\(\int \frac {27 x^2-27 x^3-45 x^4-17 x^5-2 x^6+e^6 (x^2-2 x^3)+e^4 (9 x^2-15 x^3-6 x^4)+e^2 (27 x^2-36 x^3-33 x^4-6 x^5)+e^{\frac {-125-15 e^4+e^2 (-90-30 x)-90 x-15 x^2}{9 x+e^4 x+6 x^2+x^3+e^2 (6 x+2 x^2)}} (375+15 e^6+375 x+135 x^2+15 x^3+e^4 (135+45 x)+e^2 (395+270 x+45 x^2))}{27 x^2+e^6 x^2+27 x^3+9 x^4+x^5+e^4 (9 x^2+3 x^3)+e^2 (27 x^2+18 x^3+3 x^4)} \, dx\) [1360]

Optimal result

Integrand size = 251, antiderivative size = 27 \[ \int \frac {27 x^2-27 x^3-45 x^4-17 x^5-2 x^6+e^6 \left (x^2-2 x^3\right )+e^4 \left (9 x^2-15 x^3-6 x^4\right )+e^2 \left (27 x^2-36 x^3-33 x^4-6 x^5\right )+e^{\frac {-125-15 e^4+e^2 (-90-30 x)-90 x-15 x^2}{9 x+e^4 x+6 x^2+x^3+e^2 \left (6 x+2 x^2\right )}} \left (375+15 e^6+375 x+135 x^2+15 x^3+e^4 (135+45 x)+e^2 \left (395+270 x+45 x^2\right )\right )}{27 x^2+e^6 x^2+27 x^3+9 x^4+x^5+e^4 \left (9 x^2+3 x^3\right )+e^2 \left (27 x^2+18 x^3+3 x^4\right )} \, dx=9+e^{\frac {5 \left (-3+\frac {2}{\left (3+e^2+x\right )^2}\right )}{x}}+x-x^2 \]

[Out]

exp(5*(2/(3+exp(2)+x)^2-3)/x)-x^2+9+x

Rubi [F]

\[ \int \frac {27 x^2-27 x^3-45 x^4-17 x^5-2 x^6+e^6 \left (x^2-2 x^3\right )+e^4 \left (9 x^2-15 x^3-6 x^4\right )+e^2 \left (27 x^2-36 x^3-33 x^4-6 x^5\right )+e^{\frac {-125-15 e^4+e^2 (-90-30 x)-90 x-15 x^2}{9 x+e^4 x+6 x^2+x^3+e^2 \left (6 x+2 x^2\right )}} \left (375+15 e^6+375 x+135 x^2+15 x^3+e^4 (135+45 x)+e^2 \left (395+270 x+45 x^2\right )\right )}{27 x^2+e^6 x^2+27 x^3+9 x^4+x^5+e^4 \left (9 x^2+3 x^3\right )+e^2 \left (27 x^2+18 x^3+3 x^4\right )} \, dx=\int \frac {27 x^2-27 x^3-45 x^4-17 x^5-2 x^6+e^6 \left (x^2-2 x^3\right )+e^4 \left (9 x^2-15 x^3-6 x^4\right )+e^2 \left (27 x^2-36 x^3-33 x^4-6 x^5\right )+\exp \left (\frac {-125-15 e^4+e^2 (-90-30 x)-90 x-15 x^2}{9 x+e^4 x+6 x^2+x^3+e^2 \left (6 x+2 x^2\right )}\right ) \left (375+15 e^6+375 x+135 x^2+15 x^3+e^4 (135+45 x)+e^2 \left (395+270 x+45 x^2\right )\right )}{27 x^2+e^6 x^2+27 x^3+9 x^4+x^5+e^4 \left (9 x^2+3 x^3\right )+e^2 \left (27 x^2+18 x^3+3 x^4\right )} \, dx \]

[In]

Int[(27*x^2 - 27*x^3 - 45*x^4 - 17*x^5 - 2*x^6 + E^6*(x^2 - 2*x^3) + E^4*(9*x^2 - 15*x^3 - 6*x^4) + E^2*(27*x^
2 - 36*x^3 - 33*x^4 - 6*x^5) + E^((-125 - 15*E^4 + E^2*(-90 - 30*x) - 90*x - 15*x^2)/(9*x + E^4*x + 6*x^2 + x^
3 + E^2*(6*x + 2*x^2)))*(375 + 15*E^6 + 375*x + 135*x^2 + 15*x^3 + E^4*(135 + 45*x) + E^2*(395 + 270*x + 45*x^
2)))/(27*x^2 + E^6*x^2 + 27*x^3 + 9*x^4 + x^5 + E^4*(9*x^2 + 3*x^3) + E^2*(27*x^2 + 18*x^3 + 3*x^4)),x]

