Integrand size = 73, antiderivative size = 34 \[ \int \frac {6 x-e^5 x+6 x^2-5 x^3-2 x^4-x \log \left (\frac {1}{x^2}\right )+\left (6+2 x+e^5 (3+x)+(3+x) \log \left (\frac {1}{x^2}\right )\right ) \log \left (\frac {3+x}{2}\right )}{3 x^2+x^3} \, dx=1-x^2+\frac {\left (-e^5+x-\log \left (\frac {1}{x^2}\right )\right ) \left (x+\log \left (\frac {3+x}{2}\right )\right )}{x} \] Output:
1+(ln(3/2+1/2*x)+x)*(x-exp(5)-ln(1/x^2))/x-x^2
Time = 0.35 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06 \[ \int \frac {6 x-e^5 x+6 x^2-5 x^3-2 x^4-x \log \left (\frac {1}{x^2}\right )+\left (6+2 x+e^5 (3+x)+(3+x) \log \left (\frac {1}{x^2}\right )\right ) \log \left (\frac {3+x}{2}\right )}{3 x^2+x^3} \, dx=x-x^2+2 \log (x)-\frac {\left (e^5+\log \left (\frac {1}{x^2}\right )\right ) \log \left (\frac {3+x}{2}\right )}{x}+\log (3+x) \] Input:
Integrate[(6*x - E^5*x + 6*x^2 - 5*x^3 - 2*x^4 - x*Log[x^(-2)] + (6 + 2*x + E^5*(3 + x) + (3 + x)*Log[x^(-2)])*Log[(3 + x)/2])/(3*x^2 + x^3),x]
Output:
x - x^2 + 2*Log[x] - ((E^5 + Log[x^(-2)])*Log[(3 + x)/2])/x + Log[3 + x]
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.85 (sec) , antiderivative size = 172, normalized size of antiderivative = 5.06, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.055, 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.
\(\displaystyle \int \frac {-2 x^4-5 x^3+6 x^2-x \log \left (\frac {1}{x^2}\right )+\left ((x+3) \log \left (\frac {1}{x^2}\right )+2 x+e^5 (x+3)+6\right ) \log \left (\frac {x+3}{2}\right )-e^5 x+6 x}{x^3+3 x^2} \, dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {-2 x^4-5 x^3+6 x^2-x \log \left (\frac {1}{x^2}\right )+\left ((x+3) \log \left (\frac {1}{x^2}\right )+2 x+e^5 (x+3)+6\right ) \log \left (\frac {x+3}{2}\right )+\left (6-e^5\right ) x}{x^3+3 x^2}dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {-2 x^4-5 x^3+6 x^2-x \log \left (\frac {1}{x^2}\right )+\left ((x+3) \log \left (\frac {1}{x^2}\right )+2 x+e^5 (x+3)+6\right ) \log \left (\frac {x+3}{2}\right )+\left (6-e^5\right ) x}{x^2 (x+3)}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {\log \left (\frac {x}{2}+\frac {3}{2}\right ) \left (\log \left (\frac {1}{x^2}\right )+2 \left (1+\frac {e^5}{2}\right )\right )}{x^2}+\frac {-2 x^3-5 x^2-\log \left (\frac {1}{x^2}\right )+6 x+6 \left (1-\frac {e^5}{6}\right )}{x (x+3)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 \operatorname {PolyLog}\left (2,-\frac {3}{x}\right )}{3}+\frac {2 \operatorname {PolyLog}\left (2,-\frac {x}{3}\right )}{3}-x^2+\frac {1}{3} \log \left (\frac {3}{x}+1\right ) \log \left (\frac {1}{x^2}\right )+\frac {1}{3} \left (\log \left (\frac {1}{x^2}\right )+e^5+2\right ) \log (x)-\frac {1}{3} \left (\log \left (\frac {1}{x^2}\right )+e^5+2\right ) \log (x+3)-\frac {\log \left (\frac {x}{2}+\frac {3}{2}\right ) \left (\log \left (\frac {1}{x^2}\right )+e^5+2\right )}{x}+x+\frac {\log ^2(x)}{3}-\frac {2}{3} \log (3) \log (x)+\frac {1}{3} \left (6-e^5\right ) \log (x)-\frac {2 \log (x)}{3}+\frac {1}{3} \left (3+e^5\right ) \log (x+3)+\frac {2}{3} \log (x+3)+\frac {2 \log \left (\frac {x}{2}+\frac {3}{2}\right )}{x}\) |
Input:
