Integrand size = 158, antiderivative size = 33 \[ \int \frac {e^x \left (\left (-6 x^3+21 x^4-12 x^5-12 x^6+e^8 \left (-6 x-3 x^2\right )+e^4 \left (12 x^2-18 x^3-12 x^4\right )\right ) \log (4)+\left (2 x^4-2 e^4 x^4-6 x^5-4 x^6\right ) \log ^2(4)\right )}{9 e^8+9 x^2-36 x^3+36 x^4+e^4 \left (-18 x+36 x^2\right )+\left (12 e^4 x^2-12 x^3+24 x^4\right ) \log (4)+4 x^4 \log ^2(4)} \, dx=\frac {e^x x^2}{\frac {2}{-2+\frac {-e^4+x}{x^2}}-\frac {3}{\log (4)}} \]
\[ \int \frac {e^x \left (\left (-6 x^3+21 x^4-12 x^5-12 x^6+e^8 \left (-6 x-3 x^2\right )+e^4 \left (12 x^2-18 x^3-12 x^4\right )\right ) \log (4)+\left (2 x^4-2 e^4 x^4-6 x^5-4 x^6\right ) \log ^2(4)\right )}{9 e^8+9 x^2-36 x^3+36 x^4+e^4 \left (-18 x+36 x^2\right )+\left (12 e^4 x^2-12 x^3+24 x^4\right ) \log (4)+4 x^4 \log ^2(4)} \, dx=\int \frac {e^x \left (\left (-6 x^3+21 x^4-12 x^5-12 x^6+e^8 \left (-6 x-3 x^2\right )+e^4 \left (12 x^2-18 x^3-12 x^4\right )\right ) \log (4)+\left (2 x^4-2 e^4 x^4-6 x^5-4 x^6\right ) \log ^2(4)\right )}{9 e^8+9 x^2-36 x^3+36 x^4+e^4 \left (-18 x+36 x^2\right )+\left (12 e^4 x^2-12 x^3+24 x^4\right ) \log (4)+4 x^4 \log ^2(4)} \, dx \]
Integrate[(E^x*((-6*x^3 + 21*x^4 - 12*x^5 - 12*x^6 + E^8*(-6*x - 3*x^2) + E^4*(12*x^2 - 18*x^3 - 12*x^4))*Log[4] + (2*x^4 - 2*E^4*x^4 - 6*x^5 - 4*x^ 6)*Log[4]^2))/(9*E^8 + 9*x^2 - 36*x^3 + 36*x^4 + E^4*(-18*x + 36*x^2) + (1 2*E^4*x^2 - 12*x^3 + 24*x^4)*Log[4] + 4*x^4*Log[4]^2),x]
Integrate[(E^x*((-6*x^3 + 21*x^4 - 12*x^5 - 12*x^6 + E^8*(-6*x - 3*x^2) + E^4*(12*x^2 - 18*x^3 - 12*x^4))*Log[4] + (2*x^4 - 2*E^4*x^4 - 6*x^5 - 4*x^ 6)*Log[4]^2))/(9*E^8 + 9*x^2 - 36*x^3 + 36*x^4 + E^4*(-18*x + 36*x^2) + (1 2*E^4*x^2 - 12*x^3 + 24*x^4)*Log[4] + 4*x^4*Log[4]^2), x]
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 5.20 (sec) , antiderivative size = 1779, normalized size of antiderivative = 53.91, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {6, 2463, 6, 7292, 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.
