Integrand size = 183, antiderivative size = 34 \[ \int \frac {e^6 \left (450+360 x+72 x^3-18 x^4+e \left (-150 x+90 x^3-60 x^4\right )+e^2 \left (-50 x^3-50 x^4\right )\right )+e^{6+x} \left (450 x-450 x^2-108 x^3+126 x^4-18 x^5+e \left (540 x^3+120 x^4-60 x^5\right )+e^2 \left (-50 x^3-150 x^4-50 x^5\right )\right )+e^{6+2 x} \left (-360 x^3+162 x^4-18 x^5+e \left (150 x^3+240 x^4-60 x^5\right )+e^2 \left (-50 x^4-50 x^5\right )\right )}{25 x^3} \, dx=e^4-e^6 \left (-\frac {3}{5}-e+\frac {3}{x}\right )^2 \left (1+\left (1+e^x\right ) x\right )^2 \]
Leaf count is larger than twice the leaf count of optimal. \(86\) vs. \(2(34)=68\).
Time = 2.09 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.53 \[ \int \frac {e^6 \left (450+360 x+72 x^3-18 x^4+e \left (-150 x+90 x^3-60 x^4\right )+e^2 \left (-50 x^3-50 x^4\right )\right )+e^{6+x} \left (450 x-450 x^2-108 x^3+126 x^4-18 x^5+e \left (540 x^3+120 x^4-60 x^5\right )+e^2 \left (-50 x^3-150 x^4-50 x^5\right )\right )+e^{6+2 x} \left (-360 x^3+162 x^4-18 x^5+e \left (150 x^3+240 x^4-60 x^5\right )+e^2 \left (-50 x^4-50 x^5\right )\right )}{25 x^3} \, dx=-\frac {e^6 \left (225+30 \left (12-5 e+15 e^x\right ) x+15 e^x \left (18-20 e+15 e^x\right ) x^2-2 (3+5 e) \left (1+e^x\right ) \left (12-5 e+15 e^x\right ) x^3+(3+5 e)^2 \left (1+e^x\right )^2 x^4\right )}{25 x^2} \]
Integrate[(E^6*(450 + 360*x + 72*x^3 - 18*x^4 + E*(-150*x + 90*x^3 - 60*x^ 4) + E^2*(-50*x^3 - 50*x^4)) + E^(6 + x)*(450*x - 450*x^2 - 108*x^3 + 126* x^4 - 18*x^5 + E*(540*x^3 + 120*x^4 - 60*x^5) + E^2*(-50*x^3 - 150*x^4 - 5 0*x^5)) + E^(6 + 2*x)*(-360*x^3 + 162*x^4 - 18*x^5 + E*(150*x^3 + 240*x^4 - 60*x^5) + E^2*(-50*x^4 - 50*x^5)))/(25*x^3),x]
-1/25*(E^6*(225 + 30*(12 - 5*E + 15*E^x)*x + 15*E^x*(18 - 20*E + 15*E^x)*x ^2 - 2*(3 + 5*E)*(1 + E^x)*(12 - 5*E + 15*E^x)*x^3 + (3 + 5*E)^2*(1 + E^x) ^2*x^4))/x^2
Leaf count is larger than twice the leaf count of optimal. \(285\) vs. \(2(34)=68\).
Time = 0.84 (sec) , antiderivative size = 285, normalized size of antiderivative = 8.38, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {27, 27, 2010, 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.
| \(\displaystyle \int \frac {e^6 \left (-18 x^4+72 x^3+e \left (-60 x^4+90 x^3-150 x\right )+e^2 \left (-50 x^4-50 x^3\right )+360 x+450\right )+e^{2 x+6} \left (-18 x^5+162 x^4-360 x^3+e^2 \left (-50 x^5-50 x^4\right )+e \left (-60 x^5+240 x^4+150 x^3\right )\right )+e^{x+6} \left (-18 x^5+126 x^4-108 x^3-450 x^2+e \left (-60 x^5+120 x^4+540 x^3\right )+e^2 \left (-50 x^5-150 x^4-50 x^3\right )+450 x\right )}{25 x^3} \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{25} \int \frac {2 \left (e^6 \left (-9 x^4+36 x^3+180 x-25 e^2 \left (x^4+x^3\right )-15 e \left (2 x^4-3 x^3+5 x\right )+225\right )-e^{2 x+6} \left (9 x^5-81 x^4+180 x^3-15 e \left (-2 x^5+8 x^4+5 x^3\right )+25 e^2 \left (x^5+x^4\right )\right )+e^{x+6} \left (-9 x^5+63 x^4-54 x^3-225 x^2+225 x+30 e \left (-x^5+2 x^4+9 x^3\right )-25 e^2 \left (x^5+3 x^4+x^3\right )\right )\right )}{x^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{25} \int \frac {e^6 \left (-9 x^4+36 x^3+180 x-25 e^2 \left (x^4+x^3\right )-15 e \left (2 x^4-3 x^3+5 x\right )+225\right )-e^{2 x+6} \left (9 x^5-81 x^4+180 x^3-15 e \left (-2 x^5+8 x^4+5 x^3\right )+25 e^2 \left (x^5+x^4\right )\right )+e^{x+6} \left (-9 x^5+63 x^4-54 x^3-225 x^2+225 x+30 e \left (-x^5+2 x^4+9 x^3\right )-25 e^2 \left (x^5+3 x^4+x^3\right )\right )}{x^3}dx\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle \frac {2}{25} \int \left (e^{2 x+6} (15-(3+5 e) x) ((3+5 e) x+5 e-12)+\frac {e^6 (x+1) (15-(3+5 e) x) \left ((3+5 e) x^2+15\right )}{x^3}+\frac {e^{x+6} (15-(3+5 e) x) \left ((3+5 e) x^3-3 (2-5 e) x^2-(12-5 e) x+15\right )}{x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2}{25} \left ((3+5 e)^2 \left (-e^{x+6}\right ) x^2-\frac {1}{2} (3+5 e)^2 e^{2 x+6} x^2-\frac {1}{2} e^6 (3+5 e)^2 x^2-\frac {225 e^6}{2 x^2}+\frac {1}{2} \left (81+120 e-25 e^2\right ) e^{2 x+6} x+3 \left (21+20 e-25 e^2\right ) e^{x+6} x+e^6 (36+5 (9-5 e) e) x+2 (3+5 e)^2 e^{x+6} x+\frac {1}{2} (3+5 e)^2 e^{2 x+6} x-\frac {15}{2} (12-5 e) e^{2 x+6}-\frac {1}{4} \left (81+120 e-25 e^2\right ) e^{2 x+6}-3 \left (21+20 e-25 e^2\right ) e^{x+6}-(54-5 (54-5 e) e) e^{x+6}-2 (3+5 e)^2 e^{x+6}-\frac {1}{4} (3+5 e)^2 e^{2 x+6}-\frac {225 e^{x+6}}{x}-\frac {15 (12-5 e) e^6}{x}\right )\) |
Int[(E^6*(450 + 360*x + 72*x^3 - 18*x^4 + E*(-150*x + 90*x^3 - 60*x^4) + E ^2*(-50*x^3 - 50*x^4)) + E^(6 + x)*(450*x - 450*x^2 - 108*x^3 + 126*x^4 - 18*x^5 + E*(540*x^3 + 120*x^4 - 60*x^5) + E^2*(-50*x^3 - 150*x^4 - 50*x^5) ) + E^(6 + 2*x)*(-360*x^3 + 162*x^4 - 18*x^5 + E*(150*x^3 + 240*x^4 - 60*x ^5) + E^2*(-50*x^4 - 50*x^5)))/(25*x^3),x]
(2*((-15*(12 - 5*E)*E^(6 + 2*x))/2 - 2*E^(6 + x)*(3 + 5*E)^2 - (E^(6 + 2*x )*(3 + 5*E)^2)/4 - E^(6 + x)*(54 - 5*(54 - 5*E)*E) - 3*E^(6 + x)*(21 + 20* E - 25*E^2) - (E^(6 + 2*x)*(81 + 120*E - 25*E^2))/4 - (225*E^6)/(2*x^2) - (15*(12 - 5*E)*E^6)/x - (225*E^(6 + x))/x + 2*E^(6 + x)*(3 + 5*E)^2*x + (E ^(6 + 2*x)*(3 + 5*E)^2*x)/2 + E^6*(36 + 5*(9 - 5*E)*E)*x + 3*E^(6 + x)*(21 + 20*E - 25*E^2)*x + (E^(6 + 2*x)*(81 + 120*E - 25*E^2)*x)/2 - (E^6*(3 + 5*E)^2*x^2)/2 - E^(6 + x)*(3 + 5*E)^2*x^2 - (E^(6 + 2*x)*(3 + 5*E)^2*x^2)/ 2))/25
3.11.16.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] , x] /; FreeQ[{c, m}, x] && SumQ[u] && !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
Leaf count of result is larger than twice the leaf count of optimal. \(169\) vs. \(2(32)=64\).
Time = 0.11 (sec) , antiderivative size = 170, normalized size of antiderivative = 5.00
| method | result | size |
| risch | \(-\frac {6 \,{\mathrm e}^{6} {\mathrm e} x^{2}}{5}+\frac {18 \,{\mathrm e}^{6} x \,{\mathrm e}}{5}-\frac {9 x^{2} {\mathrm e}^{6}}{25}+\frac {72 x \,{\mathrm e}^{6}}{25}-{\mathrm e}^{6} x^{2} {\mathrm e}^{2}-2 \,{\mathrm e}^{6} x \,{\mathrm e}^{2}+\frac {\left (150 \,{\mathrm e}^{7}-360 \,{\mathrm e}^{6}\right ) x -225 \,{\mathrm e}^{6}}{25 x^{2}}+\frac {\left (-25 x^{2} {\mathrm e}^{8}-30 x^{2} {\mathrm e}^{7}+150 x \,{\mathrm e}^{7}-9 x^{2} {\mathrm e}^{6}+90 x \,{\mathrm e}^{6}-225 \,{\mathrm e}^{6}\right ) {\mathrm e}^{2 x}}{25}-\frac {2 \left (25 x^{3} {\mathrm e}^{2}+25 x^{2} {\mathrm e}^{2}+30 x^{3} {\mathrm e}-120 x^{2} {\mathrm e}+9 x^{3}-150 x \,{\mathrm e}-81 x^{2}+135 x +225\right ) {\mathrm e}^{6+x}}{25 x}\) | \(170\) |
