\(\int \frac {-80 \log (\frac {1}{2 x})+(80-80 \log (\frac {1}{2 x})) \log (x)+(20-20 \log (\frac {1}{2 x})) \log ^2(x)}{16+8 \log (x)+\log ^2(x)} \, dx\) [2694]

Optimal result

Integrand size = 56, antiderivative size = 21 \[ \int \frac {-80 \log \left (\frac {1}{2 x}\right )+\left (80-80 \log \left (\frac {1}{2 x}\right )\right ) \log (x)+\left (20-20 \log \left (\frac {1}{2 x}\right )\right ) \log ^2(x)}{16+8 \log (x)+\log ^2(x)} \, dx=24-\frac {20 x \log \left (\frac {1}{2 x}\right ) \log (x)}{4+\log (x)} \]

[Out]

24-20/(ln(x)+4)*ln(x)*ln(1/2/x)*x

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 0.13 (sec) , antiderivative size = 79, normalized size of antiderivative = 3.76, number of steps used = 15, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {6820, 6874, 2332, 2334, 2336, 2209, 2408, 6617, 12} \[ \int \frac {-80 \log \left (\frac {1}{2 x}\right )+\left (80-80 \log \left (\frac {1}{2 x}\right )\right ) \log (x)+\left (20-20 \log \left (\frac {1}{2 x}\right )\right ) \log ^2(x)}{16+8 \log (x)+\log ^2(x)} \, dx=-\frac {80 \left (1-\log \left (\frac {1}{2 x}\right )\right ) \operatorname {ExpIntegralEi}(\log (x)+4)}{e^4}-\frac {80 \log \left (\frac {1}{2 x}\right ) \operatorname {ExpIntegralEi}(\log (x)+4)}{e^4}+\frac {80 \operatorname {ExpIntegralEi}(\log (x)+4)}{e^4}-20 x \log \left (\frac {1}{2 x}\right )+\frac {80 x \log \left (\frac {1}{2 x}\right )}{\log (x)+4} \]

[In]

Int[(-80*Log[1/(2*x)] + (80 - 80*Log[1/(2*x)])*Log[x] + (20 - 20*Log[1/(2*x)])*Log[x]^2)/(16 + 8*Log[x] + Log[
x]^2),x]

[Out]

(80*ExpIntegralEi[4 + Log[x]])/E^4 - (80*ExpIntegralEi[4 + Log[x]]*(1 - Log[1/(2*x)]))/E^4 - 20*x*Log[1/(2*x)]
- (80*ExpIntegralEi[4 + Log[x]]*Log[1/(2*x)])/E^4 + (80*x*Log[1/(2*x)])/(4 + Log[x])

Rule 12

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

Rule 2209

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - c*(f/d)))/d)*ExpInteg
ralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2334

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Simp[x*((a + b*Log[c*x^n])^(p + 1)/(b*n*(p + 1)))
, x] - Dist[1/(b*n*(p + 1)), Int[(a + b*Log[c*x^n])^(p + 1), x], x] /; FreeQ[{a, b, c, n}, x] && LtQ[p, -1] &&
IntegerQ[2*p]

Rule 2336

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

Rule 2408

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.) + Log[(f_.)*(x_)^(r_.)]*(e_.)), x_Symbol] :> With[{u =
IntHide[(a + b*Log[c*x^n])^p, x]}, Dist[d + e*Log[f*x^r], u, x] - Dist[e*r, Int[SimplifyIntegrand[u/x, x], x],
x]] /; FreeQ[{a, b, c, d, e, f, n, p, r}, x]