Int[(6*x - E^5*x + 6*x^2 - 5*x^3 - 2*x^4 - x*Log[x^(-2)] + (6 + 2*x + E^5* (3 + x) + (3 + x)*Log[x^(-2)])*Log[(3 + x)/2])/(3*x^2 + x^3),x]
Output:
x - x^2 + (2*Log[3/2 + x/2])/x + (Log[1 + 3/x]*Log[x^(-2)])/3 - (Log[3/2 + x/2]*(2 + E^5 + Log[x^(-2)]))/x - (2*Log[x])/3 + ((6 - E^5)*Log[x])/3 - ( 2*Log[3]*Log[x])/3 + ((2 + E^5 + Log[x^(-2)])*Log[x])/3 + Log[x]^2/3 + (2* Log[3 + x])/3 + ((3 + E^5)*Log[3 + x])/3 - ((2 + E^5 + Log[x^(-2)])*Log[3 + x])/3 + (2*PolyLog[2, -3/x])/3 + (2*PolyLog[2, -1/3*x])/3
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])
Time = 3.23 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.71
method | result | size |
parallelrisch | \(-\frac {2 x^{3}+2 \ln \left (\frac {3}{2}+\frac {x}{2}\right ) {\mathrm e}^{5}-2 x^{2}+2 x \ln \left (\frac {1}{x^{2}}\right )-2 \ln \left (\frac {3}{2}+\frac {x}{2}\right ) x +2 \ln \left (\frac {1}{x^{2}}\right ) \ln \left (\frac {3}{2}+\frac {x}{2}\right )-6 x}{2 x}\) | \(58\) |
risch | \(-\frac {\left (i \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-2 i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+2 \,{\mathrm e}^{5}-4 \ln \left (x \right )\right ) \ln \left (\frac {3}{2}+\frac {x}{2}\right )}{2 x}+2 \ln \left (x \right )+\ln \left (3+x \right )+x -x^{2}\) | \(85\) |
orering | \(\text {Expression too large to display}\) | \(1523\) |
Input:
int((((3+x)*ln(1/x^2)+(3+x)*exp(5)+2*x+6)*ln(3/2+1/2*x)-x*ln(1/x^2)-x*exp( 5)-2*x^4-5*x^3+6*x^2+6*x)/(x^3+3*x^2),x,method=_RETURNVERBOSE)
Output:
-1/2*(2*x^3+2*ln(3/2+1/2*x)*exp(5)-2*x^2+2*x*ln(1/x^2)-2*ln(3/2+1/2*x)*x+2 *ln(1/x^2)*ln(3/2+1/2*x)-6*x)/x
Time = 0.09 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.18 \[ \int \frac {6 x-e^5 x+6 x^2-5 x^3-2 x^4-x \log \left (\frac {1}{x^2}\right )+\left (6+2 x+e^5 (3+x)+(3+x) \log \left (\frac {1}{x^2}\right )\right ) \log \left (\frac {3+x}{2}\right )}{3 x^2+x^3} \, dx=-\frac {x^{3} - x^{2} - {\left (x - e^{5} - \log \left (\frac {1}{x^{2}}\right )\right )} \log \left (\frac {1}{2} \, x + \frac {3}{2}\right ) + x \log \left (\frac {1}{x^{2}}\right )}{x} \] Input:
integrate((((3+x)*log(1/x^2)+(3+x)*exp(5)+2*x+6)*log(3/2+1/2*x)-x*log(1/x^ 2)-x*exp(5)-2*x^4-5*x^3+6*x^2+6*x)/(x^3+3*x^2),x, algorithm="fricas")
Output:
-(x^3 - x^2 - (x - e^5 - log(x^(-2)))*log(1/2*x + 3/2) + x*log(x^(-2)))/x
Time = 0.34 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06 \[ \int \frac {6 x-e^5 x+6 x^2-5 x^3-2 x^4-x \log \left (\frac {1}{x^2}\right )+\left (6+2 x+e^5 (3+x)+(3+x) \log \left (\frac {1}{x^2}\right )\right ) \log \left (\frac {3+x}{2}\right )}{3 x^2+x^3} \, dx=- x^{2} + x + 2 \log {\left (x \right )} + \log {\left (x + 3 \right )} + \frac {\left (- \log {\left (\frac {1}{x^{2}} \right )} - e^{5}\right ) \log {\left (\frac {x}{2} + \frac {3}{2} \right )}}{x} \] Input:
integrate((((3+x)*ln(1/x**2)+(3+x)*exp(5)+2*x+6)*ln(3/2+1/2*x)-x*ln(1/x**2 )-x*exp(5)-2*x**4-5*x**3+6*x**2+6*x)/(x**3+3*x**2),x)
Output:
-x**2 + x + 2*log(x) + log(x + 3) + (-log(x**(-2)) - exp(5))*log(x/2 + 3/2 )/x