| \(\displaystyle \int \frac {e^x \left (\left (-4 x^6-6 x^5-2 e^4 x^4+2 x^4\right ) \log ^2(4)+\left (-12 x^6-12 x^5+21 x^4-6 x^3+e^8 \left (-3 x^2-6 x\right )+e^4 \left (-12 x^4-18 x^3+12 x^2\right )\right ) \log (4)\right )}{36 x^4+4 x^4 \log ^2(4)-36 x^3+9 x^2+e^4 \left (36 x^2-18 x\right )+\left (24 x^4-12 x^3+12 e^4 x^2\right ) \log (4)+9 e^8} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {e^x \left (\left (-4 x^6-6 x^5-2 e^4 x^4+2 x^4\right ) \log ^2(4)+\left (-12 x^6-12 x^5+21 x^4-6 x^3+e^8 \left (-3 x^2-6 x\right )+e^4 \left (-12 x^4-18 x^3+12 x^2\right )\right ) \log (4)\right )}{x^4 \left (36+4 \log ^2(4)\right )-36 x^3+9 x^2+e^4 \left (36 x^2-18 x\right )+\left (24 x^4-12 x^3+12 e^4 x^2\right ) \log (4)+9 e^8}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \frac {e^x \left (\left (-4 x^6-6 x^5-2 e^4 x^4+2 x^4\right ) \log ^2(4)+\left (-12 x^6-12 x^5+21 x^4-6 x^3+e^8 \left (-3 x^2-6 x\right )+e^4 \left (-12 x^4-18 x^3+12 x^2\right )\right ) \log (4)\right )}{\left (6 x^2+2 x^2 \log (4)-3 x+3 e^4\right )^2}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {e^x \left (\left (-4 x^6-6 x^5-2 e^4 x^4+2 x^4\right ) \log ^2(4)+\left (-12 x^6-12 x^5+21 x^4-6 x^3+e^8 \left (-3 x^2-6 x\right )+e^4 \left (-12 x^4-18 x^3+12 x^2\right )\right ) \log (4)\right )}{\left (x^2 (6+2 \log (4))-3 x+3 e^4\right )^2}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {e^x \left (\left (-4 x^6-6 x^5-2 e^4 x^4+2 x^4\right ) \log ^2(4)+\left (-12 x^6-12 x^5+21 x^4-6 x^3+e^8 \left (-3 x^2-6 x\right )+e^4 \left (-12 x^4-18 x^3+12 x^2\right )\right ) \log (4)\right )}{\left (2 x^2 (3+\log (4))-3 x+3 e^4\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {3 e^x \log ^2(4) \left (9 e^4 \left (1-2 e^4 (3+\log (4))\right )-x \left (9-24 e^4 (3+\log (4))+8 e^8 (3+\log (4))^2\right )\right )}{4 (3+\log (4))^3 \left (2 x^2 (3+\log (4))-3 x+3 e^4\right )^2}+\frac {3 e^x \log ^2(4) \left (x \left (3-4 e^4 (3+\log (4))\right )-3+e^4 (9+\log (256))+2 e^8 (3+\log (4))\right )}{4 (3+\log (4))^3 \left (2 x^2 (3+\log (4))-3 x+3 e^4\right )}-\frac {e^x x^2 \log (4)}{3+\log (4)}+\frac {e^x \log ^2(4) \left (9+2 \log (4)-2 e^4 (3+\log (4))\right )}{4 (3+\log (4))^3}-\frac {3 e^x x \log (4) (4+\log (4))}{2 (3+\log (4))^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {e^x \log (4) x^2}{3+\log (4)}-\frac {3 e^x \log (4) (4+\log (4)) x}{2 (3+\log (4))^2}+\frac {2 e^x \log (4) x}{3+\log (4)}-\frac {e^x \log ^2(4) \left (9-24 e^4 (3+\log (4))+8 e^8 (3+\log (4))^2\right ) \left (3-i \sqrt {-9+24 e^4 (3+\log (4))}\right )}{4 (3+\log (4))^3 \left (3-8 e^4 (3+\log (4))\right ) \left (-4 (3+\log (4)) x-i \sqrt {-9+24 e^4 (3+\log (4))}+3\right )}+\frac {9 e^{x+4} \log ^2(4) \left (1-2 e^4 (3+\log (4))\right )}{(3+\log (4))^2 \left (3-8 e^4 (3+\log (4))\right ) \left (-4 (3+\log (4)) x-i \sqrt {-9+24 e^4 (3+\log (4))}+3\right )}-\frac {e^x \log ^2(4) \left (9-24 e^4 (3+\log (4))+8 e^8 (3+\log (4))^2\right ) \left (3+i \sqrt {-9+24 e^4 (3+\log (4))}\right )}{4 (3+\log (4))^3 \left (3-8 e^4 (3+\log (4))\right ) \left (-4 (3+\log (4)) x+i \sqrt {-9+24 e^4 (3+\log (4))}+3\right )}+\frac {9 e^{x+4} \log ^2(4) \left (1-2 e^4 (3+\log (4))\right )}{(3+\log (4))^2 \left (3-8 e^4 (3+\log (4))\right ) \left (-4 (3+\log (4)) x+i \sqrt {-9+24 e^4 (3+\log (4))}+3\right )}+\frac {3 e^{\frac {3-i \sqrt {-9+24 e^4 (3+\log (4))}}{4 (3+\log (4))}} \operatorname {ExpIntegralEi}\left (-\frac {-4 (3+\log (4)) x-i \sqrt {-9+24 e^4 (3+\log (4))}+3}{4 (3+\log (4))}\right ) \log ^2(4) \left (3-4 e^4 (3+\log (4))+\frac {i \left (8 e^8 (3+\log (4))^2-3 (9+\log (256))+e^4 \left (72+24 \log (4)+12 \log (256)+\log ^2(256)\right )\right )}{\sqrt {-9+24 e^4 (3+\log (4))}}\right )}{16 (3+\log (4))^4}+\frac {3 e^{\frac {3+i \sqrt {-9+24 e^4 (3+\log (4))}}{4 (3+\log (4))}} \operatorname {ExpIntegralEi}\left (-\frac {-4 (3+\log (4)) x+i \sqrt {-9+24 e^4 (3+\log (4))}+3}{4 (3+\log (4))}\right ) \log ^2(4) \left (3-4 e^4 (3+\log (4))-\frac {i \left (8 e^8 (3+\log (4))^2-3 (9+\log (256))+e^4 \left (72+24 \log (4)+12 \log (256)+\log ^2(256)\right )\right )}{\sqrt {-9+24 e^4 (3+\log (4))}}\right )}{16 (3+\log (4))^4}+\frac {e^x \log ^2(4) \left (9-2 e^4 (3+\log (4))+\log (16)\right )}{4 (3+\log (4))^3}-\frac {e^{\frac {3+i \sqrt {-9+24 e^4 (3+\log (4))}}{4 (3+\log (4))}} \operatorname {ExpIntegralEi}\left (-\frac {-4 (3+\log (4)) x+i \sqrt {-9+24 e^4 (3+\log (4))}+3}{4 (3+\log (4))}\right ) \log ^2(4) \left (9-24 e^4 (3+\log (4))+8 e^8 (3+\log (4))^2\right ) \left (3+i \sqrt {-9+24 e^4 (3+\log (4))}\right )}{16 (3+\log (4))^4 \left (3-8 e^4 (3+\log (4))\right )}-\frac {e^{\frac {3-i \sqrt {-9+24 e^4 (3+\log (4))}}{4 (3+\log (4))}} \operatorname {ExpIntegralEi}\left (-\frac {-4 (3+\log (4)) x-i \sqrt {-9+24 e^4 (3+\log (4))}+3}{4 (3+\log (4))}\right ) \log ^2(4) \left (9-24 e^4 (3+\log (4))+8 e^8 (3+\log (4))^2\right ) \left (3-i \sqrt {-9+24 e^4 (3+\log (4))}\right )}{16 (3+\log (4))^4 \left (3-8 e^4 (3+\log (4))\right )}-\frac {i \sqrt {3} e^{\frac {3-i \sqrt {-9+24 e^4 (3+\log (4))}}{4 (3+\log (4))}} \operatorname {ExpIntegralEi}\left (-\frac {-4 (3+\log (4)) x-i \sqrt {-9+24 e^4 (3+\log (4))}+3}{4 (3+\log (4))}\right ) \log ^2(4) \left (9-24 e^4 (3+\log (4))+8 e^8 (3+\log (4))^2\right )}{4 (3+\log (4))^3 \left (-3+8 e^4 (3+\log (4))\right )^{3/2}}+\frac {i \sqrt {3} e^{\frac {3+i \sqrt {-9+24 e^4 (3+\log (4))}}{4 (3+\log (4))}} \operatorname {ExpIntegralEi}\left (-\frac {-4 (3+\log (4)) x+i \sqrt {-9+24 e^4 (3+\log (4))}+3}{4 (3+\log (4))}\right ) \log ^2(4) \left (9-24 e^4 (3+\log (4))+8 e^8 (3+\log (4))^2\right )}{4 (3+\log (4))^3 \left (-3+8 e^4 (3+\log (4))\right )^{3/2}}+\frac {3 i \sqrt {3} e^{\frac {51+16 \log (4)-i \sqrt {-9+24 e^4 (3+\log (4))}}{4 (3+\log (4))}} \operatorname {ExpIntegralEi}\left (-\frac {-4 (3+\log (4)) x-i \sqrt {-9+24 e^4 (3+\log (4))}+3}{4 (3+\log (4))}\right ) \log ^2(4) \left (1-2 e^4 (3+\log (4))\right )}{(3+\log (4))^2 \left (-3+8 e^4 (3+\log (4))\right )^{3/2}}-\frac {3 i \sqrt {3} e^{\frac {51+16 \log (4)+i \sqrt {-9+24 e^4 (3+\log (4))}}{4 (3+\log (4))}} \operatorname {ExpIntegralEi}\left (-\frac {-4 (3+\log (4)) x+i \sqrt {-9+24 e^4 (3+\log (4))}+3}{4 (3+\log (4))}\right ) \log ^2(4) \left (1-2 e^4 (3+\log (4))\right )}{(3+\log (4))^2 \left (-3+8 e^4 (3+\log (4))\right )^{3/2}}+\frac {9 e^{\frac {51+16 \log (4)-i \sqrt {-9+24 e^4 (3+\log (4))}}{4 (3+\log (4))}} \operatorname {ExpIntegralEi}\left (-\frac {-4 (3+\log (4)) x-i \sqrt {-9+24 e^4 (3+\log (4))}+3}{4 (3+\log (4))}\right ) \log ^2(4) \left (1-2 e^4 (3+\log (4))\right )}{4 (3+\log (4))^3 \left (3-8 e^4 (3+\log (4))\right )}+\frac {9 e^{\frac {51+16 \log (4)+i \sqrt {-9+24 e^4 (3+\log (4))}}{4 (3+\log (4))}} \operatorname {ExpIntegralEi}\left (-\frac {-4 (3+\log (4)) x+i \sqrt {-9+24 e^4 (3+\log (4))}+3}{4 (3+\log (4))}\right ) \log ^2(4) \left (1-2 e^4 (3+\log (4))\right )}{4 (3+\log (4))^3 \left (3-8 e^4 (3+\log (4))\right )}+\frac {3 e^x \log (4) (4+\log (4))}{2 (3+\log (4))^2}-\frac {2 e^x \log (4)}{3+\log (4)}\) |
Int[(E^x*((-6*x^3 + 21*x^4 - 12*x^5 - 12*x^6 + E^8*(-6*x - 3*x^2) + E^4*(1 2*x^2 - 18*x^3 - 12*x^4))*Log[4] + (2*x^4 - 2*E^4*x^4 - 6*x^5 - 4*x^6)*Log [4]^2))/(9*E^8 + 9*x^2 - 36*x^3 + 36*x^4 + E^4*(-18*x + 36*x^2) + (12*E^4* x^2 - 12*x^3 + 24*x^4)*Log[4] + 4*x^4*Log[4]^2),x]
(-2*E^x*Log[4])/(3 + Log[4]) + (2*E^x*x*Log[4])/(3 + Log[4]) - (E^x*x^2*Lo g[4])/(3 + Log[4]) + (3*E^x*Log[4]*(4 + Log[4]))/(2*(3 + Log[4])^2) - (3*E ^x*x*Log[4]*(4 + Log[4]))/(2*(3 + Log[4])^2) + (9*E^((51 + 16*Log[4] - I*S qrt[-9 + 24*E^4*(3 + Log[4])])/(4*(3 + Log[4])))*ExpIntegralEi[-1/4*(3 - 4 *x*(3 + Log[4]) - I*Sqrt[-9 + 24*E^4*(3 + Log[4])])/(3 + Log[4])]*Log[4]^2 *(1 - 2*E^4*(3 + Log[4])))/(4*(3 + Log[4])^3*(3 - 8*E^4*(3 + Log[4]))) + ( 9*E^((51 + 16*Log[4] + I*Sqrt[-9 + 24*E^4*(3 + Log[4])])/(4*(3 + Log[4]))) *ExpIntegralEi[-1/4*(3 - 4*x*(3 + Log[4]) + I*Sqrt[-9 + 24*E^4*(3 + Log[4] )])/(3 + Log[4])]*Log[4]^2*(1 - 2*E^4*(3 + Log[4])))/(4*(3 + Log[4])^3*(3 - 8*E^4*(3 + Log[4]))) + ((3*I)*Sqrt[3]*E^((51 + 16*Log[4] - I*Sqrt[-9 + 2 4*E^4*(3 + Log[4])])/(4*(3 + Log[4])))*ExpIntegralEi[-1/4*(3 - 4*x*(3 + Lo g[4]) - I*Sqrt[-9 + 24*E^4*(3 + Log[4])])/(3 + Log[4])]*Log[4]^2*(1 - 2*E^ 4*(3 + Log[4])))/((3 + Log[4])^2*(-3 + 8*E^4*(3 + Log[4]))^(3/2)) - ((3*I) *Sqrt[3]*E^((51 + 16*Log[4] + I*Sqrt[-9 + 24*E^4*(3 + Log[4])])/(4*(3 + Lo g[4])))*ExpIntegralEi[-1/4*(3 - 4*x*(3 + Log[4]) + I*Sqrt[-9 + 24*E^4*(3 + Log[4])])/(3 + Log[4])]*Log[4]^2*(1 - 2*E^4*(3 + Log[4])))/((3 + Log[4])^ 2*(-3 + 8*E^4*(3 + Log[4]))^(3/2)) - ((I/4)*Sqrt[3]*E^((3 - I*Sqrt[-9 + 24 *E^4*(3 + Log[4])])/(4*(3 + Log[4])))*ExpIntegralEi[-1/4*(3 - 4*x*(3 + Log [4]) - I*Sqrt[-9 + 24*E^4*(3 + Log[4])])/(3 + Log[4])]*Log[4]^2*(9 - 24*E^ 4*(3 + Log[4]) + 8*E^8*(3 + Log[4])^2))/((3 + Log[4])^3*(-3 + 8*E^4*(3 ...
3.12.21.3.1 Defintions of rubi rules used
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Time = 0.53 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.30
| method | result | size |
| gosper | \(-\frac {2 x^{2} \left (2 x^{2}+{\mathrm e}^{4}-x \right ) \ln \left (2\right ) {\mathrm e}^{x}}{4 x^{2} \ln \left (2\right )+6 x^{2}+3 \,{\mathrm e}^{4}-3 x}\) | \(43\) |
| risch | \(-\frac {2 x^{2} \left (2 x^{2}+{\mathrm e}^{4}-x \right ) \ln \left (2\right ) {\mathrm e}^{x}}{4 x^{2} \ln \left (2\right )+6 x^{2}+3 \,{\mathrm e}^{4}-3 x}\) | \(43\) |
| norman | \(\frac {2 x^{3} \ln \left (2\right ) {\mathrm e}^{x}-4 x^{4} \ln \left (2\right ) {\mathrm e}^{x}-2 x^{2} {\mathrm e}^{4} \ln \left (2\right ) {\mathrm e}^{x}}{4 x^{2} \ln \left (2\right )+6 x^{2}+3 \,{\mathrm e}^{4}-3 x}\) | \(54\) |
| parallelrisch | \(-\frac {12 x^{4} \ln \left (2\right ) {\mathrm e}^{x}+6 x^{2} {\mathrm e}^{4} \ln \left (2\right ) {\mathrm e}^{x}-6 x^{3} \ln \left (2\right ) {\mathrm e}^{x}}{3 \left (4 x^{2} \ln \left (2\right )+6 x^{2}+3 \,{\mathrm e}^{4}-3 x \right )}\) | \(55\) |
| default | \(\text {Expression too large to display}\) | \(42483\) |
int((4*(-2*x^4*exp(4)-4*x^6-6*x^5+2*x^4)*ln(2)^2+2*((-3*x^2-6*x)*exp(4)^2+ (-12*x^4-18*x^3+12*x^2)*exp(4)-12*x^6-12*x^5+21*x^4-6*x^3)*ln(2))*exp(x)/( 16*x^4*ln(2)^2+2*(12*x^2*exp(4)+24*x^4-12*x^3)*ln(2)+9*exp(4)^2+(36*x^2-18 *x)*exp(4)+36*x^4-36*x^3+9*x^2),x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.36 \[ \int \frac {e^x \left (\left (-6 x^3+21 x^4-12 x^5-12 x^6+e^8 \left (-6 x-3 x^2\right )+e^4 \left (12 x^2-18 x^3-12 x^4\right )\right ) \log (4)+\left (2 x^4-2 e^4 x^4-6 x^5-4 x^6\right ) \log ^2(4)\right )}{9 e^8+9 x^2-36 x^3+36 x^4+e^4 \left (-18 x+36 x^2\right )+\left (12 e^4 x^2-12 x^3+24 x^4\right ) \log (4)+4 x^4 \log ^2(4)} \, dx=-\frac {2 \, {\left (2 \, x^{4} - x^{3} + x^{2} e^{4}\right )} e^{x} \log \left (2\right )}{4 \, x^{2} \log \left (2\right ) + 6 \, x^{2} - 3 \, x + 3 \, e^{4}} \]
