Integrand size = 110, antiderivative size = 23 \begin {dmath*} \int \frac {1953125 x^3-1953125 x^4+781250 x^5-156250 x^6+15625 x^7-625 x^8+e^{\frac {\log ^4(3)}{390625 x^2-312500 x^3+93750 x^4-12500 x^5+625 x^6}} (10-6 x) \log ^4(3)}{-1953125 x^3+1953125 x^4-781250 x^5+156250 x^6-15625 x^7+625 x^8} \, dx=-4+e^{\frac {\log ^4(3)}{625 (-5+x)^4 x^2}}-x \end {dmath*}
Time = 0.17 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \begin {dmath*} \int \frac {1953125 x^3-1953125 x^4+781250 x^5-156250 x^6+15625 x^7-625 x^8+e^{\frac {\log ^4(3)}{390625 x^2-312500 x^3+93750 x^4-12500 x^5+625 x^6}} (10-6 x) \log ^4(3)}{-1953125 x^3+1953125 x^4-781250 x^5+156250 x^6-15625 x^7+625 x^8} \, dx=e^{\frac {\log ^4(3)}{625 (-5+x)^4 x^2}}-x \end {dmath*}
Integrate[(1953125*x^3 - 1953125*x^4 + 781250*x^5 - 156250*x^6 + 15625*x^7 - 625*x^8 + E^(Log[3]^4/(390625*x^2 - 312500*x^3 + 93750*x^4 - 12500*x^5 + 625*x^6))*(10 - 6*x)*Log[3]^4)/(-1953125*x^3 + 1953125*x^4 - 781250*x^5 + 156250*x^6 - 15625*x^7 + 625*x^8),x]
Leaf count is larger than twice the leaf count of optimal. \(76\) vs. \(2(23)=46\).
Time = 1.40 (sec) , antiderivative size = 76, normalized size of antiderivative = 3.30, 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.
\(\displaystyle \int \frac {-625 x^8+15625 x^7-156250 x^6+781250 x^5-1953125 x^4+1953125 x^3+(10-6 x) \log ^4(3) e^{\frac {\log ^4(3)}{625 x^6-12500 x^5+93750 x^4-312500 x^3+390625 x^2}}}{625 x^8-15625 x^7+156250 x^6-781250 x^5+1953125 x^4-1953125 x^3} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {-625 x^8+15625 x^7-156250 x^6+781250 x^5-1953125 x^4+1953125 x^3+(10-6 x) \log ^4(3) e^{\frac {\log ^4(3)}{625 x^6-12500 x^5+93750 x^4-312500 x^3+390625 x^2}}}{x^3 \left (625 x^5-15625 x^4+156250 x^3-781250 x^2+1953125 x-1953125\right )}dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {-625 x^8+15625 x^7-156250 x^6+781250 x^5-1953125 x^4+1953125 x^3+(10-6 x) \log ^4(3) e^{\frac {\log ^4(3)}{625 x^6-12500 x^5+93750 x^4-312500 x^3+390625 x^2}}}{x^3 \left (5^{4/5} x-5\ 5^{4/5}\right )^5}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {x^5}{(x-5)^5}+\frac {25 x^4}{(x-5)^5}-\frac {250 x^3}{(x-5)^5}+\frac {1250 x^2}{(x-5)^5}-\frac {2 (3 x-5) \log ^4(3) e^{\frac {\log ^4(3)}{625 (x-5)^4 x^2}}}{625 (x-5)^5 x^3}-\frac {3125 x}{(x-5)^5}+\frac {3125}{(x-5)^5}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {25 x^4}{2 (5-x)^4}+e^{\frac {\log ^4(3)}{625 (5-x)^4 x^2}}-x+\frac {250}{5-x}-\frac {1875}{(5-x)^2}+\frac {6250}{(5-x)^3}-\frac {15625}{2 (5-x)^4}\) |
Int[(1953125*x^3 - 1953125*x^4 + 781250*x^5 - 156250*x^6 + 15625*x^7 - 625 *x^8 + E^(Log[3]^4/(390625*x^2 - 312500*x^3 + 93750*x^4 - 12500*x^5 + 625* x^6))*(10 - 6*x)*Log[3]^4)/(-1953125*x^3 + 1953125*x^4 - 781250*x^5 + 1562 50*x^6 - 15625*x^7 + 625*x^8),x]
E^(Log[3]^4/(625*(5 - x)^4*x^2)) - 15625/(2*(5 - x)^4) + 6250/(5 - x)^3 - 1875/(5 - x)^2 + 250/(5 - x) - x + (25*x^4)/(2*(5 - x)^4)