| norman | \(\frac {\left (6 \,{\mathrm e} \,{\mathrm e}^{6}-\frac {72 \,{\mathrm e}^{6}}{5}\right ) x +\left (-2 \,{\mathrm e}^{2} {\mathrm e}^{6}+\frac {18 \,{\mathrm e} \,{\mathrm e}^{6}}{5}+\frac {72 \,{\mathrm e}^{6}}{25}\right ) x^{3}+\left (-{\mathrm e}^{2} {\mathrm e}^{6}-\frac {6 \,{\mathrm e} \,{\mathrm e}^{6}}{5}-\frac {9 \,{\mathrm e}^{6}}{25}\right ) x^{4}+\left (6 \,{\mathrm e} \,{\mathrm e}^{6}+\frac {18 \,{\mathrm e}^{6}}{5}\right ) x^{3} {\mathrm e}^{2 x}+\left (12 \,{\mathrm e} \,{\mathrm e}^{6}-\frac {54 \,{\mathrm e}^{6}}{5}\right ) x^{2} {\mathrm e}^{x}+\left (-2 \,{\mathrm e}^{2} {\mathrm e}^{6}-\frac {12 \,{\mathrm e} \,{\mathrm e}^{6}}{5}-\frac {18 \,{\mathrm e}^{6}}{25}\right ) x^{4} {\mathrm e}^{x}+\left (-2 \,{\mathrm e}^{2} {\mathrm e}^{6}+\frac {48 \,{\mathrm e} \,{\mathrm e}^{6}}{5}+\frac {162 \,{\mathrm e}^{6}}{25}\right ) x^{3} {\mathrm e}^{x}+\left (-{\mathrm e}^{2} {\mathrm e}^{6}-\frac {6 \,{\mathrm e} \,{\mathrm e}^{6}}{5}-\frac {9 \,{\mathrm e}^{6}}{25}\right ) x^{4} {\mathrm e}^{2 x}-9 \,{\mathrm e}^{6}-9 \,{\mathrm e}^{2 x} {\mathrm e}^{6} x^{2}-18 x \,{\mathrm e}^{6} {\mathrm e}^{x}}{x^{2}}\) | \(248\) |
| parallelrisch | \(-\frac {270 x^{2} {\mathrm e}^{6} {\mathrm e}^{x}+225 \,{\mathrm e}^{2 x} {\mathrm e}^{6} x^{2}-90 \,{\mathrm e} \,{\mathrm e}^{6} x^{3}+9 \,{\mathrm e}^{2 x} {\mathrm e}^{6} x^{4}-90 \,{\mathrm e}^{2 x} {\mathrm e}^{6} x^{3}+360 x \,{\mathrm e}^{6}-72 x^{3} {\mathrm e}^{6}+9 \,{\mathrm e}^{6} x^{4}+50 \,{\mathrm e}^{6} {\mathrm e}^{x} x^{3} {\mathrm e}^{2}+60 \,{\mathrm e}^{6} {\mathrm e}^{x} x^{4} {\mathrm e}-240 \,{\mathrm e}^{6} {\mathrm e}^{x} x^{3} {\mathrm e}-300 x^{2} {\mathrm e}^{6} {\mathrm e}^{x} {\mathrm e}+25 \,{\mathrm e}^{6} {\mathrm e}^{2 x} x^{4} {\mathrm e}^{2}+30 \,{\mathrm e}^{6} {\mathrm e}^{2 x} x^{4} {\mathrm e}-150 \,{\mathrm e}^{6} {\mathrm e}^{2 x} x^{3} {\mathrm e}+50 \,{\mathrm e}^{6} {\mathrm e}^{x} x^{4} {\mathrm e}^{2}+225 \,{\mathrm e}^{6}+18 \,{\mathrm e}^{6} {\mathrm e}^{x} x^{4}-162 \,{\mathrm e}^{6} {\mathrm e}^{x} x^{3}+450 x \,{\mathrm e}^{6} {\mathrm e}^{x}+25 \,{\mathrm e}^{2} {\mathrm e}^{6} x^{4}+50 \,{\mathrm e}^{2} {\mathrm e}^{6} x^{3}+30 \,{\mathrm e} \,{\mathrm e}^{6} x^{4}-150 \,{\mathrm e}^{6} x \,{\mathrm e}}{25 x^{2}}\) | \(292\) |
