5}-5^{4/5} x\right )-\frac {\left (1-\sqrt {5}\right ) \log \left (4\ 2^{3/5}-5\ 2^{4/5} 5^{3/10} x+10^{4/5} x+10\ 5^{3/5} x^2\right )}{10 \sqrt [5]{2} 5^{4/5}}-\frac {\left (1+\sqrt {5}\right ) \log \left (4\ 2^{3/5}+5\ 2^{4/5} 5^{3/10} x+10^{4/5} x+10\ 5^{3/5} x^2\right )}{10 \sqrt [5]{2} 5^{4/5}}-\frac {16 x \log \left (\frac {5 x}{\log (x)}\right )}{16-625 x^5}+\frac {1}{5} 2^{4/5} \int \frac {1}{2^{4/5}-5^{4/5} x} \, dx+\frac {1}{5} \left (2\ 2^{4/5}\right ) \int \frac {2^{4/5}+\frac {1}{4} 5^{4/5} \left (1-\sqrt {5}\right ) x}{2\ 2^{3/5}+\frac {5^{4/5} \left (1-\sqrt {5}\right ) x}{\sqrt [5]{2}}+5\ 5^{3/5} x^2} \, dx+\frac {1}{5} \left (2\ 2^{4/5}\right ) \int \frac {2^{4/5}+\frac {1}{4} 5^{4/5} \left (1+\sqrt {5}\right ) x}{2\ 2^{3/5}+\frac {5^{4/5} \left (1+\sqrt {5}\right ) x}{\sqrt [5]{2}}+5\ 5^{3/5} x^2} \, dx+\frac {1}{5} \left (2^{3/5} \left (5-\sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{25\ 2^{3/5} \sqrt [10]{5} \left (1-\sqrt {5}\right )-x^2} \, dx,x,\frac {5^{4/5} \left (1+\sqrt {5}\right )}{\sqrt [5]{2}}+10\ 5^{3/5} x\right )+\frac {1}{5} \left (2^{3/5} \left (5+\sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-25 2^{3/5} \left (\sqrt [10]{5}+5^{3/5}\right )-x^2} \, dx,x,\frac {5^{4/5} \left (1-\sqrt {5}\right )}{\sqrt [5]{2}}+10\ 5^{3/5} x\right )\\ &=-\frac {2^{3/10} \sqrt {1+\sqrt {5}} \tan ^{-1}\left (\frac {1-\sqrt {5}+2 \sqrt [5]{2} 5^{4/5} x}{\sqrt [4]{5} \sqrt {2 \left (1+\sqrt {5}\right )}}\right )}{5\ 5^{11/20}}-\frac {2^{3/10} \sqrt {-1+\sqrt {5}} \tan ^{-1}\left (\frac {1+\sqrt {5}+2 \sqrt [5]{2} 5^{4/5} x}{\sqrt [4]{5} \sqrt {2 \left (-1+\sqrt {5}\right )}}\right )}{5\ 5^{11/20}}-\frac {\left (1-\sqrt {5}\right ) \log \left (4\ 2^{3/5}-5\ 2^{4/5} 5^{3/10} x+10^{4/5} x+10\ 5^{3/5} x^2\right )}{10 \sqrt [5]{2} 5^{4/5}}-\frac {\left (1+\sqrt {5}\right ) \log \left (4\ 2^{3/5}+5\ 2^{4/5} 5^{3/10} x+10^{4/5} x+10\ 5^{3/5} x^2\right )}{10 \sqrt [5]{2} 5^{4/5}}-\frac {16 x \log \left (\frac {5 x}{\log (x)}\right )}{16-625 x^5}+\frac {\left (1-\sqrt {5}\right ) \int \frac {\frac {5^{4/5} \left (1-\sqrt {5}\right )}{\sqrt [5]{2}}+10\ 5^{3/5} x}{2\ 2^{3/5}+\frac {5^{4/5} \left (1-\sqrt {5}\right ) x}{\sqrt [5]{2}}+5\ 5^{3/5} x^2} \, dx}{10 \sqrt [5]{2} 5^{4/5}}+\frac {\left (5-\sqrt {5}\right ) \int \frac {1}{2\ 2^{3/5}+\frac {5^{4/5} \left (1+\sqrt {5}\right ) x}{\sqrt [5]{2}}+5\ 5^{3/5} x^2} \, dx}{5\ 2^{2/5}}+\frac {\left (1+\sqrt {5}\right ) \int \frac {\frac {5^{4/5} \left (1+\sqrt {5}\right )}{\sqrt [5]{2}}+10\ 5^{3/5} x}{2\ 2^{3/5}+\frac {5^{4/5} \left (1+\sqrt {5}\right ) x}{\sqrt [5]{2}}+5\ 5^{3/5} x^2} \, dx}{10 \sqrt [5]{2} 5^{4/5}}+\frac {\left (5+\sqrt {5}\right ) \int \frac {1}{2\ 2^{3/5}+\frac {5^{4/5} \left (1-\sqrt {5}\right ) x}{\sqrt [5]{2}}+5\ 5^{3/5} x^2} \, dx}{5\ 2^{2/5}}\\ &=-\frac {2^{3/10} \sqrt {1+\sqrt {5}} \tan ^{-1}\left (\frac {1-\sqrt {5}+2 \sqrt [5]{2} 5^{4/5} x}{\sqrt [4]{5} \sqrt {2 \left (1+\sqrt {5}\right )}}\right )}{5\ 5^{11/20}}-\frac {2^{3/10} \sqrt {-1+\sqrt {5}} \tan ^{-1}\left (\frac {1+\sqrt {5}+2 \sqrt [5]{2} 5^{4/5} x}{\sqrt [4]{5} \sqrt {2 \left (-1+\sqrt {5}\right )}}\right )}{5\ 5^{11/20}}-\frac {16 x \log \left (\frac {5 x}{\log (x)}\right )}{16-625 x^5}-\frac {1}{5} \left (2^{3/5} \left (5-\sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{25\ 2^{3/5} \sqrt [10]{5} \left (1-\sqrt {5}\right )-x^2} \, dx,x,\frac {5^{4/5} \left (1+\sqrt {5}\right )}{\sqrt [5]{2}}+10\ 5^{3/5} x\right )-\frac {1}{5} \left (2^{3/5} \left (5+\sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-25 2^{3/5} \left (\sqrt [10]{5}+5^{3/5}\right )-x^2} \, dx,x,\frac {5^{4/5} \left (1-\sqrt {5}\right )}{\sqrt [5]{2}}+10\ 5^{3/5} x\right )\\ &=-\frac {16 x \log \left (\frac {5 x}{\log (x)}\right )}{16-625 x^5}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.47, size = 20, normalized size = 0.83 \begin {gather*} \frac {16 x \log \left (\frac {5 x}{\log (x)}\right )}{-16+625 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(256 - 10000*x^5 + (-256 + 10000*x^5)*Log[x] + (-256 - 40000*x^5)*Log[x]*Log[(5*x)/Log[x]])/((256 -
20000*x^5 + 390625*x^10)*Log[x]),x]

