Integrand size = 60, antiderivative size = 18 \[ \int \frac {40+8 \log (x)+(32-24 x+8 e x+(8-6 x+2 e x) \log (x)) \log \left (\frac {4-3 x+e x}{x}\right )}{4 x^2-3 x^3+e x^3} \, dx=-\frac {2 \log \left (-3+e+\frac {4}{x}\right ) (5+\log (x))}{x} \]
Time = 0.50 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.72 \[ \int \frac {40+8 \log (x)+(32-24 x+8 e x+(8-6 x+2 e x) \log (x)) \log \left (\frac {4-3 x+e x}{x}\right )}{4 x^2-3 x^3+e x^3} \, dx=-\frac {10 \log \left (-3+e+\frac {4}{x}\right )}{x}-\frac {2 \log \left (-3+e+\frac {4}{x}\right ) \log (x)}{x} \]
Integrate[(40 + 8*Log[x] + (32 - 24*x + 8*E*x + (8 - 6*x + 2*E*x)*Log[x])* Log[(4 - 3*x + E*x)/x])/(4*x^2 - 3*x^3 + E*x^3),x]
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.70 (sec) , antiderivative size = 181, normalized size of antiderivative = 10.06, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {6, 2026, 7293, 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.
\(\displaystyle \int \frac {8 \log (x)+(8 e x-24 x+(2 e x-6 x+8) \log (x)+32) \log \left (\frac {e x-3 x+4}{x}\right )+40}{e x^3-3 x^3+4 x^2} \, dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {8 \log (x)+(8 e x-24 x+(2 e x-6 x+8) \log (x)+32) \log \left (\frac {e x-3 x+4}{x}\right )+40}{(e-3) x^3+4 x^2}dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {8 \log (x)+(8 e x-24 x+(2 e x-6 x+8) \log (x)+32) \log \left (\frac {e x-3 x+4}{x}\right )+40}{x^2 (4-(3-e) x)}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {2 \log \left (\frac {4}{x}+e-3\right ) (\log (x)+4)}{x^2}+\frac {8 (\log (x)+5)}{x^2 (4-(3-e) x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} (3-e) \operatorname {PolyLog}\left (2,1-\frac {4}{(3-e) x}\right )+\frac {1}{2} (3-e) \operatorname {PolyLog}\left (2,\frac {4}{(3-e) x}\right )+\frac {2}{x}+\frac {1}{2} \left (-\frac {4}{x}-e+3\right ) \log \left (\frac {4}{x}+e-3\right )+\frac {1}{2} (3-e) \log \left (\frac {4}{(3-e) x}\right ) \log \left (\frac {4}{x}+e-3\right )+\frac {1}{2} \left (-\frac {4}{x}-e+3\right ) (\log (x)+4) \log \left (\frac {4}{x}+e-3\right )+\frac {2 (\log (x)+4)}{x}-\frac {1}{2} (3-e) \log \left (1-\frac {4}{(3-e) x}\right ) (\log (x)+5)-\frac {2 (\log (x)+5)}{x}\) |
Int[(40 + 8*Log[x] + (32 - 24*x + 8*E*x + (8 - 6*x + 2*E*x)*Log[x])*Log[(4 - 3*x + E*x)/x])/(4*x^2 - 3*x^3 + E*x^3),x]
2/x + ((3 - E - 4/x)*Log[-3 + E + 4/x])/2 + ((3 - E)*Log[-3 + E + 4/x]*Log [4/((3 - E)*x)])/2 + (2*(4 + Log[x]))/x + ((3 - E - 4/x)*Log[-3 + E + 4/x] *(4 + Log[x]))/2 - (2*(5 + Log[x]))/x - ((3 - E)*Log[1 - 4/((3 - E)*x)]*(5 + Log[x]))/2 + ((3 - E)*PolyLog[2, 1 - 4/((3 - E)*x)])/2 + ((3 - E)*PolyL og[2, 4/((3 - E)*x)])/2
3.30.94.3.1 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[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Leaf count of result is larger than twice the leaf count of optimal. \(130\) vs. \(2(19)=38\).
