\(\int \frac {1-x}{(x^2+x \log (\frac {4}{x})) \log (x+\log (\frac {4}{x}))+(2 x^2+2 x \log (\frac {4}{x})) \log ^2(x+\log (\frac {4}{x}))} \, dx\) [6771]

Optimal result

Integrand size = 58, antiderivative size = 14 \[ \int \frac {1-x}{\left (x^2+x \log \left (\frac {4}{x}\right )\right ) \log \left (x+\log \left (\frac {4}{x}\right )\right )+\left (2 x^2+2 x \log \left (\frac {4}{x}\right )\right ) \log ^2\left (x+\log \left (\frac {4}{x}\right )\right )} \, dx=\log \left (2+\frac {1}{\log \left (x+\log \left (\frac {4}{x}\right )\right )}\right ) \]

[Out]

ln(2+1/ln(ln(4/x)+x))

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.93, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {6873, 6824, 36, 29, 31} \[ \int \frac {1-x}{\left (x^2+x \log \left (\frac {4}{x}\right )\right ) \log \left (x+\log \left (\frac {4}{x}\right )\right )+\left (2 x^2+2 x \log \left (\frac {4}{x}\right )\right ) \log ^2\left (x+\log \left (\frac {4}{x}\right )\right )} \, dx=\log \left (2 \log \left (x+\log \left (\frac {4}{x}\right )\right )+1\right )-\log \left (\log \left (x+\log \left (\frac {4}{x}\right )\right )\right ) \]

[In]

Int[(1 - x)/((x^2 + x*Log[4/x])*Log[x + Log[4/x]] + (2*x^2 + 2*x*Log[4/x])*Log[x + Log[4/x]]^2),x]

[Out]

-Log[Log[x + Log[4/x]]] + Log[1 + 2*Log[x + Log[4/x]]]

Rule 29

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

Rule 31

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

Rule 36

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

Rule 6824

Int[(u_)*((c_.) + (d_.)*(v_))^(n_.)*((a_.) + (b_.)*(y_))^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u,
x]}, Dist[q, Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, y], x] /;  !FalseQ[q]] /; FreeQ[{a, b, c, d, m, n}, x]
&& EqQ[v, y]

Rule 6873

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

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

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.93 \[ \int \frac {1-x}{\left (x^2+x \log \left (\frac {4}{x}\right )\right ) \log \left (x+\log \left (\frac {4}{x}\right )\right )+\left (2 x^2+2 x \log \left (\frac {4}{x}\right )\right ) \log ^2\left (x+\log \left (\frac {4}{x}\right )\right )} \, dx=-\log \left (\log \left (x+\log \left (\frac {4}{x}\right )\right )\right )+\log \left (1+2 \log \left (x+\log \left (\frac {4}{x}\right )\right )\right ) \]

[In]

Integrate[(1 - x)/((x^2 + x*Log[4/x])*Log[x + Log[4/x]] + (2*x^2 + 2*x*Log[4/x])*Log[x + Log[4/x]]^2),x]

[Out]

-Log[Log[x + Log[4/x]]] + Log[1 + 2*Log[x + Log[4/x]]]

Maple [A] (verified)

Time = 1.15 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.86

method	result	size
parallelrisch	\(-\ln \left (\ln \left (\ln \left (\frac {4}{x}\right )+x \right )\right )+\ln \left (\ln \left (\ln \left (\frac {4}{x}\right )+x \right )+\frac {1}{2}\right )\)	\(26\)
default	\(\ln \left (2 \ln \left (\left (\frac {2 \ln \left (2\right )}{x}+\frac {\ln \left (\frac {1}{x}\right )}{x}+1\right ) x \right )+1\right )-\ln \left (\ln \left (\left (\frac {2 \ln \left (2\right )}{x}+\frac {\ln \left (\frac {1}{x}\right )}{x}+1\right ) x \right )\right )\)	\(50\)

[In]

int((1-x)/((2*x*ln(4/x)+2*x^2)*ln(ln(4/x)+x)^2+(x*ln(4/x)+x^2)*ln(ln(4/x)+x)),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.93 \[ \int \frac {1-x}{\left (x^2+x \log \left (\frac {4}{x}\right )\right ) \log \left (x+\log \left (\frac {4}{x}\right )\right )+\left (2 x^2+2 x \log \left (\frac {4}{x}\right )\right ) \log ^2\left (x+\log \left (\frac {4}{x}\right )\right )} \, dx=\log \left (2 \, \log \left (x + \log \left (\frac {4}{x}\right )\right ) + 1\right ) - \log \left (\log \left (x + \log \left (\frac {4}{x}\right )\right )\right ) \]

