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

Optimal. Leaf size=14 \[ \log \left (2+\frac {1}{\log \left (x+\log \left (\frac {4}{x}\right )\right )}\right ) \]

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Rubi [A]  time = 0.36, antiderivative size = 27, normalized size of antiderivative = 1.93, number of steps used = 5, number of rules used = 5, integrand size = 58, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {6741, 6692, 36, 29, 31} \begin {gather*} \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 ) \end {gather*}

Antiderivative was successfully verified.

[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 6692

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 6741

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

Rubi steps

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

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Mathematica [A]  time = 0.14, size = 27, normalized size = 1.93 \begin {gather*} -\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 {gather*}

Antiderivative was successfully verified.

[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]]]

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fricas [A]  time = 0.66, size = 27, normalized size = 1.93 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x+1)/((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="frica
s")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x+1)/((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)

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {1-x}{\left (2 x \ln \left (\frac {4}{x}\right )+2 x^{2}\right ) \ln \left (\ln \left (\frac {4}{x}\right )+x \right )^{2}+\left (x \ln \left (\frac {4}{x}\right )+x^{2}\right ) \ln \left (\ln \left (\frac {4}{x}\right )+x \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[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)

[Out]

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)

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maxima [B]  time = 0.60, size = 29, normalized size = 2.07 \begin {gather*} \log \left (\log \left (x + 2 \, \log \relax (2) - \log \relax (x)\right ) + \frac {1}{2}\right ) - \log \left (\log \left (x + 2 \, \log \relax (2) - \log \relax (x)\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x+1)/((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="maxim
a")

[Out]

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

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mupad [B]  time = 5.30, size = 71, normalized size = 5.07 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: PolynomialError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x+1)/((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

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