Integrand size = 169, antiderivative size = 32 \[ \int \frac {4 x+x^2-4 (i \pi +\log (3))+\left (4 x+x^2\right ) \log (2) (i \pi +\log (3))-x (i \pi +\log (3)) \log (x)+(i \pi -x+\log (3)-x \log (2) (i \pi +\log (3))) \log (3 x)+(-x-x \log (2) (i \pi +\log (3))+(i \pi +\log (3)) \log (x)) \log (x+x \log (2) (i \pi +\log (3))-(i \pi +\log (3)) \log (x))}{-x^2-x^2 \log (2) (i \pi +\log (3))+x (i \pi +\log (3)) \log (x)} \, dx=-x+(-4+\log (3 x)) \log (x+(i \pi +\log (3)) (x \log (2)-\log (x))) \]
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(153\) vs. \(2(32)=64\).
Time = 0.25 (sec) , antiderivative size = 153, normalized size of antiderivative = 4.78 \[ \int \frac {4 x+x^2-4 (i \pi +\log (3))+\left (4 x+x^2\right ) \log (2) (i \pi +\log (3))-x (i \pi +\log (3)) \log (x)+(i \pi -x+\log (3)-x \log (2) (i \pi +\log (3))) \log (3 x)+(-x-x \log (2) (i \pi +\log (3))+(i \pi +\log (3)) \log (x)) \log (x+x \log (2) (i \pi +\log (3))-(i \pi +\log (3)) \log (x))}{-x^2-x^2 \log (2) (i \pi +\log (3))+x (i \pi +\log (3)) \log (x)} \, dx=-x+i \arctan \left (\frac {\pi (-x \log (2)+\log (x))}{x+x \log (2) \log (3)-\log (3) \log (x)}\right ) (4+\log (x)-\log (3 x))+\log (x) \log (x+x \log (2) (i \pi +\log (3))-(i \pi +\log (3)) \log (x))-\frac {1}{2} (4+\log (x)-\log (3 x)) \log \left (x^2 \left (1+\pi ^2 \log ^2(2)+\log ^2(2) \log ^2(3)+\log (3) \log (4)\right )-x \left (\pi ^2 \log (4)+\log ^2(3) \log (4)+\log (9)\right ) \log (x)+\left (\pi ^2+\log ^2(3)\right ) \log ^2(x)\right ) \]
Integrate[(4*x + x^2 - 4*(I*Pi + Log[3]) + (4*x + x^2)*Log[2]*(I*Pi + Log[ 3]) - x*(I*Pi + Log[3])*Log[x] + (I*Pi - x + Log[3] - x*Log[2]*(I*Pi + Log [3]))*Log[3*x] + (-x - x*Log[2]*(I*Pi + Log[3]) + (I*Pi + Log[3])*Log[x])* Log[x + x*Log[2]*(I*Pi + Log[3]) - (I*Pi + Log[3])*Log[x]])/(-x^2 - x^2*Lo g[2]*(I*Pi + Log[3]) + x*(I*Pi + Log[3])*Log[x]),x]
-x + I*ArcTan[(Pi*(-(x*Log[2]) + Log[x]))/(x + x*Log[2]*Log[3] - Log[3]*Lo g[x])]*(4 + Log[x] - Log[3*x]) + Log[x]*Log[x + x*Log[2]*(I*Pi + Log[3]) - (I*Pi + Log[3])*Log[x]] - ((4 + Log[x] - Log[3*x])*Log[x^2*(1 + Pi^2*Log[ 2]^2 + Log[2]^2*Log[3]^2 + Log[3]*Log[4]) - x*(Pi^2*Log[4] + Log[3]^2*Log[ 4] + Log[9])*Log[x] + (Pi^2 + Log[3]^2)*Log[x]^2])/2
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.