integrate((4*(-2*x^4*exp(4)-4*x^6-6*x^5+2*x^4)*log(2)^2+2*((-3*x^2-6*x)*ex p(4)^2+(-12*x^4-18*x^3+12*x^2)*exp(4)-12*x^6-12*x^5+21*x^4-6*x^3)*log(2))* exp(x)/(16*x^4*log(2)^2+2*(12*x^2*exp(4)+24*x^4-12*x^3)*log(2)+9*exp(4)^2+ (36*x^2-18*x)*exp(4)+36*x^4-36*x^3+9*x^2),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (24) = 48\).
Time = 0.22 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.61 \[ \int \frac {e^x \left (\left (-6 x^3+21 x^4-12 x^5-12 x^6+e^8 \left (-6 x-3 x^2\right )+e^4 \left (12 x^2-18 x^3-12 x^4\right )\right ) \log (4)+\left (2 x^4-2 e^4 x^4-6 x^5-4 x^6\right ) \log ^2(4)\right )}{9 e^8+9 x^2-36 x^3+36 x^4+e^4 \left (-18 x+36 x^2\right )+\left (12 e^4 x^2-12 x^3+24 x^4\right ) \log (4)+4 x^4 \log ^2(4)} \, dx=\frac {\left (- 4 x^{4} \log {\left (2 \right )} + 2 x^{3} \log {\left (2 \right )} - 2 x^{2} e^{4} \log {\left (2 \right )}\right ) e^{x}}{4 x^{2} \log {\left (2 \right )} + 6 x^{2} - 3 x + 3 e^{4}} \]
integrate((4*(-2*x**4*exp(4)-4*x**6-6*x**5+2*x**4)*ln(2)**2+2*((-3*x**2-6* x)*exp(4)**2+(-12*x**4-18*x**3+12*x**2)*exp(4)-12*x**6-12*x**5+21*x**4-6*x **3)*ln(2))*exp(x)/(16*x**4*ln(2)**2+2*(12*x**2*exp(4)+24*x**4-12*x**3)*ln (2)+9*exp(4)**2+(36*x**2-18*x)*exp(4)+36*x**4-36*x**3+9*x**2),x)
(-4*x**4*log(2) + 2*x**3*log(2) - 2*x**2*exp(4)*log(2))*exp(x)/(4*x**2*log (2) + 6*x**2 - 3*x + 3*exp(4))
Time = 0.33 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.45 \[ \int \frac {e^x \left (\left (-6 x^3+21 x^4-12 x^5-12 x^6+e^8 \left (-6 x-3 x^2\right )+e^4 \left (12 x^2-18 x^3-12 x^4\right )\right ) \log (4)+\left (2 x^4-2 e^4 x^4-6 x^5-4 x^6\right ) \log ^2(4)\right )}{9 e^8+9 x^2-36 x^3+36 x^4+e^4 \left (-18 x+36 x^2\right )+\left (12 e^4 x^2-12 x^3+24 x^4\right ) \log (4)+4 x^4 \log ^2(4)} \, dx=-\frac {2 \, {\left (2 \, x^{4} \log \left (2\right ) - x^{3} \log \left (2\right ) + x^{2} e^{4} \log \left (2\right )\right )} e^{x}}{2 \, x^{2} {\left (2 \, \log \left (2\right ) + 3\right )} - 3 \, x + 3 \, e^{4}} \]
integrate((4*(-2*x^4*exp(4)-4*x^6-6*x^5+2*x^4)*log(2)^2+2*((-3*x^2-6*x)*ex p(4)^2+(-12*x^4-18*x^3+12*x^2)*exp(4)-12*x^6-12*x^5+21*x^4-6*x^3)*log(2))* exp(x)/(16*x^4*log(2)^2+2*(12*x^2*exp(4)+24*x^4-12*x^3)*log(2)+9*exp(4)^2+ (36*x^2-18*x)*exp(4)+36*x^4-36*x^3+9*x^2),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 288 vs. \(2 (32) = 64\).
