Integrand size = 109, antiderivative size = 28 \[ \int \frac {2 x-6 x^2+40 x^3-108 x^4+288 x^5-648 x^6+864 x^7-1296 x^8+864 x^9+e^{2 x} \left (72-864 x+432 x^2\right )+e^x \left (-12+12 x+12 x^2+576 x^4-432 x^5+432 x^6\right )}{1+18 x^2+108 x^4+216 x^6} \, dx=x^2 \left (-2+x+\frac {x+\frac {e^x}{\frac {1}{6}+x^2}}{x}\right )^2 \]
Time = 6.46 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.18 \[ \int \frac {2 x-6 x^2+40 x^3-108 x^4+288 x^5-648 x^6+864 x^7-1296 x^8+864 x^9+e^{2 x} \left (72-864 x+432 x^2\right )+e^x \left (-12+12 x+12 x^2+576 x^4-432 x^5+432 x^6\right )}{1+18 x^2+108 x^4+216 x^6} \, dx=\frac {\left (6 e^x+x \left (-1+x-6 x^2+6 x^3\right )\right )^2}{\left (1+6 x^2\right )^2} \]
Integrate[(2*x - 6*x^2 + 40*x^3 - 108*x^4 + 288*x^5 - 648*x^6 + 864*x^7 - 1296*x^8 + 864*x^9 + E^(2*x)*(72 - 864*x + 432*x^2) + E^x*(-12 + 12*x + 12 *x^2 + 576*x^4 - 432*x^5 + 432*x^6))/(1 + 18*x^2 + 108*x^4 + 216*x^6),x]
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.
\(\displaystyle \int \frac {864 x^9-1296 x^8+864 x^7-648 x^6+288 x^5-108 x^4+40 x^3-6 x^2+e^{2 x} \left (432 x^2-864 x+72\right )+e^x \left (432 x^6-432 x^5+576 x^4+12 x^2+12 x-12\right )+2 x}{216 x^6+108 x^4+18 x^2+1} \, dx\) |
\(\Big \downarrow \) 2070 |
\(\displaystyle \int \frac {864 x^9-1296 x^8+864 x^7-648 x^6+288 x^5-108 x^4+40 x^3-6 x^2+e^{2 x} \left (432 x^2-864 x+72\right )+e^x \left (432 x^6-432 x^5+576 x^4+12 x^2+12 x-12\right )+2 x}{\left (6 x^2+1\right )^3}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {6 x^2}{\left (6 x^2+1\right )^3}+\frac {2 x}{\left (6 x^2+1\right )^3}+\frac {72 e^{2 x} \left (6 x^2-12 x+1\right )}{\left (6 x^2+1\right )^3}+\frac {864 x^9}{\left (6 x^2+1\right )^3}-\frac {1296 x^8}{\left (6 x^2+1\right )^3}+\frac {864 x^7}{\left (6 x^2+1\right )^3}-\frac {648 x^6}{\left (6 x^2+1\right )^3}+\frac {288 x^5}{\left (6 x^2+1\right )^3}-\frac {108 x^4}{\left (6 x^2+1\right )^3}+\frac {40 x^3}{\left (6 x^2+1\right )^3}+\frac {12 e^x \left (6 x^4-6 x^3+7 x^2+x-1\right )}{\left (6 x^2+1\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 24 \int \frac {e^x x}{\left (6 x^2+1\right )^2}dx-\frac {1}{6} \left (6+5 i \sqrt {6}\right ) e^{\frac {i}{\sqrt {6}}} \operatorname {ExpIntegralEi}\left (-\frac {i-\sqrt {6} x}{\sqrt {6}}\right )+i \sqrt {6} e^{\frac {i}{\sqrt {6}}} \operatorname {ExpIntegralEi}\left (-\frac {i-\sqrt {6} x}{\sqrt {6}}\right )+e^{\frac {i}{\sqrt {6}}} \operatorname {ExpIntegralEi}\left (-\frac {i-\sqrt {6} x}{\sqrt {6}}\right )-\frac {1}{6} \left (6-5 