Time = 0.17 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.35 \[ \int \frac {6 x-e^5 x+6 x^2-5 x^3-2 x^4-x \log \left (\frac {1}{x^2}\right )+\left (6+2 x+e^5 (3+x)+(3+x) \log \left (\frac {1}{x^2}\right )\right ) \log \left (\frac {3+x}{2}\right )}{3 x^2+x^3} \, dx=-\frac {x^{3} - x^{2} - e^{5} \log \left (2\right ) - {\left (x - e^{5} + 2 \, \log \left (x\right )\right )} \log \left (x + 3\right ) - 2 \, {\left (x - \log \left (2\right )\right )} \log \left (x\right )}{x} \] Input:
integrate((((3+x)*log(1/x^2)+(3+x)*exp(5)+2*x+6)*log(3/2+1/2*x)-x*log(1/x^ 2)-x*exp(5)-2*x^4-5*x^3+6*x^2+6*x)/(x^3+3*x^2),x, algorithm="maxima")
Output:
-(x^3 - x^2 - e^5*log(2) - (x - e^5 + 2*log(x))*log(x + 3) - 2*(x - log(2) )*log(x))/x
Time = 0.13 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.53 \[ \int \frac {6 x-e^5 x+6 x^2-5 x^3-2 x^4-x \log \left (\frac {1}{x^2}\right )+\left (6+2 x+e^5 (3+x)+(3+x) \log \left (\frac {1}{x^2}\right )\right ) \log \left (\frac {3+x}{2}\right )}{3 x^2+x^3} \, dx=-\frac {x^{3} - x^{2} + \log \left (2\right ) \log \left (x^{2}\right ) - x \log \left (x + 3\right ) - \log \left (x^{2}\right ) \log \left (x + 3\right ) - 2 \, x \log \left (x\right ) + e^{5} \log \left (\frac {1}{2} \, x + \frac {3}{2}\right )}{x} \] Input:
integrate((((3+x)*log(1/x^2)+(3+x)*exp(5)+2*x+6)*log(3/2+1/2*x)-x*log(1/x^ 2)-x*exp(5)-2*x^4-5*x^3+6*x^2+6*x)/(x^3+3*x^2),x, algorithm="giac")
Output:
-(x^3 - x^2 + log(2)*log(x^2) - x*log(x + 3) - log(x^2)*log(x + 3) - 2*x*l og(x) + e^5*log(1/2*x + 3/2))/x
Time = 4.60 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.26 \[ \int \frac {6 x-e^5 x+6 x^2-5 x^3-2 x^4-x \log \left (\frac {1}{x^2}\right )+\left (6+2 x+e^5 (3+x)+(3+x) \log \left (\frac {1}{x^2}\right )\right ) \log \left (\frac {3+x}{2}\right )}{3 x^2+x^3} \, dx=x+\ln \left (x+3\right )+2\,\ln \left (x\right )-x^2-\frac {\ln \left (\frac {1}{x^2}\right )\,\ln \left (\frac {x}{2}+\frac {3}{2}\right )}{x}-\frac {{\mathrm {e}}^5\,\ln \left (\frac {x}{2}+\frac {3}{2}\right )}{x} \] Input:
int(-(x*log(1/x^2) - 6*x + x*exp(5) - log(x/2 + 3/2)*(2*x + log(1/x^2)*(x + 3) + exp(5)*(x + 3) + 6) - 6*x^2 + 5*x^3 + 2*x^4)/(3*x^2 + x^3),x)
Output:
x + log(x + 3) + 2*log(x) - x^2 - (log(1/x^2)*log(x/2 + 3/2))/x - (exp(5)* log(x/2 + 3/2))/x
Time = 0.16 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.59 \[ \int \frac {6 x-e^5 x+6 x^2-5 x^3-2 x^4-x \log \left (\frac {1}{x^2}\right )+\left (6+2 x+e^5 (3+x)+(3+x) \log \left (\frac {1}{x^2}\right )\right ) \log \left (\frac {3+x}{2}\right )}{3 x^2+x^3} \, dx=\frac {3 \,\mathrm {log}\left (x^{2}\right ) \mathrm {log}\left (\frac {x}{2}+\frac {3}{2}\right )-\mathrm {log}\left (x^{2}\right ) x +\mathrm {log}\left (x +3\right ) e^{5} x +5 \,\mathrm {log}\left (x +3\right ) x -\mathrm {log}\left (\frac {x}{2}+\frac {3}{2}\right ) e^{5} x -3 \,\mathrm {log}\left (\frac {x}{2}+\frac {3}{2}\right ) e^{5}-2 \,\mathrm {log}\left (\frac {x}{2}+\frac {3}{2}\right ) x +8 \,\mathrm {log}\left (x \right ) x -3 x^{3}+3 x^{2}}{3 x} \] Input:
int((((3+x)*log(1/x^2)+(3+x)*exp(5)+2*x+6)*log(3/2+1/2*x)-x*log(1/x^2)-x*e xp(5)-2*x^4-5*x^3+6*x^2+6*x)/(x^3+3*x^2),x)
Output:
(3*log(x**2)*log((x + 3)/2) - log(x**2)*x + log(x + 3)*e**5*x + 5*log(x + 3)*x - log((x + 3)/2)*e**5*x - 3*log((x + 3)/2)*e**5 - 2*log((x + 3)/2)*x + 8*log(x)*x - 3*x**3 + 3*x**2)/(3*x)