Integrand size = 141, antiderivative size = 25 \begin {dmath*} \int \frac {2 x-7 x^2+3 x^3+e^3 \left (4 x^3-32 x^4+48 x^5-20 x^6\right )+e^6 \left (-28 x^6+84 x^7-84 x^8+28 x^9\right )+\left (-2+10 x-12 x^2+4 x^3+e^3 \left (16 x^3-48 x^4+48 x^5-16 x^6\right )\right ) \log (x)+\left (-1+3 x-3 x^2+x^3\right ) \log ^2(x)}{-1+3 x-3 x^2+x^3} \, dx=x \left (-\frac {x}{-1+x}+2 e^3 x^3-\log (x)\right )^2 \end {dmath*}
Leaf count is larger than twice the leaf count of optimal. \(121\) vs. \(2(25)=50\).
Time = 0.11 (sec) , antiderivative size = 121, normalized size of antiderivative = 4.84 \begin {dmath*} \int \frac {2 x-7 x^2+3 x^3+e^3 \left (4 x^3-32 x^4+48 x^5-20 x^6\right )+e^6 \left (-28 x^6+84 x^7-84 x^8+28 x^9\right )+\left (-2+10 x-12 x^2+4 x^3+e^3 \left (16 x^3-48 x^4+48 x^5-16 x^6\right )\right ) \log (x)+\left (-1+3 x-3 x^2+x^3\right ) \log ^2(x)}{-1+3 x-3 x^2+x^3} \, dx=-3+17 e^3-4 e^6+\frac {1}{(-1+x)^2}+\frac {3}{-1+x}-\frac {4 e^3}{-1+x}+x-4 e^3 x-4 e^3 x^2-4 e^3 x^3-4 e^3 x^4+4 e^6 x^7-2 \log (1-x)+2 \log (-1+x)+2 \log (x)+\frac {2 \log (x)}{-1+x}+2 x \log (x)-4 e^3 x^4 \log (x)+x \log ^2(x) \end {dmath*}
Integrate[(2*x - 7*x^2 + 3*x^3 + E^3*(4*x^3 - 32*x^4 + 48*x^5 - 20*x^6) + E^6*(-28*x^6 + 84*x^7 - 84*x^8 + 28*x^9) + (-2 + 10*x - 12*x^2 + 4*x^3 + E ^3*(16*x^3 - 48*x^4 + 48*x^5 - 16*x^6))*Log[x] + (-1 + 3*x - 3*x^2 + x^3)* Log[x]^2)/(-1 + 3*x - 3*x^2 + x^3),x]
-3 + 17*E^3 - 4*E^6 + (-1 + x)^(-2) + 3/(-1 + x) - (4*E^3)/(-1 + x) + x - 4*E^3*x - 4*E^3*x^2 - 4*E^3*x^3 - 4*E^3*x^4 + 4*E^6*x^7 - 2*Log[1 - x] + 2 *Log[-1 + x] + 2*Log[x] + (2*Log[x])/(-1 + x) + 2*x*Log[x] - 4*E^3*x^4*Log [x] + x*Log[x]^2
Leaf count is larger than twice the leaf count of optimal. \(115\) vs. \(2(25)=50\).
Time = 0.61 (sec) , antiderivative size = 115, normalized size of antiderivative = 4.60, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.021, Rules used = {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.