Rule 6617

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

Rule 6820

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

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {-20 \log \left (\frac {1}{2 x}\right ) (2+\log (x))^2+20 \log (x) (4+\log (x))}{(4+\log (x))^2} \, dx \\ & = \int \left (-20 \left (-1+\log \left (\frac {1}{2 x}\right )\right )-\frac {80 \log \left (\frac {1}{2 x}\right )}{(4+\log (x))^2}+\frac {80 \left (-1+\log \left (\frac {1}{2 x}\right )\right )}{4+\log (x)}\right ) \, dx \\ & = -\left (20 \int \left (-1+\log \left (\frac {1}{2 x}\right )\right ) \, dx\right )-80 \int \frac {\log \left (\frac {1}{2 x}\right )}{(4+\log (x))^2} \, dx+80 \int \frac {-1+\log \left (\frac {1}{2 x}\right )}{4+\log (x)} \, dx \\ & = 20 x-\frac {80 \text {Ei}(4+\log (x)) \left (1-\log \left (\frac {1}{2 x}\right )\right )}{e^4}-\frac {80 \text {Ei}(4+\log (x)) \log \left (\frac {1}{2 x}\right )}{e^4}+\frac {80 x \log \left (\frac {1}{2 x}\right )}{4+\log (x)}-20 \int \log \left (\frac {1}{2 x}\right ) \, dx+80 \int \frac {\text {Ei}(4+\log (x))}{e^4 x} \, dx-80 \int \left (\frac {\text {Ei}(4+\log (x))}{e^4 x}-\frac {1}{4+\log (x)}\right ) \, dx \\ & = -\frac {80 \text {Ei}(4+\log (x)) \left (1-\log \left (\frac {1}{2 x}\right )\right )}{e^4}-20 x \log \left (\frac {1}{2 x}\right )-\frac {80 \text {Ei}(4+\log (x)) \log \left (\frac {1}{2 x}\right )}{e^4}+\frac {80 x \log \left (\frac {1}{2 x}\right )}{4+\log (x)}+80 \int \frac {1}{4+\log (x)} \, dx \\ & = -\frac {80 \text {Ei}(4+\log (x)) \left (1-\log \left (\frac {1}{2 x}\right )\right )}{e^4}-20 x \log \left (\frac {1}{2 x}\right )-\frac {80 \text {Ei}(4+\log (x)) \log \left (\frac {1}{2 x}\right )}{e^4}+\frac {80 x \log \left (\frac {1}{2 x}\right )}{4+\log (x)}+80 \text {Subst}\left (\int \frac {e^x}{4+x} \, dx,x,\log (x)\right ) \\ & = \frac {80 \text {Ei}(4+\log (x))}{e^4}-\frac {80 \text {Ei}(4+\log (x)) \left (1-\log \left (\frac {1}{2 x}\right )\right )}{e^4}-20 x \log \left (\frac {1}{2 x}\right )-\frac {80 \text {Ei}(4+\log (x)) \log \left (\frac {1}{2 x}\right )}{e^4}+\frac {80 x \log \left (\frac {1}{2 x}\right )}{4+\log (x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {-80 \log \left (\frac {1}{2 x}\right )+\left (80-80 \log \left (\frac {1}{2 x}\right )\right ) \log (x)+\left (20-20 \log \left (\frac {1}{2 x}\right )\right ) \log ^2(x)}{16+8 \log (x)+\log ^2(x)} \, dx=-\frac {20 x \log \left (\frac {1}{2 x}\right ) \log (x)}{4+\log (x)} \]

[In]

Integrate[(-80*Log[1/(2*x)] + (80 - 80*Log[1/(2*x)])*Log[x] + (20 - 20*Log[1/(2*x)])*Log[x]^2)/(16 + 8*Log[x]
+ Log[x]^2),x]

[Out]

(-20*x*Log[1/(2*x)]*Log[x])/(4 + Log[x])

Maple [A] (verified)