| parts | \(-\frac {2 \,{\mathrm e}^{6} \left (\frac {225 \,{\mathrm e}^{2 x}}{2}-45 x \,{\mathrm e}^{2 x}+\frac {9 \,{\mathrm e}^{2 x} x^{2}}{2}-\frac {75 \,{\mathrm e} \,{\mathrm e}^{2 x}}{2}-120 \,{\mathrm e} \left (\frac {x \,{\mathrm e}^{2 x}}{2}-\frac {{\mathrm e}^{2 x}}{4}\right )+25 \,{\mathrm e}^{2} \left (\frac {x \,{\mathrm e}^{2 x}}{2}-\frac {{\mathrm e}^{2 x}}{4}\right )+30 \,{\mathrm e} \left (\frac {{\mathrm e}^{2 x} x^{2}}{2}-\frac {x \,{\mathrm e}^{2 x}}{2}+\frac {{\mathrm e}^{2 x}}{4}\right )+25 \,{\mathrm e}^{2} \left (\frac {{\mathrm e}^{2 x} x^{2}}{2}-\frac {x \,{\mathrm e}^{2 x}}{2}+\frac {{\mathrm e}^{2 x}}{4}\right )\right )}{25}-\frac {2 \,{\mathrm e}^{6} \left (\frac {25 x^{2} {\mathrm e}^{2}}{2}+15 x^{2} {\mathrm e}+25 \,{\mathrm e}^{2} x -45 x \,{\mathrm e}+\frac {9 x^{2}}{2}-36 x -\frac {75 \,{\mathrm e}-180}{x}+\frac {225}{2 x^{2}}\right )}{25}-\frac {2 \,{\mathrm e}^{6} \left (\frac {225 \,{\mathrm e}^{x}}{x}-81 \,{\mathrm e}^{x} x +135 \,{\mathrm e}^{x}+9 \,{\mathrm e}^{x} x^{2}-270 \,{\mathrm e} \,{\mathrm e}^{x}+25 \,{\mathrm e}^{2} {\mathrm e}^{x}-60 \,{\mathrm e} \left ({\mathrm e}^{x} x -{\mathrm e}^{x}\right )+75 \,{\mathrm e}^{2} \left ({\mathrm e}^{x} x -{\mathrm e}^{x}\right )+30 \,{\mathrm e} \left ({\mathrm e}^{x} x^{2}-2 \,{\mathrm e}^{x} x +2 \,{\mathrm e}^{x}\right )+25 \,{\mathrm e}^{2} \left ({\mathrm e}^{x} x^{2}-2 \,{\mathrm e}^{x} x +2 \,{\mathrm e}^{x}\right )\right )}{25}\) | \(302\) |
| default | \(-\frac {9 x^{2} {\mathrm e}^{6}}{25}+\frac {72 x \,{\mathrm e}^{6}}{25}-\frac {108 \,{\mathrm e}^{6} {\mathrm e}^{x}}{25}+18 \,{\mathrm e}^{6} \operatorname {Ei}_{1}\left (-x \right )-\frac {36 \,{\mathrm e}^{6} {\mathrm e}^{2 x}}{5}+\frac {126 \,{\mathrm e}^{6} \left ({\mathrm e}^{x} x -{\mathrm e}^{x}\right )}{25}-\frac {72 \,{\mathrm e}^{6}}{5 x}-\frac {9 \,{\mathrm e}^{6}}{x^{2}}-\frac {18 \,{\mathrm e}^{6} \left ({\mathrm e}^{x} x^{2}-2 \,{\mathrm e}^{x} x +2 \,{\mathrm e}^{x}\right )}{25}+18 \,{\mathrm e}^{6} \left (-\frac {{\mathrm e}^{x}}{x}-\operatorname {Ei}_{1}\left (-x \right )\right )-\frac {18 \,{\mathrm e}^{6} \left (\frac {{\mathrm e}^{2 x} x^{2}}{2}-\frac {x \,{\mathrm e}^{2 x}}{2}+\frac {{\mathrm e}^{2 x}}{4}\right )}{25}+\frac {162 \,{\mathrm e}^{6} \left (\frac {x \,{\mathrm e}^{2 x}}{2}-\frac {{\mathrm e}^{2 x}}{4}\right )}{25}-\frac {6 \,{\mathrm e}^{6} {\mathrm e} x^{2}}{5}+\frac {18 \,{\mathrm e}^{6} x \,{\mathrm e}}{5}+3 \,{\mathrm e}^{6} {\mathrm e}^{2 x} {\mathrm e}-2 \,{\mathrm e}^{6} {\mathrm e}^{x} {\mathrm e}^{2}+\frac {108 \,{\mathrm e}^{6} {\mathrm e}^{x} {\mathrm e}}{5}+\frac {24 \,{\mathrm e} \,{\mathrm e}^{6} \left ({\mathrm e}^{x} x -{\mathrm e}^{x}\right )}{5}-6 \,{\mathrm e}^{6} {\mathrm e}^{2} \left ({\mathrm e}^{x} x -{\mathrm e}^{x}\right )+\frac {48 \,{\mathrm e} \,{\mathrm e}^{6} \left (\frac {x \,{\mathrm e}^{2 x}}{2}-\frac {{\mathrm e}^{2 x}}{4}\right )}{5}-2 \,{\mathrm e}^{6} {\mathrm e}^{2} \left (\frac {x \,{\mathrm e}^{2 x}}{2}-\frac {{\mathrm e}^{2 x}}{4}\right )-\frac {12 \,{\mathrm e} \,{\mathrm e}^{6} \left ({\mathrm e}^{x} x^{2}-2 \,{\mathrm e}^{x} x +2 \,{\mathrm e}^{x}\right )}{5}-2 \,{\mathrm e}^{6} {\mathrm e}^{2} \left ({\mathrm e}^{x} x^{2}-2 \,{\mathrm e}^{x} x +2 \,{\mathrm e}^{x}\right )+\frac {6 \,{\mathrm e} \,{\mathrm e}^{6}}{x}-\frac {12 \,{\mathrm e} \,{\mathrm e}^{6} \left (\frac {{\mathrm e}^{2 x} x^{2}}{2}-\frac {x \,{\mathrm e}^{2 x}}{2}+\frac {{\mathrm e}^{2 x}}{4}\right )}{5}-2 \,{\mathrm e}^{6} {\mathrm e}^{2} \left (\frac {{\mathrm e}^{2 x} x^{2}}{2}-\frac {x \,{\mathrm e}^{2 x}}{2}+\frac {{\mathrm e}^{2 x}}{4}\right )-{\mathrm e}^{6} x^{2} {\mathrm e}^{2}-2 \,{\mathrm e}^{6} x \,{\mathrm e}^{2}\) | \(457\) |