[Out]

-17*x - 6*E^2*x + 6*(3 + E^2)*x - x^2 - 27/(2*(3 + E^2 + x)^2) + (45*(3 + E^2)^2)/(2*(3 + E^2 + x)^2) - (17*(3
+ E^2)^3)/(2*(3 + E^2 + x)^2) + (3 + E^2)^4/(3 + E^2 + x)^2 - (E^6*(1 - 2*x)^2)/(2*(7 + 2*E^2)*(3 + E^2 + x)^
2) - (27*x^2)/(2*(3 + E^2)*(3 + E^2 + x)^2) - (90*(3 + E^2))/(3 + E^2 + x) + (51*(3 + E^2)^2)/(3 + E^2 + x) -
(8*(3 + E^2)^3)/(3 + E^2 + x) + (6*E^4*(7 + 3*E^2))/(3 + E^2 + x) - (3*E^4*(7 + 4*E^2))/(3 + E^2 + x) - 45*Log
[3 + E^2 + x] - 6*E^4*Log[3 + E^2 + x] + 51*(3 + E^2)*Log[3 + E^2 + x] - 12*(3 + E^2)^2*Log[3 + E^2 + x] + 3*E
^2*(7 + 6*E^2)*Log[3 + E^2 + x] + (5*(25 + 18*E^2 + 3*E^4)*Defer[Int][1/(E^((5*(25 + 18*E^2 + 3*E^4 + 6*(3 + E
^2)*x + 3*x^2))/(x*(3 + E^2 + x)^2))*x^2), x])/(3 + E^2)^2 + (20*Defer[Int][1/(E^((5*(25 + 18*E^2 + 3*E^4 + 6*
(3 + E^2)*x + 3*x^2))/(x*(3 + E^2 + x)^2))*(3 + E^2 + x)^3), x])/(3 + E^2) + (10*Defer[Int][1/(E^((5*(25 + 18*
E^2 + 3*E^4 + 6*(3 + E^2)*x + 3*x^2))/(x*(3 + E^2 + x)^2))*(3 + E^2 + x)^2), x])/(3 + E^2)^2