[In]

integrate((1-x)/((2*x*log(4/x)+2*x^2)*log(log(4/x)+x)^2+(x*log(4/x)+x^2)*log(log(4/x)+x)),x, algorithm="fricas
")

[Out]

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

Sympy [F(-2)]

Exception generated. \[ \int \frac {1-x}{\left (x^2+x \log \left (\frac {4}{x}\right )\right ) \log \left (x+\log \left (\frac {4}{x}\right )\right )+\left (2 x^2+2 x \log \left (\frac {4}{x}\right )\right ) \log ^2\left (x+\log \left (\frac {4}{x}\right )\right )} \, dx=\text {Exception raised: PolynomialError} \]

[In]

integrate((1-x)/((2*x*ln(4/x)+2*x**2)*ln(ln(4/x)+x)**2+(x*ln(4/x)+x**2)*ln(ln(4/x)+x)),x)

[Out]

Exception raised: PolynomialError >> 1/(_t0*x + x**2) contains an element of the set of generators.

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 29 vs. \(2 (14) = 28\).

Time = 0.32 (sec) , antiderivative size = 29, normalized size of antiderivative = 2.07 \[ \int \frac {1-x}{\left (x^2+x \log \left (\frac {4}{x}\right )\right ) \log \left (x+\log \left (\frac {4}{x}\right )\right )+\left (2 x^2+2 x \log \left (\frac {4}{x}\right )\right ) \log ^2\left (x+\log \left (\frac {4}{x}\right )\right )} \, dx=\log \left (\log \left (x + 2 \, \log \left (2\right ) - \log \left (x\right )\right ) + \frac {1}{2}\right ) - \log \left (\log \left (x + 2 \, \log \left (2\right ) - \log \left (x\right )\right )\right ) \]

[In]

integrate((1-x)/((2*x*log(4/x)+2*x^2)*log(log(4/x)+x)^2+(x*log(4/x)+x^2)*log(log(4/x)+x)),x, algorithm="maxima
")

[Out]

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

Giac [F]

\[ \int \frac {1-x}{\left (x^2+x \log \left (\frac {4}{x}\right )\right ) \log \left (x+\log \left (\frac {4}{x}\right )\right )+\left (2 x^2+2 x \log \left (\frac {4}{x}\right )\right ) \log ^2\left (x+\log \left (\frac {4}{x}\right )\right )} \, dx=\int { -\frac {x - 1}{2 \, {\left (x^{2} + x \log \left (\frac {4}{x}\right )\right )} \log \left (x + \log \left (\frac {4}{x}\right )\right )^{2} + {\left (x^{2} + x \log \left (\frac {4}{x}\right )\right )} \log \left (x + \log \left (\frac {4}{x}\right )\right )} \,d x } \]

[In]

integrate((1-x)/((2*x*log(4/x)+2*x^2)*log(log(4/x)+x)^2+(x*log(4/x)+x^2)*log(log(4/x)+x)),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 14.05 (sec) , antiderivative size = 71, normalized size of antiderivative = 5.07 \[ \int \frac {1-x}{\left (x^2+x \log \left (\frac {4}{x}\right )\right ) \log \left (x+\log \left (\frac {4}{x}\right )\right )+\left (2 x^2+2 x \log \left (\frac {4}{x}\right )\right ) \log ^2\left (x+\log \left (\frac {4}{x}\right )\right )} \, dx=\ln \left (\frac {\left (2\,\ln \left (x+\ln \left (\frac {4}{x}\right )\right )+1\right )\,\left (x-1\right )}{\ln \left (2^{2\,x}\right )+x\,\ln \left (\frac {1}{x}\right )+x^2}\right )-\ln \left (\frac {\ln \left (x+\ln \left (\frac {4}{x}\right )\right )\,\left (x-1\right )}{\ln \left (2^{2\,x}\right )+x\,\ln \left (\frac {1}{x}\right )+x^2}\right ) \]

[In]

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

[Out]

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