\(\int \frac {-x+(4+x+\log (4)) \log (\frac {4+x+\log (4)}{\log (2)})}{(4+x+\log (4)) \log ^2(\frac {4+x+\log (4)}{\log (2)})} \, dx\) [6460]

Optimal result

Integrand size = 42, antiderivative size = 15 \[ \int \frac {-x+(4+x+\log (4)) \log \left (\frac {4+x+\log (4)}{\log (2)}\right )}{(4+x+\log (4)) \log ^2\left (\frac {4+x+\log (4)}{\log (2)}\right )} \, dx=\frac {x}{\log \left (\frac {4+x+\log (4)}{\log (2)}\right )} \]

[Out]

x/ln((2*ln(2)+4+x)/ln(2))

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(75\) vs. \(2(15)=30\).

Time = 0.35 (sec) , antiderivative size = 75, normalized size of antiderivative = 5.00, number of steps used = 22, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.262, Rules used = {6873, 6874, 2458, 12, 2395, 2334, 2335, 2339, 30, 29, 2437} \[ \int \frac {-x+(4+x+\log (4)) \log \left (\frac {4+x+\log (4)}{\log (2)}\right )}{(4+x+\log (4)) \log ^2\left (\frac {4+x+\log (4)}{\log (2)}\right )} \, dx=\frac {x+4+\log (4)}{\log \left (\frac {x+4+\log (4)}{\log (2)}\right )}-(4+\log (4)) \log \left (\log \left (\frac {x+4+\log (4)}{\log (2)}\right )\right )+2 (2+\log (2)) \log \left (\log \left (\frac {x+4+\log (4)}{\log (2)}\right )\right )-\frac {4+\log (4)}{\log \left (\frac {x+4+\log (4)}{\log (2)}\right )} \]

[In]

Int[(-x + (4 + x + Log[4])*Log[(4 + x + Log[4])/Log[2]])/((4 + x + Log[4])*Log[(4 + x + Log[4])/Log[2]]^2),x]

[Out]

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

Rule 12

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

Rule 29

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 2335

Int[Log[(c_.)*(x_)]^(-1), x_Symbol] :> Simp[LogIntegral[c*x]/c, x] /; FreeQ[c, x]

Rule 2339

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 2395

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol]
:> With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[
{a, b, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[m] && IntegerQ[r
]))

Rule 2437

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[(f*(x/d))^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 2458

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

Rule 6873

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

Rule 6874

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

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

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {-x+(4+x+\log (4)) \log \left (\frac {4+x+\log (4)}{\log (2)}\right )}{(4+x+\log (4)) \log ^2\left (\frac {4+x+\log (4)}{\log (2)}\right )} \, dx=\frac {x}{\log \left (\frac {4+x+\log (4)}{\log (2)}\right )} \]

[In]

Integrate[(-x + (4 + x + Log[4])*Log[(4 + x + Log[4])/Log[2]])/((4 + x + Log[4])*Log[(4 + x + Log[4])/Log[2]]^
2),x]

[Out]

x/Log[(4 + x + Log[4])/Log[2]]

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.20

method	result	size
norman	\(\frac {x}{\ln \left (\frac {2 \ln \left (2\right )+4+x}{\ln \left (2\right )}\right )}\)	\(18\)
risch	\(\frac {x}{\ln \left (\frac {2 \ln \left (2\right )+4+x}{\ln \left (2\right )}\right )}\)	\(18\)
parallelrisch	\(\frac {x}{\ln \left (\frac {2 \ln \left (2\right )+4+x}{\ln \left (2\right )}\right )}\)	\(18\)
parts	\(\frac {x}{\ln \left (\frac {x}{\ln \left (2\right )}+\frac {4+2 \ln \left (2\right )}{\ln \left (2\right )}\right )}\)	\(24\)
derivativedivides	\(\ln \left (2\right ) \left (\frac {\frac {x}{\ln \left (2\right )}+\frac {4+2 \ln \left (2\right )}{\ln \left (2\right )}}{\ln \left (\frac {x}{\ln \left (2\right )}+\frac {4+2 \ln \left (2\right )}{\ln \left (2\right )}\right )}-\frac {2}{\ln \left (\frac {x}{\ln \left (2\right )}+\frac {4+2 \ln \left (2\right )}{\ln \left (2\right )}\right )}-\frac {4}{\ln \left (2\right ) \ln \left (\frac {x}{\ln \left (2\right )}+\frac {4+2 \ln \left (2\right )}{\ln \left (2\right )}\right )}\right )\)	\(95\)
default	\(\ln \left (2\right ) \left (\frac {\frac {x}{\ln \left (2\right )}+\frac {4+2 \ln \left (2\right )}{\ln \left (2\right )}}{\ln \left (\frac {x}{\ln \left (2\right )}+\frac {4+2 \ln \left (2\right )}{\ln \left (2\right )}\right )}-\frac {2}{\ln \left (\frac {x}{\ln \left (2\right )}+\frac {4+2 \ln \left (2\right )}{\ln \left (2\right )}\right )}-\frac {4}{\ln \left (2\right ) \ln \left (\frac {x}{\ln \left (2\right )}+\frac {4+2 \ln \left (2\right )}{\ln \left (2\right )}\right )}\right )\)	\(95\)