[Out]

(16*x*Log[(5*x)/Log[x]])/(-16 + 625*x^5)

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fricas [A]  time = 0.64, size = 20, normalized size = 0.83 \begin {gather*} \frac {16 \, x \log \left (\frac {5 \, x}{\log \relax (x)}\right )}{625 \, x^{5} - 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-40000*x^5-256)*log(x)*log(5*x/log(x))+(10000*x^5-256)*log(x)-10000*x^5+256)/(390625*x^10-20000*x^
5+256)/log(x),x, algorithm="fricas")

[Out]

16*x*log(5*x/log(x))/(625*x^5 - 16)

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giac [A]  time = 2.97, size = 44, normalized size = 1.83 \begin {gather*} \frac {16 \, x \log \relax (5)}{625 \, x^{5} - 16} + \frac {16 \, x \log \relax (x)}{625 \, x^{5} - 16} - \frac {16 \, x \log \left (\log \relax (x)\right )}{625 \, x^{5} - 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-40000*x^5-256)*log(x)*log(5*x/log(x))+(10000*x^5-256)*log(x)-10000*x^5+256)/(390625*x^10-20000*x^
5+256)/log(x),x, algorithm="giac")

[Out]

16*x*log(5)/(625*x^5 - 16) + 16*x*log(x)/(625*x^5 - 16) - 16*x*log(log(x))/(625*x^5 - 16)

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maple [C]  time = 0.29, size = 122, normalized size = 5.08

method	result	size
risch	\(-\frac {16 x \ln \left (\ln \relax (x )\right )}{625 x^{5}-16}+\frac {8 x \left (-i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right ) \mathrm {csgn}\left (\frac {i x}{\ln \relax (x )}\right )+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i x}{\ln \relax (x )}\right )^{2}+i \pi \,\mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right ) \mathrm {csgn}\left (\frac {i x}{\ln \relax (x )}\right )^{2}-i \pi \mathrm {csgn}\left (\frac {i x}{\ln \relax (x )}\right )^{3}+2 \ln \relax (5)+2 \ln \relax (x )\right )}{625 x^{5}-16}\)	\(122\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-40000*x^5-256)*ln(x)*ln(5*x/ln(x))+(10000*x^5-256)*ln(x)-10000*x^5+256)/(390625*x^10-20000*x^5+256)/ln(
x),x,method=_RETURNVERBOSE)

[Out]

-16*x/(625*x^5-16)*ln(ln(x))+8*x*(-I*Pi*csgn(I*x)*csgn(I/ln(x))*csgn(I*x/ln(x))+I*Pi*csgn(I*x)*csgn(I*x/ln(x))
^2+I*Pi*csgn(I/ln(x))*csgn(I*x/ln(x))^2-I*Pi*csgn(I*x/ln(x))^3+2*ln(5)+2*ln(x))/(625*x^5-16)

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maxima [A]  time = 0.49, size = 26, normalized size = 1.08 \begin {gather*} \frac {16 \, {\left (x \log \relax (5) + x \log \relax (x) - x \log \left (\log \relax (x)\right )\right )}}{625 \, x^{5} - 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-40000*x^5-256)*log(x)*log(5*x/log(x))+(10000*x^5-256)*log(x)-10000*x^5+256)/(390625*x^10-20000*x^
5+256)/log(x),x, algorithm="maxima")

[Out]

16*(x*log(5) + x*log(x) - x*log(log(x)))/(625*x^5 - 16)

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mupad [B]  time = 1.30, size = 20, normalized size = 0.83 \begin {gather*} \frac {16\,x\,\ln \left (\frac {5\,x}{\ln \relax (x)}\right )}{625\,\left (x^5-\frac {16}{625}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(10000*x^5 - log(x)*(10000*x^5 - 256) + log((5*x)/log(x))*log(x)*(40000*x^5 + 256) - 256)/(log(x)*(390625
*x^10 - 20000*x^5 + 256)),x)

[Out]

(16*x*log((5*x)/log(x)))/(625*(x^5 - 16/625))

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sympy [A]  time = 0.34, size = 17, normalized size = 0.71 \begin {gather*} \frac {16 x \log {\left (\frac {5 x}{\log {\relax (x )}} \right )}}{625 x^{5} - 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-40000*x**5-256)*ln(x)*ln(5*x/ln(x))+(10000*x**5-256)*ln(x)-10000*x**5+256)/(390625*x**10-20000*x*
*5+256)/ln(x),x)

[Out]

16*x*log(5*x/log(x))/(625*x**5 - 16)

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