Time = 0.38 (sec) , antiderivative size = 288, normalized size of antiderivative = 8.73 \[ \int \frac {e^x \left (\left (-6 x^3+21 x^4-12 x^5-12 x^6+e^8 \left (-6 x-3 x^2\right )+e^4 \left (12 x^2-18 x^3-12 x^4\right )\right ) \log (4)+\left (2 x^4-2 e^4 x^4-6 x^5-4 x^6\right ) \log ^2(4)\right )}{9 e^8+9 x^2-36 x^3+36 x^4+e^4 \left (-18 x+36 x^2\right )+\left (12 e^4 x^2-12 x^3+24 x^4\right ) \log (4)+4 x^4 \log ^2(4)} \, dx=-\frac {32 \, x^{4} e^{x} \log \left (2\right )^{4} + 144 \, x^{4} e^{x} \log \left (2\right )^{3} - 16 \, x^{3} e^{x} \log \left (2\right )^{4} + 216 \, x^{4} e^{x} \log \left (2\right )^{2} - 72 \, x^{3} e^{x} \log \left (2\right )^{3} + 16 \, x^{2} e^{\left (x + 4\right )} \log \left (2\right )^{4} + 108 \, x^{4} e^{x} \log \left (2\right ) - 108 \, x^{3} e^{x} \log \left (2\right )^{2} + 72 \, x^{2} e^{\left (x + 4\right )} \log \left (2\right )^{3} - 54 \, x^{3} e^{x} \log \left (2\right ) + 108 \, x^{2} e^{\left (x + 4\right )} \log \left (2\right )^{2} + 24 \, x e^{\left (x + 4\right )} \log \left (2\right )^{3} + 54 \, x^{2} e^{\left (x + 4\right )} \log \left (2\right ) + 36 \, x e^{\left (x + 4\right )} \log \left (2\right )^{2} - 9 \, x e^{x} \log \left (2\right )^{2} - 12 \, e^{\left (x + 8\right )} \log \left (2\right )^{3} - 18 \, e^{\left (x + 8\right )} \log \left (2\right )^{2} + 9 \, e^{\left (x + 4\right )} \log \left (2\right )^{2}}{32 \, x^{2} \log \left (2\right )^{4} + 192 \, x^{2} \log \left (2\right )^{3} + 432 \, x^{2} \log \left (2\right )^{2} - 24 \, x \log \left (2\right )^{3} + 24 \, e^{4} \log \left (2\right )^{3} + 432 \, x^{2} \log \left (2\right ) - 108 \, x \log \left (2\right )^{2} + 108 \, e^{4} \log \left (2\right )^{2} + 162 \, x^{2} - 162 \, x \log \left (2\right ) + 162 \, e^{4} \log \left (2\right ) - 81 \, x + 81 \, e^{4}} \]
integrate((4*(-2*x^4*exp(4)-4*x^6-6*x^5+2*x^4)*log(2)^2+2*((-3*x^2-6*x)*ex p(4)^2+(-12*x^4-18*x^3+12*x^2)*exp(4)-12*x^6-12*x^5+21*x^4-6*x^3)*log(2))* exp(x)/(16*x^4*log(2)^2+2*(12*x^2*exp(4)+24*x^4-12*x^3)*log(2)+9*exp(4)^2+ (36*x^2-18*x)*exp(4)+36*x^4-36*x^3+9*x^2),x, algorithm=\