| \(\displaystyle \int \frac {x^2+\left (x^2+4 x\right ) \log (2) (\log (3)+i \pi )+4 x-x (\log (3)+i \pi ) \log (x)+(-x+x (-\log (2)) (\log (3)+i \pi )+i \pi +\log (3)) \log (3 x)+(-x+x (-\log (2)) (\log (3)+i \pi )+(\log (3)+i \pi ) \log (x)) \log (x+x \log (2) (\log (3)+i \pi )-(\log (3)+i \pi ) \log (x))-4 (\log (3)+i \pi )}{-x^2+x^2 (-\log (2)) (\log (3)+i \pi )+x (\log (3)+i \pi ) \log (x)} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {x^2+\left (x^2+4 x\right ) \log (2) (\log (3)+i \pi )+4 x-x (\log (3)+i \pi ) \log (x)+(-x+x (-\log (2)) (\log (3)+i \pi )+i \pi +\log (3)) \log (3 x)+(-x+x (-\log (2)) (\log (3)+i \pi )+(\log (3)+i \pi ) \log (x)) \log (x+x \log (2) (\log (3)+i \pi )-(\log (3)+i \pi ) \log (x))-4 (\log (3)+i \pi )}{x^2 (-1-\log (2) (\log (3)+i \pi ))+x (\log (3)+i \pi ) \log (x)}dx\) |
| \(\Big \downarrow \) 3041 |
| \(\displaystyle \int \frac {x^2+\left (x^2+4 x\right ) \log (2) (\log (3)+i \pi )+4 x-x (\log (3)+i \pi ) \log (x)+(-x+x (-\log (2)) (\log (3)+i \pi )+i \pi +\log (3)) \log (3 x)+(-x+x (-\log (2)) (\log (3)+i \pi )+(\log (3)+i \pi ) \log (x)) \log (x+x \log (2) (\log (3)+i \pi )-(\log (3)+i \pi ) \log (x))-4 (\log (3)+i \pi )}{x (x (-1-\log (2) (\log (3)+i \pi ))+(\log (3)+i \pi ) \log (x))}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {i x}{-i x (1+\log (2) (\log (3)+i \pi ))-\pi \left (1-\frac {i \log (3)}{\pi }\right ) \log (x)}+\frac {4 i}{-i x (1+\log (2) (\log (3)+i \pi ))-\pi \left (1-\frac {i \log (3)}{\pi }\right ) \log (x)}+\frac {(-\pi +i \log (3)) \log (x)}{i x (1+\log (2) (\log (3)+i \pi ))+\pi \left (1-\frac {i \log (3)}{\pi }\right ) \log (x)}+\frac {(x+4) \log (2) (\pi -i \log (3))}{i x (1+\log (2) (\log (3)+i \pi ))+\pi \left (1-\frac {i \log (3)}{\pi }\right ) \log (x)}+\frac {(-(x (\pi \log (2)-i (1+\log (2) \log (3))))+\pi -i \log (3)) \log (3 x)}{x \left (i x (1+\log (2) (\log (3)+i \pi ))+\pi \left (1-\frac {i \log (3)}{\pi }\right ) \log (x)\right )}+\frac {\log (x (1+\log (2) (\log (3)+i \pi ))-(\log (3)+i \pi ) \log (x))}{x}+\frac {4 (-\pi +i \log (3))}{x \left (i x (1+\log (2) (\log (3)+i \pi ))+\pi \left (1-\frac {i \log (3)}{\pi }\right ) \log (x)\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 4 i \int \frac {1}{-i (1+\log (2) (i \pi +\log (3))) x-\pi \left (1-\frac {i \log (3)}{\pi }\right ) \log (x)}dx+i \int \frac {x}{-i (1+\log (2) (i \pi +\log (3))) x-\pi \left (1-\frac {i \log (3)}{\pi }\right ) \log (x)}dx+4 \log (2) (\pi -i \log (3)) \int \frac {1}{i (1+\log (2) (i \pi +\log (3))) x+\pi \left (1-\frac {i \log (3)}{\pi }\right ) \log (x)}dx-4 (\pi -i \log (3)) \int \frac {1}{x \left (i (1+\log (2) (i \pi +\log (3))) x+\pi \left (1-\frac {i \log (3)}{\pi }\right ) \log (x)\right )}dx-(\pi \log (2)-i (1+\log (2) \log (3))) \int \frac {x}{i (1+\log (2) (i \pi +\log (3))) x+\pi \left (1-\frac {i \log (3)}{\pi }\right ) \log (x)}dx+\log (2) (\pi -i \log (3)) \int \frac {x}{i (1+\log (2) (i \pi +\log (3))) x+\pi \left (1-\frac {i \log (3)}{\pi }\right ) \log (x)}dx-i (1+\log (2) (\log (3)+i \pi )) \int \frac {\log (3 x)}{-i (1+\log (2) (i \pi +\log (3))) x-\pi \left (1-\frac {i \log (3)}{\pi }\right ) \log (x)}dx+(\pi -i \log (3)) \int \frac {\log (3 x)}{x \left (i (1+\log (2) (i \pi +\log (3))) x+\pi \left (1-\frac {i \log (3)}{\pi }\right ) \log (x)\right )}dx+\int \frac {\log (x (1+\log (2) (i \pi +\log (3)))-(i \pi +\log (3)) \log (x))}{x}dx-x\) |
Int[(4*x + x^2 - 4*(I*Pi + Log[3]) + (4*x + x^2)*Log[2]*(I*Pi + Log[3]) - x*(I*Pi + Log[3])*Log[x] + (I*Pi - x + Log[3] - x*Log[2]*(I*Pi + Log[3]))* Log[3*x] + (-x - x*Log[2]*(I*Pi + Log[3]) + (I*Pi + Log[3])*Log[x])*Log[x + x*Log[2]*(I*Pi + Log[3]) - (I*Pi + Log[3])*Log[x]])/(-x^2 - x^2*Log[2]*( I*Pi + Log[3]) + x*(I*Pi + Log[3])*Log[x]),x]
3.20.8.3.1 Defintions of rubi rules used
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Int[(u_.)*((a_.)*(x_)^(m_.) + Log[(c_.)*(x_)^(n_.)]^(q_.)*(b_.)*(x_)^(r_.)) ^(p_.), x_Symbol] :> Int[u*x^(p*r)*(a*x^(m - r) + b*Log[c*x^n]^q)^p, x] /; FreeQ[{a, b, c, m, n, p, q, r}, x] && IntegerQ[p]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (31 ) = 62\).
Time = 0.81 (sec) , antiderivative size = 98, normalized size of antiderivative = 3.06
| method | result | size |
| risch | \(\ln \left (x \right ) \ln \left (-\left (\ln \left (3\right )+i \pi \right ) \ln \left (x \right )+x \left (\ln \left (3\right )+i \pi \right ) \ln \left (2\right )+x \right )-x +\ln \left (3\right ) \ln \left (\ln \left (x \right )-\frac {x \left (-i \ln \left (2\right ) \ln \left (3\right )+\pi \ln \left (2\right )-i\right )}{\pi -i \ln \left (3\right )}\right )-4 \ln \left (\ln \left (x \right )-\frac {x \left (-i \ln \left (2\right ) \ln \left (3\right )+\pi \ln \left (2\right )-i\right )}{\pi -i \ln \left (3\right )}\right )\) | \(98\) |
int((((ln(3)+I*Pi)*ln(x)-x*(ln(3)+I*Pi)*ln(2)-x)*ln(-(ln(3)+I*Pi)*ln(x)+x* (ln(3)+I*Pi)*ln(2)+x)+(-x*(ln(3)+I*Pi)*ln(2)+ln(3)+I*Pi-x)*ln(3*x)-x*(ln(3 )+I*Pi)*ln(x)+(x^2+4*x)*(ln(3)+I*Pi)*ln(2)-4*ln(3)-4*I*Pi+x^2+4*x)/(x*(ln( 3)+I*Pi)*ln(x)-x^2*(ln(3)+I*Pi)*ln(2)-x^2),x,method=_RETURNVERBOSE)
