Integrand size = 45, antiderivative size = 27 \[ \int \frac {20+4 x+\left (5-6 x+x^2\right ) \log \left (\frac {-5+x}{(-1+x) (i \pi +\log (3))}\right )}{5-6 x+x^2} \, dx=(5+x) \log \left (\frac {1+\frac {4}{1-x}}{i \pi +\log (3)}\right ) \] Output:
ln((4/(1-x)+1)/(ln(3)+I*Pi))*(5+x)
Time = 0.03 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.52 \[ \int \frac {20+4 x+\left (5-6 x+x^2\right ) \log \left (\frac {-5+x}{(-1+x) (i \pi +\log (3))}\right )}{5-6 x+x^2} \, dx=-10 \log (1-x)+10 \log (5-x)+(-5+x) \log \left (\frac {-5+x}{(-1+x) (i \pi +\log (3))}\right ) \] Input:
Integrate[(20 + 4*x + (5 - 6*x + x^2)*Log[(-5 + x)/((-1 + x)*(I*Pi + Log[3 ]))])/(5 - 6*x + x^2),x]
Output:
-10*Log[1 - x] + 10*Log[5 - x] + (-5 + x)*Log[(-5 + x)/((-1 + x)*(I*Pi + L og[3]))]
Time = 0.32 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.78, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.044, Rules used = {7279, 2009}
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.
\(\displaystyle \int \frac {\left (x^2-6 x+5\right ) \log \left (\frac {x-5}{(x-1) (\log (3)+i \pi )}\right )+4 x+20}{x^2-6 x+5} \, dx\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle \int \left (\frac {4 (x+5)}{x^2-6 x+5}+\log \left (\frac {x-5}{(x-1) (\log (3)+i \pi )}\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -10 \log (1-x)+10 \log (5-x)-(5-x) \log \left (\frac {5-x}{(1-x) (\log (3)+i \pi )}\right )\) |
Input:
Int[(20 + 4*x + (5 - 6*x + x^2)*Log[(-5 + x)/((-1 + x)*(I*Pi + Log[3]))])/ (5 - 6*x + x^2),x]
Output:
-10*Log[1 - x] + 10*Log[5 - x] - (5 - x)*Log[(5 - x)/((1 - x)*(I*Pi + Log[ 3]))]
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Time = 0.57 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.30
method | result | size |
risch | \(\ln \left (\frac {-5+x}{\left (-1+x \right ) \left (\ln \left (3\right )+i \pi \right )}\right ) x -5 \ln \left (-1+x \right )+5 \ln \left (-5+x \right )\) | \(35\) |
norman | \(\ln \left (\frac {-5+x}{\left (-1+x \right ) \left (\ln \left (3\right )+i \pi \right )}\right ) x +5 \ln \left (\frac {-5+x}{\left (-1+x \right ) \left (\ln \left (3\right )+i \pi \right )}\right )\) | \(44\) |
parallelrisch | \(\ln \left (\frac {-5+x}{\left (-1+x \right ) \left (\ln \left (3\right )+i \pi \right )}\right ) x +5 \ln \left (\frac {-5+x}{\left (-1+x \right ) \left (\ln \left (3\right )+i \pi \right )}\right )\) | \(44\) |
orering | \(\frac {\left (5+x \right ) \left (\left (x^{2}-6 x +5\right ) \ln \left (\frac {-5+x}{\left (-1+x \right ) \left (\ln \left (3\right )+i \pi \right )}\right )+20+4 x \right )}{x^{2}-6 x +5}+\frac {\left (5+4 x \right ) \left (-1+x \right ) \left (-5+x \right ) \left (\frac {\left (2 x -6\right ) \ln \left (\frac {-5+x}{\left (-1+x \right ) \left (\ln \left (3\right )+i \pi \right )}\right )+\frac {\left (x^{2}-6 x +5\right ) \left (\frac {1}{\left (-1+x \right ) \left (\ln \left (3\right )+i \pi \right )}-\frac {-5+x}{\left (-1+x \right )^{2} \left (\ln \left (3\right )+i \pi \right )}\right ) \left (-1+x \right ) \left (\ln \left (3\right )+i \pi \right )}{-5+x}+4}{x^{2}-6 x +5}-\frac {\left (\left (x^{2}-6 x +5\right ) \ln \left (\frac {-5+x}{\left (-1+x \right ) \left (\ln \left (3\right )+i \pi \right )}\right )+20+4 x \right ) \left (2 x -6\right )}{\left (x^{2}-6 x +5\right )^{2}}\right )}{4 x -10}\) | \(217\) |
