Integrand size = 104, antiderivative size = 22 \[ \int \frac {-45 x^2-3 x^3+\left (30 x+6 x^2\right ) \log (2)+e^{2 x} \left (-15-39 x+6 x^2+(42-6 x) \log (2)\right )+e^x \left (-60 x-42 x^2+6 x^3+\left (30+48 x-6 x^2\right ) \log (2)\right )}{-3125+3125 x-1250 x^2+250 x^3-25 x^4+x^5} \, dx=\frac {3 \left (e^x+x\right )^2 (x-\log (2))}{(5-x)^4} \] Output:
(exp(x)+x)^2*(3*x-3*ln(2))/(5-x)^4
Leaf count is larger than twice the leaf count of optimal. \(51\) vs. \(2(22)=44\).
Time = 1.42 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.32 \[ \int \frac {-45 x^2-3 x^3+\left (30 x+6 x^2\right ) \log (2)+e^{2 x} \left (-15-39 x+6 x^2+(42-6 x) \log (2)\right )+e^x \left (-60 x-42 x^2+6 x^3+\left (30+48 x-6 x^2\right ) \log (2)\right )}{-3125+3125 x-1250 x^2+250 x^3-25 x^4+x^5} \, dx=\frac {8 e^x x (6 x-\log (64))+2 x^2 (12 x-\log (4096))+e^{2 x} (24 x-\log (16777216))}{8 (-5+x)^4} \] Input:
Integrate[(-45*x^2 - 3*x^3 + (30*x + 6*x^2)*Log[2] + E^(2*x)*(-15 - 39*x + 6*x^2 + (42 - 6*x)*Log[2]) + E^x*(-60*x - 42*x^2 + 6*x^3 + (30 + 48*x - 6 *x^2)*Log[2]))/(-3125 + 3125*x - 1250*x^2 + 250*x^3 - 25*x^4 + x^5),x]
Output:
(8*E^x*x*(6*x - Log[64]) + 2*x^2*(12*x - Log[4096]) + E^(2*x)*(24*x - Log[ 16777216]))/(8*(-5 + x)^4)
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 1.71 (sec) , antiderivative size = 575, normalized size of antiderivative = 26.14, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2007, 7292, 27, 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.
| \(\displaystyle \int \frac {-3 x^3-45 x^2+e^{2 x} \left (6 x^2-39 x+(42-6 x) \log (2)-15\right )+\left (6 x^2+30 x\right ) \log (2)+e^x \left (6 x^3-42 x^2+\left (-6 x^2+48 x+30\right ) \log (2)-60 x\right )}{x^5-25 x^4+250 x^3-1250 x^2+3125 x-3125} \, dx\) |
| \(\Big \downarrow \) 2007 |
| \(\displaystyle \int \frac {-3 x^3-45 x^2+e^{2 x} \left (6 x^2-39 x+(42-6 x) \log (2)-15\right )+\left (6 x^2+30 x\right ) \log (2)+e^x \left (6 x^3-42 x^2+\left (-6 x^2+48 x+30\right ) \log (2)-60 x\right )}{(x-5)^5}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {3 \left (x+e^x\right ) \left (-2 e^x x^2+x^2+13 e^x x \left (1+\frac {2 \log (2)}{13}\right )+15 x \left (1-\frac {2 \log (2)}{15}\right )+5 e^x \left (1-\frac {14 \log (2)}{5}\right )-\log (1024)\right )}{(5-x)^5}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 3 \int \frac {\left (x+e^x\right ) \left (-2 e^x x^2+x^2+e^x (13+\log (4)) x+(15-\log (4)) x-\log (1024)+e^x (5-14 \log (2))\right )}{(5-x)^5}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 3 \int \left (-\frac {x^3}{(x-5)^5}+\frac {(-15+\log (4)) x^2}{(x-5)^5}+\frac {\log (1024) x}{(x-5)^5}+\frac {e^{2 x} \left (-2 x^2+(13+\log (4)) x-14 \log (2)+5\right )}{(5-x)^5}+\frac {e^x \left (-2 x^3+(14+\log (4)) x^2+(20-\log (65536)) x-\log (1024)\right )}{(5-x)^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 3 \left (4 e^{10} \operatorname {ExpIntegralEi}(-2 (5-x))+2 e^5 \operatorname {ExpIntegralEi}(x-5)-\frac {1}{24} e^5 (200-\log (1099511627776)) \operatorname {ExpIntegralEi}(x-5)-\frac {1}{6} e^5 (10+\log (16)) \operatorname {ExpIntegralEi}(x-5)-\frac {2}{3} e^{10} (20-\log (16)) \operatorname {ExpIntegralEi}(-2 (5-x))+\frac {1}{2} e^5 (16-\log (4)) \operatorname {ExpIntegralEi}(x-5)+\frac {4}{3} e^{10} (7-\log (4)) \operatorname {ExpIntegralEi}(-2 (5-x))+\frac {x^4}{20 (5-x)^4}+\frac {2 e^x}{5-x}+\frac {2 e^{2 x}}{5-x}-\frac {e^{2 x}}{(5-x)^2}-\frac {e^x (200-\log (1099511627776))}{24 (5-x)}+\frac {e^x (200-\log (1099511627776))}{24 (5-x)^2}-\frac {e^x (200-\log (1099511627776))}{12 (5-x)^3}+\frac {e^x (200-\log (1099511627776))}{4 (5-x)^4}+\frac {\log (1024)}{3 (5-x)^3}-\frac {5 \log (1024)}{4 (5-x)^4}-\frac {e^x (10+\log (16))}{6 (5-x)}+\frac {e^x (10+\log (16))}{6 (5-x)^2}-\frac {e^x (10+\log (16))}{3 (5-x)^3}-\frac {e^{2 x} (20-\log (16))}{3 (5-x)}+\frac {e^{2 x} (20-\log (16))}{6 (5-x)^2}-\frac {e^{2 x} (20-\log (16))}{6 (5-x)^3}+\frac {e^{2 x} (20-\log (16))}{4 (5-x)^4}+\frac {e^x (16-\log (4))}{2 (5-x)}-\frac {e^x (16-\log (4))}{2 (5-x)^2}+\frac {15-\log (4)}{2 (5-x)^2}-\frac {10 (15-\log (4))}{3 (5-x)^3}+\frac {25 (15-\log (4))}{4 (5-x)^4}+\frac {2 e^{2 x} (7-\log (4))}{3 (5-x)}-\frac {e^{2 x} (7-\log (4))}{3 (5-x)^2}+\frac {e^{2 x} (7-\log (4))}{3 (5-x)^3}\right )\) |
Input:
Int[(-45*x^2 - 3*x^3 + (30*x + 6*x^2)*Log[2] + E^(2*x)*(-15 - 39*x + 6*x^2 + (42 - 6*x)*Log[2]) + E^x*(-60*x - 42*x^2 + 6*x^3 + (30 + 48*x - 6*x^2)* Log[2]))/(-3125 + 3125*x - 1250*x^2 + 250*x^3 - 25*x^4 + x^5),x]
Output:
3*(-(E^(2*x)/(5 - x)^2) + (2*E^x)/(5 - x) + (2*E^(2*x))/(5 - x) + x^4/(20* (5 - x)^4) + 4*E^10*ExpIntegralEi[-2*(5 - x)] + 2*E^5*ExpIntegralEi[-5 + x ] + (E^(2*x)*(7 - Log[4]))/(3*(5 - x)^3) - (E^(2*x)*(7 - Log[4]))/(3*(5 - x)^2) + (2*E^(2*x)*(7 - Log[4]))/(3*(5 - x)) + (4*E^10*ExpIntegralEi[-2*(5 - x)]*(7 - Log[4]))/3 + (25*(15 - Log[4]))/(4*(5 - x)^4) - (10*(15 - Log[ 4]))/(3*(5 - x)^3) + (15 - Log[4])/(2*(5 - x)^2) - (E^x*(16 - Log[4]))/(2* (5 - x)^2) + (E^x*(16 - Log[4]))/(2*(5 - x)) + (E^5*ExpIntegralEi[-5 + x]* (16 - Log[4]))/2 + (E^(2*x)*(20 - Log[16]))/(4*(5 - x)^4) - (E^(2*x)*(20 - Log[16]))/(6*(5 - x)^3) + (E^(2*x)*(20 - Log[16]))/(6*(5 - x)^2) - (E^(2* x)*(20 - Log[16]))/(3*(5 - x)) - (2*E^10*ExpIntegralEi[-2*(5 - x)]*(20 - L og[16]))/3 - (E^x*(10 + Log[16]))/(3*(5 - x)^3) + (E^x*(10 + Log[16]))/(6* (5 - x)^2) - (E^x*(10 + Log[16]))/(6*(5 - x)) - (E^5*ExpIntegralEi[-5 + x] *(10 + Log[16]))/6 - (5*Log[1024])/(4*(5 - x)^4) + Log[1024]/(3*(5 - x)^3) + (E^x*(200 - Log[1099511627776]))/(4*(5 - x)^4) - (E^x*(200 - Log[109951 1627776]))/(12*(5 - x)^3) + (E^x*(200 - Log[1099511627776]))/(24*(5 - x)^2 ) - (E^x*(200 - Log[1099511627776]))/(24*(5 - x)) - (E^5*ExpIntegralEi[-5 + x]*(200 - Log[1099511627776]))/24)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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. \(48\) vs. \(2(22)=44\).
Time = 1.47 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.23
| method | result | size |
| norman | \(\frac {3 x^{3}-3 x^{2} \ln \left (2\right )+3 x \,{\mathrm e}^{2 x}+6 \,{\mathrm e}^{x} x^{2}-3 \ln \left (2\right ) {\mathrm e}^{2 x}-6 x \ln \left (2\right ) {\mathrm e}^{x}}{\left (-5+x \right )^{4}}\) | \(49\) |
| parallelrisch | \(-\frac {3 x^{2} \ln \left (2\right )+6 x \ln \left (2\right ) {\mathrm e}^{x}+3 \ln \left (2\right ) {\mathrm e}^{2 x}-3 x^{3}-6 \,{\mathrm e}^{x} x^{2}-3 x \,{\mathrm e}^{2 x}}{x^{4}-20 x^{3}+150 x^{2}-500 x +625}\) | \(65\) |
| risch | \(\frac {-3 x^{2} \ln \left (2\right )+3 x^{3}}{x^{4}-20 x^{3}+150 x^{2}-500 x +625}-\frac {3 \left (\ln \left (2\right )-x \right ) {\mathrm e}^{2 x}}{\left (-5+x \right )^{4}}-\frac {6 x \left (\ln \left (2\right )-x \right ) {\mathrm e}^{x}}{\left (-5+x \right )^{4}}\) | \(69\) |
| parts | \(-\frac {3 \left (2 \ln \left (2\right )-30\right )}{2 \left (-5+x \right )^{2}}+\frac {3}{-5+x}-\frac {3 \left (100 \ln \left (2\right )-500\right )}{4 \left (-5+x \right )^{4}}-\frac {30 \ln \left (2\right )-225}{\left (-5+x \right )^{3}}+\frac {15 \,{\mathrm e}^{2 x}}{\left (-5+x \right )^{4}}+\frac {3 \,{\mathrm e}^{2 x}}{\left (-5+x \right )^{3}}-\frac {3 \ln \left (2\right ) {\mathrm e}^{2 x}}{\left (-5+x \right )^{4}}-\frac {30 \ln \left (2\right ) {\mathrm e}^{x}}{\left (-5+x \right )^{4}}-\frac {6 \ln \left (2\right ) {\mathrm e}^{x}}{\left (-5+x \right )^{3}}+\frac {150 \,{\mathrm e}^{x}}{\left (-5+x \right )^{4}}+\frac {60 \,{\mathrm e}^{x}}{\left (-5+x \right )^{3}}+\frac {6 \,{\mathrm e}^{x}}{\left (-5+x \right )^{2}}\) | \(132\) |