i \sqrt {6}\right ) e^{-\frac {i}{\sqrt {6}}} \operatorname {ExpIntegralEi}\left (\frac {\sqrt {6} x+i}{\sqrt {6}}\right )-i \sqrt {6} e^{-\frac {i}{\sqrt {6}}} \operatorname {ExpIntegralEi}\left (\frac {\sqrt {6} x+i}{\sqrt {6}}\right )+e^{-\frac {i}{\sqrt {6}}} \operatorname {ExpIntegralEi}\left (\frac {\sqrt {6} x+i}{\sqrt {6}}\right )+x^4-\frac {35 x^3}{4}+x^2+\frac {x}{6 x^2+1}+\frac {x}{4 \left (6 x^2+1\right )^2}+\frac {5}{9 \left (6 x^2+1\right )}+\frac {36 e^{2 x}}{\left (6 x^2+1\right )^2}-\frac {5}{18 \left (6 x^2+1\right )^2}+\frac {54 x^7}{\left (6 x^2+1\right )^2}+\frac {63 x^5}{2 \left (6 x^2+1\right )}+\frac {27 x^5}{\left (6 x^2+1\right )^2}+\frac {10 x^4}{\left (6 x^2+1\right )^2}+\frac {45 x^3}{4 \left (6 x^2+1\right )}+\frac {9 x^3}{2 \left (6 x^2+1\right )^2}-\frac {5 x}{4}+2 e^x+\frac {6 e^x}{-6 x+i \sqrt {6}}-\frac {\sqrt {6} e^x}{\sqrt {6} x+i}\) |
Int[(2*x - 6*x^2 + 40*x^3 - 108*x^4 + 288*x^5 - 648*x^6 + 864*x^7 - 1296*x ^8 + 864*x^9 + E^(2*x)*(72 - 864*x + 432*x^2) + E^x*(-12 + 12*x + 12*x^2 + 576*x^4 - 432*x^5 + 432*x^6))/(1 + 18*x^2 + 108*x^4 + 216*x^6),x]
3.16.25.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x^2, 0], Expon[Px , x^2]], b = Rt[Coeff[Px, x^2, Expon[Px, x^2]], Expon[Px, x^2]]}, Int[u*(a + b*x^2)^(Expon[Px, x^2]*p), x] /; EqQ[Px, (a + b*x^2)^Expon[Px, x^2]]] /; IntegerQ[p] && PolyQ[Px, x^2] && GtQ[Expon[Px, x^2], 1] && NeQ[Coeff[Px, x^ 2, 0], 0]
Time = 0.23 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.61
method | result | size |
risch | \(x^{4}-2 x^{3}+x^{2}+\frac {36 \,{\mathrm e}^{2 x}}{\left (6 x^{2}+1\right )^{2}}+\frac {12 x \left (-1+x \right ) {\mathrm e}^{x}}{6 x^{2}+1}\) | \(45\) |
norman | \(\frac {-\frac {10 x^{2}}{3}-2 x^{3}-24 x^{5}+48 x^{6}-72 x^{7}+36 x^{8}+36 \,{\mathrm e}^{2 x}-12 \,{\mathrm e}^{x} x +12 \,{\mathrm e}^{x} x^{2}-72 \,{\mathrm e}^{x} x^{3}+72 \,{\mathrm e}^{x} x^{4}-\frac {13}{36}}{\left (6 x^{2}+1\right )^{2}}\) | \(75\) |
parallelrisch | \(\frac {1296 x^{8}-2592 x^{7}+1728 x^{6}-864 x^{5}+2592 \,{\mathrm e}^{x} x^{4}-13-2592 \,{\mathrm e}^{x} x^{3}-72 x^{3}+432 \,{\mathrm e}^{x} x^{2}-120 x^{2}-432 \,{\mathrm e}^{x} x +1296 \,{\mathrm e}^{2 x}}{1296 x^{4}+432 x^{2}+36}\) | \(81\) |
parts | \(x^{4}-2 x^{3}+x^{2}+\frac {3 \,{\mathrm e}^{2 x} \left (54 x^{3}+6 x^{2}+15 x +1\right )}{36 x^{4}+12 x^{2}+1}-\frac {36 \,{\mathrm e}^{2 x} \left (6 x^{3}+x -1\right )}{36 x^{4}+12 x^{2}+1}+\frac {3 \,{\mathrm e}^{2 x} \left (18 x^{3}-6 x^{2}-3 x -1\right )}{36 x^{4}+12 x^{2}+1}-\frac {2 \,{\mathrm e}^{x}}{6 x^{2}+1}-\frac {12 \,{\mathrm e}^{x} x}{6 x^{2}+1}+2 \,{\mathrm e}^{x}\) | \(142\) |
