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

Optimal. Leaf size=15 \[ \frac {x}{\log \left (\frac {4+x+\log (4)}{\log (2)}\right )} \]

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Rubi [B]  time = 0.55, antiderivative size = 75, normalized size of antiderivative = 5.00, number of steps used = 22, number of rules used = 11, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.262, Rules used = {6741, 6742, 2411, 12, 2353, 2297, 2298, 2302, 30, 29, 2390} \begin {gather*} \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 )} \end {gather*}

Antiderivative was successfully verified.

[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 2297

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 2298

Int[Log[(c_.)*(x_)]^(-1), x_Symbol] :> Simp[LogIntegral[c*x]/c, x] /; FreeQ[c, 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 2353

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 2390

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 2411

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 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 {-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) \operatorname {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) \operatorname {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))) \operatorname {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))) \operatorname {Subst}\left (\int \frac {1}{x \log (x)} \, dx,x,\frac {x}{\log (2)}+\frac {4+\log (4)}{\log (2)}\right )-\operatorname {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 )+\operatorname {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))) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (\frac {4+x+\log (4)}{\log (2)}\right )\right )-\operatorname {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 )+\operatorname {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) \operatorname {Subst}\left (\int \frac {1}{\log ^2(x)} \, dx,x,\frac {x}{\log (2)}+\frac {4+\log (4)}{\log (2)}\right )+\log (2) \operatorname {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,\frac {x}{\log (2)}+\frac {4+\log (4)}{\log (2)}\right )-(-4-\log (4)) \operatorname {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)) \operatorname {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) \operatorname {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,\frac {x}{\log (2)}+\frac {4+\log (4)}{\log (2)}\right )-(-4-\log (4)) \operatorname {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log \left (\frac {4+x+\log (4)}{\log (2)}\right )\right )+(-4-\log (4)) \operatorname {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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.15, size = 15, normalized size = 1.00 \begin {gather*} \frac {x}{\log \left (\frac {4+x+\log (4)}{\log (2)}\right )} \end {gather*}

Antiderivative was successfully verified.

[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]]

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fricas [A]  time = 0.80, size = 17, normalized size = 1.13 \begin {gather*} \frac {x}{\log \left (\frac {x + 2 \, \log \relax (2) + 4}{\log \relax (2)}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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giac [A]  time = 0.16, size = 18, normalized size = 1.20 \begin {gather*} \frac {x}{\log \left (x + 2 \, \log \relax (2) + 4\right ) - \log \left (\log \relax (2)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)))

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maple [A]  time = 0.39, size = 18, normalized size = 1.20

method	result	size
norman	\(\frac {x}{\ln \left (\frac {2 \ln \relax (2)+4+x}{\ln \relax (2)}\right )}\)	\(18\)
risch	\(\frac {x}{\ln \left (\frac {2 \ln \relax (2)+4+x}{\ln \relax (2)}\right )}\)	\(18\)
derivativedivides	\(\ln \relax (2) \left (\frac {\frac {x}{\ln \relax (2)}+\frac {4+2 \ln \relax (2)}{\ln \relax (2)}}{\ln \left (\frac {x}{\ln \relax (2)}+\frac {4+2 \ln \relax (2)}{\ln \relax (2)}\right )}-\frac {2}{\ln \left (\frac {x}{\ln \relax (2)}+\frac {4+2 \ln \relax (2)}{\ln \relax (2)}\right )}-\frac {4}{\ln \relax (2) \ln \left (\frac {x}{\ln \relax (2)}+\frac {4+2 \ln \relax (2)}{\ln \relax (2)}\right )}\right )\)	\(91\)
default	\(\ln \relax (2) \left (\frac {\frac {x}{\ln \relax (2)}+\frac {4+2 \ln \relax (2)}{\ln \relax (2)}}{\ln \left (\frac {x}{\ln \relax (2)}+\frac {4+2 \ln \relax (2)}{\ln \relax (2)}\right )}-\frac {2}{\ln \left (\frac {x}{\ln \relax (2)}+\frac {4+2 \ln \relax (2)}{\ln \relax (2)}\right )}-\frac {4}{\ln \relax (2) \ln \left (\frac {x}{\ln \relax (2)}+\frac {4+2 \ln \relax (2)}{\ln \relax (2)}\right )}\right )\)	\(91\)

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maxima [A]  time = 0.49, size = 18, normalized size = 1.20 \begin {gather*} \frac {x}{\log \left (x + 2 \, \log \relax (2) + 4\right ) - \log \left (\log \relax (2)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)))

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mupad [B]  time = 4.31, size = 15, normalized size = 1.00 \begin {gather*} \frac {x}{\ln \left (\frac {x+\ln \relax (4)+4}{\ln \relax (2)}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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sympy [A]  time = 0.12, size = 14, normalized size = 0.93 \begin {gather*} \frac {x}{\log {\left (\frac {x + 2 \log {\relax (2 )} + 4}{\log {\relax (2 )}} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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