\(\displaystyle \int \frac {3 x^3-7 x^2+\left (x^3-3 x^2+3 x-1\right ) \log ^2(x)+e^6 \left (28 x^9-84 x^8+84 x^7-28 x^6\right )+e^3 \left (-20 x^6+48 x^5-32 x^4+4 x^3\right )+\left (4 x^3-12 x^2+e^3 \left (-16 x^6+48 x^5-48 x^4+16 x^3\right )+10 x-2\right ) \log (x)+2 x}{x^3-3 x^2+3 x-1} \, dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {3 x^3-7 x^2+\left (x^3-3 x^2+3 x-1\right ) \log ^2(x)+e^6 \left (28 x^9-84 x^8+84 x^7-28 x^6\right )+e^3 \left (-20 x^6+48 x^5-32 x^4+4 x^3\right )+\left (4 x^3-12 x^2+e^3 \left (-16 x^6+48 x^5-48 x^4+16 x^3\right )+10 x-2\right ) \log (x)+2 x}{(x-1)^3}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (28 e^6 x^6+\frac {3 x^3}{(x-1)^3}-\frac {7 x^2}{(x-1)^3}-\frac {4 e^3 \left (5 x^2-7 x+1\right ) x^3}{(x-1)^2}-\frac {2 \left (8 e^3 x^5-16 e^3 x^4+8 e^3 x^3-2 x^2+4 x-1\right ) \log (x)}{(x-1)^2}+\frac {2 x}{(x-1)^3}+\log ^2(x)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 4 e^6 x^7-4 e^3 x^4-4 e^3 x^4 \log (x)-4 e^3 x^3-\frac {x^2}{(1-x)^2}-4 e^3 x^2-4 e^3 x+x+\frac {4 e^3}{1-x}-\frac {5}{1-x}+\frac {2}{(1-x)^2}+x \log ^2(x)-\frac {2 x \log (x)}{1-x}+2 x \log (x)\) |
Int[(2*x - 7*x^2 + 3*x^3 + E^3*(4*x^3 - 32*x^4 + 48*x^5 - 20*x^6) + E^6*(- 28*x^6 + 84*x^7 - 84*x^8 + 28*x^9) + (-2 + 10*x - 12*x^2 + 4*x^3 + E^3*(16 *x^3 - 48*x^4 + 48*x^5 - 16*x^6))*Log[x] + (-1 + 3*x - 3*x^2 + x^3)*Log[x] ^2)/(-1 + 3*x - 3*x^2 + x^3),x]
2/(1 - x)^2 - 5/(1 - x) + (4*E^3)/(1 - x) + x - 4*E^3*x - 4*E^3*x^2 - x^2/ (1 - x)^2 - 4*E^3*x^3 - 4*E^3*x^4 + 4*E^6*x^7 + 2*x*Log[x] - (2*x*Log[x])/ (1 - x) - 4*E^3*x^4*Log[x] + x*Log[x]^2
3.9.49.3.1 Defintions of rubi rules used
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]
Leaf count of result is larger than twice the leaf count of optimal. \(83\) vs. \(2(24)=48\).
Time = 0.43 (sec) , antiderivative size = 84, normalized size of antiderivative = 3.36
method | result | size |
default | \(x \ln \left (x \right )^{2}+2 x \ln \left (x \right )+x +4 \,{\mathrm e}^{6} x^{7}-4 x^{4} {\mathrm e}^{3}-4 x^{3} {\mathrm e}^{3}-4 x^{2} {\mathrm e}^{3}-4 x \,{\mathrm e}^{3}+\frac {1}{\left (-1+x \right )^{2}}-\frac {4 \,{\mathrm e}^{3}-3}{-1+x}-4 \ln \left (x \right ) {\mathrm e}^{3} x^{4}+\frac {2 \ln \left (x \right ) x}{-1+x}\) | \(84\) |
parts | \(x \ln \left (x \right )^{2}+2 x \ln \left (x \right )+x +4 \,{\mathrm e}^{6} x^{7}-4 x^{4} {\mathrm e}^{3}-4 x^{3} {\mathrm e}^{3}-4 x^{2} {\mathrm e}^{3}-4 x \,{\mathrm e}^{3}+\frac {1}{\left (-1+x \right )^{2}}-\frac {4 \,{\mathrm e}^{3}-3}{-1+x}-4 \ln \left (x \right ) {\mathrm e}^{3} x^{4}+\frac {2 \ln \left (x \right ) x}{-1+x}\) | \(84\) |