int(1/25*(((-50*x^5-50*x^4)*exp(1)^2+(-60*x^5+240*x^4+150*x^3)*exp(1)-18*x ^5+162*x^4-360*x^3)*exp(3)^2*exp(x)^2+((-50*x^5-150*x^4-50*x^3)*exp(1)^2+( -60*x^5+120*x^4+540*x^3)*exp(1)-18*x^5+126*x^4-108*x^3-450*x^2+450*x)*exp( 3)^2*exp(x)+((-50*x^4-50*x^3)*exp(1)^2+(-60*x^4+90*x^3-150*x)*exp(1)-18*x^ 4+72*x^3+360*x+450)*exp(3)^2)/x^3,x,method=_RETURNVERBOSE)
-6/5*exp(6)*exp(1)*x^2+18/5*exp(6)*x*exp(1)-9/25*x^2*exp(6)+72/25*x*exp(6) -exp(6)*x^2*exp(1)^2-2*exp(6)*x*exp(1)^2+1/25*((150*exp(7)-360*exp(6))*x-2 25*exp(6))/x^2+1/25*(-25*x^2*exp(8)-30*x^2*exp(7)+150*x*exp(7)-9*x^2*exp(6 )+90*x*exp(6)-225*exp(6))*exp(2*x)-2/25/x*(25*x^3*exp(2)+25*x^2*exp(2)+30* x^3*exp(1)-120*x^2*exp(1)+9*x^3-150*x*exp(1)-81*x^2+135*x+225)*exp(6+x)
Leaf count of result is larger than twice the leaf count of optimal. 154 vs. \(2 (30) = 60\).
Time = 0.25 (sec) , antiderivative size = 154, normalized size of antiderivative = 4.53 \[ \int \frac {e^6 \left (450+360 x+72 x^3-18 x^4+e \left (-150 x+90 x^3-60 x^4\right )+e^2 \left (-50 x^3-50 x^4\right )\right )+e^{6+x} \left (450 x-450 x^2-108 x^3+126 x^4-18 x^5+e \left (540 x^3+120 x^4-60 x^5\right )+e^2 \left (-50 x^3-150 x^4-50 x^5\right )\right )+e^{6+2 x} \left (-360 x^3+162 x^4-18 x^5+e \left (150 x^3+240 x^4-60 x^5\right )+e^2 \left (-50 x^4-50 x^5\right )\right )}{25 x^3} \, dx=-\frac {{\left (25 \, {\left (x^{4} + 2 \, x^{3}\right )} e^{14} + 30 \, {\left (x^{4} - 3 \, x^{3} - 5 \, x\right )} e^{13} + 9 \, {\left (x^{4} - 8 \, x^{3} + 40 \, x + 25\right )} e^{12} + {\left (25 \, x^{4} e^{2} + 9 \, x^{4} - 90 \, x^{3} + 225 \, x^{2} + 30 \, {\left (x^{4} - 5 \, x^{3}\right )} e\right )} e^{\left (2 \, x + 12\right )} + 2 \, {\left (25 \, {\left (x^{4} + x^{3}\right )} e^{8} + 30 \, {\left (x^{4} - 4 \, x^{3} - 5 \, x^{2}\right )} e^{7} + 9 \, {\left (x^{4} - 9 \, x^{3} + 15 \, x^{2} + 25 \, x\right )} e^{6}\right )} e^{\left (x + 6\right )}\right )} e^{\left (-6\right )}}{25 \, x^{2}} \]
integrate(1/25*(((-50*x^5-50*x^4)*exp(1)^2+(-60*x^5+240*x^4+150*x^3)*exp(1 )-18*x^5+162*x^4-360*x^3)*exp(3)^2*exp(x)^2+((-50*x^5-150*x^4-50*x^3)*exp( 1)^2+(-60*x^5+120*x^4+540*x^3)*exp(1)-18*x^5+126*x^4-108*x^3-450*x^2+450*x )*exp(3)^2*exp(x)+((-50*x^4-50*x^3)*exp(1)^2+(-60*x^4+90*x^3-150*x)*exp(1) -18*x^4+72*x^3+360*x+450)*exp(3)^2)/x^3,x, algorithm=\
-1/25*(25*(x^4 + 2*x^3)*e^14 + 30*(x^4 - 3*x^3 - 5*x)*e^13 + 9*(x^4 - 8*x^ 3 + 40*x + 25)*e^12 + (25*x^4*e^2 + 9*x^4 - 90*x^3 + 225*x^2 + 30*(x^4 - 5 *x^3)*e)*e^(2*x + 12) + 2*(25*(x^4 + x^3)*e^8 + 30*(x^4 - 4*x^3 - 5*x^2)*e ^7 + 9*(x^4 - 9*x^3 + 15*x^2 + 25*x)*e^6)*e^(x + 6))*e^(-6)/x^2
Leaf count of result is larger than twice the leaf count of optimal. 190 vs. \(2 (27) = 54\).