Rubi steps \begin{align*} \text {integral}& = \int \frac {27 x^2-27 x^3-45 x^4-17 x^5-2 x^6+e^6 \left (x^2-2 x^3\right )+e^4 \left (9 x^2-15 x^3-6 x^4\right )+e^2 \left (27 x^2-36 x^3-33 x^4-6 x^5\right )+\exp \left (\frac {-125-15 e^4+e^2 (-90-30 x)-90 x-15 x^2}{9 x+e^4 x+6 x^2+x^3+e^2 \left (6 x+2 x^2\right )}\right ) \left (375+15 e^6+375 x+135 x^2+15 x^3+e^4 (135+45 x)+e^2 \left (395+270 x+45 x^2\right )\right )}{\left (27+e^6\right ) x^2+27 x^3+9 x^4+x^5+e^4 \left (9 x^2+3 x^3\right )+e^2 \left (27 x^2+18 x^3+3 x^4\right )} \, dx \\ & = \int \frac {27 x^2-27 x^3-45 x^4-17 x^5-2 x^6-3 e^2 x^2 (3+x)^2 (-1+2 x)-3 e^4 x^2 \left (-3+5 x+2 x^2\right )+e^6 \left (x^2-2 x^3\right )+5 \exp \left (-\frac {5 \left (25+3 e^4+18 x+3 x^2+6 e^2 (3+x)\right )}{x \left (3+e^2+x\right )^2}\right ) \left (3 e^6+9 e^4 (3+x)+e^2 \left (79+54 x+9 x^2\right )+3 \left (25+25 x+9 x^2+x^3\right )\right )}{x^2 \left (3+e^2+x\right )^3} \, dx \\ & = \int \left (\frac {27}{\left (3+e^2+x\right )^3}-\frac {27 x}{\left (3+e^2+x\right )^3}-\frac {45 x^2}{\left (3+e^2+x\right )^3}-\frac {17 x^3}{\left (3+e^2+x\right )^3}-\frac {2 x^4}{\left (3+e^2+x\right )^3}-\frac {e^6 (-1+2 x)}{\left (3+e^2+x\right )^3}-\frac {3 e^4 (3+x) (-1+2 x)}{\left (3+e^2+x\right )^3}-\frac {3 e^2 (3+x)^2 (-1+2 x)}{\left (3+e^2+x\right )^3}+\frac {5 \exp \left (-\frac {5 \left (25+18 e^2+3 e^4+6 \left (3+e^2\right ) x+3 x^2\right )}{x \left (3+e^2+x\right )^2}\right ) \left (75+79 e^2+27 e^4+3 e^6+3 \left (25+18 e^2+3 e^4\right ) x+9 \left (3+e^2\right ) x^2+3 x^3\right )}{x^2 \left (3+e^2+x\right )^3}\right ) \, dx \\ & = -\frac {27}{2 \left (3+e^2+x\right )^2}-2 \int \frac {x^4}{\left (3+e^2+x\right )^3} \, dx+5 \int \frac {\exp \left (-\frac {5 \left (25+18 e^2+3 e^4+6 \left (3+e^2\right ) x+3 x^2\right )}{x \left (3+e^2+x\right )^2}\right ) \left (75+79 e^2+27 e^4+3 e^6+3 \left (25+18 e^2+3 e^4\right ) x+9 \left (3+e^2\right ) x^2+3 x^3\right )}{x^2 \left (3+e^2+x\right )^3} \, dx-17 \int \frac {x^3}{\left (3+e^2+x\right )^3} \, dx-27 \int \frac {x}{\left (3+e^2+x\right )^3} \, dx-45 \int \frac {x^2}{\left (3+e^2+x\right )^3} \, dx-\left (3 e^2\right ) \int \frac {(3+x)^2 (-1+2 x)}{\left (3+e^2+x\right )^3} \, dx-\left (3 e^4\right ) \int \frac {(3+x) (-1+2 x)}{\left (3+e^2+x\right )^3} \, dx-e^6 \int \frac {-1+2 x}{\left (3+e^2+x\right )^3} \, dx \\ & = -\frac {27}{2 \left (3+e^2+x\right )^2}-\frac {e^6 (1-2 x)^2}{2 \left (7+2 e^2\right ) \left (3+e^2+x\right )^2}-\frac {27 x^2}{2 \left (3+e^2\right ) \left (3+e^2+x\right )^2}-2 \int \left (-3 \left (3+e^2\right )+x+\frac {\left (3+e^2\right )^4}{\left (3+e^2+x\right )^3}-\frac {4 \left (3+e^2\right )^3}{\left (3+e^2+x\right )^2}+\frac {6 \left (3+e^2\right )^2}{3+e^2+x}\right ) \, dx+5 \int \left (\frac {\exp \left (-\frac {5 \left (25+18 e^2+3 e^4+6 \left (3+e^2\right ) x+3 x^2\right )}{x \left (3+e^2+x\right )^2}\right ) \left (25+18 e^2+3 e^4\right )}{\left (3+e^2\right )^2 x^2}+\frac {4 \exp \left (-\frac {5 \left (25+18 e^2+3 e^4+6 \left (3+e^2\right ) x+3 x^2\right )}{x \left (3+e^2+x\right )^2}\right )}{\left (3+e^2\right ) \left (3+e^2+x\right )^3}+\frac {2 \exp \left (-\frac {5 \left (25+18 e^2+3 e^4+6 \left (3+e^2\right ) x+3 x^2\right )}{x \left (3+e^2+x\right )^2}\right )}{\left (3+e^2\right )^2 \left (3+e^2+x\right )^2}\right ) \, dx-17 \int \left (1-\frac {\left (3+e^2\right )^3}{\left (3+e^2+x\right )^3}+\frac {3 \left (3+e^2\right )^2}{\left (3+e^2+x\right )^2}-\frac {3 \left (3+e^2\right )}{3+e^2+x}\right ) \, dx-45 \int \left (\frac {\left (3+e^2\right )^2}{\left (3+e^2+x\right )^3}-\frac {2 \left (3+e^2\right )}{\left (3+e^2+x\right )^2}+\frac {1}{3+e^2+x}\right ) \, dx-\left (3 e^2\right ) \int \left (2-\frac {e^4 \left (7+2 e^2\right )}{\left (3+e^2+x\right )^3}+\frac {2 e^2 \left (7+3 e^2\right )}{\left (3+e^2+x\right )^2}+\frac {-7-6 e^2}{3+e^2+x}\right ) \, dx-\left (3 e^4\right ) \int \left (\frac {e^2 \left (7+2 e^2\right )}{\left (3+e^2+x\right )^3}+\frac {-7-4 e^2}{\left (3+e^2+x\right )^2}+\frac {2}{3+e^2+x}\right ) \, dx \\ & = -17 x-6 e^2 x+6 \left (3+e^2\right ) x-x^2-\frac {27}{2 \left (3+e^2+x\right )^2}+\frac {45 \left (3+e^2\right )^2}{2 \left (3+e^2+x\right )^2}-\frac {17 \left (3+e^2\right )^3}{2 \left (3+e^2+x\right )^2}+\frac {\left (3+e^2\right )^4}{\left (3+e^2+x\right )^2}-\frac {e^6 (1-2 x)^2}{2 \left (7+2 e^2\right ) \left (3+e^2+x\right )^2}-\frac {27 x^2}{2 \left (3+e^2\right ) \left (3+e^2+x\right )^2}-\frac {90 \left (3+e^2\right )}{3+e^2+x}+\frac {51 \left (3+e^2\right )^2}{3+e^2+x}-\frac {8 \left (3+e^2\right )^3}{3+e^2+x}+\frac {6 e^4 \left (7+3 e^2\right )}{3+e^2+x}-\frac {3 e^4 \left (7+4 e^2\right )}{3+e^2+x}-45 \log \left (3+e^2+x\right )-6 e^4 \log \left (3+e^2+x\right )+51 \left (3+e^2\right ) \log \left (3+e^2+x\right )-12 \left (3+e^2\right )^2 \log \left (3+e^2+x\right )+3 e^2 \left (7+6 e^2\right ) \log \left (3+e^2+x\right )+\frac {10 \int \frac {\exp \left (-\frac {5 \left (25+18 e^2+3 e^4+6 \left (3+e^2\right ) x+3 x^2\right )}{x \left (3+e^2+x\right )^2}\right )}{\left (3+e^2+x\right )^2} \, dx}{\left (3+e^2\right )^2}+\frac {20 \int \frac {\exp \left (-\frac {5 \left (25+18 e^2+3 e^4+6 \left (3+e^2\right ) x+3 x^2\right )}{x \left (3+e^2+x\right )^2}\right )}{\left (3+e^2+x\right )^3} \, dx}{3+e^2}+\frac {\left (5 \left (25+18 e^2+3 e^4\right )\right ) \int \frac {\exp \left (-\frac {5 \left (25+18 e^2+3 e^4+6 \left (3+e^2\right ) x+3 x^2\right )}{x \left (3+e^2+x\right )^2}\right )}{x^2} \, dx}{\left (3+e^2\right )^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.67 \[ \int \frac {27 x^2-27 x^3-45 x^4-17 x^5-2 x^6+e^6 \left (x^2-2 x^3\right )+e^4 \left (9 x^2-15 x^3-6 x^4\right )+e^2 \left (27 x^2-36 x^3-33 x^4-6 x^5\right )+e^{\frac {-125-15 e^4+e^2 (-90-30 x)-90 x-15 x^2}{9 x+e^4 x+6 x^2+x^3+e^2 \left (6 x+2 x^2\right )}} \left (375+15 e^6+375 x+135 x^2+15 x^3+e^4 (135+45 x)+e^2 \left (395+270 x+45 x^2\right )\right )}{27 x^2+e^6 x^2+27 x^3+9 x^4+x^5+e^4 \left (9 x^2+3 x^3\right )+e^2 \left (27 x^2+18 x^3+3 x^4\right )} \, dx=e^{-\frac {5 \left (25+3 e^4+18 x+3 x^2+6 e^2 (3+x)\right )}{x \left (3+e^2+x\right )^2}}+x-x^2 \]