Time = 1.00 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86

method	result	size
norman	\(-\frac {20 \ln \left (x \right ) \ln \left (\frac {1}{2 x}\right ) x}{\ln \left (x \right )+4}\)	\(18\)
parallelrisch	\(-\frac {20 \ln \left (x \right ) \ln \left (\frac {1}{2 x}\right ) x}{\ln \left (x \right )+4}\)	\(18\)
risch	\(20 x \ln \left (x \right )+20 x \ln \left (2\right )-80 x -\frac {40 x \left (2 \ln \left (2\right )-8\right )}{\ln \left (x \right )+4}\)	\(30\)
default	\(100 x +80 \,{\mathrm e}^{-4} \operatorname {Ei}_{1}\left (-\ln \left (x \right )-4\right )+\frac {20 \ln \left (2\right ) \ln \left (x \right ) x}{\ln \left (x \right )+4}-20 \left (3 \ln \left (\frac {1}{x}\right )+2 \ln \left (x \right )+9\right ) x -\frac {20 x \left (\left (\ln \left (x \right )+\ln \left (\frac {1}{x}\right )\right )^{3}+64+12 \left (\ln \left (x \right )+\ln \left (\frac {1}{x}\right )\right )^{2}+48 \ln \left (\frac {1}{x}\right )+48 \ln \left (x \right )\right )}{-\ln \left (x \right )-4}-20 \left (-\left (\ln \left (x \right )+\ln \left (\frac {1}{x}\right )\right )^{3}-16-9 \left (\ln \left (x \right )+\ln \left (\frac {1}{x}\right )\right )^{2}-24 \ln \left (\frac {1}{x}\right )-24 \ln \left (x \right )\right ) {\mathrm e}^{-\ln \left (x \right )-\ln \left (\frac {1}{x}\right )-4} \operatorname {Ei}_{1}\left (-\ln \left (x \right )-4\right )+20 \left (\ln \left (x \right )+\ln \left (\frac {1}{x}\right )\right )^{2} \left (-\frac {x \left (\ln \left (x \right )+\ln \left (\frac {1}{x}\right )+4\right )}{-\ln \left (x \right )-4}-\left (-\ln \left (\frac {1}{x}\right )-\ln \left (x \right )-3\right ) {\mathrm e}^{-\ln \left (x \right )-\ln \left (\frac {1}{x}\right )-4} \operatorname {Ei}_{1}\left (-\ln \left (x \right )-4\right )\right )-\frac {80 x \left (\ln \left (x \right )+\ln \left (\frac {1}{x}\right )+4\right )}{-\ln \left (x \right )-4}-80 \left (-\ln \left (\frac {1}{x}\right )-\ln \left (x \right )-3\right ) {\mathrm e}^{-\ln \left (x \right )-\ln \left (\frac {1}{x}\right )-4} \operatorname {Ei}_{1}\left (-\ln \left (x \right )-4\right )+\frac {80 x \left (\left (\ln \left (x \right )+\ln \left (\frac {1}{x}\right )\right )^{2}+16+8 \ln \left (\frac {1}{x}\right )+8 \ln \left (x \right )\right )}{-\ln \left (x \right )-4}+80 \left (-\left (\ln \left (x \right )+\ln \left (\frac {1}{x}\right )\right )^{2}-8-6 \ln \left (\frac {1}{x}\right )-6 \ln \left (x \right )\right ) {\mathrm e}^{-\ln \left (x \right )-\ln \left (\frac {1}{x}\right )-4} \operatorname {Ei}_{1}\left (-\ln \left (x \right )-4\right )+80 \left (\ln \left (x \right )+\ln \left (\frac {1}{x}\right )\right ) \left (-\frac {x \left (\ln \left (x \right )+\ln \left (\frac {1}{x}\right )+4\right )}{-\ln \left (x \right )-4}-\left (-\ln \left (\frac {1}{x}\right )-\ln \left (x \right )-3\right ) {\mathrm e}^{-\ln \left (x \right )-\ln \left (\frac {1}{x}\right )-4} \operatorname {Ei}_{1}\left (-\ln \left (x \right )-4\right )\right )-40 \left (\ln \left (x \right )+\ln \left (\frac {1}{x}\right )\right ) \left (-x -\frac {x \left (\left (\ln \left (x \right )+\ln \left (\frac {1}{x}\right )\right )^{2}+16+8 \ln \left (\frac {1}{x}\right )+8 \ln \left (x \right )\right )}{-\ln \left (x \right )-4}-\left (-\left (\ln \left (x \right )+\ln \left (\frac {1}{x}\right )\right )^{2}-8-6 \ln \left (\frac {1}{x}\right )-6 \ln \left (x \right )\right ) {\mathrm e}^{-\ln \left (x \right )-\ln \left (\frac {1}{x}\right )-4} \operatorname {Ei}_{1}\left (-\ln \left (x \right )-4\right )\right )\)	\(498\)

[In]

int(((-20*ln(1/2/x)+20)*ln(x)^2+(-80*ln(1/2/x)+80)*ln(x)-80*ln(1/2/x))/(ln(x)^2+8*ln(x)+16),x,method=_RETURNVE
RBOSE)

[Out]

