Integrand size = 178, antiderivative size = 32 \[ \int \frac {e^x \left (-54+\sqrt {e} (6-6 x)+66 x+12 x^2-6 x^3+(-36+36 x) \log (2)+(-6+6 x) \log ^2(2)\right )}{81 x^2+e x^2-18 x^3-17 x^4+2 x^5+x^6+\sqrt {e} \left (-18 x^2+2 x^3+2 x^4\right )+\left (108 x^2-12 \sqrt {e} x^2-12 x^3-12 x^4\right ) \log (2)+\left (54 x^2-2 \sqrt {e} x^2-2 x^3-2 x^4\right ) \log ^2(2)+12 x^2 \log ^3(2)+x^2 \log ^4(2)} \, dx=\frac {6 e^x}{x \left (-\sqrt {e}-x-x^2+(3+\log (2))^2\right )} \]
\[ \int \frac {e^x \left (-54+\sqrt {e} (6-6 x)+66 x+12 x^2-6 x^3+(-36+36 x) \log (2)+(-6+6 x) \log ^2(2)\right )}{81 x^2+e x^2-18 x^3-17 x^4+2 x^5+x^6+\sqrt {e} \left (-18 x^2+2 x^3+2 x^4\right )+\left (108 x^2-12 \sqrt {e} x^2-12 x^3-12 x^4\right ) \log (2)+\left (54 x^2-2 \sqrt {e} x^2-2 x^3-2 x^4\right ) \log ^2(2)+12 x^2 \log ^3(2)+x^2 \log ^4(2)} \, dx=\int \frac {e^x \left (-54+\sqrt {e} (6-6 x)+66 x+12 x^2-6 x^3+(-36+36 x) \log (2)+(-6+6 x) \log ^2(2)\right )}{81 x^2+e x^2-18 x^3-17 x^4+2 x^5+x^6+\sqrt {e} \left (-18 x^2+2 x^3+2 x^4\right )+\left (108 x^2-12 \sqrt {e} x^2-12 x^3-12 x^4\right ) \log (2)+\left (54 x^2-2 \sqrt {e} x^2-2 x^3-2 x^4\right ) \log ^2(2)+12 x^2 \log ^3(2)+x^2 \log ^4(2)} \, dx \]
Integrate[(E^x*(-54 + Sqrt[E]*(6 - 6*x) + 66*x + 12*x^2 - 6*x^3 + (-36 + 3 6*x)*Log[2] + (-6 + 6*x)*Log[2]^2))/(81*x^2 + E*x^2 - 18*x^3 - 17*x^4 + 2* x^5 + x^6 + Sqrt[E]*(-18*x^2 + 2*x^3 + 2*x^4) + (108*x^2 - 12*Sqrt[E]*x^2 - 12*x^3 - 12*x^4)*Log[2] + (54*x^2 - 2*Sqrt[E]*x^2 - 2*x^3 - 2*x^4)*Log[2 ]^2 + 12*x^2*Log[2]^3 + x^2*Log[2]^4),x]
Integrate[(E^x*(-54 + Sqrt[E]*(6 - 6*x) + 66*x + 12*x^2 - 6*x^3 + (-36 + 3 6*x)*Log[2] + (-6 + 6*x)*Log[2]^2))/(81*x^2 + E*x^2 - 18*x^3 - 17*x^4 + 2* x^5 + x^6 + Sqrt[E]*(-18*x^2 + 2*x^3 + 2*x^4) + (108*x^2 - 12*Sqrt[E]*x^2 - 12*x^3 - 12*x^4)*Log[2] + (54*x^2 - 2*Sqrt[E]*x^2 - 2*x^3 - 2*x^4)*Log[2 ]^2 + 12*x^2*Log[2]^3 + x^2*Log[2]^4), x]
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 18.67 (sec) , antiderivative size = 1969, normalized size of antiderivative = 61.53, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {6, 6, 6, 2026, 2463, 7239, 27, 25, 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.
