Integrand size = 102, antiderivative size = 27 \[ \int \frac {16+4 e^{2 x}+\left (-36-72 x-24 x^2\right ) \log (3)+e^x \left (-16+\left (18+18 x-6 x^2-4 x^3\right ) \log (3)-4 \log (4)\right )+\left (8+\left (-9-18 x-6 x^2\right ) \log (3)\right ) \log (4)+\log ^2(4)}{16+4 e^{2 x}+e^x (-16-4 \log (4))+8 \log (4)+\log ^2(4)} \, dx=x+\frac {x (3+x) (3+2 x) \log (3)}{-4+2 e^x-\log (4)} \] Output:
x+(3+2*x)/(2*exp(x)-2*ln(2)-4)*x*ln(3)*(3+x)
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 5.28 (sec) , antiderivative size = 445, normalized size of antiderivative = 16.48 \[ \int \frac {16+4 e^{2 x}+\left (-36-72 x-24 x^2\right ) \log (3)+e^x \left (-16+\left (18+18 x-6 x^2-4 x^3\right ) \log (3)-4 \log (4)\right )+\left (8+\left (-9-18 x-6 x^2\right ) \log (3)\right ) \log (4)+\log ^2(4)}{16+4 e^{2 x}+e^x (-16-4 \log (4))+8 \log (4)+\log ^2(4)} \, dx=-\frac {-16 x+8 e^x x-\log ^2(4)+x^3 \log (4) \log (9)+\log (4) \log (16)-x \log (256)+\log (4) \log \left (\frac {19683}{4}\right )+x \log (4) \log \left (\frac {19683}{4}\right )+x^3 \log (6561)-\log (4) \log (19683)+x^2 \log (4) \log (19683)-e^x x \log \left (\frac {387420489}{16}\right )+e^x x \log (387420489)+x \log \left (\frac {150094635296999121}{256}\right )+x^2 \log (150094635296999121)-x \log (16) \log (27) \log (4+\log (4))-x^2 \log (9) \log (64) \log (4+\log (4))+x \log (4) \log (729) \log (4+\log (4))+x^2 \log (4) \log (729) \log (4+\log (4))-\log ^2(4) \log \left (4-2 e^x+\log (4)\right )+e^x \log (16) \log \left (4-2 e^x+\log (4)\right )+x \log (16) \log (27) \log \left (4-2 e^x+\log (4)\right )+x^2 \log (9) \log (64) \log \left (4-2 e^x+\log (4)\right )-\log (256) \log \left (4-2 e^x+\log (4)\right )-x \log (4) \log (729) \log \left (4-2 e^x+\log (4)\right )-x^2 \log (4) \log (729) \log \left (4-2 e^x+\log (4)\right )-\log (4) \log \left (\frac {19683}{4}\right ) \log \left (4-2 e^x+\log (4)\right )+\log (4) \log (19683) \log \left (4-2 e^x+\log (4)\right )+e^x \log \left (\frac {387420489}{16}\right ) \log \left (4-2 e^x+\log (4)\right )-e^x \log (387420489) \log \left (4-2 e^x+\log (4)\right )-\log \left (\frac {150094635296999121}{256}\right ) \log \left (4-2 e^x+\log (4)\right )+\log (150094635296999121) \log \left (4-2 e^x+\log (4)\right )+(\log (16) \log (27)-\log (4) \log (729)+x \log (9) \log (4096)-x \log (4) \log (531441)) \operatorname {PolyLog}\left (2,\frac {2 e^x}{4+\log (4)}\right )+(-\log (9) \log (4096)+\log (4) \log (531441)) \operatorname {PolyLog}\left (3,\frac {2 e^x}{4+\log (4)}\right )}{(4+\log (4)) \left (4-2 e^x+\log (4)\right )} \] Input:
Integrate[(16 + 4*E^(2*x) + (-36 - 72*x - 24*x^2)*Log[3] + E^x*(-16 + (18 + 18*x - 6*x^2 - 4*x^3)*Log[3] - 4*Log[4]) + (8 + (-9 - 18*x - 6*x^2)*Log[ 3])*Log[4] + Log[4]^2)/(16 + 4*E^(2*x) + E^x*(-16 - 4*Log[4]) + 8*Log[4] + Log[4]^2),x]
Output:
-((-16*x + 8*E^x*x - Log[4]^2 + x^3*Log[4]*Log[9] + Log[4]*Log[16] - x*Log [256] + Log[4]*Log[19683/4] + x*Log[4]*Log[19683/4] + x^3*Log[6561] - Log[ 4]*Log[19683] + x^2*Log[4]*Log[19683] - E^x*x*Log[387420489/16] + E^x*x*Lo g[387420489] + x*Log[150094635296999121/256] + x^2*Log[150094635296999121] - x*Log[16]*Log[27]*Log[4 + Log[4]] - x^2*Log[9]*Log[64]*Log[4 + Log[4]] + x*Log[4]*Log[729]*Log[4 + Log[4]] + x^2*Log[4]*Log[729]*Log[4 + Log[4]] - Log[4]^2*Log[4 - 2*E^x + Log[4]] + E^x*Log[16]*Log[4 - 2*E^x + Log[4]] + x*Log[16]*Log[27]*Log[4 - 2*E^x + Log[4]] + x^2*Log[9]*Log[64]*Log[4 - 2* E^x + Log[4]] - Log[256]*Log[4 - 2*E^x + Log[4]] - x*Log[4]*Log[729]*Log[4 - 2*E^x + Log[4]] - x^2*Log[4]*Log[729]*Log[4 - 2*E^x + Log[4]] - Log[4]* Log[19683/4]*Log[4 - 2*E^x + Log[4]] + Log[4]*Log[19683]*Log[4 - 2*E^x + L og[4]] + E^x*Log[387420489/16]*Log[4 - 2*E^x + Log[4]] - E^x*Log[387420489 ]*Log[4 - 2*E^x + Log[4]] - Log[150094635296999121/256]*Log[4 - 2*E^x + Lo g[4]] + Log[150094635296999121]*Log[4 - 2*E^x + Log[4]] + (Log[16]*Log[27] - Log[4]*Log[729] + x*Log[9]*Log[4096] - x*Log[4]*Log[531441])*PolyLog[2, (2*E^x)/(4 + Log[4])] + (-(Log[9]*Log[4096]) + Log[4]*Log[531441])*PolyLo g[3, (2*E^x)/(4 + Log[4])])/((4 + Log[4])*(4 - 2*E^x + Log[4])))
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 4.24 (sec) , antiderivative size = 897, normalized size of antiderivative = 33.22, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {7292, 7292, 27, 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.