3.13.52.3.1 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]
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 = 1.13 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87
method | result | size |
risch | \(-x +{\mathrm e}^{\frac {\ln \left (3\right )^{4}}{625 x^{2} \left (-5+x \right )^{4}}}\) | \(20\) |
parallelrisch | \(-x +{\mathrm e}^{\frac {\ln \left (3\right )^{4}}{625 x^{2} \left (x^{4}-20 x^{3}+150 x^{2}-500 x +625\right )}}-50\) | \(36\) |
parts | \(-x +\frac {x^{6} {\mathrm e}^{\frac {\ln \left (3\right )^{4}}{625 x^{6}-12500 x^{5}+93750 x^{4}-312500 x^{3}+390625 x^{2}}}+625 x^{2} {\mathrm e}^{\frac {\ln \left (3\right )^{4}}{625 x^{6}-12500 x^{5}+93750 x^{4}-312500 x^{3}+390625 x^{2}}}-500 x^{3} {\mathrm e}^{\frac {\ln \left (3\right )^{4}}{625 x^{6}-12500 x^{5}+93750 x^{4}-312500 x^{3}+390625 x^{2}}}+150 x^{4} {\mathrm e}^{\frac {\ln \left (3\right )^{4}}{625 x^{6}-12500 x^{5}+93750 x^{4}-312500 x^{3}+390625 x^{2}}}-20 x^{5} {\mathrm e}^{\frac {\ln \left (3\right )^{4}}{625 x^{6}-12500 x^{5}+93750 x^{4}-312500 x^{3}+390625 x^{2}}}}{x^{2} \left (-5+x \right )^{4}}\) | \(209\) |
int(((-6*x+10)*ln(3)^4*exp(ln(3)^4/(625*x^6-12500*x^5+93750*x^4-312500*x^3 +390625*x^2))-625*x^8+15625*x^7-156250*x^6+781250*x^5-1953125*x^4+1953125* x^3)/(625*x^8-15625*x^7+156250*x^6-781250*x^5+1953125*x^4-1953125*x^3),x,m ethod=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.61 \begin {dmath*} \int \frac {1953125 x^3-1953125 x^4+781250 x^5-156250 x^6+15625 x^7-625 x^8+e^{\frac {\log ^4(3)}{390625 x^2-312500 x^3+93750 x^4-12500 x^5+625 x^6}} (10-6 x) \log ^4(3)}{-1953125 x^3+1953125 x^4-781250 x^5+156250 x^6-15625 x^7+625 x^8} \, dx=-x + e^{\left (\frac {\log \left (3\right )^{4}}{625 \, {\left (x^{6} - 20 \, x^{5} + 150 \, x^{4} - 500 \, x^{3} + 625 \, x^{2}\right )}}\right )} \end {dmath*}
integrate(((-6*x+10)*log(3)^4*exp(log(3)^4/(625*x^6-12500*x^5+93750*x^4-31 2500*x^3+390625*x^2))-625*x^8+15625*x^7-156250*x^6+781250*x^5-1953125*x^4+ 1953125*x^3)/(625*x^8-15625*x^7+156250*x^6-781250*x^5+1953125*x^4-1953125* x^3),x, algorithm=\
Time = 0.27 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.39 \begin {dmath*} \int \frac {1953125 x^3-1953125 x^4+781250 x^5-156250 x^6+15625 x^7-625 x^8+e^{\frac {\log ^4(3)}{390625 x^2-312500 x^3+93750 x^4-12500 x^5+625 x^6}} (10-6 x) \log ^4(3)}{-1953125 x^3+1953125 x^4-781250 x^5+156250 x^6-15625 x^7+625 x^8} \, dx=- x + e^{\frac {\log {\left (3 \right )}^{4}}{625 x^{6} - 12500 x^{5} + 93750 x^{4} - 312500 x^{3} + 390625 x^{2}}} \end {dmath*}
integrate(((-6*x+10)*ln(3)**4*exp(ln(3)**4/(625*x**6-12500*x**5+93750*x**4 -312500*x**3+390625*x**2))-625*x**8+15625*x**7-156250*x**6+781250*x**5-195 3125*x**4+1953125*x**3)/(625*x**8-15625*x**7+156250*x**6-781250*x**5+19531 25*x**4-1953125*x**3),x)
Leaf count of result is larger than twice the leaf count of optimal. 290 vs. \(2 (20) = 40\).