| \(\displaystyle \int \frac {e^x \left (-6 x^3+12 x^2+66 x+\sqrt {e} (6-6 x)+(6 x-6) \log ^2(2)+(36 x-36) \log (2)-54\right )}{x^6+2 x^5-17 x^4-18 x^3+e x^2+81 x^2+x^2 \log ^4(2)+12 x^2 \log ^3(2)+\sqrt {e} \left (2 x^4+2 x^3-18 x^2\right )+\left (-2 x^4-2 x^3-2 \sqrt {e} x^2+54 x^2\right ) \log ^2(2)+\left (-12 x^4-12 x^3-12 \sqrt {e} x^2+108 x^2\right ) \log (2)} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {e^x \left (-6 x^3+12 x^2+66 x+\sqrt {e} (6-6 x)+(6 x-6) \log ^2(2)+(36 x-36) \log (2)-54\right )}{x^6+2 x^5-17 x^4-18 x^3+(81+e) x^2+x^2 \log ^4(2)+12 x^2 \log ^3(2)+\sqrt {e} \left (2 x^4+2 x^3-18 x^2\right )+\left (-2 x^4-2 x^3-2 \sqrt {e} x^2+54 x^2\right ) \log ^2(2)+\left (-12 x^4-12 x^3-12 \sqrt {e} x^2+108 x^2\right ) \log (2)}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {e^x \left (-6 x^3+12 x^2+66 x+\sqrt {e} (6-6 x)+(6 x-6) \log ^2(2)+(36 x-36) \log (2)-54\right )}{x^6+2 x^5-17 x^4-18 x^3+x^2 \log ^4(2)+x^2 \left (81+e+12 \log ^3(2)\right )+\sqrt {e} \left (2 x^4+2 x^3-18 x^2\right )+\left (-2 x^4-2 x^3-2 \sqrt {e} x^2+54 x^2\right ) \log ^2(2)+\left (-12 x^4-12 x^3-12 \sqrt {e} x^2+108 x^2\right ) \log (2)}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {e^x \left (-6 x^3+12 x^2+66 x+\sqrt {e} (6-6 x)+(6 x-6) \log ^2(2)+(36 x-36) \log (2)-54\right )}{x^6+2 x^5-17 x^4-18 x^3+x^2 \left (81+e+\log ^4(2)+12 \log ^3(2)\right )+\sqrt {e} \left (2 x^4+2 x^3-18 x^2\right )+\left (-2 x^4-2 x^3-2 \sqrt {e} x^2+54 x^2\right ) \log ^2(2)+\left (-12 x^4-12 x^3-12 \sqrt {e} x^2+108 x^2\right ) \log (2)}dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {e^x \left (-6 x^3+12 x^2+66 x+\sqrt {e} (6-6 x)+(6 x-6) \log ^2(2)+(36 x-36) \log (2)-54\right )}{x^2 \left (x^4+2 x^3-x^2 \left (17-2 \sqrt {e}+2 \log ^2(2)+12 \log (2)\right )+2 x \left (\sqrt {e}-(3+\log (2))^2\right )+\left (\sqrt {e}-(3+\log (2))^2\right )^2\right )}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {4 e^x \left (-6 x^3+12 x^2+66 x+\sqrt {e} (6-6 x)+(6 x-6) \log ^2(2)+(36 x-36) \log (2)-54\right )}{x^2 \left (37-4 \sqrt {e}+4 \log ^2(2)+24 \log (2)\right )^{3/2} \left (-2 x-1+\sqrt {37-4 \sqrt {e}+4 \log ^2(2)+24 \log (2)}\right )}+\frac {4 e^x \left (-6 x^3+12 x^2+66 x+\sqrt {e} (6-6 x)+(6 x-6) \log ^2(2)+(36 x-36) \log (2)-54\right )}{x^2 \left (37-4 \sqrt {e}+4 \log ^2(2)+24 \log (2)\right )^{3/2} \left (2 x+1+\sqrt {37-4 \sqrt {e}+4 \log ^2(2)+24 \log (2)}\right )}+\frac {4 e^x \left (-6 x^3+12 x^2+66 x+\sqrt {e} (6-6 x)+(6 x-6) \log ^2(2)+(36 x-36) \log (2)-54\right )}{x^2 \left (37-4 \sqrt {e}+4 \log ^2(2)+24 \log (2)\right ) \left (-2 x-1+\sqrt {37-4 \sqrt {e}+4 \log ^2(2)+24 \log (2)}\right )^2}+\frac {4 e^x \left (-6 x^3+12 x^2+66 x+\sqrt {e} (6-6 x)+(6 x-6) \log ^2(2)+(36 x-36) \log (2)-54\right )}{x^2 \left (37-4 \sqrt {e}+4 \log ^2(2)+24 \log (2)\right ) \left (2 x+1+\sqrt {37-4 \sqrt {e}+4 \log ^2(2)+24 \log (2)}\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {6 e^x \left (-x^3+2 x^2+x \left (11-\sqrt {e}+\log ^2(2)+\log (64)\right )+\sqrt {e}-9-\log ^2(2)-\log (64)\right )}{x^2 \left (x^2+x+\sqrt {e}-(3+\log (2))^2\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 6 \int -\frac {e^x \left (x^3-2 x^2-\left (11-\sqrt {e}+\log ^2(2)+\log (64)\right ) x+\log (64)+\log ^2(2)-\sqrt {e}+9\right )}{x^2 \left (x^2+x-(3+\log (2))^2+\sqrt {e}\right )^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -6 \int \frac {e^x \left (x^3-2 x^2-\left (11-\sqrt {e}+\log ^2(2)+\log (64)\right ) x+\log (64)+\log ^2(2)-\sqrt {e}+9\right )}{x^2 \left (x^2+x-(3+\log (2))^2+\sqrt {e}\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -6 \int \left (\frac {e^x \left (-81-e-66 \log (2)-6 \log ^3(2)-\log ^4(2)-7 \log (64)-\log ^2(64)-\log ^2(2) (18+\log (64))+2 \sqrt {e} \left (9+\log ^2(2)+\log (64)\right )\right ) x}{\left (\sqrt {e}-(3+\log (2))^2\right )^3 \left (x^2+x-(3+\log (2))^2+\sqrt {e}\right )}+\frac {e^x \left (-\left (\left (9-\sqrt {e}-6 \log ^3(2)-\log ^2(2) (35-\log (64))+\log (64)+\log ^2(64)\right ) x\right )+\sqrt {e} \left (37+4 \log ^2(2)+\log (16777216)\right )+\log (64)-2 \log ^4(2)-24 \log ^3(2)-109 \log ^2(2)-228 \log (2)-2 e-171\right )}{\left (\sqrt {e}-(3+\log (2))^2\right )^2 \left (x^2+x-(3+\log (2))^2+\sqrt {e}\right )^2}+\frac {e^x \left (81+e+66 \log (2)+6 \log ^3(2)+\log ^4(2)+7 \log (64)+\log ^2(64)+\log ^2(2) (18+\log (64))-2 \sqrt {e} \left (9+\log ^2(2)+\log (64)\right )\right )}{\left (\sqrt {e}-(3+\log (2))^2\right )^3 x}+\frac {e^x \left (9-\sqrt {e}+\log ^2(2)+\log (64)\right )}{\left (\sqrt {e}-(3+\log (2))^2\right )^2 x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -6 \left (\frac {\left (81+e+66 \log (2)+6 \log ^3(2)+\log ^4(2)+7 \log (64)+\log ^2(64)+\log ^2(2) (18+\log (64))-2 \sqrt {e} \left (9+\log ^2(2)+\log (64)\right )\right ) \operatorname {ExpIntegralEi}(x)}{\left (\sqrt {e}-(3+\log (2))^2\right )^3}-\frac {\operatorname {ExpIntegralEi}(x)}{\sqrt {e}-(3+\log (2))^2}+\frac {e^x \left (1-\sqrt {37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)}\right )}{\left (37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)\right ) \left (\sqrt {e}-(3+\log (2))^2\right ) \left (2 x-\sqrt {37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)}+1\right )}+\frac {e^x \left (1+\sqrt {37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)}\right )}{\left (37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)\right ) \left (\sqrt {e}-(3+\log (2))^2\right ) \left (2 x+\sqrt {37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)}+1\right )}+\frac {2 e^x \left (171+2 e+228 \log (2)+109 \log ^2(2)+24 \log ^3(2)+2 \log ^4(2)-\log (64)-\sqrt {e} \left (37+4 \log ^2(2)+\log (16777216)\right )\right )}{\left (37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)\right ) \left (\sqrt {e}-(3+\log (2))^2\right )^2 \left (2 x-\sqrt {37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)}+1\right )}+\frac {2 e^x \left (171+2 e+228 \log (2)+109 \log ^2(2)+24 \log ^3(2)+2 \log ^4(2)-\log (64)-\sqrt {e} \left (37+4 \log ^2(2)+\log (16777216)\right )\right )}{\left (37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)\right ) \left (\sqrt {e}-(3+\log (2))^2\right )^2 \left (2 x+\sqrt {37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)}+1\right )}-\frac {e^{\frac {1}{2} \left (-1+\sqrt {37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (2 x-\sqrt {37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)}+1\right )\right ) \left (171+2 e+228 \log (2)+109 \log ^2(2)+24 \log ^3(2)+2 \log ^4(2)-\log (64)-\sqrt {e} \left (37+4 \log ^2(2)+\log (16777216)\right )\right )}{\left (37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)\right ) \left (\sqrt {e}-(3+\log (2))^2\right )^2}-\frac {e^{\frac {1}{2} \left (-1-\sqrt {37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (2 x+\sqrt {37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)}+1\right )\right ) \left (171+2 e+228 \log (2)+109 \log ^2(2)+24 \log ^3(2)+2 \log ^4(2)-\log (64)-\sqrt {e} \left (37+4 \log ^2(2)+\log (16777216)\right )\right )}{\left (37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)\right ) \left (\sqrt {e}-(3+\log (2))^2\right )^2}+\frac {2 e^{\frac {1}{2} \left (-1+\sqrt {37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (2 x-\sqrt {37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)}+1\right )\right ) \left (171+2 e+228 \log (2)+109 \log ^2(2)+24 \log ^3(2)+2 \log ^4(2)-\log (64)-\sqrt {e} \left (37+4 \log ^2(2)+\log (16777216)\right )\right )}{\left (37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)\right )^{3/2} \left (\sqrt {e}-(3+\log (2))^2\right )^2}-\frac {2 e^{\frac {1}{2} \left (-1-\sqrt {37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (2 x+\sqrt {37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)}+1\right )\right ) \left (171+2 e+228 \log (2)+109 \log ^2(2)+24 \log ^3(2)+2 \log ^4(2)-\log (64)-\sqrt {e} \left (37+4 \log ^2(2)+\log (16777216)\right )\right )}{\left (37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)\right )^{3/2} \left (\sqrt {e}-(3+\log (2))^2\right )^2}-\frac {e^{\frac {1}{2} \left (-1-\sqrt {37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (2 x+\sqrt {37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)}+1\right )\right ) \left (1+\frac {1}{\sqrt {37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)}}\right ) \left (81+e+66 \log (2)+6 \log ^3(2)+\log ^4(2)+7 \log (64)+\log ^2(64)+\log ^2(2) (18+\log (64))-2 \sqrt {e} \left (9+\log ^2(2)+\log (64)\right )\right )}{2 \left (\sqrt {e}-(3+\log (2))^2\right )^3}-\frac {e^{\frac {1}{2} \left (-1+\sqrt {37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (2 x-\sqrt {37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)}+1\right )\right ) \left (1-\frac {1}{\sqrt {37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)}}\right ) \left (81+e+66 \log (2)+6 \log ^3(2)+\log ^4(2)+7 \log (64)+\log ^2(64)+\log ^2(2) (18+\log (64))-2 \sqrt {e} \left (9+\log ^2(2)+\log (64)\right )\right )}{2 \left (\sqrt {e}-(3+\log (2))^2\right )^3}-\frac {e^{\frac {1}{2} \left (-1-\sqrt {37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (2 x+\sqrt {37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)}+1\right )\right ) \left (1+\sqrt {37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)}\right )}{2 \left (37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)\right ) \left (\sqrt {e}-(3+\log (2))^2\right )}-\frac {e^{\frac {1}{2} \left (-1+\sqrt {37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (2 x-\sqrt {37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)}+1\right )\right ) \left (1-\sqrt {37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)}\right )}{2 \left (37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)\right ) \left (\sqrt {e}-(3+\log (2))^2\right )}+\frac {e^x}{x \left (\sqrt {e}-(3+\log (2))^2\right )}+\frac {e^{\frac {1}{2} \left (-1+\sqrt {37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (2 x-\sqrt {37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)}+1\right )\right )}{\left (37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)\right )^{3/2} \left (\sqrt {e}-(3+\log (2))^2\right )}-\frac {e^{\frac {1}{2} \left (-1-\sqrt {37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (2 x+\sqrt {37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)}+1\right )\right )}{\left (37-4 \sqrt {e}+24 \log (2)+4 \log ^2(2)\right )^{3/2} \left (\sqrt {e}-(3+\log (2))^2\right )}\right )\) |