[In]

int(((2*ln(2)+4+x)*ln((2*ln(2)+4+x)/ln(2))-x)/(2*ln(2)+4+x)/ln((2*ln(2)+4+x)/ln(2))^2,x,method=_RETURNVERBOSE)

[Out]

x/ln((2*ln(2)+4+x)/ln(2))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int \frac {-x+(4+x+\log (4)) \log \left (\frac {4+x+\log (4)}{\log (2)}\right )}{(4+x+\log (4)) \log ^2\left (\frac {4+x+\log (4)}{\log (2)}\right )} \, dx=\frac {x}{\log \left (\frac {x + 2 \, \log \left (2\right ) + 4}{\log \left (2\right )}\right )} \]

[In]

integrate(((2*log(2)+4+x)*log((2*log(2)+4+x)/log(2))-x)/(2*log(2)+4+x)/log((2*log(2)+4+x)/log(2))^2,x, algorit
hm="fricas")

[Out]

x/log((x + 2*log(2) + 4)/log(2))

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93 \[ \int \frac {-x+(4+x+\log (4)) \log \left (\frac {4+x+\log (4)}{\log (2)}\right )}{(4+x+\log (4)) \log ^2\left (\frac {4+x+\log (4)}{\log (2)}\right )} \, dx=\frac {x}{\log {\left (\frac {x + 2 \log {\left (2 \right )} + 4}{\log {\left (2 \right )}} \right )}} \]

[In]

integrate(((2*ln(2)+4+x)*ln((2*ln(2)+4+x)/ln(2))-x)/(2*ln(2)+4+x)/ln((2*ln(2)+4+x)/ln(2))**2,x)

[Out]

x/log((x + 2*log(2) + 4)/log(2))

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.20 \[ \int \frac {-x+(4+x+\log (4)) \log \left (\frac {4+x+\log (4)}{\log (2)}\right )}{(4+x+\log (4)) \log ^2\left (\frac {4+x+\log (4)}{\log (2)}\right )} \, dx=\frac {x}{\log \left (x + 2 \, \log \left (2\right ) + 4\right ) - \log \left (\log \left (2\right )\right )} \]

[In]

integrate(((2*log(2)+4+x)*log((2*log(2)+4+x)/log(2))-x)/(2*log(2)+4+x)/log((2*log(2)+4+x)/log(2))^2,x, algorit
hm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.20 \[ \int \frac {-x+(4+x+\log (4)) \log \left (\frac {4+x+\log (4)}{\log (2)}\right )}{(4+x+\log (4)) \log ^2\left (\frac {4+x+\log (4)}{\log (2)}\right )} \, dx=\frac {x}{\log \left (x + 2 \, \log \left (2\right ) + 4\right ) - \log \left (\log \left (2\right )\right )} \]

[In]

integrate(((2*log(2)+4+x)*log((2*log(2)+4+x)/log(2))-x)/(2*log(2)+4+x)/log((2*log(2)+4+x)/log(2))^2,x, algorit
hm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 11.46 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {-x+(4+x+\log (4)) \log \left (\frac {4+x+\log (4)}{\log (2)}\right )}{(4+x+\log (4)) \log ^2\left (\frac {4+x+\log (4)}{\log (2)}\right )} \, dx=\frac {x}{\ln \left (\frac {x+\ln \left (4\right )+4}{\ln \left (2\right )}\right )} \]

[In]

int(-(x - log((x + 2*log(2) + 4)/log(2))*(x + 2*log(2) + 4))/(log((x + 2*log(2) + 4)/log(2))^2*(x + 2*log(2) +
4)),x)

[Out]

x/log((x + log(4) + 4)/log(2))