Time = 1.04 (sec) , antiderivative size = 131, normalized size of antiderivative = 7.28
method | result | size |
parallelrisch | \(-\frac {80 \,{\mathrm e} \ln \left (\frac {x \,{\mathrm e}+4-3 x}{{\mathrm e}-3}\right ) x -80 x \,{\mathrm e} \ln \left (x \right )-80 \ln \left (\frac {x \,{\mathrm e}+4-3 x}{x}\right ) {\mathrm e} x -240 \ln \left (\frac {x \,{\mathrm e}+4-3 x}{{\mathrm e}-3}\right ) x +240 x \ln \left (x \right )+240 \ln \left (\frac {x \,{\mathrm e}+4-3 x}{x}\right ) x +32 \ln \left (\frac {x \,{\mathrm e}+4-3 x}{x}\right ) \ln \left (x \right )+160 \ln \left (\frac {x \,{\mathrm e}+4-3 x}{x}\right )}{16 x}\) | \(131\) |
default | \(\frac {2+2 \left (-\frac {{\mathrm e}}{4}+\frac {3}{4}\right ) x \ln \left (\frac {x \,{\mathrm e}+4-3 x}{x}\right )-2 \ln \left (\frac {x \,{\mathrm e}+4-3 x}{x}\right ) \ln \left (x \right )-2 \ln \left (\frac {x \,{\mathrm e}+4-3 x}{x}\right )}{x}-\frac {10}{x}+40 \left (-\frac {{\mathrm e}}{16}+\frac {3}{16}\right ) \ln \left (x \right )-\frac {5 \left ({\mathrm e}-3\right )^{2} \ln \left (-x \,{\mathrm e}-4+3 x \right )}{2 \left (3-{\mathrm e}\right )}-\frac {2 \ln \left (\frac {x \,{\mathrm e}+4-3 x}{x}\right ) \left (x \,{\mathrm e}+4-3 x \right )}{x}+\frac {2 x \,{\mathrm e}-6 x +8}{x}\) | \(149\) |
parts | \(\frac {2+2 \left (-\frac {{\mathrm e}}{4}+\frac {3}{4}\right ) x \ln \left (\frac {x \,{\mathrm e}+4-3 x}{x}\right )-2 \ln \left (\frac {x \,{\mathrm e}+4-3 x}{x}\right ) \ln \left (x \right )-2 \ln \left (\frac {x \,{\mathrm e}+4-3 x}{x}\right )}{x}-\frac {10}{x}+40 \left (-\frac {{\mathrm e}}{16}+\frac {3}{16}\right ) \ln \left (x \right )-\frac {5 \left ({\mathrm e}-3\right )^{2} \ln \left (-x \,{\mathrm e}-4+3 x \right )}{2 \left (3-{\mathrm e}\right )}-\frac {2 \ln \left (\frac {x \,{\mathrm e}+4-3 x}{x}\right ) \left (x \,{\mathrm e}+4-3 x \right )}{x}+\frac {2 x \,{\mathrm e}-6 x +8}{x}\) | \(149\) |
risch | \(-\frac {2 \left (5+\ln \left (x \right )\right ) \ln \left (x \,{\mathrm e}+4-3 x \right )}{x}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (i \left (x \,{\mathrm e}+4-3 x \right )\right ) \operatorname {csgn}\left (\frac {i \left (x \,{\mathrm e}+4-3 x \right )}{x}\right ) \ln \left (x \right )-i \pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (\frac {i \left (x \,{\mathrm e}+4-3 x \right )}{x}\right )^{2} \ln \left (x \right )-i \pi \,\operatorname {csgn}\left (i \left (x \,{\mathrm e}+4-3 x \right )\right ) \operatorname {csgn}\left (\frac {i \left (x \,{\mathrm e}+4-3 x \right )}{x}\right )^{2} \ln \left (x \right )+i \pi \operatorname {csgn}\left (\frac {i \left (x \,{\mathrm e}+4-3 x \right )}{x}\right )^{3} \ln \left (x \right )+5 i \pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (i \left (x \,{\mathrm e}+4-3 x \right )\right ) \operatorname {csgn}\left (\frac {i \left (x \,{\mathrm e}+4-3 x \right )}{x}\right )-5 i \pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (\frac {i \left (x \,{\mathrm e}+4-3 x \right )}{x}\right )^{2}-5 i \pi \,\operatorname {csgn}\left (i \left (x \,{\mathrm e}+4-3 x \right )\right ) \operatorname {csgn}\left (\frac {i \left (x \,{\mathrm e}+4-3 x \right )}{x}\right )^{2}+5 i \pi \operatorname {csgn}\left (\frac {i \left (x \,{\mathrm e}+4-3 x \right )}{x}\right )^{3}+2 \ln \left (x \right )^{2}+10 \ln \left (x \right )}{x}\) | \(296\) |
int((((2*x*exp(1)-6*x+8)*ln(x)+8*x*exp(1)-24*x+32)*ln((x*exp(1)+4-3*x)/x)+ 8*ln(x)+40)/(x^3*exp(1)-3*x^3+4*x^2),x,method=_RETURNVERBOSE)
-1/16*(80*exp(1)*ln((x*exp(1)+4-3*x)/(exp(1)-3))*x-80*x*exp(1)*ln(x)-80*ln ((x*exp(1)+4-3*x)/x)*exp(1)*x-240*ln((x*exp(1)+4-3*x)/(exp(1)-3))*x+240*x* ln(x)+240*ln((x*exp(1)+4-3*x)/x)*x+32*ln((x*exp(1)+4-3*x)/x)*ln(x)+160*ln( (x*exp(1)+4-3*x)/x))/x
Time = 0.25 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.28 \[ \int \frac {40+8 \log (x)+(32-24 x+8 e x+(8-6 x+2 e x) \log (x)) \log \left (\frac {4-3 x+e x}{x}\right )}{4 x^2-3 x^3+e x^3} \, dx=-\frac {2 \, {\left (\log \left (x\right ) + 5\right )} \log \left (\frac {x e - 3 \, x + 4}{x}\right )}{x} \]
integrate((((2*x*exp(1)-6*x+8)*log(x)+8*x*exp(1)-24*x+32)*log((x*exp(1)+4- 3*x)/x)+8*log(x)+40)/(x^3*exp(1)-3*x^3+4*x^2),x, algorithm=\
Time = 0.36 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {40+8 \log (x)+(32-24 x+8 e x+(8-6 x+2 e x) \log (x)) \log \left (\frac {4-3 x+e x}{x}\right )}{4 x^2-3 x^3+e x^3} \, dx=\frac {\left (- 2 \log {\left (x \right )} - 10\right ) \log {\left (\frac {- 3 x + e x + 4}{x} \right )}}{x} \]
integrate((((2*x*exp(1)-6*x+8)*ln(x)+8*x*exp(1)-24*x+32)*ln((x*exp(1)+4-3* x)/x)+8*ln(x)+40)/(x**3*exp(1)-3*x**3+4*x**2),x)
Leaf count of result is larger than twice the leaf count of optimal. 304 vs. \(2 (19) = 38\).