parallelrisch | \(\frac {4 \,{\mathrm e}^{6} x^{9}-8 \,{\mathrm e}^{6} x^{8}+4 \,{\mathrm e}^{6} x^{7}-4 \ln \left (x \right ) {\mathrm e}^{3} x^{6}-4 x^{6} {\mathrm e}^{3}+8 \ln \left (x \right ) {\mathrm e}^{3} x^{5}+4 x^{5} {\mathrm e}^{3}-4 \ln \left (x \right ) {\mathrm e}^{3} x^{4}+x^{3} \ln \left (x \right )^{2}+2 x^{3} \ln \left (x \right )-2 x^{2} \ln \left (x \right )^{2}+x^{3}-2 x^{2} \ln \left (x \right )+x \ln \left (x \right )^{2}}{x^{2}-2 x +1}\) | \(121\) |
risch | \(x \ln \left (x \right )^{2}-\frac {2 \left (2 x^{5} {\mathrm e}^{3}-2 x^{4} {\mathrm e}^{3}-x^{2}+x -1\right ) \ln \left (x \right )}{-1+x}+\frac {4 \,{\mathrm e}^{6} x^{9}-8 \,{\mathrm e}^{6} x^{8}+4 \,{\mathrm e}^{6} x^{7}-4 x^{6} {\mathrm e}^{3}+4 x^{5} {\mathrm e}^{3}+2 x^{2} \ln \left (x \right )+4 x^{2} {\mathrm e}^{3}+x^{3}-4 x \ln \left (x \right )-8 x \,{\mathrm e}^{3}-2 x^{2}+2 \ln \left (x \right )+4 \,{\mathrm e}^{3}+4 x -2}{\left (-1+x \right )^{2}}\) | \(125\) |
int(((x^3-3*x^2+3*x-1)*ln(x)^2+((-16*x^6+48*x^5-48*x^4+16*x^3)*exp(3)+4*x^ 3-12*x^2+10*x-2)*ln(x)+(28*x^9-84*x^8+84*x^7-28*x^6)*exp(3)^2+(-20*x^6+48* x^5-32*x^4+4*x^3)*exp(3)+3*x^3-7*x^2+2*x)/(x^3-3*x^2+3*x-1),x,method=_RETU RNVERBOSE)
x*ln(x)^2+2*x*ln(x)+x+4*exp(6)*x^7-4*x^4*exp(3)-4*x^3*exp(3)-4*x^2*exp(3)- 4*x*exp(3)+1/(-1+x)^2-(4*exp(3)-3)/(-1+x)-4*ln(x)*exp(3)*x^4+2*ln(x)*x/(-1 +x)
Leaf count of result is larger than twice the leaf count of optimal. 106 vs. \(2 (24) = 48\).
Time = 0.24 (sec) , antiderivative size = 106, normalized size of antiderivative = 4.24 \begin {dmath*} \int \frac {2 x-7 x^2+3 x^3+e^3 \left (4 x^3-32 x^4+48 x^5-20 x^6\right )+e^6 \left (-28 x^6+84 x^7-84 x^8+28 x^9\right )+\left (-2+10 x-12 x^2+4 x^3+e^3 \left (16 x^3-48 x^4+48 x^5-16 x^6\right )\right ) \log (x)+\left (-1+3 x-3 x^2+x^3\right ) \log ^2(x)}{-1+3 x-3 x^2+x^3} \, dx=\frac {x^{3} + {\left (x^{3} - 2 \, x^{2} + x\right )} \log \left (x\right )^{2} - 2 \, x^{2} + 4 \, {\left (x^{9} - 2 \, x^{8} + x^{7}\right )} e^{6} - 4 \, {\left (x^{6} - x^{5} - x^{2} + 2 \, x - 1\right )} e^{3} + 2 \, {\left (x^{3} - x^{2} - 2 \, {\left (x^{6} - 2 \, x^{5} + x^{4}\right )} e^{3}\right )} \log \left (x\right ) + 4 \, x - 2}{x^{2} - 2 \, x + 1} \end {dmath*}
integrate(((x^3-3*x^2+3*x-1)*log(x)^2+((-16*x^6+48*x^5-48*x^4+16*x^3)*exp( 3)+4*x^3-12*x^2+10*x-2)*log(x)+(28*x^9-84*x^8+84*x^7-28*x^6)*exp(3)^2+(-20 *x^6+48*x^5-32*x^4+4*x^3)*exp(3)+3*x^3-7*x^2+2*x)/(x^3-3*x^2+3*x-1),x, alg orithm=\
(x^3 + (x^3 - 2*x^2 + x)*log(x)^2 - 2*x^2 + 4*(x^9 - 2*x^8 + x^7)*e^6 - 4* (x^6 - x^5 - x^2 + 2*x - 1)*e^3 + 2*(x^3 - x^2 - 2*(x^6 - 2*x^5 + x^4)*e^3 )*log(x) + 4*x - 2)/(x^2 - 2*x + 1)
Leaf count of result is larger than twice the leaf count of optimal. 110 vs. \(2 (19) = 38\).
