Integrand size = 110, antiderivative size = 39 \[ \int \frac {36+372 x-1075 x^2+1250 x^3-777 x^4+262 x^5-42 x^6+2 x^7+e^x \left (-32 x^2+32 x^3-8 x^5+2 x^6\right )+\left (-2-5 x+11 x^2-6 x^3+x^4\right ) \log (x)-x \log ^2(x)}{-16 x+24 x^2-12 x^3+2 x^4} \, dx=\left (1+e^x-2 x\right ) x^2+\frac {1}{4} \left (-(3-x)^2+\frac {\log (x)}{2-x}\right )^2 \] Output:
1/4*(ln(x)/(2-x)-(3-x)^2)^2+x^2*(exp(x)-2*x+1)
Time = 0.06 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.28 \[ \int \frac {36+372 x-1075 x^2+1250 x^3-777 x^4+262 x^5-42 x^6+2 x^7+e^x \left (-32 x^2+32 x^3-8 x^5+2 x^6\right )+\left (-2-5 x+11 x^2-6 x^3+x^4\right ) \log (x)-x \log ^2(x)}{-16 x+24 x^2-12 x^3+2 x^4} \, dx=\frac {1}{4} \left (x \left (-108+\left (58+4 e^x\right ) x-20 x^2+x^3\right )+\frac {2 (-3+x)^2 \log (x)}{-2+x}+\frac {\log ^2(x)}{(-2+x)^2}\right ) \] Input:
Integrate[(36 + 372*x - 1075*x^2 + 1250*x^3 - 777*x^4 + 262*x^5 - 42*x^6 + 2*x^7 + E^x*(-32*x^2 + 32*x^3 - 8*x^5 + 2*x^6) + (-2 - 5*x + 11*x^2 - 6*x ^3 + x^4)*Log[x] - x*Log[x]^2)/(-16*x + 24*x^2 - 12*x^3 + 2*x^4),x]
Output:
(x*(-108 + (58 + 4*E^x)*x - 20*x^2 + x^3) + (2*(-3 + x)^2*Log[x])/(-2 + x) + Log[x]^2/(-2 + x)^2)/4
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 1.43 (sec) , antiderivative size = 164, normalized size of antiderivative = 4.21, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {2026, 2007, 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.
\(\displaystyle \int \frac {2 x^7-42 x^6+262 x^5-777 x^4+1250 x^3-1075 x^2+\left (x^4-6 x^3+11 x^2-5 x-2\right ) \log (x)+e^x \left (2 x^6-8 x^5+32 x^3-32 x^2\right )+372 x-x \log ^2(x)+36}{2 x^4-12 x^3+24 x^2-16 x} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {2 x^7-42 x^6+262 x^5-777 x^4+1250 x^3-1075 x^2+\left (x^4-6 x^3+11 x^2-5 x-2\right ) \log (x)+e^x \left (2 x^6-8 x^5+32 x^3-32 x^2\right )+372 x-x \log ^2(x)+36}{x \left (2 x^3-12 x^2+24 x-16\right )}dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {2 x^7-42 x^6+262 x^5-777 x^4+1250 x^3-1075 x^2+\left (x^4-6 x^3+11 x^2-5 x-2\right ) \log (x)+e^x \left (2 x^6-8 x^5+32 x^3-32 x^2\right )+372 x-x \log ^2(x)+36}{x \left (\sqrt [3]{2} x-2 \sqrt [3]{2}\right )^3}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {x^6}{(x-2)^3}-\frac {21 x^5}{(x-2)^3}+\frac {131 x^4}{(x-2)^3}-\frac {777 x^3}{2 (x-2)^3}+\frac {625 x^2}{(x-2)^3}+\frac {\left (x^3-4 x^2+3 x+1\right ) \log (x)}{2 (x-2)^2 x}+e^x (x+2) x-\frac {1075 x}{2 (x-2)^3}+\frac {186}{(x-2)^3}+\frac {18}{(x-2)^3 x}-\frac {\log ^2(x)}{2 (x-2)^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{8} \operatorname {PolyLog}\left (2,1-\frac {x}{2}\right )-\frac {\operatorname {PolyLog}\left (2,\frac {2}{x}\right )}{8}+\frac {x^4}{4}-5 x^3+e^x x^2+\frac {1075 x^2}{8 (2-x)^2}+\frac {29 x^2}{2}-27 x+\frac {1075}{2 (2-x)}-\frac {1075}{2 (2-x)^2}+\frac {\log ^2(x)}{4 (2-x)^2}+\frac {\log ^2(x)}{16}-\frac {x \log (x)}{4 (2-x)}+\frac {1}{2} x \log (x)-\frac {1}{8} \log (2) \log (x-2)+\frac {1}{8} \log \left (1-\frac {2}{x}\right ) \log (x)-\frac {9 \log (x)}{4}\) |
Input:
Int[(36 + 372*x - 1075*x^2 + 1250*x^3 - 777*x^4 + 262*x^5 - 42*x^6 + 2*x^7 + E^x*(-32*x^2 + 32*x^3 - 8*x^5 + 2*x^6) + (-2 - 5*x + 11*x^2 - 6*x^3 + x ^4)*Log[x] - x*Log[x]^2)/(-16*x + 24*x^2 - 12*x^3 + 2*x^4),x]
Output:
-1075/(2*(2 - x)^2) + 1075/(2*(2 - x)) - 27*x + (29*x^2)/2 + E^x*x^2 + (10 75*x^2)/(8*(2 - x)^2) - 5*x^3 + x^4/4 - (Log[2]*Log[-2 + x])/8 - (9*Log[x] )/4 + (x*Log[x])/2 - (x*Log[x])/(4*(2 - x)) + (Log[1 - 2/x]*Log[x])/8 + Lo g[x]^2/16 + Log[x]^2/(4*(2 - x)^2) + PolyLog[2, 1 - x/2]/8 - PolyLog[2, 2/ x]/8
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^(Ex pon[Px, x]*p), x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; IntegerQ[p] && Pol yQ[Px, x] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
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 = 5.11 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.62
method | result | size |
risch | \(\frac {\ln \left (x \right )^{2}}{4 x^{2}-16 x +16}+\frac {\left (x^{2}-2 x +1\right ) \ln \left (x \right )}{2 x -4}+\frac {x^{4}}{4}-5 x^{3}+\frac {29 x^{2}}{2}-27 x -2 \ln \left (x \right )+{\mathrm e}^{x} x^{2}\) | \(63\) |
parallelrisch | \(-\frac {864-8 \,{\mathrm e}^{x} x^{4}-32 \,{\mathrm e}^{x} x^{2}+32 \,{\mathrm e}^{x} x^{3}-84 x \ln \left (x \right )-4 x^{3} \ln \left (x \right )+32 x^{2} \ln \left (x \right )+72 \ln \left (x \right )-2 \ln \left (x \right )^{2}-284 x^{4}+840 x^{3}-1112 x^{2}-2 x^{6}+48 x^{5}}{8 \left (x^{2}-4 x +4\right )}\) | \(90\) |
orering | \(\text {Expression too large to display}\) | \(2732\) |
Input:
int((-x*ln(x)^2+(x^4-6*x^3+11*x^2-5*x-2)*ln(x)+(2*x^6-8*x^5+32*x^3-32*x^2) *exp(x)+2*x^7-42*x^6+262*x^5-777*x^4+1250*x^3-1075*x^2+372*x+36)/(2*x^4-12 *x^3+24*x^2-16*x),x,method=_RETURNVERBOSE)
Output:
1/4/(x^2-4*x+4)*ln(x)^2+1/2*(x^2-2*x+1)/(-2+x)*ln(x)+1/4*x^4-5*x^3+29/2*x^ 2-27*x-2*ln(x)+exp(x)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (33) = 66\).