| \(\displaystyle \int \frac {\left (-24 x^2-72 x-36\right ) \log (3)+\log (4) \left (\left (-6 x^2-18 x-9\right ) \log (3)+8\right )+e^x \left (\left (-4 x^3-6 x^2+18 x+18\right ) \log (3)-16-4 \log (4)\right )+4 e^{2 x}+16+\log ^2(4)}{4 e^{2 x}+e^x (-16-4 \log (4))+16+\log ^2(4)+8 \log (4)} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {\left (-24 x^2-72 x-36\right ) \log (3)+\log (4) \left (\left (-6 x^2-18 x-9\right ) \log (3)+8\right )+e^x \left (\left (-4 x^3-6 x^2+18 x+18\right ) \log (3)-16-4 \log (4)\right )+4 e^{2 x}+16 \left (1+\frac {\log ^2(4)}{16}\right )}{\left (2 e^x-4 \left (1+\frac {\log (2)}{2}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {\left (-24 x^2-72 x-36\right ) \log (3)+\log (4) \left (\left (-6 x^2-18 x-9\right ) \log (3)+8\right )+e^x \left (\left (-4 x^3-6 x^2+18 x+18\right ) \log (3)-16-4 \log (4)\right )+4 e^{2 x}+16 \left (1+\frac {\log ^2(4)}{16}\right )}{4 \left (e^x-2 \left (1+\frac {\log (2)}{2}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \int \frac {-12 \log (3) \left (2 x^2+6 x+3\right )+4 e^{2 x}+2 e^x \left (\left (-2 x^3-3 x^2+9 x+9\right ) \log (3)-2 (4+\log (4))\right )+\log ^2(4)+\left (8-3 \left (2 x^2+6 x+3\right ) \log (3)\right ) \log (4)+16}{\left (2-e^x+\log (2)\right )^2}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \frac {1}{4} \int \frac {-12 \log (3) \left (2 x^2+6 x+3\right )+4 e^{2 x}+2 e^x \left (\left (-2 x^3-3 x^2+9 x+9\right ) \log (3)-2 (4+\log (4))\right )+16 \left (1+\frac {\log ^2(4)}{16}\right )+\left (8-3 \left (2 x^2+6 x+3\right ) \log (3)\right ) \log (4)}{\left (e^x-2 \left (1+\frac {\log (2)}{2}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{4} \int \left (\frac {2 x \log (3) \left (-\left ((4+\log (4)) x^2\right )-3 (6+\log (8)) x-\log (512)-18\right )}{\left (e^x-2 \left (1+\frac {\log (2)}{2}\right )\right )^2}+\frac {2 \left (-\log (9) x^3-\log (27) x^2+\log (19683) x+\log (19683)\right )}{e^x-2 \left (1+\frac {\log (2)}{2}\right )}+4\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{4} \left (\frac {\log (9) x^4}{2 (2+\log (2))}-\frac {\log (3) (4+\log (4)) x^4}{2 (2+\log (2))^2}-\frac {2 \log (9) \log \left (1-\frac {e^x}{2+\log (2)}\right ) x^3}{2+\log (2)}+\frac {2 \log (3) (4+\log (4)) \log \left (1-\frac {e^x}{2+\log (2)}\right ) x^3}{(2+\log (2))^2}+\frac {2 \log (27) x^3}{3 (2+\log (2))}-\frac {2 \log (3) (6+\log (8)) x^3}{(2+\log (2))^2}-\frac {2 \log (3) (4+\log (4)) x^3}{(2+\log (2)) \left (2-e^x+\log (2)\right )}+\frac {2 \log (3) (4+\log (4)) x^3}{(2+\log (2))^2}-\frac {2 \log (27) \log \left (1-\frac {e^x}{2+\log (2)}\right ) x^2}{2+\log (2)}+\frac {6 \log (3) (6+\log (8)) \log \left (1-\frac {e^x}{2+\log (2)}\right ) x^2}{(2+\log (2))^2}-\frac {6 \log (3) (4+\log (4)) \log \left (1-\frac {e^x}{2+\log (2)}\right ) x^2}{(2+\log (2))^2}-\frac {6 \log (9) \operatorname {PolyLog}\left (2,\frac {e^x}{2+\log (2)}\right ) x^2}{2+\log (2)}+\frac {6 \log (3) (4+\log (4)) \operatorname {PolyLog}\left (2,\frac {e^x}{2+\log (2)}\right ) x^2}{(2+\log (2))^2}-\frac {\log (19683) x^2}{2+\log (2)}-\frac {\log (3) (18+\log (512)) x^2}{(2+\log (2))^2}-\frac {6 \log (3) (6+\log (8)) x^2}{(2+\log (2)) \left (2-e^x+\log (2)\right )}+\frac {6 \log (3) (6+\log (8)) x^2}{(2+\log (2))^2}+\frac {2 \log (19683) \log \left (1-\frac {e^x}{2+\log (2)}\right ) x}{2+\log (2)}+\frac {2 \log (3) (18+\log (512)) \log \left (1-\frac {e^x}{2+\log (2)}\right ) x}{(2+\log (2))^2}-\frac {12 \log (3) (6+\log (8)) \log \left (1-\frac {e^x}{2+\log (2)}\right ) x}{(2+\log (2))^2}-\frac {4 \log (27) \operatorname {PolyLog}\left (2,\frac {e^x}{2+\log (2)}\right ) x}{2+\log (2)}+\frac {12 \log (3) (6+\log (8)) \operatorname {PolyLog}\left (2,\frac {e^x}{2+\log (2)}\right ) x}{(2+\log (2))^2}-\frac {12 \log (3) (4+\log (4)) \operatorname {PolyLog}\left (2,\frac {e^x}{2+\log (2)}\right ) x}{(2+\log (2))^2}+\frac {12 \log (9) \operatorname {PolyLog}\left (3,\frac {e^x}{2+\log (2)}\right ) x}{2+\log (2)}-\frac {12 \log (3) (4+\log (4)) \operatorname {PolyLog}\left (3,\frac {e^x}{2+\log (2)}\right ) x}{(2+\log (2))^2}-\frac {2 \log (19683) x}{2+\log (2)}-\frac {2 \log (3) (18+\log (512)) x}{(2+\log (2)) \left (2-e^x+\log (2)\right )}+\frac {2 \log (3) (18+\log (512)) x}{(2+\log (2))^2}+4 x+\frac {2 \log (19683) \log \left (2-e^x+\log (2)\right )}{2+\log (2)}-\frac {2 \log (3) (18+\log (512)) \log \left (2-e^x+\log (2)\right )}{(2+\log (2))^2}+\frac {2 \log (19683) \operatorname {PolyLog}\left (2,\frac {e^x}{2+\log (2)}\right )}{2+\log (2)}+\frac {2 \log (3) (18+\log (512)) \operatorname {PolyLog}\left (2,\frac {e^x}{2+\log (2)}\right )}{(2+\log (2))^2}-\frac {12 \log (3) (6+\log (8)) \operatorname {PolyLog}\left (2,\frac {e^x}{2+\log (2)}\right )}{(2+\log (2))^2}+\frac {4 \log (27) \operatorname {PolyLog}\left (3,\frac {e^x}{2+\log (2)}\right )}{2+\log (2)}-\frac {12 \log (3) (6+\log (8)) \operatorname {PolyLog}\left (3,\frac {e^x}{2+\log (2)}\right )}{(2+\log (2))^2}+\frac {12 \log (3) (4+\log (4)) \operatorname {PolyLog}\left (3,\frac {e^x}{2+\log (2)}\right )}{(2+\log (2))^2}-\frac {12 \log (9) \operatorname {PolyLog}\left (4,\frac {e^x}{2+\log (2)}\right )}{2+\log (2)}+\frac {12 \log (3) (4+\log (4)) \operatorname {PolyLog}\left (4,\frac {e^x}{2+\log (2)}\right )}{(2+\log (2))^2}\right )\) |
