Integrand size = 58, antiderivative size = 14 \[ \int \frac {1-x}{\left (x^2+x \log \left (\frac {4}{x}\right )\right ) \log \left (x+\log \left (\frac {4}{x}\right )\right )+\left (2 x^2+2 x \log \left (\frac {4}{x}\right )\right ) \log ^2\left (x+\log \left (\frac {4}{x}\right )\right )} \, dx=\log \left (2+\frac {1}{\log \left (x+\log \left (\frac {4}{x}\right )\right )}\right ) \] Output:
ln(2+1/ln(ln(4/x)+x))
Time = 0.07 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.93 \[ \int \frac {1-x}{\left (x^2+x \log \left (\frac {4}{x}\right )\right ) \log \left (x+\log \left (\frac {4}{x}\right )\right )+\left (2 x^2+2 x \log \left (\frac {4}{x}\right )\right ) \log ^2\left (x+\log \left (\frac {4}{x}\right )\right )} \, dx=-\log \left (\log \left (x+\log \left (\frac {4}{x}\right )\right )\right )+\log \left (1+2 \log \left (x+\log \left (\frac {4}{x}\right )\right )\right ) \] Input:
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]
Output:
-Log[Log[x + Log[4/x]]] + Log[1 + 2*Log[x + Log[4/x]]]
Time = 0.54 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.93, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {7292, 7243, 47, 14, 16}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1-x}{\left (2 x^2+2 x \log \left (\frac {4}{x}\right )\right ) \log ^2\left (x+\log \left (\frac {4}{x}\right )\right )+\left (x^2+x \log \left (\frac {4}{x}\right )\right ) \log \left (x+\log \left (\frac {4}{x}\right )\right )} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {1-x}{x \left (x+\log \left (\frac {4}{x}\right )\right ) \log \left (x+\log \left (\frac {4}{x}\right )\right ) \left (2 \log \left (x+\log \left (\frac {4}{x}\right )\right )+1\right )}dx\) |
\(\Big \downarrow \) 7243 |
\(\displaystyle -\int \frac {1}{\log \left (x+\log \left (\frac {4}{x}\right )\right ) \left (2 \log \left (x+\log \left (\frac {4}{x}\right )\right )+1\right )}d\log \left (x+\log \left (\frac {4}{x}\right )\right )\) |
\(\Big \downarrow \) 47 |
\(\displaystyle 2 \int \frac {1}{2 \log \left (x+\log \left (\frac {4}{x}\right )\right )+1}d\log \left (x+\log \left (\frac {4}{x}\right )\right )-\int \frac {1}{\log \left (x+\log \left (\frac {4}{x}\right )\right )}d\log \left (x+\log \left (\frac {4}{x}\right )\right )\) |
\(\Big \downarrow \) 14 |
\(\displaystyle 2 \int \frac {1}{2 \log \left (x+\log \left (\frac {4}{x}\right )\right )+1}d\log \left (x+\log \left (\frac {4}{x}\right )\right )-\log \left (\log \left (x+\log \left (\frac {4}{x}\right )\right )\right )\) |
\(\Big \downarrow \) 16 |
\(\displaystyle \log \left (2 \log \left (x+\log \left (\frac {4}{x}\right )\right )+1\right )-\log \left (\log \left (x+\log \left (\frac {4}{x}\right )\right )\right )\) |
Input:
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]
Output:
-Log[Log[x + Log[4/x]]] + Log[1 + 2*Log[x + Log[4/x]]]
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Simp[b/(b*c - a*d) Int[1/(a + b*x), x], x] - Simp[d/(b*c - a*d) Int[1/(c + d*x), x ], x] /; FreeQ[{a, b, c, d}, x]
Int[(u_)*((c_.) + (d_.)*(v_))^(n_.)*((a_.) + (b_.)*(y_))^(m_.), x_Symbol] : > With[{q = DerivativeDivides[y, u, x]}, Simp[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]
Time = 1.06 (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 | \(\ln \left (2 \ln \left (\left (\frac {2 \ln \left (2\right )}{x}+\frac {\ln \left (\frac {1}{x}\right )}{x}+1\right ) x \right )+1\right )-\ln \left (\ln \left (\left (\frac {2 \ln \left (2\right )}{x}+\frac {\ln \left (\frac {1}{x}\right )}{x}+1\right ) x \right )\right )\) | \(50\) |
Input:
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)
Output:
-ln(ln(ln(4/x)+x))+ln(ln(ln(4/x)+x)+1/2)
Time = 0.08 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.93 \[ \int \frac {1-x}{\left (x^2+x \log \left (\frac {4}{x}\right )\right ) \log \left (x+\log \left (\frac {4}{x}\right )\right )+\left (2 x^2+2 x \log \left (\frac {4}{x}\right )\right ) \log ^2\left (x+\log \left (\frac {4}{x}\right )\right )} \, dx=\log \left (2 \, \log \left (x + \log \left (\frac {4}{x}\right )\right ) + 1\right ) - \log \left (\log \left (x + \log \left (\frac {4}{x}\right )\right )\right ) \] Input:
integrate((1-x)/((2*x*log(4/x)+2*x^2)*log(log(4/x)+x)^2+(x*log(4/x)+x^2)*l og(log(4/x)+x)),x, algorithm="fricas")
Output:
log(2*log(x + log(4/x)) + 1) - log(log(x + log(4/x)))
Exception generated. \[ \int \frac {1-x}{\left (x^2+x \log \left (\frac {4}{x}\right )\right ) \log \left (x+\log \left (\frac {4}{x}\right )\right )+\left (2 x^2+2 x \log \left (\frac {4}{x}\right )\right ) \log ^2\left (x+\log \left (\frac {4}{x}\right )\right )} \, dx=\text {Exception raised: PolynomialError} \] Input:
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)
Output:
Exception raised: PolynomialError >> 1/(_t0*x + x**2) contains an element of the set of generators.