Time = 0.07 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.00 \[ \int \frac {36+372 x-1075 x^2+1250 x^3-777 x^4+262 x^5-42 x^6+2 x^7+e^x \left (-32 x^2+32 x^3-8 x^5+2 x^6\right )+\left (-2-5 x+11 x^2-6 x^3+x^4\right ) \log (x)-x \log ^2(x)}{-16 x+24 x^2-12 x^3+2 x^4} \, dx=\frac {x^{6} - 24 \, x^{5} + 142 \, x^{4} - 420 \, x^{3} + 664 \, x^{2} + 4 \, {\left (x^{4} - 4 \, x^{3} + 4 \, x^{2}\right )} e^{x} + 2 \, {\left (x^{3} - 8 \, x^{2} + 21 \, x - 18\right )} \log \left (x\right ) + \log \left (x\right )^{2} - 432 \, x}{4 \, {\left (x^{2} - 4 \, x + 4\right )}} \] Input:
integrate((-x*log(x)^2+(x^4-6*x^3+11*x^2-5*x-2)*log(x)+(2*x^6-8*x^5+32*x^3 -32*x^2)*exp(x)+2*x^7-42*x^6+262*x^5-777*x^4+1250*x^3-1075*x^2+372*x+36)/( 2*x^4-12*x^3+24*x^2-16*x),x, algorithm="fricas")
Output:
1/4*(x^6 - 24*x^5 + 142*x^4 - 420*x^3 + 664*x^2 + 4*(x^4 - 4*x^3 + 4*x^2)* e^x + 2*(x^3 - 8*x^2 + 21*x - 18)*log(x) + log(x)^2 - 432*x)/(x^2 - 4*x + 4)
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (26) = 52\).
Time = 0.23 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.62 \[ \int \frac {36+372 x-1075 x^2+1250 x^3-777 x^4+262 x^5-42 x^6+2 x^7+e^x \left (-32 x^2+32 x^3-8 x^5+2 x^6\right )+\left (-2-5 x+11 x^2-6 x^3+x^4\right ) \log (x)-x \log ^2(x)}{-16 x+24 x^2-12 x^3+2 x^4} \, dx=\frac {x^{4}}{4} - 5 x^{3} + x^{2} e^{x} + \frac {29 x^{2}}{2} - 27 x - 2 \log {\left (x \right )} + \frac {\log {\left (x \right )}^{2}}{4 x^{2} - 16 x + 16} + \frac {\left (x^{2} - 2 x + 1\right ) \log {\left (x \right )}}{2 x - 4} \] Input:
integrate((-x*ln(x)**2+(x**4-6*x**3+11*x**2-5*x-2)*ln(x)+(2*x**6-8*x**5+32 *x**3-32*x**2)*exp(x)+2*x**7-42*x**6+262*x**5-777*x**4+1250*x**3-1075*x**2 +372*x+36)/(2*x**4-12*x**3+24*x**2-16*x),x)
Output:
x**4/4 - 5*x**3 + x**2*exp(x) + 29*x**2/2 - 27*x - 2*log(x) + log(x)**2/(4 *x**2 - 16*x + 16) + (x**2 - 2*x + 1)*log(x)/(2*x - 4)
Leaf count of result is larger than twice the leaf count of optimal. 217 vs. \(2 (33) = 66\).