Int[(E^x*(-54 + Sqrt[E]*(6 - 6*x) + 66*x + 12*x^2 - 6*x^3 + (-36 + 36*x)*L og[2] + (-6 + 6*x)*Log[2]^2))/(81*x^2 + E*x^2 - 18*x^3 - 17*x^4 + 2*x^5 + x^6 + Sqrt[E]*(-18*x^2 + 2*x^3 + 2*x^4) + (108*x^2 - 12*Sqrt[E]*x^2 - 12*x ^3 - 12*x^4)*Log[2] + (54*x^2 - 2*Sqrt[E]*x^2 - 2*x^3 - 2*x^4)*Log[2]^2 + 12*x^2*Log[2]^3 + x^2*Log[2]^4),x]
-6*(E^x/(x*(Sqrt[E] - (3 + Log[2])^2)) - ExpIntegralEi[x]/(Sqrt[E] - (3 + Log[2])^2) + (E^((-1 + Sqrt[37 - 4*Sqrt[E] + 24*Log[2] + 4*Log[2]^2])/2)*E xpIntegralEi[(1 + 2*x - Sqrt[37 - 4*Sqrt[E] + 24*Log[2] + 4*Log[2]^2])/2]) /((37 - 4*Sqrt[E] + 24*Log[2] + 4*Log[2]^2)^(3/2)*(Sqrt[E] - (3 + Log[2])^ 2)) - (E^((-1 - Sqrt[37 - 4*Sqrt[E] + 24*Log[2] + 4*Log[2]^2])/2)*ExpInteg ralEi[(1 + 2*x + Sqrt[37 - 4*Sqrt[E] + 24*Log[2] + 4*Log[2]^2])/2])/((37 - 4*Sqrt[E] + 24*Log[2] + 4*Log[2]^2)^(3/2)*(Sqrt[E] - (3 + Log[2])^2)) - ( E^((-1 + Sqrt[37 - 4*Sqrt[E] + 24*Log[2] + 4*Log[2]^2])/2)*ExpIntegralEi[( 1 + 2*x - Sqrt[37 - 4*Sqrt[E] + 24*Log[2] + 4*Log[2]^2])/2]*(1 - Sqrt[37 - 4*Sqrt[E] + 24*Log[2] + 4*Log[2]^2]))/(2*(37 - 4*Sqrt[E] + 24*Log[2] + 4* Log[2]^2)*(Sqrt[E] - (3 + Log[2])^2)) + (E^x*(1 - Sqrt[37 - 4*Sqrt[E] + 24 *Log[2] + 4*Log[2]^2]))/((37 - 4*Sqrt[E] + 24*Log[2] + 4*Log[2]^2)*(Sqrt[E ] - (3 + Log[2])^2)*(1 + 2*x - Sqrt[37 - 4*Sqrt[E] + 24*Log[2] + 4*Log[2]^ 2])) - (E^((-1 - Sqrt[37 - 4*Sqrt[E] + 24*Log[2] + 4*Log[2]^2])/2)*ExpInte gralEi[(1 + 2*x + Sqrt[37 - 4*Sqrt[E] + 24*Log[2] + 4*Log[2]^2])/2]*(1 + S qrt[37 - 4*Sqrt[E] + 24*Log[2] + 4*Log[2]^2]))/(2*(37 - 4*Sqrt[E] + 24*Log [2] + 4*Log[2]^2)*(Sqrt[E] - (3 + Log[2])^2)) + (E^x*(1 + Sqrt[37 - 4*Sqrt [E] + 24*Log[2] + 4*Log[2]^2]))/((37 - 4*Sqrt[E] + 24*Log[2] + 4*Log[2]^2) *(Sqrt[E] - (3 + Log[2])^2)*(1 + 2*x + Sqrt[37 - 4*Sqrt[E] + 24*Log[2] + 4 *Log[2]^2])) + (ExpIntegralEi[x]*(81 + E + 66*Log[2] + 6*Log[2]^3 + Log...
3.24.8.3.1 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[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
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]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.43 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88
| method | result | size |
| gosper | \(-\frac {6 \,{\mathrm e}^{x}}{x \left (-\ln \left (2\right )^{2}+x^{2}+{\mathrm e}^{\frac {1}{2}}-6 \ln \left (2\right )+x -9\right )}\) | \(28\) |