Time = 0.24 (sec) , antiderivative size = 304, normalized size of antiderivative = 16.89 \[ \int \frac {40+8 \log (x)+(32-24 x+8 e x+(8-6 x+2 e x) \log (x)) \log \left (\frac {4-3 x+e x}{x}\right )}{4 x^2-3 x^3+e x^3} \, dx=-2 \, {\left (\log \left (x {\left (e - 3\right )} + 4\right ) - \log \left (x\right )\right )} e \log \left (\frac {4}{x} + e - 3\right ) + {\left (\log \left (x {\left (e - 3\right )} + 4\right )^{2} - 2 \, \log \left (x {\left (e - 3\right )} + 4\right ) \log \left (x\right ) + \log \left (x\right )^{2}\right )} e + \frac {5}{2} \, {\left (e - 3\right )} \log \left (x {\left (e - 3\right )} + 4\right ) - 3 \, \log \left (x {\left (e - 3\right )} + 4\right )^{2} - \frac {5}{2} \, {\left (e - 3\right )} \log \left (x\right ) + 6 \, \log \left (x {\left (e - 3\right )} + 4\right ) \log \left (x\right ) - 3 \, \log \left (x\right )^{2} + 2 \, {\left ({\left (e - 3\right )} \log \left (x {\left (e - 3\right )} + 4\right ) - {\left (e - 3\right )} \log \left (x\right ) - \frac {4}{x}\right )} \log \left (\frac {4}{x} + e - 3\right ) + 6 \, {\left (\log \left (x {\left (e - 3\right )} + 4\right ) - \log \left (x\right )\right )} \log \left (\frac {4}{x} + e - 3\right ) - \frac {x {\left (e - 3\right )} \log \left (x {\left (e - 3\right )} + 4\right )^{2} + x {\left (e - 3\right )} \log \left (x\right )^{2} - 2 \, x {\left (e - 3\right )} \log \left (x\right ) - 2 \, {\left (x {\left (e - 3\right )} \log \left (x\right ) - x {\left (e - 3\right )}\right )} \log \left (x {\left (e - 3\right )} + 4\right ) - 8}{x} - \frac {{\left (x {\left (e - 3\right )} + 4 \, \log \left (x\right ) + 4\right )} \log \left (x {\left (e - 3\right )} + 4\right ) - {\left (x {\left (e - 3\right )} + 4\right )} \log \left (x\right ) - 4 \, \log \left (x\right )^{2} - 4}{2 \, x} - \frac {10}{x} \]
integrate((((2*x*exp(1)-6*x+8)*log(x)+8*x*exp(1)-24*x+32)*log((x*exp(1)+4- 3*x)/x)+8*log(x)+40)/(x^3*exp(1)-3*x^3+4*x^2),x, algorithm=\
-2*(log(x*(e - 3) + 4) - log(x))*e*log(4/x + e - 3) + (log(x*(e - 3) + 4)^ 2 - 2*log(x*(e - 3) + 4)*log(x) + log(x)^2)*e + 5/2*(e - 3)*log(x*(e - 3) + 4) - 3*log(x*(e - 3) + 4)^2 - 5/2*(e - 3)*log(x) + 6*log(x*(e - 3) + 4)* log(x) - 3*log(x)^2 + 2*((e - 3)*log(x*(e - 3) + 4) - (e - 3)*log(x) - 4/x )*log(4/x + e - 3) + 6*(log(x*(e - 3) + 4) - log(x))*log(4/x + e - 3) - (x *(e - 3)*log(x*(e - 3) + 4)^2 + x*(e - 3)*log(x)^2 - 2*x*(e - 3)*log(x) - 2*(x*(e - 3)*log(x) - x*(e - 3))*log(x*(e - 3) + 4) - 8)/x - 1/2*((x*(e - 3) + 4*log(x) + 4)*log(x*(e - 3) + 4) - (x*(e - 3) + 4)*log(x) - 4*log(x)^ 2 - 4)/x - 10/x
Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (19) = 38\).
Time = 0.29 (sec) , antiderivative size = 41, normalized size of antiderivative = 2.28 \[ \int \frac {40+8 \log (x)+(32-24 x+8 e x+(8-6 x+2 e x) \log (x)) \log \left (\frac {4-3 x+e x}{x}\right )}{4 x^2-3 x^3+e x^3} \, dx=-\frac {2 \, {\left (\log \left (x e - 3 \, x + 4\right ) \log \left (x\right ) - \log \left (x\right )^{2} + 5 \, \log \left (x e - 3 \, x + 4\right ) - 5 \, \log \left (x\right )\right )}}{x} \]
integrate((((2*x*exp(1)-6*x+8)*log(x)+8*x*exp(1)-24*x+32)*log((x*exp(1)+4- 3*x)/x)+8*log(x)+40)/(x^3*exp(1)-3*x^3+4*x^2),x, algorithm=\
Time = 12.08 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.28 \[ \int \frac {40+8 \log (x)+(32-24 x+8 e x+(8-6 x+2 e x) \log (x)) \log \left (\frac {4-3 x+e x}{x}\right )}{4 x^2-3 x^3+e x^3} \, dx=-\frac {2\,\ln \left (\frac {x\,\mathrm {e}-3\,x+4}{x}\right )\,\left (\ln \left (x\right )+5\right )}{x} \]