ln(x)*ln(-(ln(3)+I*Pi)*ln(x)+x*(ln(3)+I*Pi)*ln(2)+x)-x+ln(3)*ln(ln(x)-x*(- I*ln(2)*ln(3)+Pi*ln(2)-I)/(Pi-I*ln(3)))-4*ln(ln(x)-x*(-I*ln(2)*ln(3)+Pi*ln (2)-I)/(Pi-I*ln(3)))
Time = 0.27 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.16 \[ \int \frac {4 x+x^2-4 (i \pi +\log (3))+\left (4 x+x^2\right ) \log (2) (i \pi +\log (3))-x (i \pi +\log (3)) \log (x)+(i \pi -x+\log (3)-x \log (2) (i \pi +\log (3))) \log (3 x)+(-x-x \log (2) (i \pi +\log (3))+(i \pi +\log (3)) \log (x)) \log (x+x \log (2) (i \pi +\log (3))-(i \pi +\log (3)) \log (x))}{-x^2-x^2 \log (2) (i \pi +\log (3))+x (i \pi +\log (3)) \log (x)} \, dx={\left (\log \left (3\right ) + \log \left (x\right ) - 4\right )} \log \left (i \, \pi x \log \left (2\right ) + x \log \left (3\right ) \log \left (2\right ) + {\left (-i \, \pi - \log \left (3\right )\right )} \log \left (x\right ) + x\right ) - x \]
integrate((((log(3)+I*pi)*log(x)-x*(log(3)+I*pi)*log(2)-x)*log(-(log(3)+I* pi)*log(x)+x*(log(3)+I*pi)*log(2)+x)+(-x*(log(3)+I*pi)*log(2)+log(3)+I*pi- x)*log(3*x)-x*(log(3)+I*pi)*log(x)+(x^2+4*x)*(log(3)+I*pi)*log(2)-4*log(3) -4*I*pi+x^2+4*x)/(x*(log(3)+I*pi)*log(x)-x^2*(log(3)+I*pi)*log(2)-x^2),x, algorithm=\
(log(3) + log(x) - 4)*log(I*pi*x*log(2) + x*log(3)*log(2) + (-I*pi - log(3 ))*log(x) + x) - x
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (26) = 52\).
Time = 10.66 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.34 \[ \int \frac {4 x+x^2-4 (i \pi +\log (3))+\left (4 x+x^2\right ) \log (2) (i \pi +\log (3))-x (i \pi +\log (3)) \log (x)+(i \pi -x+\log (3)-x \log (2) (i \pi +\log (3))) \log (3 x)+(-x-x \log (2) (i \pi +\log (3))+(i \pi +\log (3)) \log (x)) \log (x+x \log (2) (i \pi +\log (3))-(i \pi +\log (3)) \log (x))}{-x^2-x^2 \log (2) (i \pi +\log (3))+x (i \pi +\log (3)) \log (x)} \, dx=- x + \log {\left (x \right )} \log {\left (x \log {\left (2 \right )} \log {\left (3 \right )} + x + i \pi x \log {\left (2 \right )} - \log {\left (3 \right )} \log {\left (x \right )} - i \pi \log {\left (x \right )} \right )} + \left (-4 + \log {\left (3 \right )}\right ) \log {\left (\frac {- x - x \log {\left (2 \right )} \log {\left (3 \right )} - i \pi x \log {\left (2 \right )}}{\log {\left (3 \right )} + i \pi } + \log {\left (x \right )} \right )} \]
integrate((((ln(3)+I*pi)*ln(x)-x*(ln(3)+I*pi)*ln(2)-x)*ln(-(ln(3)+I*pi)*ln (x)+x*(ln(3)+I*pi)*ln(2)+x)+(-x*(ln(3)+I*pi)*ln(2)+ln(3)+I*pi-x)*ln(3*x)-x *(ln(3)+I*pi)*ln(x)+(x**2+4*x)*(ln(3)+I*pi)*ln(2)-4*ln(3)-4*I*pi+x**2+4*x) /(x*(ln(3)+I*pi)*ln(x)-x**2*(ln(3)+I*pi)*ln(2)-x**2),x)