| norman | \(-\frac {6 \,{\mathrm e}^{x}}{x \left (-\ln \left (2\right )^{2}+x^{2}+{\mathrm e}^{\frac {1}{2}}-6 \ln \left (2\right )+x -9\right )}\) | \(28\) |
| risch | \(-\frac {6 \,{\mathrm e}^{x}}{x \left (-\ln \left (2\right )^{2}+x^{2}+{\mathrm e}^{\frac {1}{2}}-6 \ln \left (2\right )+x -9\right )}\) | \(28\) |
| parallelrisch | \(-\frac {6 \,{\mathrm e}^{x}}{x \left (-\ln \left (2\right )^{2}+x^{2}+{\mathrm e}^{\frac {1}{2}}-6 \ln \left (2\right )+x -9\right )}\) | \(28\) |
| default | \(\text {Expression too large to display}\) | \(40397\) |
int(((6*x-6)*ln(2)^2+(36*x-36)*ln(2)+(6-6*x)*exp(1/2)-6*x^3+12*x^2+66*x-54 )*exp(x)/(x^2*ln(2)^4+12*x^2*ln(2)^3+(-2*x^2*exp(1/2)-2*x^4-2*x^3+54*x^2)* ln(2)^2+(-12*x^2*exp(1/2)-12*x^4-12*x^3+108*x^2)*ln(2)+x^2*exp(1/2)^2+(2*x ^4+2*x^3-18*x^2)*exp(1/2)+x^6+2*x^5-17*x^4-18*x^3+81*x^2),x,method=_RETURN VERBOSE)
Time = 0.25 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \frac {e^x \left (-54+\sqrt {e} (6-6 x)+66 x+12 x^2-6 x^3+(-36+36 x) \log (2)+(-6+6 x) \log ^2(2)\right )}{81 x^2+e x^2-18 x^3-17 x^4+2 x^5+x^6+\sqrt {e} \left (-18 x^2+2 x^3+2 x^4\right )+\left (108 x^2-12 \sqrt {e} x^2-12 x^3-12 x^4\right ) \log (2)+\left (54 x^2-2 \sqrt {e} x^2-2 x^3-2 x^4\right ) \log ^2(2)+12 x^2 \log ^3(2)+x^2 \log ^4(2)} \, dx=-\frac {6 \, e^{x}}{x^{3} - x \log \left (2\right )^{2} + x^{2} + x e^{\frac {1}{2}} - 6 \, x \log \left (2\right ) - 9 \, x} \]
integrate(((6*x-6)*log(2)^2+(36*x-36)*log(2)+(6-6*x)*exp(1/2)-6*x^3+12*x^2 +66*x-54)*exp(x)/(x^2*log(2)^4+12*x^2*log(2)^3+(-2*x^2*exp(1/2)-2*x^4-2*x^ 3+54*x^2)*log(2)^2+(-12*x^2*exp(1/2)-12*x^4-12*x^3+108*x^2)*log(2)+x^2*exp (1/2)^2+(2*x^4+2*x^3-18*x^2)*exp(1/2)+x^6+2*x^5-17*x^4-18*x^3+81*x^2),x, a lgorithm=\
Timed out. \[ \int \frac {e^x \left (-54+\sqrt {e} (6-6 x)+66 x+12 x^2-6 x^3+(-36+36 x) \log (2)+(-6+6 x) \log ^2(2)\right )}{81 x^2+e x^2-18 x^3-17 x^4+2 x^5+x^6+\sqrt {e} \left (-18 x^2+2 x^3+2 x^4\right )+\left (108 x^2-12 \sqrt {e} x^2-12 x^3-12 x^4\right ) \log (2)+\left (54 x^2-2 \sqrt {e} x^2-2 x^3-2 x^4\right ) \log ^2(2)+12 x^2 \log ^3(2)+x^2 \log ^4(2)} \, dx=\text {Timed out} \]
integrate(((6*x-6)*ln(2)**2+(36*x-36)*ln(2)+(6-6*x)*exp(1/2)-6*x**3+12*x** 2+66*x-54)*exp(x)/(x**2*ln(2)**4+12*x**2*ln(2)**3+(-2*x**2*exp(1/2)-2*x**4 -2*x**3+54*x**2)*ln(2)**2+(-12*x**2*exp(1/2)-12*x**4-12*x**3+108*x**2)*ln( 2)+x**2*exp(1/2)**2+(2*x**4+2*x**3-18*x**2)*exp(1/2)+x**6+2*x**5-17*x**4-1 8*x**3+81*x**2),x)