Leaf count of result is larger than twice the leaf count of optimal. 29 vs. \(2 (14) = 28\).
Time = 0.17 (sec) , antiderivative size = 29, normalized size of antiderivative = 2.07 \[ \int \frac {1-x}{\left (x^2+x \log \left (\frac {4}{x}\right )\right ) \log \left (x+\log \left (\frac {4}{x}\right )\right )+\left (2 x^2+2 x \log \left (\frac {4}{x}\right )\right ) \log ^2\left (x+\log \left (\frac {4}{x}\right )\right )} \, dx=\log \left (\log \left (x + 2 \, \log \left (2\right ) - \log \left (x\right )\right ) + \frac {1}{2}\right ) - \log \left (\log \left (x + 2 \, \log \left (2\right ) - \log \left (x\right )\right )\right ) \] Input:
integrate((1-x)/((2*x*log(4/x)+2*x^2)*log(log(4/x)+x)^2+(x*log(4/x)+x^2)*l og(log(4/x)+x)),x, algorithm="maxima")
Output:
log(log(x + 2*log(2) - log(x)) + 1/2) - log(log(x + 2*log(2) - log(x)))
\[ \int \frac {1-x}{\left (x^2+x \log \left (\frac {4}{x}\right )\right ) \log \left (x+\log \left (\frac {4}{x}\right )\right )+\left (2 x^2+2 x \log \left (\frac {4}{x}\right )\right ) \log ^2\left (x+\log \left (\frac {4}{x}\right )\right )} \, dx=\int { -\frac {x - 1}{2 \, {\left (x^{2} + x \log \left (\frac {4}{x}\right )\right )} \log \left (x + \log \left (\frac {4}{x}\right )\right )^{2} + {\left (x^{2} + x \log \left (\frac {4}{x}\right )\right )} \log \left (x + \log \left (\frac {4}{x}\right )\right )} \,d x } \] Input:
integrate((1-x)/((2*x*log(4/x)+2*x^2)*log(log(4/x)+x)^2+(x*log(4/x)+x^2)*l og(log(4/x)+x)),x, algorithm="giac")
Output:
integrate(-(x - 1)/(2*(x^2 + x*log(4/x))*log(x + log(4/x))^2 + (x^2 + x*lo g(4/x))*log(x + log(4/x))), x)
Time = 3.26 (sec) , antiderivative size = 71, normalized size of antiderivative = 5.07 \[ \int \frac {1-x}{\left (x^2+x \log \left (\frac {4}{x}\right )\right ) \log \left (x+\log \left (\frac {4}{x}\right )\right )+\left (2 x^2+2 x \log \left (\frac {4}{x}\right )\right ) \log ^2\left (x+\log \left (\frac {4}{x}\right )\right )} \, dx=\ln \left (\frac {\left (2\,\ln \left (x+\ln \left (\frac {4}{x}\right )\right )+1\right )\,\left (x-1\right )}{\ln \left (2^{2\,x}\right )+x\,\ln \left (\frac {1}{x}\right )+x^2}\right )-\ln \left (\frac {\ln \left (x+\ln \left (\frac {4}{x}\right )\right )\,\left (x-1\right )}{\ln \left (2^{2\,x}\right )+x\,\ln \left (\frac {1}{x}\right )+x^2}\right ) \] Input:
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)
Output:
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))
Time = 0.18 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.93 \[ \int \frac {1-x}{\left (x^2+x \log \left (\frac {4}{x}\right )\right ) \log \left (x+\log \left (\frac {4}{x}\right )\right )+\left (2 x^2+2 x \log \left (\frac {4}{x}\right )\right ) \log ^2\left (x+\log \left (\frac {4}{x}\right )\right )} \, dx=-\mathrm {log}\left (\mathrm {log}\left (\mathrm {log}\left (\frac {4}{x}\right )+x \right )\right )+\mathrm {log}\left (2 \,\mathrm {log}\left (\mathrm {log}\left (\frac {4}{x}\right )+x \right )+1\right ) \] Input:
int((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)
Output:
- log(log(log(4/x) + x)) + log(2*log(log(4/x) + x) + 1)