Input:
Int[(16 + 4*E^(2*x) + (-36 - 72*x - 24*x^2)*Log[3] + E^x*(-16 + (18 + 18*x - 6*x^2 - 4*x^3)*Log[3] - 4*Log[4]) + (8 + (-9 - 18*x - 6*x^2)*Log[3])*Lo g[4] + Log[4]^2)/(16 + 4*E^(2*x) + E^x*(-16 - 4*Log[4]) + 8*Log[4] + Log[4 ]^2),x]
Output:
(4*x + (2*x^3*Log[3]*(4 + Log[4]))/(2 + Log[2])^2 - (x^4*Log[3]*(4 + Log[4 ]))/(2*(2 + Log[2])^2) - (2*x^3*Log[3]*(4 + Log[4]))/((2 + Log[2])*(2 - E^ x + Log[2])) + (6*x^2*Log[3]*(6 + Log[8]))/(2 + Log[2])^2 - (2*x^3*Log[3]* (6 + Log[8]))/(2 + Log[2])^2 - (6*x^2*Log[3]*(6 + Log[8]))/((2 + Log[2])*( 2 - E^x + Log[2])) + (x^4*Log[9])/(2*(2 + Log[2])) + (2*x^3*Log[27])/(3*(2 + Log[2])) + (2*x*Log[3]*(18 + Log[512]))/(2 + Log[2])^2 - (x^2*Log[3]*(1 8 + Log[512]))/(2 + Log[2])^2 - (2*x*Log[3]*(18 + Log[512]))/((2 + Log[2]) *(2 - E^x + Log[2])) - (2*x*Log[19683])/(2 + Log[2]) - (x^2*Log[19683])/(2 + Log[2]) - (2*Log[3]*(18 + Log[512])*Log[2 - E^x + Log[2]])/(2 + Log[2]) ^2 + (2*Log[19683]*Log[2 - E^x + Log[2]])/(2 + Log[2]) - (6*x^2*Log[3]*(4 + Log[4])*Log[1 - E^x/(2 + Log[2])])/(2 + Log[2])^2 + (2*x^3*Log[3]*(4 + L og[4])*Log[1 - E^x/(2 + Log[2])])/(2 + Log[2])^2 - (12*x*Log[3]*(6 + Log[8 ])*Log[1 - E^x/(2 + Log[2])])/(2 + Log[2])^2 + (6*x^2*Log[3]*(6 + Log[8])* Log[1 - E^x/(2 + Log[2])])/(2 + Log[2])^2 - (2*x^3*Log[9]*Log[1 - E^x/(2 + Log[2])])/(2 + Log[2]) - (2*x^2*Log[27]*Log[1 - E^x/(2 + Log[2])])/(2 + L og[2]) + (2*x*Log[3]*(18 + Log[512])*Log[1 - E^x/(2 + Log[2])])/(2 + Log[2 ])^2 + (2*x*Log[19683]*Log[1 - E^x/(2 + Log[2])])/(2 + Log[2]) - (12*x*Log [3]*(4 + Log[4])*PolyLog[2, E^x/(2 + Log[2])])/(2 + Log[2])^2 + (6*x^2*Log [3]*(4 + Log[4])*PolyLog[2, E^x/(2 + Log[2])])/(2 + Log[2])^2 - (12*Log[3] *(6 + Log[8])*PolyLog[2, E^x/(2 + Log[2])])/(2 + Log[2])^2 + (12*x*Log[...
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Time = 0.71 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04
| method | result | size |
| risch | \(x -\frac {\left (2 x^{2}+9 x +9\right ) \ln \left (3\right ) x}{2 \left (-{\mathrm e}^{x}+\ln \left (2\right )+2\right )}\) | \(28\) |