default | \(x^{4}-2 x^{3}+x^{2}+2 \,{\mathrm e}^{x}+\frac {3 \,{\mathrm e}^{2 x} \left (54 x^{3}+6 x^{2}+15 x +1\right )}{36 x^{4}+12 x^{2}+1}+\frac {3 \,{\mathrm e}^{2 x} \left (18 x^{3}-6 x^{2}-3 x -1\right )}{36 x^{4}+12 x^{2}+1}+\frac {{\mathrm e}^{x} \left (324 x^{3}-6 x^{2}+42 x -1\right )}{864 x^{4}+288 x^{2}+24}-\frac {{\mathrm e}^{x} \left (6 x^{3}+48 x^{2}+x +6\right )}{4 \left (36 x^{4}+12 x^{2}+1\right )}-\frac {36 \,{\mathrm e}^{2 x} \left (6 x^{3}+x -1\right )}{36 x^{4}+12 x^{2}+1}-\frac {{\mathrm e}^{x} \left (108 x^{3}+6 x^{2}+30 x +1\right )}{4 \left (36 x^{4}+12 x^{2}+1\right )}+\frac {{\mathrm e}^{x} \left (6 x^{3}+x -2\right )}{144 x^{4}+48 x^{2}+4}-\frac {108 \left (-\frac {13}{144} x^{3}-\frac {11}{864} x \right )}{\left (6 x^{2}+1\right )^{2}}-\frac {216 \left (\frac {1}{288} x^{3}-\frac {1}{1728} x \right )}{\left (6 x^{2}+1\right )^{2}}+\frac {-\frac {81}{4} x^{3}-\frac {21}{8} x}{\left (6 x^{2}+1\right )^{2}}-\frac {{\mathrm e}^{x} \left (180 x^{3}-6 x^{2}+18 x -1\right )}{3 \left (36 x^{4}+12 x^{2}+1\right )}+\frac {{\mathrm e}^{x} \left (36 x^{3}-6 x^{2}-6 x -1\right )}{864 x^{4}+288 x^{2}+24}-\frac {3888 \left (-\frac {5}{1728} x^{3}-\frac {1}{3456} x \right )}{\left (6 x^{2}+1\right )^{2}}\) | \(384\) |
int(((432*x^2-864*x+72)*exp(x)^2+(432*x^6-432*x^5+576*x^4+12*x^2+12*x-12)* exp(x)+864*x^9-1296*x^8+864*x^7-648*x^6+288*x^5-108*x^4+40*x^3-6*x^2+2*x)/ (216*x^6+108*x^4+18*x^2+1),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (28) = 56\).
Time = 0.28 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.71 \[ \int \frac {2 x-6 x^2+40 x^3-108 x^4+288 x^5-648 x^6+864 x^7-1296 x^8+864 x^9+e^{2 x} \left (72-864 x+432 x^2\right )+e^x \left (-12+12 x+12 x^2+576 x^4-432 x^5+432 x^6\right )}{1+18 x^2+108 x^4+216 x^6} \, dx=\frac {36 \, x^{8} - 72 \, x^{7} + 48 \, x^{6} - 24 \, x^{5} + 13 \, x^{4} - 2 \, x^{3} + x^{2} + 12 \, {\left (6 \, x^{4} - 6 \, x^{3} + x^{2} - x\right )} e^{x} + 36 \, e^{\left (2 \, x\right )}}{36 \, x^{4} + 12 \, x^{2} + 1} \]
integrate(((432*x^2-864*x+72)*exp(x)^2+(432*x^6-432*x^5+576*x^4+12*x^2+12* x-12)*exp(x)+864*x^9-1296*x^8+864*x^7-648*x^6+288*x^5-108*x^4+40*x^3-6*x^2 +2*x)/(216*x^6+108*x^4+18*x^2+1),x, algorithm=\
(36*x^8 - 72*x^7 + 48*x^6 - 24*x^5 + 13*x^4 - 2*x^3 + x^2 + 12*(6*x^4 - 6* x^3 + x^2 - x)*e^x + 36*e^(2*x))/(36*x^4 + 12*x^2 + 1)
Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (20) = 40\).