Time = 0.43 (sec) , antiderivative size = 110, normalized size of antiderivative = 4.40 \begin {dmath*} \int \frac {2 x-7 x^2+3 x^3+e^3 \left (4 x^3-32 x^4+48 x^5-20 x^6\right )+e^6 \left (-28 x^6+84 x^7-84 x^8+28 x^9\right )+\left (-2+10 x-12 x^2+4 x^3+e^3 \left (16 x^3-48 x^4+48 x^5-16 x^6\right )\right ) \log (x)+\left (-1+3 x-3 x^2+x^3\right ) \log ^2(x)}{-1+3 x-3 x^2+x^3} \, dx=4 x^{7} e^{6} - 4 x^{4} e^{3} - 4 x^{3} e^{3} - 4 x^{2} e^{3} + x \log {\left (x \right )}^{2} + x \left (1 - 4 e^{3}\right ) + 2 \log {\left (x \right )} + \frac {x \left (3 - 4 e^{3}\right ) - 2 + 4 e^{3}}{x^{2} - 2 x + 1} + \frac {\left (- 4 x^{5} e^{3} + 4 x^{4} e^{3} + 2 x^{2} - 2 x + 2\right ) \log {\left (x \right )}}{x - 1} \end {dmath*}
integrate(((x**3-3*x**2+3*x-1)*ln(x)**2+((-16*x**6+48*x**5-48*x**4+16*x**3 )*exp(3)+4*x**3-12*x**2+10*x-2)*ln(x)+(28*x**9-84*x**8+84*x**7-28*x**6)*ex p(3)**2+(-20*x**6+48*x**5-32*x**4+4*x**3)*exp(3)+3*x**3-7*x**2+2*x)/(x**3- 3*x**2+3*x-1),x)
4*x**7*exp(6) - 4*x**4*exp(3) - 4*x**3*exp(3) - 4*x**2*exp(3) + x*log(x)** 2 + x*(1 - 4*exp(3)) + 2*log(x) + (x*(3 - 4*exp(3)) - 2 + 4*exp(3))/(x**2 - 2*x + 1) + (-4*x**5*exp(3) + 4*x**4*exp(3) + 2*x**2 - 2*x + 2)*log(x)/(x - 1)
Leaf count of result is larger than twice the leaf count of optimal. 557 vs. \(2 (24) = 48\).
Time = 0.26 (sec) , antiderivative size = 557, normalized size of antiderivative = 22.28 \begin {dmath*} \int \frac {2 x-7 x^2+3 x^3+e^3 \left (4 x^3-32 x^4+48 x^5-20 x^6\right )+e^6 \left (-28 x^6+84 x^7-84 x^8+28 x^9\right )+\left (-2+10 x-12 x^2+4 x^3+e^3 \left (16 x^3-48 x^4+48 x^5-16 x^6\right )\right ) \log (x)+\left (-1+3 x-3 x^2+x^3\right ) \log ^2(x)}{-1+3 x-3 x^2+x^3} \, dx =\text {Too large to display} \end {dmath*}
integrate(((x^3-3*x^2+3*x-1)*log(x)^2+((-16*x^6+48*x^5-48*x^4+16*x^3)*exp( 3)+4*x^3-12*x^2+10*x-2)*log(x)+(28*x^9-84*x^8+84*x^7-28*x^6)*exp(3)^2+(-20 *x^6+48*x^5-32*x^4+4*x^3)*exp(3)+3*x^3-7*x^2+2*x)/(x^3-3*x^2+3*x-1),x, alg orithm=\
2/5*(10*x^7 + 35*x^6 + 84*x^5 + 175*x^4 + 350*x^3 + 735*x^2 + 1960*x - 35* (18*x - 17)/(x^2 - 2*x + 1) + 2520*log(x - 1))*e^6 - 14/5*(5*x^6 + 18*x^5 + 45*x^4 + 100*x^3 + 225*x^2 + 630*x - 15*(16*x - 15)/(x^2 - 2*x + 1) + 84 0*log(x - 1))*e^6 + 21/5*(4*x^5 + 15*x^4 + 40*x^3 + 100*x^2 + 300*x - 10*( 14*x - 13)/(x^2 - 2*x + 1) + 420*log(x - 1))*e^6 - 7*(x^4 + 4*x^3 + 12*x^2 + 40*x - 2*(12*x - 11)/(x^2 - 2*x + 1) + 60*log(x - 1))*e^6 - 5*(x^4 + 4* x^3 + 12*x^2 + 40*x - 2*(12*x - 11)/(x^2 - 2*x + 1) + 60*log(x - 1))*e^3 + 8*(2*x^3 + 9*x^2 + 36*x - 3*(10*x - 9)/(x^2 - 2*x + 1) + 60*log(x - 1))*e ^3 - 16*(x^2 + 6*x - (8*x - 7)/(x^2 - 2*x + 1) + 12*log(x - 1))*e^3 + 2*(2 *x - (6*x - 5)/(x^2 - 2*x + 1) + 6*log(x - 1))*e^3 + 3*x - 5*(2*x - 1)*log (x)/(x^2 - 2*x + 1) + (x^6*e^3 - 2*x^5*e^3 + x^4*e^3 - 2*x^3 + (x^3 - 2*x^ 2 + x)*log(x)^2 + 4*x^2 - 2*(2*x^6*e^3 - 4*x^5*e^3 + 2*x^4*e^3 - x^3 - 2*x ^2 + x)*log(x) + 2*x - 4)/(x^2 - 2*x + 1) - 3/2*(6*x - 5)/(x^2 - 2*x + 1) + 7/2*(4*x - 3)/(x^2 - 2*x + 1) - (2*x - 1)/(x^2 - 2*x + 1) + log(x)/(x^2 - 2*x + 1) - 4/(x - 1) - 6*log(x)
Leaf count of result is larger than twice the leaf count of optimal. 139 vs. \(2 (24) = 48\).