Time = 0.63 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.94 \[ \int \frac {e^x \left (-54+\sqrt {e} (6-6 x)+66 x+12 x^2-6 x^3+(-36+36 x) \log (2)+(-6+6 x) \log ^2(2)\right )}{81 x^2+e x^2-18 x^3-17 x^4+2 x^5+x^6+\sqrt {e} \left (-18 x^2+2 x^3+2 x^4\right )+\left (108 x^2-12 \sqrt {e} x^2-12 x^3-12 x^4\right ) \log (2)+\left (54 x^2-2 \sqrt {e} x^2-2 x^3-2 x^4\right ) \log ^2(2)+12 x^2 \log ^3(2)+x^2 \log ^4(2)} \, dx=-\frac {6 \, e^{x}}{x^{3} - {\left (\log \left (2\right )^{2} - e^{\frac {1}{2}} + 6 \, \log \left (2\right ) + 9\right )} x + x^{2}} \]
integrate(((6*x-6)*log(2)^2+(36*x-36)*log(2)+(6-6*x)*exp(1/2)-6*x^3+12*x^2 +66*x-54)*exp(x)/(x^2*log(2)^4+12*x^2*log(2)^3+(-2*x^2*exp(1/2)-2*x^4-2*x^ 3+54*x^2)*log(2)^2+(-12*x^2*exp(1/2)-12*x^4-12*x^3+108*x^2)*log(2)+x^2*exp (1/2)^2+(2*x^4+2*x^3-18*x^2)*exp(1/2)+x^6+2*x^5-17*x^4-18*x^3+81*x^2),x, a lgorithm=\
Time = 0.46 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \frac {e^x \left (-54+\sqrt {e} (6-6 x)+66 x+12 x^2-6 x^3+(-36+36 x) \log (2)+(-6+6 x) \log ^2(2)\right )}{81 x^2+e x^2-18 x^3-17 x^4+2 x^5+x^6+\sqrt {e} \left (-18 x^2+2 x^3+2 x^4\right )+\left (108 x^2-12 \sqrt {e} x^2-12 x^3-12 x^4\right ) \log (2)+\left (54 x^2-2 \sqrt {e} x^2-2 x^3-2 x^4\right ) \log ^2(2)+12 x^2 \log ^3(2)+x^2 \log ^4(2)} \, dx=-\frac {12 \, e^{x}}{x^{3} - x \log \left (2\right )^{2} + x^{2} + x e^{\frac {1}{2}} - 6 \, x \log \left (2\right ) - 9 \, x} \]
integrate(((6*x-6)*log(2)^2+(36*x-36)*log(2)+(6-6*x)*exp(1/2)-6*x^3+12*x^2 +66*x-54)*exp(x)/(x^2*log(2)^4+12*x^2*log(2)^3+(-2*x^2*exp(1/2)-2*x^4-2*x^ 3+54*x^2)*log(2)^2+(-12*x^2*exp(1/2)-12*x^4-12*x^3+108*x^2)*log(2)+x^2*exp (1/2)^2+(2*x^4+2*x^3-18*x^2)*exp(1/2)+x^6+2*x^5-17*x^4-18*x^3+81*x^2),x, a lgorithm=\
Time = 15.58 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88 \[ \int \frac {e^x \left (-54+\sqrt {e} (6-6 x)+66 x+12 x^2-6 x^3+(-36+36 x) \log (2)+(-6+6 x) \log ^2(2)\right )}{81 x^2+e x^2-18 x^3-17 x^4+2 x^5+x^6+\sqrt {e} \left (-18 x^2+2 x^3+2 x^4\right )+\left (108 x^2-12 \sqrt {e} x^2-12 x^3-12 x^4\right ) \log (2)+\left (54 x^2-2 \sqrt {e} x^2-2 x^3-2 x^4\right ) \log ^2(2)+12 x^2 \log ^3(2)+x^2 \log ^4(2)} \, dx=-\frac {6\,{\mathrm {e}}^x}{x^3+x^2+\left (\sqrt {\mathrm {e}}-\ln \left (64\right )-{\ln \left (2\right )}^2-9\right )\,x} \]
int((exp(x)*(66*x + log(2)*(36*x - 36) + log(2)^2*(6*x - 6) + 12*x^2 - 6*x ^3 - exp(1/2)*(6*x - 6) - 54))/(12*x^2*log(2)^3 + x^2*log(2)^4 - log(2)*(1 2*x^2*exp(1/2) - 108*x^2 + 12*x^3 + 12*x^4) + x^2*exp(1) + exp(1/2)*(2*x^3 - 18*x^2 + 2*x^4) - log(2)^2*(2*x^2*exp(1/2) - 54*x^2 + 2*x^3 + 2*x^4) + 81*x^2 - 18*x^3 - 17*x^4 + 2*x^5 + x^6),x)