Time = 0.08 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.54 \[ \int \frac {2 x-6 x^2+40 x^3-108 x^4+288 x^5-648 x^6+864 x^7-1296 x^8+864 x^9+e^{2 x} \left (72-864 x+432 x^2\right )+e^x \left (-12+12 x+12 x^2+576 x^4-432 x^5+432 x^6\right )}{1+18 x^2+108 x^4+216 x^6} \, dx=x^{4} - 2 x^{3} + x^{2} + \frac {\left (216 x^{2} + 36\right ) e^{2 x} + \left (432 x^{6} - 432 x^{5} + 144 x^{4} - 144 x^{3} + 12 x^{2} - 12 x\right ) e^{x}}{216 x^{6} + 108 x^{4} + 18 x^{2} + 1} \]
integrate(((432*x**2-864*x+72)*exp(x)**2+(432*x**6-432*x**5+576*x**4+12*x* *2+12*x-12)*exp(x)+864*x**9-1296*x**8+864*x**7-648*x**6+288*x**5-108*x**4+ 40*x**3-6*x**2+2*x)/(216*x**6+108*x**4+18*x**2+1),x)
x**4 - 2*x**3 + x**2 + ((216*x**2 + 36)*exp(2*x) + (432*x**6 - 432*x**5 + 144*x**4 - 144*x**3 + 12*x**2 - 12*x)*exp(x))/(216*x**6 + 108*x**4 + 18*x* *2 + 1)
Leaf count of result is larger than twice the leaf count of optimal. 260 vs. \(2 (28) = 56\).
Time = 0.32 (sec) , antiderivative size = 260, normalized size of antiderivative = 9.29 \[ \int \frac {2 x-6 x^2+40 x^3-108 x^4+288 x^5-648 x^6+864 x^7-1296 x^8+864 x^9+e^{2 x} \left (72-864 x+432 x^2\right )+e^x \left (-12+12 x+12 x^2+576 x^4-432 x^5+432 x^6\right )}{1+18 x^2+108 x^4+216 x^6} \, dx=x^{4} - 2 \, x^{3} + x^{2} + \frac {78 \, x^{3} + 11 \, x}{8 \, {\left (36 \, x^{4} + 12 \, x^{2} + 1\right )}} - \frac {3 \, {\left (54 \, x^{3} + 7 \, x\right )}}{8 \, {\left (36 \, x^{4} + 12 \, x^{2} + 1\right )}} + \frac {9 \, {\left (10 \, x^{3} + x\right )}}{8 \, {\left (36 \, x^{4} + 12 \, x^{2} + 1\right )}} - \frac {6 \, x^{3} - x}{8 \, {\left (36 \, x^{4} + 12 \, x^{2} + 1\right )}} + \frac {48 \, x^{2} + 7}{36 \, {\left (36 \, x^{4} + 12 \, x^{2} + 1\right )}} - \frac {36 \, x^{2} + 5}{6 \, {\left (36 \, x^{4} + 12 \, x^{2} + 1\right )}} - \frac {5 \, {\left (12 \, x^{2} + 1\right )}}{18 \, {\left (36 \, x^{4} + 12 \, x^{2} + 1\right )}} + \frac {8 \, x^{2} + 1}{36 \, x^{4} + 12 \, x^{2} + 1} + \frac {12 \, {\left ({\left (6 \, x^{4} - 6 \, x^{3} + x^{2} - x\right )} e^{x} + 3 \, e^{\left (2 \, x\right )}\right )}}{36 \, x^{4} + 12 \, x^{2} + 1} - \frac {1}{12 \, {\left (36 \, x^{4} + 12 \, x^{2} + 1\right )}} \]