| norman | \(\frac {\left (\ln \left (2\right )+2-\frac {9 \ln \left (3\right )}{2}\right ) x -\frac {9 x^{2} \ln \left (3\right )}{2}-x^{3} \ln \left (3\right )-{\mathrm e}^{x} x}{-{\mathrm e}^{x}+\ln \left (2\right )+2}\) | \(42\) |
| parallelrisch | \(\frac {-2 x^{3} \ln \left (3\right )-9 x^{2} \ln \left (3\right )+2 x \ln \left (2\right )-9 x \ln \left (3\right )-2 \,{\mathrm e}^{x} x +4 x}{-2 \,{\mathrm e}^{x}+2 \ln \left (2\right )+4}\) | \(46\) |
Input:
int((4*exp(x)^2+(-8*ln(2)+(-4*x^3-6*x^2+18*x+18)*ln(3)-16)*exp(x)+4*ln(2)^ 2+2*((-6*x^2-18*x-9)*ln(3)+8)*ln(2)+(-24*x^2-72*x-36)*ln(3)+16)/(4*exp(x)^ 2+(-8*ln(2)-16)*exp(x)+4*ln(2)^2+16*ln(2)+16),x,method=_RETURNVERBOSE)
Output:
x-1/2*(2*x^2+9*x+9)*ln(3)*x/(-exp(x)+ln(2)+2)
Time = 0.07 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.59 \[ \int \frac {16+4 e^{2 x}+\left (-36-72 x-24 x^2\right ) \log (3)+e^x \left (-16+\left (18+18 x-6 x^2-4 x^3\right ) \log (3)-4 \log (4)\right )+\left (8+\left (-9-18 x-6 x^2\right ) \log (3)\right ) \log (4)+\log ^2(4)}{16+4 e^{2 x}+e^x (-16-4 \log (4))+8 \log (4)+\log ^2(4)} \, dx=\frac {2 \, x e^{x} + {\left (2 \, x^{3} + 9 \, x^{2} + 9 \, x\right )} \log \left (3\right ) - 2 \, x \log \left (2\right ) - 4 \, x}{2 \, {\left (e^{x} - \log \left (2\right ) - 2\right )}} \] Input:
integrate((4*exp(x)^2+(-8*log(2)+(-4*x^3-6*x^2+18*x+18)*log(3)-16)*exp(x)+ 4*log(2)^2+2*((-6*x^2-18*x-9)*log(3)+8)*log(2)+(-24*x^2-72*x-36)*log(3)+16 )/(4*exp(x)^2+(-8*log(2)-16)*exp(x)+4*log(2)^2+16*log(2)+16),x, algorithm= "fricas")
Output:
1/2*(2*x*e^x + (2*x^3 + 9*x^2 + 9*x)*log(3) - 2*x*log(2) - 4*x)/(e^x - log (2) - 2)
Time = 0.09 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.33 \[ \int \frac {16+4 e^{2 x}+\left (-36-72 x-24 x^2\right ) \log (3)+e^x \left (-16+\left (18+18 x-6 x^2-4 x^3\right ) \log (3)-4 \log (4)\right )+\left (8+\left (-9-18 x-6 x^2\right ) \log (3)\right ) \log (4)+\log ^2(4)}{16+4 e^{2 x}+e^x (-16-4 \log (4))+8 \log (4)+\log ^2(4)} \, dx=x + \frac {2 x^{3} \log {\left (3 \right )} + 9 x^{2} \log {\left (3 \right )} + 9 x \log {\left (3 \right )}}{2 e^{x} - 4 - 2 \log {\left (2 \right )}} \] Input:
integrate((4*exp(x)**2+(-8*ln(2)+(-4*x**3-6*x**2+18*x+18)*ln(3)-16)*exp(x) +4*ln(2)**2+2*((-6*x**2-18*x-9)*ln(3)+8)*ln(2)+(-24*x**2-72*x-36)*ln(3)+16 )/(4*exp(x)**2+(-8*ln(2)-16)*exp(x)+4*ln(2)**2+16*ln(2)+16),x)
Output:
x + (2*x**3*log(3) + 9*x**2*log(3) + 9*x*log(3))/(2*exp(x) - 4 - 2*log(2))
Leaf count of result is larger than twice the leaf count of optimal. 434 vs. \(2 (25) = 50\).