| default | \(\frac {45}{\left (-5+x \right )^{2}}+\frac {375}{\left (-5+x \right )^{4}}+\frac {225}{\left (-5+x \right )^{3}}+\frac {3}{-5+x}+\frac {15 \,{\mathrm e}^{2 x}}{\left (-5+x \right )^{4}}+\frac {3 \,{\mathrm e}^{2 x}}{\left (-5+x \right )^{3}}-\frac {75 \ln \left (2\right )}{2 \left (-5+x \right )^{4}}-\frac {10 \ln \left (2\right )}{\left (-5+x \right )^{3}}+6 \ln \left (2\right ) \left (-\frac {1}{2 \left (-5+x \right )^{2}}-\frac {25}{4 \left (-5+x \right )^{4}}-\frac {10}{3 \left (-5+x \right )^{3}}\right )+\frac {150 \,{\mathrm e}^{x}}{\left (-5+x \right )^{4}}+\frac {60 \,{\mathrm e}^{x}}{\left (-5+x \right )^{3}}+\frac {6 \,{\mathrm e}^{x}}{\left (-5+x \right )^{2}}+30 \ln \left (2\right ) \left (-\frac {{\mathrm e}^{x}}{4 \left (-5+x \right )^{4}}-\frac {{\mathrm e}^{x}}{12 \left (-5+x \right )^{3}}-\frac {{\mathrm e}^{x}}{24 \left (-5+x \right )^{2}}-\frac {{\mathrm e}^{x}}{24 \left (-5+x \right )}-\frac {{\mathrm e}^{5} \operatorname {expIntegral}_{1}\left (5-x \right )}{24}\right )+42 \ln \left (2\right ) \left (-\frac {{\mathrm e}^{2 x}}{4 \left (-5+x \right )^{4}}-\frac {{\mathrm e}^{2 x}}{6 \left (-5+x \right )^{3}}-\frac {{\mathrm e}^{2 x}}{6 \left (-5+x \right )^{2}}-\frac {{\mathrm e}^{2 x}}{3 \left (-5+x \right )}-\frac {2 \,{\mathrm e}^{10} \operatorname {expIntegral}_{1}\left (-2 x +10\right )}{3}\right )+48 \ln \left (2\right ) \left (-\frac {5 \,{\mathrm e}^{x}}{4 \left (-5+x \right )^{4}}-\frac {3 \,{\mathrm e}^{x}}{4 \left (-5+x \right )^{3}}-\frac {3 \,{\mathrm e}^{x}}{8 \left (-5+x \right )^{2}}-\frac {3 \,{\mathrm e}^{x}}{8 \left (-5+x \right )}-\frac {3 \,{\mathrm e}^{5} \operatorname {expIntegral}_{1}\left (5-x \right )}{8}\right )-6 \ln \left (2\right ) \left (-\frac {77 \,{\mathrm e}^{x}}{24 \left (-5+x \right )^{2}}-\frac {77 \,{\mathrm e}^{x}}{24 \left (-5+x \right )}-\frac {77 \,{\mathrm e}^{5} \operatorname {expIntegral}_{1}\left (5-x \right )}{24}-\frac {25 \,{\mathrm e}^{x}}{4 \left (-5+x \right )^{4}}-\frac {65 \,{\mathrm e}^{x}}{12 \left (-5+x \right )^{3}}\right )-6 \ln \left (2\right ) \left (-\frac {5 \,{\mathrm e}^{2 x}}{4 \left (-5+x \right )^{4}}-\frac {7 \,{\mathrm e}^{2 x}}{6 \left (-5+x \right )^{3}}-\frac {7 \,{\mathrm e}^{2 x}}{6 \left (-5+x \right )^{2}}-\frac {7 \,{\mathrm e}^{2 x}}{3 \left (-5+x \right )}-\frac {14 \,{\mathrm e}^{10} \operatorname {expIntegral}_{1}\left (-2 x +10\right )}{3}\right )\) | \(399\) |
Input:
int((((-6*x+42)*ln(2)+6*x^2-39*x-15)*exp(x)^2+((-6*x^2+48*x+30)*ln(2)+6*x^ 3-42*x^2-60*x)*exp(x)+(6*x^2+30*x)*ln(2)-3*x^3-45*x^2)/(x^5-25*x^4+250*x^3 -1250*x^2+3125*x-3125),x,method=_RETURNVERBOSE)
Output:
(3*x^3-3*x^2*ln(2)+3*x*exp(x)^2+6*exp(x)*x^2-3*ln(2)*exp(x)^2-6*x*ln(2)*ex p(x))/(-5+x)^4
Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (19) = 38\).