integrate(((432*x^2-864*x+72)*exp(x)^2+(432*x^6-432*x^5+576*x^4+12*x^2+12* x-12)*exp(x)+864*x^9-1296*x^8+864*x^7-648*x^6+288*x^5-108*x^4+40*x^3-6*x^2 +2*x)/(216*x^6+108*x^4+18*x^2+1),x, algorithm=\
x^4 - 2*x^3 + x^2 + 1/8*(78*x^3 + 11*x)/(36*x^4 + 12*x^2 + 1) - 3/8*(54*x^ 3 + 7*x)/(36*x^4 + 12*x^2 + 1) + 9/8*(10*x^3 + x)/(36*x^4 + 12*x^2 + 1) - 1/8*(6*x^3 - x)/(36*x^4 + 12*x^2 + 1) + 1/36*(48*x^2 + 7)/(36*x^4 + 12*x^2 + 1) - 1/6*(36*x^2 + 5)/(36*x^4 + 12*x^2 + 1) - 5/18*(12*x^2 + 1)/(36*x^4 + 12*x^2 + 1) + (8*x^2 + 1)/(36*x^4 + 12*x^2 + 1) + 12*((6*x^4 - 6*x^3 + x^2 - x)*e^x + 3*e^(2*x))/(36*x^4 + 12*x^2 + 1) - 1/12/(36*x^4 + 12*x^2 + 1)
Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (28) = 56\).
Time = 0.26 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.89 \[ \int \frac {2 x-6 x^2+40 x^3-108 x^4+288 x^5-648 x^6+864 x^7-1296 x^8+864 x^9+e^{2 x} \left (72-864 x+432 x^2\right )+e^x \left (-12+12 x+12 x^2+576 x^4-432 x^5+432 x^6\right )}{1+18 x^2+108 x^4+216 x^6} \, dx=\frac {36 \, x^{8} - 72 \, x^{7} + 48 \, x^{6} - 24 \, x^{5} + 72 \, x^{4} e^{x} + 13 \, x^{4} - 72 \, x^{3} e^{x} - 2 \, x^{3} + 12 \, x^{2} e^{x} + x^{2} - 12 \, x e^{x} + 36 \, e^{\left (2 \, x\right )}}{36 \, x^{4} + 12 \, x^{2} + 1} \]
integrate(((432*x^2-864*x+72)*exp(x)^2+(432*x^6-432*x^5+576*x^4+12*x^2+12* x-12)*exp(x)+864*x^9-1296*x^8+864*x^7-648*x^6+288*x^5-108*x^4+40*x^3-6*x^2 +2*x)/(216*x^6+108*x^4+18*x^2+1),x, algorithm=\
(36*x^8 - 72*x^7 + 48*x^6 - 24*x^5 + 72*x^4*e^x + 13*x^4 - 72*x^3*e^x - 2* x^3 + 12*x^2*e^x + x^2 - 12*x*e^x + 36*e^(2*x))/(36*x^4 + 12*x^2 + 1)
Time = 8.55 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.75 \[ \int \frac {2 x-6 x^2+40 x^3-108 x^4+288 x^5-648 x^6+864 x^7-1296 x^8+864 x^9+e^{2 x} \left (72-864 x+432 x^2\right )+e^x \left (-12+12 x+12 x^2+576 x^4-432 x^5+432 x^6\right )}{1+18 x^2+108 x^4+216 x^6} \, dx=\frac {{\mathrm {e}}^{2\,x}}{x^4+\frac {x^2}{3}+\frac {1}{36}}+x^2-2\,x^3+x^4-\frac {{\mathrm {e}}^x\,\left (2\,x-2\,x^2\right )}{x^2+\frac {1}{6}} \]