Time = 0.16 (sec) , antiderivative size = 434, normalized size of antiderivative = 16.07 \[ \int \frac {16+4 e^{2 x}+\left (-36-72 x-24 x^2\right ) \log (3)+e^x \left (-16+\left (18+18 x-6 x^2-4 x^3\right ) \log (3)-4 \log (4)\right )+\left (8+\left (-9-18 x-6 x^2\right ) \log (3)\right ) \log (4)+\log ^2(4)}{16+4 e^{2 x}+e^x (-16-4 \log (4))+8 \log (4)+\log ^2(4)} \, dx =\text {Too large to display} \] Input:
integrate((4*exp(x)^2+(-8*log(2)+(-4*x^3-6*x^2+18*x+18)*log(3)-16)*exp(x)+ 4*log(2)^2+2*((-6*x^2-18*x-9)*log(3)+8)*log(2)+(-24*x^2-72*x-36)*log(3)+16 )/(4*exp(x)^2+(-8*log(2)-16)*exp(x)+4*log(2)^2+16*log(2)+16),x, algorithm= "maxima")
Output:
-9/2*(x/(log(2)^2 + 4*log(2) + 4) - log(e^x - log(2) - 2)/(log(2)^2 + 4*lo g(2) + 4) - 1/((log(2) + 2)*e^x - log(2)^2 - 4*log(2) - 4))*log(3)*log(2) + (x/(log(2)^2 + 4*log(2) + 4) - log(e^x - log(2) - 2)/(log(2)^2 + 4*log(2 ) + 4) - 1/((log(2) + 2)*e^x - log(2)^2 - 4*log(2) - 4))*log(2)^2 - 9*(x/( log(2)^2 + 4*log(2) + 4) - log(e^x - log(2) - 2)/(log(2)^2 + 4*log(2) + 4) - 1/((log(2) + 2)*e^x - log(2)^2 - 4*log(2) - 4))*log(3) + 4*(x/(log(2)^2 + 4*log(2) + 4) - log(e^x - log(2) - 2)/(log(2)^2 + 4*log(2) + 4) - 1/((l og(2) + 2)*e^x - log(2)^2 - 4*log(2) - 4))*log(2) - 1/2*(9*log(3) - 2*log( 2) - 4)*log(e^x - log(2) - 2)/(log(2) + 2) + 1/2*(2*(log(3)*log(2) + 2*log (3))*x^3 + 9*(log(3)*log(2) + 2*log(3))*x^2 + 9*x*e^x*log(3) - (9*log(3) - 8)*log(2) + 2*log(2)^2 - 18*log(3) + 8)/((log(2) + 2)*e^x - log(2)^2 - 4* log(2) - 4) + 4*x/(log(2)^2 + 4*log(2) + 4) - 4*log(e^x - log(2) - 2)/(log (2)^2 + 4*log(2) + 4) - 4/((log(2) + 2)*e^x - log(2)^2 - 4*log(2) - 4)
Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (25) = 50\).
Time = 0.11 (sec) , antiderivative size = 119, normalized size of antiderivative = 4.41 \[ \int \frac {16+4 e^{2 x}+\left (-36-72 x-24 x^2\right ) \log (3)+e^x \left (-16+\left (18+18 x-6 x^2-4 x^3\right ) \log (3)-4 \log (4)\right )+\left (8+\left (-9-18 x-6 x^2\right ) \log (3)\right ) \log (4)+\log ^2(4)}{16+4 e^{2 x}+e^x (-16-4 \log (4))+8 \log (4)+\log ^2(4)} \, dx=\frac {2 \, x^{3} \log \left (3\right ) + 9 \, x^{2} \log \left (3\right ) + 2 \, x e^{x} + 9 \, x \log \left (3\right ) - 2 \, x \log \left (2\right ) + 2 \, e^{x} \log \left (e^{x} - \log \left (2\right ) - 2\right ) - 2 \, \log \left (2\right ) \log \left (e^{x} - \log \left (2\right ) - 2\right ) - 2 \, e^{x} \log \left (-e^{x} + \log \left (2\right ) + 2\right ) + 2 \, \log \left (2\right ) \log \left (-e^{x} + \log \left (2\right ) + 2\right ) - 4 \, x - 4 \, \log \left (e^{x} - \log \left (2\right ) - 2\right ) + 4 \, \log \left (-e^{x} + \log \left (2\right ) + 2\right )}{2 \, {\left (e^{x} - \log \left (2\right ) - 2\right )}} \] Input:
integrate((4*exp(x)^2+(-8*log(2)+(-4*x^3-6*x^2+18*x+18)*log(3)-16)*exp(x)+ 4*log(2)^2+2*((-6*x^2-18*x-9)*log(3)+8)*log(2)+(-24*x^2-72*x-36)*log(3)+16 )/(4*exp(x)^2+(-8*log(2)-16)*exp(x)+4*log(2)^2+16*log(2)+16),x, algorithm= "giac")
Output:
1/2*(2*x^3*log(3) + 9*x^2*log(3) + 2*x*e^x + 9*x*log(3) - 2*x*log(2) + 2*e ^x*log(e^x - log(2) - 2) - 2*log(2)*log(e^x - log(2) - 2) - 2*e^x*log(-e^x + log(2) + 2) + 2*log(2)*log(-e^x + log(2) + 2) - 4*x - 4*log(e^x - log(2 ) - 2) + 4*log(-e^x + log(2) + 2))/(e^x - log(2) - 2)
Time = 1.86 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.30 \[ \int \frac {16+4 e^{2 x}+\left (-36-72 x-24 x^2\right ) \log (3)+e^x \left (-16+\left (18+18 x-6 x^2-4 x^3\right ) \log (3)-4 \log (4)\right )+\left (8+\left (-9-18 x-6 x^2\right ) \log (3)\right ) \log (4)+\log ^2(4)}{16+4 e^{2 x}+e^x (-16-4 \log (4))+8 \log (4)+\log ^2(4)} \, dx=x-\frac {x\,\left (\ln \left (9\right )\,x^2+9\,\ln \left (3\right )\,x+2\,\ln \left (2\right )+\ln \left (\frac {19683}{4}\right )\right )}{2\,\left (\ln \left (2\right )-{\mathrm {e}}^x+2\right )} \] Input:
int((4*exp(2*x) - log(3)*(72*x + 24*x^2 + 36) - 2*log(2)*(log(3)*(18*x + 6 *x^2 + 9) - 8) + 4*log(2)^2 - exp(x)*(8*log(2) - log(3)*(18*x - 6*x^2 - 4* x^3 + 18) + 16) + 16)/(4*exp(2*x) + 16*log(2) - exp(x)*(8*log(2) + 16) + 4 *log(2)^2 + 16),x)
Output:
x - (x*(2*log(2) + log(19683/4) + 9*x*log(3) + x^2*log(9)))/(2*(log(2) - e xp(x) + 2))
Time = 0.18 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.56 \[ \int \frac {16+4 e^{2 x}+\left (-36-72 x-24 x^2\right ) \log (3)+e^x \left (-16+\left (18+18 x-6 x^2-4 x^3\right ) \log (3)-4 \log (4)\right )+\left (8+\left (-9-18 x-6 x^2\right ) \log (3)\right ) \log (4)+\log ^2(4)}{16+4 e^{2 x}+e^x (-16-4 \log (4))+8 \log (4)+\log ^2(4)} \, dx=\frac {x \left (2 e^{x}+2 \,\mathrm {log}\left (3\right ) x^{2}+9 \,\mathrm {log}\left (3\right ) x +9 \,\mathrm {log}\left (3\right )-2 \,\mathrm {log}\left (2\right )-4\right )}{2 e^{x}-2 \,\mathrm {log}\left (2\right )-4} \] Input:
int((4*exp(x)^2+(-8*log(2)+(-4*x^3-6*x^2+18*x+18)*log(3)-16)*exp(x)+4*log( 2)^2+2*((-6*x^2-18*x-9)*log(3)+8)*log(2)+(-24*x^2-72*x-36)*log(3)+16)/(4*e xp(x)^2+(-8*log(2)-16)*exp(x)+4*log(2)^2+16*log(2)+16),x)
Output:
(x*(2*e**x + 2*log(3)*x**2 + 9*log(3)*x + 9*log(3) - 2*log(2) - 4))/(2*(e* *x - log(2) - 2))