parts | \(-6 \ln \left (-1+x \right )+10 \ln \left (-5+x \right )-\frac {4 \left (-\ln \left (3\right )-i \pi \right ) \left (\frac {i \ln \left (i+\left (-i \ln \left (3\right )+\pi \right ) \left (-\frac {4}{\left (-1+x \right ) \left (\ln \left (3\right )+i \pi \right )}+\frac {1}{\ln \left (3\right )+i \pi }\right )\right )}{-i \ln \left (3\right )+\pi }-\frac {i \ln \left (-\frac {4}{\left (-1+x \right ) \left (\ln \left (3\right )+i \pi \right )}+\frac {1}{\ln \left (3\right )+i \pi }\right ) \left (-\frac {4}{\left (-1+x \right ) \left (\ln \left (3\right )+i \pi \right )}+\frac {1}{\ln \left (3\right )+i \pi }\right )}{-i \ln \left (3\right ) \left (-\frac {4}{\left (-1+x \right ) \left (\ln \left (3\right )+i \pi \right )}+\frac {1}{\ln \left (3\right )+i \pi }\right )+\pi \left (-\frac {4}{\left (-1+x \right ) \left (\ln \left (3\right )+i \pi \right )}+\frac {1}{\ln \left (3\right )+i \pi }\right )+i}\right ) \left (-2 i \pi \ln \left (3\right )+\pi ^{2}-\ln \left (3\right )^{2}\right )}{\left (\ln \left (3\right )+i \pi \right )^{2}}\) | \(221\) |
derivativedivides | \(-\frac {4 \left (-\ln \left (3\right )-i \pi \right ) \left (\left (2 i \pi \ln \left (3\right )-\pi ^{2}+\ln \left (3\right )^{2}\right ) \left (-\frac {i \ln \left (i+\left (-i \ln \left (3\right )+\pi \right ) \left (-\frac {4}{\left (-1+x \right ) \left (\ln \left (3\right )+i \pi \right )}+\frac {1}{\ln \left (3\right )+i \pi }\right )\right )}{-i \ln \left (3\right )+\pi }+\frac {i \ln \left (-\frac {4}{\left (-1+x \right ) \left (\ln \left (3\right )+i \pi \right )}+\frac {1}{\ln \left (3\right )+i \pi }\right ) \left (-\frac {4}{\left (-1+x \right ) \left (\ln \left (3\right )+i \pi \right )}+\frac {1}{\ln \left (3\right )+i \pi }\right )}{-i \ln \left (3\right ) \left (-\frac {4}{\left (-1+x \right ) \left (\ln \left (3\right )+i \pi \right )}+\frac {1}{\ln \left (3\right )+i \pi }\right )+\pi \left (-\frac {4}{\left (-1+x \right ) \left (\ln \left (3\right )+i \pi \right )}+\frac {1}{\ln \left (3\right )+i \pi }\right )+i}\right )-\frac {i \left (-3 \pi \ln \left (3\right )^{2}-3 i \ln \left (3\right ) \pi ^{2}+i \ln \left (3\right )^{3}+\pi ^{3}\right ) \ln \left (\frac {\left (-i \ln \left (3\right )+\pi \right ) \left (-i \ln \left (3\right ) \left (-\frac {4}{\left (-1+x \right ) \left (\ln \left (3\right )+i \pi \right )}+\frac {1}{\ln \left (3\right )+i \pi }\right )+\pi \left (-\frac {4}{\left (-1+x \right ) \left (\ln \left (3\right )+i \pi \right )}+\frac {1}{\ln \left (3\right )+i \pi }\right )+i\right )}{\sqrt {\ln \left (3\right )^{2}+\pi ^{2}}}\right )}{\left (-i \ln \left (3\right )+\pi \right )^{2}}-\frac {\left (-5 i \pi -5 \ln \left (3\right )\right ) \ln \left (-\frac {4}{\left (-1+x \right ) \left (\ln \left (3\right )+i \pi \right )}+\frac {1}{\ln \left (3\right )+i \pi }\right )}{2}\right )}{\left (\ln \left (3\right )+i \pi \right )^{2}}\) | \(367\) |