Time = 0.07 (sec) , antiderivative size = 217, normalized size of antiderivative = 5.56 \[ \int \frac {36+372 x-1075 x^2+1250 x^3-777 x^4+262 x^5-42 x^6+2 x^7+e^x \left (-32 x^2+32 x^3-8 x^5+2 x^6\right )+\left (-2-5 x+11 x^2-6 x^3+x^4\right ) \log (x)-x \log ^2(x)}{-16 x+24 x^2-12 x^3+2 x^4} \, dx=\frac {1}{4} \, x^{4} - 5 \, x^{3} + \frac {29}{2} \, x^{2} - \frac {53}{2} \, x - \frac {2 \, x^{3} - 8 \, x^{2} - 4 \, {\left (x^{4} - 4 \, x^{3} + 4 \, x^{2}\right )} e^{x} - 2 \, {\left (x^{3} - 4 \, x^{2} + 5 \, x - 2\right )} \log \left (x\right ) - \log \left (x\right )^{2} + 8 \, x}{4 \, {\left (x^{2} - 4 \, x + 4\right )}} - \frac {32 \, {\left (6 \, x - 11\right )}}{x^{2} - 4 \, x + 4} + \frac {336 \, {\left (5 \, x - 9\right )}}{x^{2} - 4 \, x + 4} - \frac {1048 \, {\left (4 \, x - 7\right )}}{x^{2} - 4 \, x + 4} + \frac {1554 \, {\left (3 \, x - 5\right )}}{x^{2} - 4 \, x + 4} - \frac {1250 \, {\left (2 \, x - 3\right )}}{x^{2} - 4 \, x + 4} + \frac {1075 \, {\left (x - 1\right )}}{2 \, {\left (x^{2} - 4 \, x + 4\right )}} + \frac {9 \, {\left (x - 3\right )}}{2 \, {\left (x^{2} - 4 \, x + 4\right )}} - \frac {93}{x^{2} - 4 \, x + 4} - 2 \, \log \left (x\right ) \] Input:
integrate((-x*log(x)^2+(x^4-6*x^3+11*x^2-5*x-2)*log(x)+(2*x^6-8*x^5+32*x^3 -32*x^2)*exp(x)+2*x^7-42*x^6+262*x^5-777*x^4+1250*x^3-1075*x^2+372*x+36)/( 2*x^4-12*x^3+24*x^2-16*x),x, algorithm="maxima")
Output:
1/4*x^4 - 5*x^3 + 29/2*x^2 - 53/2*x - 1/4*(2*x^3 - 8*x^2 - 4*(x^4 - 4*x^3 + 4*x^2)*e^x - 2*(x^3 - 4*x^2 + 5*x - 2)*log(x) - log(x)^2 + 8*x)/(x^2 - 4 *x + 4) - 32*(6*x - 11)/(x^2 - 4*x + 4) + 336*(5*x - 9)/(x^2 - 4*x + 4) - 1048*(4*x - 7)/(x^2 - 4*x + 4) + 1554*(3*x - 5)/(x^2 - 4*x + 4) - 1250*(2* x - 3)/(x^2 - 4*x + 4) + 1075/2*(x - 1)/(x^2 - 4*x + 4) + 9/2*(x - 3)/(x^2 - 4*x + 4) - 93/(x^2 - 4*x + 4) - 2*log(x)
Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (33) = 66\).
Time = 0.11 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.23 \[ \int \frac {36+372 x-1075 x^2+1250 x^3-777 x^4+262 x^5-42 x^6+2 x^7+e^x \left (-32 x^2+32 x^3-8 x^5+2 x^6\right )+\left (-2-5 x+11 x^2-6 x^3+x^4\right ) \log (x)-x \log ^2(x)}{-16 x+24 x^2-12 x^3+2 x^4} \, dx=\frac {x^{6} - 24 \, x^{5} + 4 \, x^{4} e^{x} + 142 \, x^{4} - 16 \, x^{3} e^{x} + 2 \, x^{3} \log \left (x\right ) - 420 \, x^{3} + 16 \, x^{2} e^{x} - 16 \, x^{2} \log \left (x\right ) + 664 \, x^{2} + 42 \, x \log \left (x\right ) + \log \left (x\right )^{2} - 432 \, x - 36 \, \log \left (x\right )}{4 \, {\left (x^{2} - 4 \, x + 4\right )}} \] Input:
integrate((-x*log(x)^2+(x^4-6*x^3+11*x^2-5*x-2)*log(x)+(2*x^6-8*x^5+32*x^3 -32*x^2)*exp(x)+2*x^7-42*x^6+262*x^5-777*x^4+1250*x^3-1075*x^2+372*x+36)/( 2*x^4-12*x^3+24*x^2-16*x),x, algorithm="giac")
Output:
1/4*(x^6 - 24*x^5 + 4*x^4*e^x + 142*x^4 - 16*x^3*e^x + 2*x^3*log(x) - 420* x^3 + 16*x^2*e^x - 16*x^2*log(x) + 664*x^2 + 42*x*log(x) + log(x)^2 - 432* x - 36*log(x))/(x^2 - 4*x + 4)
Time = 1.86 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.79 \[ \int \frac {36+372 x-1075 x^2+1250 x^3-777 x^4+262 x^5-42 x^6+2 x^7+e^x \left (-32 x^2+32 x^3-8 x^5+2 x^6\right )+\left (-2-5 x+11 x^2-6 x^3+x^4\right ) \log (x)-x \log ^2(x)}{-16 x+24 x^2-12 x^3+2 x^4} \, dx=\frac {{\ln \left (x\right )}^2}{4\,\left (x^2-4\,x+4\right )}-3\,\ln \left (x\right )-27\,x+x^2\,{\mathrm {e}}^x-\frac {3\,\ln \left (x\right )}{2\,\left (x-2\right )}+\frac {29\,x^2}{2}-5\,x^3+\frac {x^4}{4}+\frac {x^2\,\ln \left (x\right )}{2\,\left (x-2\right )} \] Input:
int((x*log(x)^2 - 372*x + log(x)*(5*x - 11*x^2 + 6*x^3 - x^4 + 2) + exp(x) *(32*x^2 - 32*x^3 + 8*x^5 - 2*x^6) + 1075*x^2 - 1250*x^3 + 777*x^4 - 262*x ^5 + 42*x^6 - 2*x^7 - 36)/(16*x - 24*x^2 + 12*x^3 - 2*x^4),x)
Output:
log(x)^2/(4*(x^2 - 4*x + 4)) - 3*log(x) - 27*x + x^2*exp(x) - (3*log(x))/( 2*(x - 2)) + (29*x^2)/2 - 5*x^3 + x^4/4 + (x^2*log(x))/(2*(x - 2))
Time = 0.21 (sec) , antiderivative size = 92, normalized size of antiderivative = 2.36 \[ \int \frac {36+372 x-1075 x^2+1250 x^3-777 x^4+262 x^5-42 x^6+2 x^7+e^x \left (-32 x^2+32 x^3-8 x^5+2 x^6\right )+\left (-2-5 x+11 x^2-6 x^3+x^4\right ) \log (x)-x \log ^2(x)}{-16 x+24 x^2-12 x^3+2 x^4} \, dx=\frac {4 e^{x} x^{4}-16 e^{x} x^{3}+16 e^{x} x^{2}+\mathrm {log}\left (x \right )^{2}+2 \,\mathrm {log}\left (x \right ) x^{3}-16 \,\mathrm {log}\left (x \right ) x^{2}+42 \,\mathrm {log}\left (x \right ) x -36 \,\mathrm {log}\left (x \right )+x^{6}-24 x^{5}+142 x^{4}-420 x^{3}+662 x^{2}-424 x -8}{4 x^{2}-16 x +16} \] Input:
int((-x*log(x)^2+(x^4-6*x^3+11*x^2-5*x-2)*log(x)+(2*x^6-8*x^5+32*x^3-32*x^ 2)*exp(x)+2*x^7-42*x^6+262*x^5-777*x^4+1250*x^3-1075*x^2+372*x+36)/(2*x^4- 12*x^3+24*x^2-16*x),x)
Output:
(4*e**x*x**4 - 16*e**x*x**3 + 16*e**x*x**2 + log(x)**2 + 2*log(x)*x**3 - 1 6*log(x)*x**2 + 42*log(x)*x - 36*log(x) + x**6 - 24*x**5 + 142*x**4 - 420* x**3 + 662*x**2 - 424*x - 8)/(4*(x**2 - 4*x + 4))