Time = 0.25 (sec) , antiderivative size = 190, normalized size of antiderivative = 5.59 \[ \int \frac {e^6 \left (450+360 x+72 x^3-18 x^4+e \left (-150 x+90 x^3-60 x^4\right )+e^2 \left (-50 x^3-50 x^4\right )\right )+e^{6+x} \left (450 x-450 x^2-108 x^3+126 x^4-18 x^5+e \left (540 x^3+120 x^4-60 x^5\right )+e^2 \left (-50 x^3-150 x^4-50 x^5\right )\right )+e^{6+2 x} \left (-360 x^3+162 x^4-18 x^5+e \left (150 x^3+240 x^4-60 x^5\right )+e^2 \left (-50 x^4-50 x^5\right )\right )}{25 x^3} \, dx=- \frac {x^{2} \cdot \left (9 e^{6} + 30 e^{7} + 25 e^{8}\right )}{25} - \frac {x \left (- 90 e^{7} - 72 e^{6} + 50 e^{8}\right )}{25} + \frac {\left (- 625 x^{3} e^{8} - 750 x^{3} e^{7} - 225 x^{3} e^{6} + 2250 x^{2} e^{6} + 3750 x^{2} e^{7} - 5625 x e^{6}\right ) e^{2 x} + \left (- 1250 x^{3} e^{8} - 1500 x^{3} e^{7} - 450 x^{3} e^{6} - 1250 x^{2} e^{8} + 4050 x^{2} e^{6} + 6000 x^{2} e^{7} - 6750 x e^{6} + 7500 x e^{7} - 11250 e^{6}\right ) e^{x}}{625 x} - \frac {x \left (- 150 e^{7} + 360 e^{6}\right ) + 225 e^{6}}{25 x^{2}} \]
integrate(1/25*(((-50*x**5-50*x**4)*exp(1)**2+(-60*x**5+240*x**4+150*x**3) *exp(1)-18*x**5+162*x**4-360*x**3)*exp(3)**2*exp(x)**2+((-50*x**5-150*x**4 -50*x**3)*exp(1)**2+(-60*x**5+120*x**4+540*x**3)*exp(1)-18*x**5+126*x**4-1 08*x**3-450*x**2+450*x)*exp(3)**2*exp(x)+((-50*x**4-50*x**3)*exp(1)**2+(-6 0*x**4+90*x**3-150*x)*exp(1)-18*x**4+72*x**3+360*x+450)*exp(3)**2)/x**3,x)
-x**2*(9*exp(6) + 30*exp(7) + 25*exp(8))/25 - x*(-90*exp(7) - 72*exp(6) + 50*exp(8))/25 + ((-625*x**3*exp(8) - 750*x**3*exp(7) - 225*x**3*exp(6) + 2 250*x**2*exp(6) + 3750*x**2*exp(7) - 5625*x*exp(6))*exp(2*x) + (-1250*x**3 *exp(8) - 1500*x**3*exp(7) - 450*x**3*exp(6) - 1250*x**2*exp(8) + 4050*x** 2*exp(6) + 6000*x**2*exp(7) - 6750*x*exp(6) + 7500*x*exp(7) - 11250*exp(6) )*exp(x))/(625*x) - (x*(-150*exp(7) + 360*exp(6)) + 225*exp(6))/(25*x**2)
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.23 (sec) , antiderivative size = 317, normalized size of antiderivative = 9.32 \[ \int \frac {e^6 \left (450+360 x+72 x^3-18 x^4+e \left (-150 x+90 x^3-60 x^4\right )+e^2 \left (-50 x^3-50 x^4\right )\right )+e^{6+x} \left (450 x-450 x^2-108 x^3+126 x^4-18 x^5+e \left (540 x^3+120 x^4-60 x^5\right )+e^2 \left (-50 x^3-150 x^4-50 x^5\right )\right )+e^{6+2 x} \left (-360 x^3+162 x^4-18 x^5+e \left (150 x^3+240 x^4-60 x^5\right )+e^2 \left (-50 x^4-50 x^5\right )\right )}{25 x^3} \, dx=-x^{2} e^{8} - \frac {6}{5} \, x^{2} e^{7} - \frac {9}{25} \, x^{2} e^{6} - 2 \, x e^{8} + \frac {18}{5} \, x e^{7} + \frac {72}{25} \, x e^{6} - 18 \, {\rm Ei}\left (x\right ) e^{6} - \frac {1}{2} \, {\left (2 \, x^{2} e^{8} - 2 \, x e^{8} + e^{8}\right )} e^{\left (2 \, x\right )} - \frac {3}{5} \, {\left (2 \, x^{2} e^{7} - 2 \, x e^{7} + e^{7}\right )} e^{\left (2 \, x\right )} - \frac {9}{50} \, {\left (2 \, x^{2} e^{6} - 2 \, x e^{6} + e^{6}\right )} e^{\left (2 \, x\right )} - \frac {1}{2} \, {\left (2 \, x e^{8} - e^{8}\right )} e^{\left (2 \, x\right )} + \frac {12}{5} \, {\left (2 \, x e^{7} - e^{7}\right )} e^{\left (2 \, x\right )} + \frac {81}{50} \, {\left (2 \, x e^{6} - e^{6}\right )} e^{\left (2 \, x\right )} - 2 \, {\left (x^{2} e^{8} - 2 \, x e^{8} + 2 \, e^{8}\right )} e^{x} - \frac {12}{5} \, {\left (x^{2} e^{7} - 2 \, x e^{7} + 2 \, e^{7}\right )} e^{x} - \frac {18}{25} \, {\left (x^{2} e^{6} - 2 \, x e^{6} + 2 \, e^{6}\right )} e^{x} - 6 \, {\left (x e^{8} - e^{8}\right )} e^{x} + \frac {24}{5} \, {\left (x e^{7} - e^{7}\right )} e^{x} + \frac {126}{25} \, {\left (x e^{6} - e^{6}\right )} e^{x} + 18 \, e^{6} \Gamma \left (-1, -x\right ) + \frac {6 \, e^{7}}{x} - \frac {72 \, e^{6}}{5 \, x} - \frac {9 \, e^{6}}{x^{2}} + 3 \, e^{\left (2 \, x + 7\right )} - \frac {36}{5} \, e^{\left (2 \, x + 6\right )} - 2 \, e^{\left (x + 8\right )} + \frac {108}{5} \, e^{\left (x + 7\right )} - \frac {108}{25} \, e^{\left (x + 6\right )} \]
integrate(1/25*(((-50*x^5-50*x^4)*exp(1)^2+(-60*x^5+240*x^4+150*x^3)*exp(1 )-18*x^5+162*x^4-360*x^3)*exp(3)^2*exp(x)^2+((-50*x^5-150*x^4-50*x^3)*exp( 1)^2+(-60*x^5+120*x^4+540*x^3)*exp(1)-18*x^5+126*x^4-108*x^3-450*x^2+450*x )*exp(3)^2*exp(x)+((-50*x^4-50*x^3)*exp(1)^2+(-60*x^4+90*x^3-150*x)*exp(1) -18*x^4+72*x^3+360*x+450)*exp(3)^2)/x^3,x, algorithm=\
-x^2*e^8 - 6/5*x^2*e^7 - 9/25*x^2*e^6 - 2*x*e^8 + 18/5*x*e^7 + 72/25*x*e^6 - 18*Ei(x)*e^6 - 1/2*(2*x^2*e^8 - 2*x*e^8 + e^8)*e^(2*x) - 3/5*(2*x^2*e^7 - 2*x*e^7 + e^7)*e^(2*x) - 9/50*(2*x^2*e^6 - 2*x*e^6 + e^6)*e^(2*x) - 1/2 *(2*x*e^8 - e^8)*e^(2*x) + 12/5*(2*x*e^7 - e^7)*e^(2*x) + 81/50*(2*x*e^6 - e^6)*e^(2*x) - 2*(x^2*e^8 - 2*x*e^8 + 2*e^8)*e^x - 12/5*(x^2*e^7 - 2*x*e^ 7 + 2*e^7)*e^x - 18/25*(x^2*e^6 - 2*x*e^6 + 2*e^6)*e^x - 6*(x*e^8 - e^8)*e ^x + 24/5*(x*e^7 - e^7)*e^x + 126/25*(x*e^6 - e^6)*e^x + 18*e^6*gamma(-1, -x) + 6*e^7/x - 72/5*e^6/x - 9*e^6/x^2 + 3*e^(2*x + 7) - 36/5*e^(2*x + 6) - 2*e^(x + 8) + 108/5*e^(x + 7) - 108/25*e^(x + 6)
Leaf count of result is larger than twice the leaf count of optimal. 453 vs. \(2 (30) = 60\).
Time = 0.29 (sec) , antiderivative size = 453, normalized size of antiderivative = 13.32 \[ \int \frac {e^6 \left (450+360 x+72 x^3-18 x^4+e \left (-150 x+90 x^3-60 x^4\right )+e^2 \left (-50 x^3-50 x^4\right )\right )+e^{6+x} \left (450 x-450 x^2-108 x^3+126 x^4-18 x^5+e \left (540 x^3+120 x^4-60 x^5\right )+e^2 \left (-50 x^3-150 x^4-50 x^5\right )\right )+e^{6+2 x} \left (-360 x^3+162 x^4-18 x^5+e \left (150 x^3+240 x^4-60 x^5\right )+e^2 \left (-50 x^4-50 x^5\right )\right )}{25 x^3} \, dx =\text {Too large to display} \]
integrate(1/25*(((-50*x^5-50*x^4)*exp(1)^2+(-60*x^5+240*x^4+150*x^3)*exp(1 )-18*x^5+162*x^4-360*x^3)*exp(3)^2*exp(x)^2+((-50*x^5-150*x^4-50*x^3)*exp( 1)^2+(-60*x^5+120*x^4+540*x^3)*exp(1)-18*x^5+126*x^4-108*x^3-450*x^2+450*x )*exp(3)^2*exp(x)+((-50*x^4-50*x^3)*exp(1)^2+(-60*x^4+90*x^3-150*x)*exp(1) -18*x^4+72*x^3+360*x+450)*exp(3)^2)/x^3,x, algorithm=\