-x + log(x)*log(x*log(2)*log(3) + x + I*pi*x*log(2) - log(3)*log(x) - I*pi *log(x)) + (-4 + log(3))*log((-x - x*log(2)*log(3) - I*pi*x*log(2))/(log(3 ) + I*pi) + log(x))
Time = 0.38 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.19 \[ \int \frac {4 x+x^2-4 (i \pi +\log (3))+\left (4 x+x^2\right ) \log (2) (i \pi +\log (3))-x (i \pi +\log (3)) \log (x)+(i \pi -x+\log (3)-x \log (2) (i \pi +\log (3))) \log (3 x)+(-x-x \log (2) (i \pi +\log (3))+(i \pi +\log (3)) \log (x)) \log (x+x \log (2) (i \pi +\log (3))-(i \pi +\log (3)) \log (x))}{-x^2-x^2 \log (2) (i \pi +\log (3))+x (i \pi +\log (3)) \log (x)} \, dx={\left (\log \left (3\right ) + \log \left (x\right ) - 4\right )} \log \left ({\left (i \, \pi \log \left (2\right ) + \log \left (3\right ) \log \left (2\right ) + 1\right )} x + {\left (-i \, \pi - \log \left (3\right )\right )} \log \left (x\right )\right ) - x \]
integrate((((log(3)+I*pi)*log(x)-x*(log(3)+I*pi)*log(2)-x)*log(-(log(3)+I* pi)*log(x)+x*(log(3)+I*pi)*log(2)+x)+(-x*(log(3)+I*pi)*log(2)+log(3)+I*pi- x)*log(3*x)-x*(log(3)+I*pi)*log(x)+(x^2+4*x)*(log(3)+I*pi)*log(2)-4*log(3) -4*I*pi+x^2+4*x)/(x*(log(3)+I*pi)*log(x)-x^2*(log(3)+I*pi)*log(2)-x^2),x, algorithm=\
(log(3) + log(x) - 4)*log((I*pi*log(2) + log(3)*log(2) + 1)*x + (-I*pi - l og(3))*log(x)) - x
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (30) = 60\).
Time = 0.35 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.28 \[ \int \frac {4 x+x^2-4 (i \pi +\log (3))+\left (4 x+x^2\right ) \log (2) (i \pi +\log (3))-x (i \pi +\log (3)) \log (x)+(i \pi -x+\log (3)-x \log (2) (i \pi +\log (3))) \log (3 x)+(-x-x \log (2) (i \pi +\log (3))+(i \pi +\log (3)) \log (x)) \log (x+x \log (2) (i \pi +\log (3))-(i \pi +\log (3)) \log (x))}{-x^2-x^2 \log (2) (i \pi +\log (3))+x (i \pi +\log (3)) \log (x)} \, dx={\left (\log \left (3\right ) - 4\right )} \log \left (\pi x \log \left (2\right ) - i \, x \log \left (3\right ) \log \left (2\right ) - \pi \log \left (x\right ) + i \, \log \left (3\right ) \log \left (x\right ) - i \, x\right ) + \frac {1}{2} i \, \pi \log \left (x\right ) + \log \left (\pi x \log \left (2\right ) - i \, x \log \left (3\right ) \log \left (2\right ) - \pi \log \left (x\right ) + i \, \log \left (3\right ) \log \left (x\right ) - i \, x\right ) \log \left (x\right ) - x \]