Time = 0.44 (sec) , antiderivative size = 290, normalized size of antiderivative = 12.61 \begin {dmath*} \int \frac {1953125 x^3-1953125 x^4+781250 x^5-156250 x^6+15625 x^7-625 x^8+e^{\frac {\log ^4(3)}{390625 x^2-312500 x^3+93750 x^4-12500 x^5+625 x^6}} (10-6 x) \log ^4(3)}{-1953125 x^3+1953125 x^4-781250 x^5+156250 x^6-15625 x^7+625 x^8} \, dx=-x - \frac {125 \, {\left (48 \, x^{3} - 540 \, x^{2} + 2200 \, x - 3125\right )}}{12 \, {\left (x^{4} - 20 \, x^{3} + 150 \, x^{2} - 500 \, x + 625\right )}} + \frac {125 \, {\left (24 \, x^{3} - 300 \, x^{2} + 1300 \, x - 1925\right )}}{12 \, {\left (x^{4} - 20 \, x^{3} + 150 \, x^{2} - 500 \, x + 625\right )}} + \frac {125 \, {\left (4 \, x^{3} - 30 \, x^{2} + 100 \, x - 125\right )}}{2 \, {\left (x^{4} - 20 \, x^{3} + 150 \, x^{2} - 500 \, x + 625\right )}} - \frac {625 \, {\left (6 \, x^{2} - 20 \, x + 25\right )}}{6 \, {\left (x^{4} - 20 \, x^{3} + 150 \, x^{2} - 500 \, x + 625\right )}} + \frac {3125 \, {\left (4 \, x - 5\right )}}{12 \, {\left (x^{4} - 20 \, x^{3} + 150 \, x^{2} - 500 \, x + 625\right )}} - \frac {3125}{4 \, {\left (x^{4} - 20 \, x^{3} + 150 \, x^{2} - 500 \, x + 625\right )}} + e^{\left (\frac {\log \left (3\right )^{4}}{15625 \, {\left (x^{4} - 20 \, x^{3} + 150 \, x^{2} - 500 \, x + 625\right )}} - \frac {2 \, \log \left (3\right )^{4}}{78125 \, {\left (x^{3} - 15 \, x^{2} + 75 \, x - 125\right )}} + \frac {3 \, \log \left (3\right )^{4}}{390625 \, {\left (x^{2} - 10 \, x + 25\right )}} - \frac {4 \, \log \left (3\right )^{4}}{1953125 \, {\left (x - 5\right )}} + \frac {4 \, \log \left (3\right )^{4}}{1953125 \, x} + \frac {\log \left (3\right )^{4}}{390625 \, x^{2}}\right )} \end {dmath*}
integrate(((-6*x+10)*log(3)^4*exp(log(3)^4/(625*x^6-12500*x^5+93750*x^4-31 2500*x^3+390625*x^2))-625*x^8+15625*x^7-156250*x^6+781250*x^5-1953125*x^4+ 1953125*x^3)/(625*x^8-15625*x^7+156250*x^6-781250*x^5+1953125*x^4-1953125* x^3),x, algorithm=\
-x - 125/12*(48*x^3 - 540*x^2 + 2200*x - 3125)/(x^4 - 20*x^3 + 150*x^2 - 5 00*x + 625) + 125/12*(24*x^3 - 300*x^2 + 1300*x - 1925)/(x^4 - 20*x^3 + 15 0*x^2 - 500*x + 625) + 125/2*(4*x^3 - 30*x^2 + 100*x - 125)/(x^4 - 20*x^3 + 150*x^2 - 500*x + 625) - 625/6*(6*x^2 - 20*x + 25)/(x^4 - 20*x^3 + 150*x ^2 - 500*x + 625) + 3125/12*(4*x - 5)/(x^4 - 20*x^3 + 150*x^2 - 500*x + 62 5) - 3125/4/(x^4 - 20*x^3 + 150*x^2 - 500*x + 625) + e^(1/15625*log(3)^4/( x^4 - 20*x^3 + 150*x^2 - 500*x + 625) - 2/78125*log(3)^4/(x^3 - 15*x^2 + 7 5*x - 125) + 3/390625*log(3)^4/(x^2 - 10*x + 25) - 4/1953125*log(3)^4/(x - 5) + 4/1953125*log(3)^4/x + 1/390625*log(3)^4/x^2)
Time = 0.31 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.61 \begin {dmath*} \int \frac {1953125 x^3-1953125 x^4+781250 x^5-156250 x^6+15625 x^7-625 x^8+e^{\frac {\log ^4(3)}{390625 x^2-312500 x^3+93750 x^4-12500 x^5+625 x^6}} (10-6 x) \log ^4(3)}{-1953125 x^3+1953125 x^4-781250 x^5+156250 x^6-15625 x^7+625 x^8} \, dx=-x + e^{\left (\frac {\log \left (3\right )^{4}}{625 \, {\left (x^{6} - 20 \, x^{5} + 150 \, x^{4} - 500 \, x^{3} + 625 \, x^{2}\right )}}\right )} \end {dmath*}
integrate(((-6*x+10)*log(3)^4*exp(log(3)^4/(625*x^6-12500*x^5+93750*x^4-31 2500*x^3+390625*x^2))-625*x^8+15625*x^7-156250*x^6+781250*x^5-1953125*x^4+ 1953125*x^3)/(625*x^8-15625*x^7+156250*x^6-781250*x^5+1953125*x^4-1953125* x^3),x, algorithm=\
Time = 14.78 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.65 \begin {dmath*} \int \frac {1953125 x^3-1953125 x^4+781250 x^5-156250 x^6+15625 x^7-625 x^8+e^{\frac {\log ^4(3)}{390625 x^2-312500 x^3+93750 x^4-12500 x^5+625 x^6}} (10-6 x) \log ^4(3)}{-1953125 x^3+1953125 x^4-781250 x^5+156250 x^6-15625 x^7+625 x^8} \, dx={\mathrm {e}}^{\frac {{\ln \left (3\right )}^4}{625\,x^6-12500\,x^5+93750\,x^4-312500\,x^3+390625\,x^2}}-x \end {dmath*}