-(32*x^4*e^x*log(2)^4 + 144*x^4*e^x*log(2)^3 - 16*x^3*e^x*log(2)^4 + 216*x ^4*e^x*log(2)^2 - 72*x^3*e^x*log(2)^3 + 16*x^2*e^(x + 4)*log(2)^4 + 108*x^ 4*e^x*log(2) - 108*x^3*e^x*log(2)^2 + 72*x^2*e^(x + 4)*log(2)^3 - 54*x^3*e ^x*log(2) + 108*x^2*e^(x + 4)*log(2)^2 + 24*x*e^(x + 4)*log(2)^3 + 54*x^2* e^(x + 4)*log(2) + 36*x*e^(x + 4)*log(2)^2 - 9*x*e^x*log(2)^2 - 12*e^(x + 8)*log(2)^3 - 18*e^(x + 8)*log(2)^2 + 9*e^(x + 4)*log(2)^2)/(32*x^2*log(2) ^4 + 192*x^2*log(2)^3 + 432*x^2*log(2)^2 - 24*x*log(2)^3 + 24*e^4*log(2)^3 + 432*x^2*log(2) - 108*x*log(2)^2 + 108*e^4*log(2)^2 + 162*x^2 - 162*x*lo g(2) + 162*e^4*log(2) - 81*x + 81*e^4)
Timed out. \[ \int \frac {e^x \left (\left (-6 x^3+21 x^4-12 x^5-12 x^6+e^8 \left (-6 x-3 x^2\right )+e^4 \left (12 x^2-18 x^3-12 x^4\right )\right ) \log (4)+\left (2 x^4-2 e^4 x^4-6 x^5-4 x^6\right ) \log ^2(4)\right )}{9 e^8+9 x^2-36 x^3+36 x^4+e^4 \left (-18 x+36 x^2\right )+\left (12 e^4 x^2-12 x^3+24 x^4\right ) \log (4)+4 x^4 \log ^2(4)} \, dx=\int -\frac {{\mathrm {e}}^x\,\left (2\,\ln \left (2\right )\,\left ({\mathrm {e}}^8\,\left (3\,x^2+6\,x\right )+{\mathrm {e}}^4\,\left (12\,x^4+18\,x^3-12\,x^2\right )+6\,x^3-21\,x^4+12\,x^5+12\,x^6\right )+4\,{\ln \left (2\right )}^2\,\left (2\,x^4\,{\mathrm {e}}^4-2\,x^4+6\,x^5+4\,x^6\right )\right )}{9\,{\mathrm {e}}^8+16\,x^4\,{\ln \left (2\right )}^2-{\mathrm {e}}^4\,\left (18\,x-36\,x^2\right )+2\,\ln \left (2\right )\,\left (24\,x^4-12\,x^3+12\,{\mathrm {e}}^4\,x^2\right )+9\,x^2-36\,x^3+36\,x^4} \,d x \]
int(-(exp(x)*(2*log(2)*(exp(8)*(6*x + 3*x^2) + exp(4)*(18*x^3 - 12*x^2 + 1 2*x^4) + 6*x^3 - 21*x^4 + 12*x^5 + 12*x^6) + 4*log(2)^2*(2*x^4*exp(4) - 2* x^4 + 6*x^5 + 4*x^6)))/(9*exp(8) + 16*x^4*log(2)^2 - exp(4)*(18*x - 36*x^2 ) + 2*log(2)*(12*x^2*exp(4) - 12*x^3 + 24*x^4) + 9*x^2 - 36*x^3 + 36*x^4), x)
int(-(exp(x)*(2*log(2)*(exp(8)*(6*x + 3*x^2) + exp(4)*(18*x^3 - 12*x^2 + 1 2*x^4) + 6*x^3 - 21*x^4 + 12*x^5 + 12*x^6) + 4*log(2)^2*(2*x^4*exp(4) - 2* x^4 + 6*x^5 + 4*x^6)))/(9*exp(8) + 16*x^4*log(2)^2 - exp(4)*(18*x - 36*x^2 ) + 2*log(2)*(12*x^2*exp(4) - 12*x^3 + 24*x^4) + 9*x^2 - 36*x^3 + 36*x^4), x)