integrate((((log(3)+I*pi)*log(x)-x*(log(3)+I*pi)*log(2)-x)*log(-(log(3)+I* pi)*log(x)+x*(log(3)+I*pi)*log(2)+x)+(-x*(log(3)+I*pi)*log(2)+log(3)+I*pi- x)*log(3*x)-x*(log(3)+I*pi)*log(x)+(x^2+4*x)*(log(3)+I*pi)*log(2)-4*log(3) -4*I*pi+x^2+4*x)/(x*(log(3)+I*pi)*log(x)-x^2*(log(3)+I*pi)*log(2)-x^2),x, algorithm=\
(log(3) - 4)*log(pi*x*log(2) - I*x*log(3)*log(2) - pi*log(x) + I*log(3)*lo g(x) - I*x) + 1/2*I*pi*log(x) + log(pi*x*log(2) - I*x*log(3)*log(2) - pi*l og(x) + I*log(3)*log(x) - I*x)*log(x) - x
Timed out. \[ \int \frac {4 x+x^2-4 (i \pi +\log (3))+\left (4 x+x^2\right ) \log (2) (i \pi +\log (3))-x (i \pi +\log (3)) \log (x)+(i \pi -x+\log (3)-x \log (2) (i \pi +\log (3))) \log (3 x)+(-x-x \log (2) (i \pi +\log (3))+(i \pi +\log (3)) \log (x)) \log (x+x \log (2) (i \pi +\log (3))-(i \pi +\log (3)) \log (x))}{-x^2-x^2 \log (2) (i \pi +\log (3))+x (i \pi +\log (3)) \log (x)} \, dx=\int \frac {\Pi \,4{}\mathrm {i}-4\,x+4\,\ln \left (3\right )+\ln \left (x-\ln \left (x\right )\,\left (\ln \left (3\right )+\Pi \,1{}\mathrm {i}\right )+x\,\ln \left (2\right )\,\left (\ln \left (3\right )+\Pi \,1{}\mathrm {i}\right )\right )\,\left (x-\ln \left (x\right )\,\left (\ln \left (3\right )+\Pi \,1{}\mathrm {i}\right )+x\,\ln \left (2\right )\,\left (\ln \left (3\right )+\Pi \,1{}\mathrm {i}\right )\right )-\ln \left (3\,x\right )\,\left (\Pi \,1{}\mathrm {i}-x+\ln \left (3\right )-x\,\ln \left (2\right )\,\left (\ln \left (3\right )+\Pi \,1{}\mathrm {i}\right )\right )-x^2+x\,\ln \left (x\right )\,\left (\ln \left (3\right )+\Pi \,1{}\mathrm {i}\right )-\ln \left (2\right )\,\left (x^2+4\,x\right )\,\left (\ln \left (3\right )+\Pi \,1{}\mathrm {i}\right )}{x^2+x^2\,\ln \left (2\right )\,\left (\ln \left (3\right )+\Pi \,1{}\mathrm {i}\right )-x\,\ln \left (x\right )\,\left (\ln \left (3\right )+\Pi \,1{}\mathrm {i}\right )} \,d x \]
int((Pi*4i - 4*x + 4*log(3) + log(x - log(x)*(Pi*1i + log(3)) + x*log(2)*( Pi*1i + log(3)))*(x - log(x)*(Pi*1i + log(3)) + x*log(2)*(Pi*1i + log(3))) - log(3*x)*(Pi*1i - x + log(3) - x*log(2)*(Pi*1i + log(3))) - x^2 + x*log (x)*(Pi*1i + log(3)) - log(2)*(4*x + x^2)*(Pi*1i + log(3)))/(x^2 + x^2*log (2)*(Pi*1i + log(3)) - x*log(x)*(Pi*1i + log(3))),x)
int((Pi*4i - 4*x + 4*log(3) + log(x - log(x)*(Pi*1i + log(3)) + x*log(2)*( Pi*1i + log(3)))*(x - log(x)*(Pi*1i + log(3)) + x*log(2)*(Pi*1i + log(3))) - log(3*x)*(Pi*1i - x + log(3) - x*log(2)*(Pi*1i + log(3))) - x^2 + x*log (x)*(Pi*1i + log(3)) - log(2)*(4*x + x^2)*(Pi*1i + log(3)))/(x^2 + x^2*log (2)*(Pi*1i + log(3)) - x*log(x)*(Pi*1i + log(3))), x)