Time = 0.09 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.59 \[ \int \frac {-45 x^2-3 x^3+\left (30 x+6 x^2\right ) \log (2)+e^{2 x} \left (-15-39 x+6 x^2+(42-6 x) \log (2)\right )+e^x \left (-60 x-42 x^2+6 x^3+\left (30+48 x-6 x^2\right ) \log (2)\right )}{-3125+3125 x-1250 x^2+250 x^3-25 x^4+x^5} \, dx=\frac {3 \, {\left (x^{3} - x^{2} \log \left (2\right ) + {\left (x - \log \left (2\right )\right )} e^{\left (2 \, x\right )} + 2 \, {\left (x^{2} - x \log \left (2\right )\right )} e^{x}\right )}}{x^{4} - 20 \, x^{3} + 150 \, x^{2} - 500 \, x + 625} \] Input:
integrate((((-6*x+42)*log(2)+6*x^2-39*x-15)*exp(x)^2+((-6*x^2+48*x+30)*log (2)+6*x^3-42*x^2-60*x)*exp(x)+(6*x^2+30*x)*log(2)-3*x^3-45*x^2)/(x^5-25*x^ 4+250*x^3-1250*x^2+3125*x-3125),x, algorithm="fricas")
Output:
3*(x^3 - x^2*log(2) + (x - log(2))*e^(2*x) + 2*(x^2 - x*log(2))*e^x)/(x^4 - 20*x^3 + 150*x^2 - 500*x + 625)
Leaf count of result is larger than twice the leaf count of optimal. 206 vs. \(2 (19) = 38\).
Time = 0.42 (sec) , antiderivative size = 206, normalized size of antiderivative = 9.36 \[ \int \frac {-45 x^2-3 x^3+\left (30 x+6 x^2\right ) \log (2)+e^{2 x} \left (-15-39 x+6 x^2+(42-6 x) \log (2)\right )+e^x \left (-60 x-42 x^2+6 x^3+\left (30+48 x-6 x^2\right ) \log (2)\right )}{-3125+3125 x-1250 x^2+250 x^3-25 x^4+x^5} \, dx=- \frac {- 3 x^{3} + 3 x^{2} \log {\left (2 \right )}}{x^{4} - 20 x^{3} + 150 x^{2} - 500 x + 625} + \frac {\left (3 x^{5} - 60 x^{4} - 3 x^{4} \log {\left (2 \right )} + 60 x^{3} \log {\left (2 \right )} + 450 x^{3} - 1500 x^{2} - 450 x^{2} \log {\left (2 \right )} + 1500 x \log {\left (2 \right )} + 1875 x - 1875 \log {\left (2 \right )}\right ) e^{2 x} + \left (6 x^{6} - 120 x^{5} - 6 x^{5} \log {\left (2 \right )} + 120 x^{4} \log {\left (2 \right )} + 900 x^{4} - 3000 x^{3} - 900 x^{3} \log {\left (2 \right )} + 3000 x^{2} \log {\left (2 \right )} + 3750 x^{2} - 3750 x \log {\left (2 \right )}\right ) e^{x}}{x^{8} - 40 x^{7} + 700 x^{6} - 7000 x^{5} + 43750 x^{4} - 175000 x^{3} + 437500 x^{2} - 625000 x + 390625} \] Input:
integrate((((-6*x+42)*ln(2)+6*x**2-39*x-15)*exp(x)**2+((-6*x**2+48*x+30)*l n(2)+6*x**3-42*x**2-60*x)*exp(x)+(6*x**2+30*x)*ln(2)-3*x**3-45*x**2)/(x**5 -25*x**4+250*x**3-1250*x**2+3125*x-3125),x)
Output:
-(-3*x**3 + 3*x**2*log(2))/(x**4 - 20*x**3 + 150*x**2 - 500*x + 625) + ((3 *x**5 - 60*x**4 - 3*x**4*log(2) + 60*x**3*log(2) + 450*x**3 - 1500*x**2 - 450*x**2*log(2) + 1500*x*log(2) + 1875*x - 1875*log(2))*exp(2*x) + (6*x**6 - 120*x**5 - 6*x**5*log(2) + 120*x**4*log(2) + 900*x**4 - 3000*x**3 - 900 *x**3*log(2) + 3000*x**2*log(2) + 3750*x**2 - 3750*x*log(2))*exp(x))/(x**8 - 40*x**7 + 700*x**6 - 7000*x**5 + 43750*x**4 - 175000*x**3 + 437500*x**2 - 625000*x + 390625)
\[ \int \frac {-45 x^2-3 x^3+\left (30 x+6 x^2\right ) \log (2)+e^{2 x} \left (-15-39 x+6 x^2+(42-6 x) \log (2)\right )+e^x \left (-60 x-42 x^2+6 x^3+\left (30+48 x-6 x^2\right ) \log (2)\right )}{-3125+3125 x-1250 x^2+250 x^3-25 x^4+x^5} \, dx=\int { -\frac {3 \, {\left (x^{3} + 15 \, x^{2} - {\left (2 \, x^{2} - 2 \, {\left (x - 7\right )} \log \left (2\right ) - 13 \, x - 5\right )} e^{\left (2 \, x\right )} - 2 \, {\left (x^{3} - 7 \, x^{2} - {\left (x^{2} - 8 \, x - 5\right )} \log \left (2\right ) - 10 \, x\right )} e^{x} - 2 \, {\left (x^{2} + 5 \, x\right )} \log \left (2\right )\right )}}{x^{5} - 25 \, x^{4} + 250 \, x^{3} - 1250 \, x^{2} + 3125 \, x - 3125} \,d x } \] Input:
integrate((((-6*x+42)*log(2)+6*x^2-39*x-15)*exp(x)^2+((-6*x^2+48*x+30)*log (2)+6*x^3-42*x^2-60*x)*exp(x)+(6*x^2+30*x)*log(2)-3*x^3-45*x^2)/(x^5-25*x^ 4+250*x^3-1250*x^2+3125*x-3125),x, algorithm="maxima")
Output:
-30*integrate(e^x/(x^5 - 25*x^4 + 250*x^3 - 1250*x^2 + 3125*x - 3125), x)* log(2) - 1/2*(6*x^2 - 20*x + 25)*log(2)/(x^4 - 20*x^3 + 150*x^2 - 500*x + 625) - 5/2*(4*x - 5)*log(2)/(x^4 - 20*x^3 + 150*x^2 - 500*x + 625) + 3/4*( 4*x^3 - 30*x^2 + 100*x - 125)/(x^4 - 20*x^3 + 150*x^2 - 500*x + 625) + 15/ 4*(6*x^2 - 20*x + 25)/(x^4 - 20*x^3 + 150*x^2 - 500*x + 625) + 3*((x - log (2))*e^(2*x) + 2*(x^2 - x*log(2))*e^x)/(x^4 - 20*x^3 + 150*x^2 - 500*x + 6 25) - 30*e^5*exp_integral_e(5, -x + 5)*log(2)/(x - 5)^4
Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (19) = 38\).