-20/(ln(x)+4)*ln(x)*ln(1/2/x)*x

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.67 \[ \int \frac {-80 \log \left (\frac {1}{2 x}\right )+\left (80-80 \log \left (\frac {1}{2 x}\right )\right ) \log (x)+\left (20-20 \log \left (\frac {1}{2 x}\right )\right ) \log ^2(x)}{16+8 \log (x)+\log ^2(x)} \, dx=-\frac {20 \, {\left (x \log \left (2\right ) \log \left (\frac {1}{2 \, x}\right ) + x \log \left (\frac {1}{2 \, x}\right )^{2}\right )}}{\log \left (2\right ) + \log \left (\frac {1}{2 \, x}\right ) - 4} \]

[In]

integrate(((-20*log(1/2/x)+20)*log(x)^2+(-80*log(1/2/x)+80)*log(x)-80*log(1/2/x))/(log(x)^2+8*log(x)+16),x, al
gorithm="fricas")

[Out]

-20*(x*log(2)*log(1/2/x) + x*log(1/2/x)^2)/(log(2) + log(1/2/x) - 4)

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.38 \[ \int \frac {-80 \log \left (\frac {1}{2 x}\right )+\left (80-80 \log \left (\frac {1}{2 x}\right )\right ) \log (x)+\left (20-20 \log \left (\frac {1}{2 x}\right )\right ) \log ^2(x)}{16+8 \log (x)+\log ^2(x)} \, dx=20 x \log {\left (x \right )} + x \left (-80 + 20 \log {\left (2 \right )}\right ) + \frac {- 80 x \log {\left (2 \right )} + 320 x}{\log {\left (x \right )} + 4} \]

[In]

integrate(((-20*ln(1/2/x)+20)*ln(x)**2+(-80*ln(1/2/x)+80)*ln(x)-80*ln(1/2/x))/(ln(x)**2+8*ln(x)+16),x)

[Out]

20*x*log(x) + x*(-80 + 20*log(2)) + (-80*x*log(2) + 320*x)/(log(x) + 4)

Maxima [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {-80 \log \left (\frac {1}{2 x}\right )+\left (80-80 \log \left (\frac {1}{2 x}\right )\right ) \log (x)+\left (20-20 \log \left (\frac {1}{2 x}\right )\right ) \log ^2(x)}{16+8 \log (x)+\log ^2(x)} \, dx=\frac {20 \, {\left (x \log \left (2\right ) \log \left (x\right ) + x \log \left (x\right )^{2}\right )}}{\log \left (x\right ) + 4} \]

[In]

integrate(((-20*log(1/2/x)+20)*log(x)^2+(-80*log(1/2/x)+80)*log(x)-80*log(1/2/x))/(log(x)^2+8*log(x)+16),x, al
gorithm="maxima")

[Out]

20*(x*log(2)*log(x) + x*log(x)^2)/(log(x) + 4)

Giac [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.29 \[ \int \frac {-80 \log \left (\frac {1}{2 x}\right )+\left (80-80 \log \left (\frac {1}{2 x}\right )\right ) \log (x)+\left (20-20 \log \left (\frac {1}{2 x}\right )\right ) \log ^2(x)}{16+8 \log (x)+\log ^2(x)} \, dx=\frac {20 \, x \log \left (2\right ) \log \left (x\right )}{\log \left (x\right ) + 4} + \frac {20 \, x \log \left (x\right )^{2}}{\log \left (x\right ) + 4} \]

[In]

integrate(((-20*log(1/2/x)+20)*log(x)^2+(-80*log(1/2/x)+80)*log(x)-80*log(1/2/x))/(log(x)^2+8*log(x)+16),x, al
gorithm="giac")

[Out]

20*x*log(2)*log(x)/(log(x) + 4) + 20*x*log(x)^2/(log(x) + 4)

Mupad [B] (verification not implemented)

Time = 8.95 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {-80 \log \left (\frac {1}{2 x}\right )+\left (80-80 \log \left (\frac {1}{2 x}\right )\right ) \log (x)+\left (20-20 \log \left (\frac {1}{2 x}\right )\right ) \log ^2(x)}{16+8 \log (x)+\log ^2(x)} \, dx=-\frac {20\,x\,\ln \left (\frac {1}{2\,x}\right )\,\ln \left (x\right )}{\ln \left (x\right )+4} \]

[In]

int(-(80*log(1/(2*x)) + log(x)^2*(20*log(1/(2*x)) - 20) + log(x)*(80*log(1/(2*x)) - 80))/(8*log(x) + log(x)^2
+ 16),x)

[Out]

-(20*x*log(1/(2*x))*log(x))/(log(x) + 4)