Time = 0.27 (sec) , antiderivative size = 139, normalized size of antiderivative = 5.56 \begin {dmath*} \int \frac {2 x-7 x^2+3 x^3+e^3 \left (4 x^3-32 x^4+48 x^5-20 x^6\right )+e^6 \left (-28 x^6+84 x^7-84 x^8+28 x^9\right )+\left (-2+10 x-12 x^2+4 x^3+e^3 \left (16 x^3-48 x^4+48 x^5-16 x^6\right )\right ) \log (x)+\left (-1+3 x-3 x^2+x^3\right ) \log ^2(x)}{-1+3 x-3 x^2+x^3} \, dx=\frac {4 \, x^{9} e^{6} - 8 \, x^{8} e^{6} + 4 \, x^{7} e^{6} - 4 \, x^{6} e^{3} \log \left (x\right ) - 4 \, x^{6} e^{3} + 8 \, x^{5} e^{3} \log \left (x\right ) + 4 \, x^{5} e^{3} - 4 \, x^{4} e^{3} \log \left (x\right ) + x^{3} \log \left (x\right )^{2} + 2 \, x^{3} \log \left (x\right ) - 2 \, x^{2} \log \left (x\right )^{2} + x^{3} + 4 \, x^{2} e^{3} - 2 \, x^{2} \log \left (x\right ) + x \log \left (x\right )^{2} - 2 \, x^{2} - 8 \, x e^{3} + 4 \, x + 4 \, e^{3} - 2}{x^{2} - 2 \, x + 1} \end {dmath*}
integrate(((x^3-3*x^2+3*x-1)*log(x)^2+((-16*x^6+48*x^5-48*x^4+16*x^3)*exp( 3)+4*x^3-12*x^2+10*x-2)*log(x)+(28*x^9-84*x^8+84*x^7-28*x^6)*exp(3)^2+(-20 *x^6+48*x^5-32*x^4+4*x^3)*exp(3)+3*x^3-7*x^2+2*x)/(x^3-3*x^2+3*x-1),x, alg orithm=\
(4*x^9*e^6 - 8*x^8*e^6 + 4*x^7*e^6 - 4*x^6*e^3*log(x) - 4*x^6*e^3 + 8*x^5* e^3*log(x) + 4*x^5*e^3 - 4*x^4*e^3*log(x) + x^3*log(x)^2 + 2*x^3*log(x) - 2*x^2*log(x)^2 + x^3 + 4*x^2*e^3 - 2*x^2*log(x) + x*log(x)^2 - 2*x^2 - 8*x *e^3 + 4*x + 4*e^3 - 2)/(x^2 - 2*x + 1)
Time = 14.11 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32 \begin {dmath*} \int \frac {2 x-7 x^2+3 x^3+e^3 \left (4 x^3-32 x^4+48 x^5-20 x^6\right )+e^6 \left (-28 x^6+84 x^7-84 x^8+28 x^9\right )+\left (-2+10 x-12 x^2+4 x^3+e^3 \left (16 x^3-48 x^4+48 x^5-16 x^6\right )\right ) \log (x)+\left (-1+3 x-3 x^2+x^3\right ) \log ^2(x)}{-1+3 x-3 x^2+x^3} \, dx=\frac {x\,{\left (x-\ln \left (x\right )+2\,x^3\,{\mathrm {e}}^3-2\,x^4\,{\mathrm {e}}^3+x\,\ln \left (x\right )\right )}^2}{{\left (x-1\right )}^2} \end {dmath*}