default | \(-\frac {4 \left (-\ln \left (3\right )-i \pi \right ) \left (\left (2 i \pi \ln \left (3\right )-\pi ^{2}+\ln \left (3\right )^{2}\right ) \left (-\frac {i \ln \left (i+\left (-i \ln \left (3\right )+\pi \right ) \left (-\frac {4}{\left (-1+x \right ) \left (\ln \left (3\right )+i \pi \right )}+\frac {1}{\ln \left (3\right )+i \pi }\right )\right )}{-i \ln \left (3\right )+\pi }+\frac {i \ln \left (-\frac {4}{\left (-1+x \right ) \left (\ln \left (3\right )+i \pi \right )}+\frac {1}{\ln \left (3\right )+i \pi }\right ) \left (-\frac {4}{\left (-1+x \right ) \left (\ln \left (3\right )+i \pi \right )}+\frac {1}{\ln \left (3\right )+i \pi }\right )}{-i \ln \left (3\right ) \left (-\frac {4}{\left (-1+x \right ) \left (\ln \left (3\right )+i \pi \right )}+\frac {1}{\ln \left (3\right )+i \pi }\right )+\pi \left (-\frac {4}{\left (-1+x \right ) \left (\ln \left (3\right )+i \pi \right )}+\frac {1}{\ln \left (3\right )+i \pi }\right )+i}\right )-\frac {i \left (-3 \pi \ln \left (3\right )^{2}-3 i \ln \left (3\right ) \pi ^{2}+i \ln \left (3\right )^{3}+\pi ^{3}\right ) \ln \left (\frac {\left (-i \ln \left (3\right )+\pi \right ) \left (-i \ln \left (3\right ) \left (-\frac {4}{\left (-1+x \right ) \left (\ln \left (3\right )+i \pi \right )}+\frac {1}{\ln \left (3\right )+i \pi }\right )+\pi \left (-\frac {4}{\left (-1+x \right ) \left (\ln \left (3\right )+i \pi \right )}+\frac {1}{\ln \left (3\right )+i \pi }\right )+i\right )}{\sqrt {\ln \left (3\right )^{2}+\pi ^{2}}}\right )}{\left (-i \ln \left (3\right )+\pi \right )^{2}}-\frac {\left (-5 i \pi -5 \ln \left (3\right )\right ) \ln \left (-\frac {4}{\left (-1+x \right ) \left (\ln \left (3\right )+i \pi \right )}+\frac {1}{\ln \left (3\right )+i \pi }\right )}{2}\right )}{\left (\ln \left (3\right )+i \pi \right )^{2}}\) | \(367\) |
Input:
int(((x^2-6*x+5)*ln((-5+x)/(-1+x)/(ln(3)+I*Pi))+20+4*x)/(x^2-6*x+5),x,meth od=_RETURNVERBOSE)
Output:
ln((-5+x)/(-1+x)/(ln(3)+I*Pi))*x-5*ln(-1+x)+5*ln(-5+x)
Time = 0.07 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {20+4 x+\left (5-6 x+x^2\right ) \log \left (\frac {-5+x}{(-1+x) (i \pi +\log (3))}\right )}{5-6 x+x^2} \, dx={\left (x + 5\right )} \log \left (\frac {x - 5}{-i \, \pi + i \, \pi x + {\left (x - 1\right )} \log \left (3\right )}\right ) \] Input:
integrate(((x^2-6*x+5)*log((-5+x)/(-1+x)/(log(3)+I*pi))+20+4*x)/(x^2-6*x+5 ),x, algorithm="fricas")
Output:
(x + 5)*log((x - 5)/(-I*pi + I*pi*x + (x - 1)*log(3)))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (17) = 34\).