Time = 0.12 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.77 \[ \int \frac {-45 x^2-3 x^3+\left (30 x+6 x^2\right ) \log (2)+e^{2 x} \left (-15-39 x+6 x^2+(42-6 x) \log (2)\right )+e^x \left (-60 x-42 x^2+6 x^3+\left (30+48 x-6 x^2\right ) \log (2)\right )}{-3125+3125 x-1250 x^2+250 x^3-25 x^4+x^5} \, dx=\frac {3 \, {\left (x^{3} + 2 \, x^{2} e^{x} - x^{2} \log \left (2\right ) - 2 \, x e^{x} \log \left (2\right ) + x e^{\left (2 \, x\right )} - e^{\left (2 \, x\right )} \log \left (2\right )\right )}}{x^{4} - 20 \, x^{3} + 150 \, x^{2} - 500 \, x + 625} \] Input:
integrate((((-6*x+42)*log(2)+6*x^2-39*x-15)*exp(x)^2+((-6*x^2+48*x+30)*log (2)+6*x^3-42*x^2-60*x)*exp(x)+(6*x^2+30*x)*log(2)-3*x^3-45*x^2)/(x^5-25*x^ 4+250*x^3-1250*x^2+3125*x-3125),x, algorithm="giac")
Output:
3*(x^3 + 2*x^2*e^x - x^2*log(2) - 2*x*e^x*log(2) + x*e^(2*x) - e^(2*x)*log (2))/(x^4 - 20*x^3 + 150*x^2 - 500*x + 625)
Time = 7.97 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.82 \[ \int \frac {-45 x^2-3 x^3+\left (30 x+6 x^2\right ) \log (2)+e^{2 x} \left (-15-39 x+6 x^2+(42-6 x) \log (2)\right )+e^x \left (-60 x-42 x^2+6 x^3+\left (30+48 x-6 x^2\right ) \log (2)\right )}{-3125+3125 x-1250 x^2+250 x^3-25 x^4+x^5} \, dx=-\frac {{\mathrm {e}}^{2\,x}\,\ln \left (8\right )+x^2\,\left (\ln \left (8\right )-6\,{\mathrm {e}}^x\right )-x\,\left (3\,{\mathrm {e}}^{2\,x}-{\mathrm {e}}^x\,\ln \left (64\right )\right )-3\,x^3}{x^4-20\,x^3+150\,x^2-500\,x+625} \] Input:
int(-(exp(2*x)*(39*x + log(2)*(6*x - 42) - 6*x^2 + 15) + exp(x)*(60*x - lo g(2)*(48*x - 6*x^2 + 30) + 42*x^2 - 6*x^3) - log(2)*(30*x + 6*x^2) + 45*x^ 2 + 3*x^3)/(3125*x - 1250*x^2 + 250*x^3 - 25*x^4 + x^5 - 3125),x)
Output:
-(exp(2*x)*log(8) + x^2*(log(8) - 6*exp(x)) - x*(3*exp(2*x) - exp(x)*log(6 4)) - 3*x^3)/(150*x^2 - 500*x - 20*x^3 + x^4 + 625)
Time = 0.15 (sec) , antiderivative size = 77, normalized size of antiderivative = 3.50 \[ \int \frac {-45 x^2-3 x^3+\left (30 x+6 x^2\right ) \log (2)+e^{2 x} \left (-15-39 x+6 x^2+(42-6 x) \log (2)\right )+e^x \left (-60 x-42 x^2+6 x^3+\left (30+48 x-6 x^2\right ) \log (2)\right )}{-3125+3125 x-1250 x^2+250 x^3-25 x^4+x^5} \, dx=\frac {-60 e^{2 x} \mathrm {log}\left (2\right )+60 e^{2 x} x -120 e^{x} \mathrm {log}\left (2\right ) x +120 e^{x} x^{2}-60 \,\mathrm {log}\left (2\right ) x^{2}+3 x^{4}+450 x^{2}-1500 x +1875}{20 x^{4}-400 x^{3}+3000 x^{2}-10000 x +12500} \] Input:
int((((-6*x+42)*log(2)+6*x^2-39*x-15)*exp(x)^2+((-6*x^2+48*x+30)*log(2)+6* x^3-42*x^2-60*x)*exp(x)+(6*x^2+30*x)*log(2)-3*x^3-45*x^2)/(x^5-25*x^4+250* x^3-1250*x^2+3125*x-3125),x)
Output:
(3*( - 20*e**(2*x)*log(2) + 20*e**(2*x)*x - 40*e**x*log(2)*x + 40*e**x*x** 2 - 20*log(2)*x**2 + x**4 + 150*x**2 - 500*x + 625))/(20*(x**4 - 20*x**3 + 150*x**2 - 500*x + 625))