-1/25*(25*(x + 6)^4*e^23 + 30*(x + 6)^4*e^22 + 9*(x + 6)^4*e^21 + 25*(x + 6)^4*e^(2*x + 23) + 30*(x + 6)^4*e^(2*x + 22) + 9*(x + 6)^4*e^(2*x + 21) + 50*(x + 6)^4*e^(x + 23) + 60*(x + 6)^4*e^(x + 22) + 18*(x + 6)^4*e^(x + 2 1) - 550*(x + 6)^3*e^23 - 810*(x + 6)^3*e^22 - 288*(x + 6)^3*e^21 - 600*(x + 6)^3*e^(2*x + 23) - 870*(x + 6)^3*e^(2*x + 22) - 306*(x + 6)^3*e^(2*x + 21) - 1150*(x + 6)^3*e^(x + 23) - 1680*(x + 6)^3*e^(x + 22) - 594*(x + 6) ^3*e^(x + 21) + 3900*(x + 6)^2*e^23 + 6480*(x + 6)^2*e^22 + 2484*(x + 6)^2 *e^21 + 5400*(x + 6)^2*e^(2*x + 23) + 9180*(x + 6)^2*e^(2*x + 22) + 3789*( x + 6)^2*e^(2*x + 21) + 9900*(x + 6)^2*e^(x + 23) + 16980*(x + 6)^2*e^(x + 22) + 7074*(x + 6)^2*e^(x + 21) - 9000*(x + 6)*e^23 - 16350*(x + 6)*e^22 - 6120*(x + 6)*e^21 - 21600*(x + 6)*e^(2*x + 23) - 42120*(x + 6)*e^(2*x + 22) - 20196*(x + 6)*e^(2*x + 21) - 37800*(x + 6)*e^(x + 23) - 74160*(x + 6 )*e^(x + 22) - 35838*(x + 6)*e^(x + 21) + 900*e^22 - 1935*e^21 + 32400*e^( 2*x + 23) + 71280*e^(2*x + 22) + 39204*e^(2*x + 21) + 54000*e^(x + 23) + 1 18800*e^(x + 22) + 65340*e^(x + 21))/((x + 6)^2*e^15 - 12*(x + 6)*e^15 + 3 6*e^15)
Time = 9.43 (sec) , antiderivative size = 139, normalized size of antiderivative = 4.09 \[ \int \frac {e^6 \left (450+360 x+72 x^3-18 x^4+e \left (-150 x+90 x^3-60 x^4\right )+e^2 \left (-50 x^3-50 x^4\right )\right )+e^{6+x} \left (450 x-450 x^2-108 x^3+126 x^4-18 x^5+e \left (540 x^3+120 x^4-60 x^5\right )+e^2 \left (-50 x^3-150 x^4-50 x^5\right )\right )+e^{6+2 x} \left (-360 x^3+162 x^4-18 x^5+e \left (150 x^3+240 x^4-60 x^5\right )+e^2 \left (-50 x^4-50 x^5\right )\right )}{25 x^3} \, dx={\mathrm {e}}^{x+6}\,\left (12\,\mathrm {e}-\frac {54}{5}\right )-x^2\,\left (\frac {9\,{\mathrm {e}}^6}{25}+\frac {6\,{\mathrm {e}}^7}{5}+{\mathrm {e}}^8+{\mathrm {e}}^{2\,x+6}\,\left (\frac {6\,\mathrm {e}}{5}+{\mathrm {e}}^2+\frac {9}{25}\right )+{\mathrm {e}}^{x+6}\,\left (\frac {12\,\mathrm {e}}{5}+2\,{\mathrm {e}}^2+\frac {18}{25}\right )\right )-\frac {9\,{\mathrm {e}}^6+x\,\left (18\,{\mathrm {e}}^{x+6}+\frac {72\,{\mathrm {e}}^6}{5}-6\,{\mathrm {e}}^7\right )}{x^2}-9\,{\mathrm {e}}^{2\,x+6}+x\,\left (\frac {72\,{\mathrm {e}}^6}{25}+\frac {18\,{\mathrm {e}}^7}{5}-2\,{\mathrm {e}}^8+{\mathrm {e}}^{x+6}\,\left (\frac {48\,\mathrm {e}}{5}-2\,{\mathrm {e}}^2+\frac {162}{25}\right )+{\mathrm {e}}^{2\,x+12}\,\left (6\,{\mathrm {e}}^{-5}+\frac {18\,{\mathrm {e}}^{-6}}{5}\right )\right ) \]
int(-((exp(6)*exp(x)*(exp(2)*(50*x^3 + 150*x^4 + 50*x^5) - 450*x - exp(1)* (540*x^3 + 120*x^4 - 60*x^5) + 450*x^2 + 108*x^3 - 126*x^4 + 18*x^5))/25 - (exp(6)*(360*x - exp(1)*(150*x - 90*x^3 + 60*x^4) - exp(2)*(50*x^3 + 50*x ^4) + 72*x^3 - 18*x^4 + 450))/25 + (exp(2*x)*exp(6)*(exp(2)*(50*x^4 + 50*x ^5) - exp(1)*(150*x^3 + 240*x^4 - 60*x^5) + 360*x^3 - 162*x^4 + 18*x^5))/2 5)/x^3,x)
exp(x + 6)*(12*exp(1) - 54/5) - x^2*((9*exp(6))/25 + (6*exp(7))/5 + exp(8) + exp(2*x + 6)*((6*exp(1))/5 + exp(2) + 9/25) + exp(x + 6)*((12*exp(1))/5 + 2*exp(2) + 18/25)) - (9*exp(6) + x*(18*exp(x + 6) + (72*exp(6))/5 - 6*e xp(7)))/x^2 - 9*exp(2*x + 6) + x*((72*exp(6))/25 + (18*exp(7))/5 - 2*exp(8 ) + exp(x + 6)*((48*exp(1))/5 - 2*exp(2) + 162/25) + exp(2*x + 12)*(6*exp( -5) + (18*exp(-6))/5))