Time = 0.11 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.96 \[ \int \frac {20+4 x+\left (5-6 x+x^2\right ) \log \left (\frac {-5+x}{(-1+x) (i \pi +\log (3))}\right )}{5-6 x+x^2} \, dx=x \log {\left (\frac {x}{x \log {\left (3 \right )} + i \pi x - \log {\left (3 \right )} - i \pi } - \frac {5}{x \log {\left (3 \right )} + i \pi x - \log {\left (3 \right )} - i \pi } \right )} + 5 \log {\left (x - 5 \right )} - 5 \log {\left (x - 1 \right )} \] Input:
integrate(((x**2-6*x+5)*ln((-5+x)/(-1+x)/(ln(3)+I*pi))+20+4*x)/(x**2-6*x+5 ),x)
Output:
x*log(x/(x*log(3) + I*pi*x - log(3) - I*pi) - 5/(x*log(3) + I*pi*x - log(3 ) - I*pi)) + 5*log(x - 5) - 5*log(x - 1)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 141 vs. \(2 (24) = 48\).
Time = 0.20 (sec) , antiderivative size = 141, normalized size of antiderivative = 5.22 \[ \int \frac {20+4 x+\left (5-6 x+x^2\right ) \log \left (\frac {-5+x}{(-1+x) (i \pi +\log (3))}\right )}{5-6 x+x^2} \, dx=-x \log \left (i \, \pi + \log \left (3\right )\right ) - \frac {1}{4} \, {\left (4 \, x + 5 \, \log \left (i \, \pi + \log \left (3\right )\right ) - 5 \, \log \left (x - 5\right ) - 4\right )} \log \left (x - 1\right ) - \frac {5}{4} \, \log \left (x - 1\right )^{2} + \frac {1}{4} \, {\left (4 \, x + 5 i - 20\right )} \log \left (x - 5\right ) + \frac {5}{4} \, \log \left (x - 1\right ) \log \left (x - 5\right ) - \frac {5}{4} \, \log \left (x - 5\right )^{2} - \frac {5}{4} \, {\left (\log \left (x - 1\right ) - \log \left (x - 5\right )\right )} \log \left (\frac {x}{-i \, \pi + i \, \pi x + x \log \left (3\right ) - \log \left (3\right )} - \frac {5}{-i \, \pi + i \, \pi x + x \log \left (3\right ) - \log \left (3\right )}\right ) - 6 \, \log \left (x - 1\right ) + 10 \, \log \left (x - 5\right ) \] Input:
integrate(((x^2-6*x+5)*log((-5+x)/(-1+x)/(log(3)+I*pi))+20+4*x)/(x^2-6*x+5 ),x, algorithm="maxima")
Output:
-x*log(I*pi + log(3)) - 1/4*(4*x + 5*log(I*pi + log(3)) - 5*log(x - 5) - 4 )*log(x - 1) - 5/4*log(x - 1)^2 + 1/4*(4*x + 5*I - 20)*log(x - 5) + 5/4*lo g(x - 1)*log(x - 5) - 5/4*log(x - 5)^2 - 5/4*(log(x - 1) - log(x - 5))*log (x/(-I*pi + I*pi*x + x*log(3) - log(3)) - 5/(-I*pi + I*pi*x + x*log(3) - l og(3))) - 6*log(x - 1) + 10*log(x - 5)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 163 vs. \(2 (24) = 48\).
Time = 0.16 (sec) , antiderivative size = 163, normalized size of antiderivative = 6.04 \[ \int \frac {20+4 x+\left (5-6 x+x^2\right ) \log \left (\frac {-5+x}{(-1+x) (i \pi +\log (3))}\right )}{5-6 x+x^2} \, dx=2 \, {\left (-i \, \pi - \log \left (3\right )\right )} {\left (\frac {3 i \, \log \left (-\frac {-i \, x + 5 i}{\pi - \pi x + i \, x \log \left (3\right ) - i \, \log \left (3\right )}\right )}{\pi - i \, \log \left (3\right )} + \frac {2 \, \log \left (-\frac {-i \, x + 5 i}{\pi - \pi x + i \, x \log \left (3\right ) - i \, \log \left (3\right )}\right )}{-i \, \pi - \frac {\pi ^{2} {\left (i \, x - 5 i\right )}}{\pi - \pi x + i \, x \log \left (3\right ) - i \, \log \left (3\right )} - \frac {2 \, \pi {\left (x - 5\right )} \log \left (3\right )}{\pi - \pi x + i \, x \log \left (3\right ) - i \, \log \left (3\right )} - \frac {{\left (-i \, x + 5 i\right )} \log \left (3\right )^{2}}{\pi - \pi x + i \, x \log \left (3\right ) - i \, \log \left (3\right )} - \log \left (3\right )}\right )} \] Input:
integrate(((x^2-6*x+5)*log((-5+x)/(-1+x)/(log(3)+I*pi))+20+4*x)/(x^2-6*x+5 ),x, algorithm="giac")
Output:
2*(-I*pi - log(3))*(3*I*log(-(-I*x + 5*I)/(pi - pi*x + I*x*log(3) - I*log( 3)))/(pi - I*log(3)) + 2*log(-(-I*x + 5*I)/(pi - pi*x + I*x*log(3) - I*log (3)))/(-I*pi - pi^2*(I*x - 5*I)/(pi - pi*x + I*x*log(3) - I*log(3)) - 2*pi *(x - 5)*log(3)/(pi - pi*x + I*x*log(3) - I*log(3)) - (-I*x + 5*I)*log(3)^ 2/(pi - pi*x + I*x*log(3) - I*log(3)) - log(3)))
Time = 2.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.11 \[ \int \frac {20+4 x+\left (5-6 x+x^2\right ) \log \left (\frac {-5+x}{(-1+x) (i \pi +\log (3))}\right )}{5-6 x+x^2} \, dx=-10\,\mathrm {atanh}\left (\frac {x}{2}-\frac {3}{2}\right )+x\,\ln \left (\frac {x-5}{\left (\ln \left (3\right )+\Pi \,1{}\mathrm {i}\right )\,\left (x-1\right )}\right ) \] Input:
int((4*x + log((x - 5)/((Pi*1i + log(3))*(x - 1)))*(x^2 - 6*x + 5) + 20)/( x^2 - 6*x + 5),x)
Output:
x*log((x - 5)/((Pi*1i + log(3))*(x - 1))) - 10*atanh(x/2 - 3/2)
Time = 0.24 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.41 \[ \int \frac {20+4 x+\left (5-6 x+x^2\right ) \log \left (\frac {-5+x}{(-1+x) (i \pi +\log (3))}\right )}{5-6 x+x^2} \, dx=6 \,\mathrm {log}\left (-5+x \right )-6 \,\mathrm {log}\left (x -1\right )+\mathrm {log}\left (\frac {-5+x}{\mathrm {log}\left (3\right ) x -\mathrm {log}\left (3\right )+i \pi x -i \pi }\right ) x -\mathrm {log}\left (\frac {-5+x}{\mathrm {log}\left (3\right ) x -\mathrm {log}\left (3\right )+i \pi x -i \pi }\right ) \] Input:
int(((x^2-6*x+5)*log((-5+x)/(-1+x)/(log(3)+I*Pi))+20+4*x)/(x^2-6*x+5),x)
Output:
6*log(x - 5) - 6*log(x - 1) + log((x - 5)/(log(3)*x - log(3) + i*pi*x - i* pi))*x - log((x - 5)/(log(3)*x - log(3) + i*pi*x - i*pi))