[In]

Integrate[(27*x^2 - 27*x^3 - 45*x^4 - 17*x^5 - 2*x^6 + E^6*(x^2 - 2*x^3) + E^4*(9*x^2 - 15*x^3 - 6*x^4) + E^2*
(27*x^2 - 36*x^3 - 33*x^4 - 6*x^5) + E^((-125 - 15*E^4 + E^2*(-90 - 30*x) - 90*x - 15*x^2)/(9*x + E^4*x + 6*x^
2 + x^3 + E^2*(6*x + 2*x^2)))*(375 + 15*E^6 + 375*x + 135*x^2 + 15*x^3 + E^4*(135 + 45*x) + E^2*(395 + 270*x +
45*x^2)))/(27*x^2 + E^6*x^2 + 27*x^3 + 9*x^4 + x^5 + E^4*(9*x^2 + 3*x^3) + E^2*(27*x^2 + 18*x^3 + 3*x^4)),x]

[Out]

E^((-5*(25 + 3*E^4 + 18*x + 3*x^2 + 6*E^2*(3 + x)))/(x*(3 + E^2 + x)^2)) + x - x^2

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(57\) vs. \(2(25)=50\).

Time = 1.50 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.15

method	result	size
risch	\(-x^{2}+x +{\mathrm e}^{-\frac {5 \left (6 \,{\mathrm e}^{2} x +3 x^{2}+18 \,{\mathrm e}^{2}+3 \,{\mathrm e}^{4}+18 x +25\right )}{x \left (2 \,{\mathrm e}^{2} x +x^{2}+6 \,{\mathrm e}^{2}+{\mathrm e}^{4}+6 x +9\right )}}\)	\(58\)
parallelrisch	\(15 \,{\mathrm e}^{4}-x^{2}+84 \,{\mathrm e}^{2}+x +{\mathrm e}^{\frac {-15 \,{\mathrm e}^{4}+\left (-30 x -90\right ) {\mathrm e}^{2}-15 x^{2}-90 x -125}{x \left (2 \,{\mathrm e}^{2} x +x^{2}+6 \,{\mathrm e}^{2}+{\mathrm e}^{4}+6 x +9\right )}}+117\)	\(71\)
parts	\(-x^{2}+x +\frac {x^{3} {\mathrm e}^{\frac {-15 \,{\mathrm e}^{4}+\left (-30 x -90\right ) {\mathrm e}^{2}-15 x^{2}-90 x -125}{x \,{\mathrm e}^{4}+\left (2 x^{2}+6 x \right ) {\mathrm e}^{2}+x^{3}+6 x^{2}+9 x}}+\left (6+2 \,{\mathrm e}^{2}\right ) x^{2} {\mathrm e}^{\frac {-15 \,{\mathrm e}^{4}+\left (-30 x -90\right ) {\mathrm e}^{2}-15 x^{2}-90 x -125}{x \,{\mathrm e}^{4}+\left (2 x^{2}+6 x \right ) {\mathrm e}^{2}+x^{3}+6 x^{2}+9 x}}+\left (6 \,{\mathrm e}^{2}+{\mathrm e}^{4}+9\right ) x \,{\mathrm e}^{\frac {-15 \,{\mathrm e}^{4}+\left (-30 x -90\right ) {\mathrm e}^{2}-15 x^{2}-90 x -125}{x \,{\mathrm e}^{4}+\left (2 x^{2}+6 x \right ) {\mathrm e}^{2}+x^{3}+6 x^{2}+9 x}}}{x \left (3+{\mathrm e}^{2}+x \right )^{2}}\)	\(220\)
norman	\(\frac {x^{3} {\mathrm e}^{\frac {-15 \,{\mathrm e}^{4}+\left (-30 x -90\right ) {\mathrm e}^{2}-15 x^{2}-90 x -125}{x \,{\mathrm e}^{4}+\left (2 x^{2}+6 x \right ) {\mathrm e}^{2}+x^{3}+6 x^{2}+9 x}}+\left (-5-2 \,{\mathrm e}^{2}\right ) x^{4}+\left (2 \,{\mathrm e}^{6}+15 \,{\mathrm e}^{4}+36 \,{\mathrm e}^{2}+27\right ) x^{2}+\left ({\mathrm e}^{8}+10 \,{\mathrm e}^{6}+36 \,{\mathrm e}^{4}+54 \,{\mathrm e}^{2}+27\right ) x +\left (6+2 \,{\mathrm e}^{2}\right ) x^{2} {\mathrm e}^{\frac {-15 \,{\mathrm e}^{4}+\left (-30 x -90\right ) {\mathrm e}^{2}-15 x^{2}-90 x -125}{x \,{\mathrm e}^{4}+\left (2 x^{2}+6 x \right ) {\mathrm e}^{2}+x^{3}+6 x^{2}+9 x}}+\left (6 \,{\mathrm e}^{2}+{\mathrm e}^{4}+9\right ) x \,{\mathrm e}^{\frac {-15 \,{\mathrm e}^{4}+\left (-30 x -90\right ) {\mathrm e}^{2}-15 x^{2}-90 x -125}{x \,{\mathrm e}^{4}+\left (2 x^{2}+6 x \right ) {\mathrm e}^{2}+x^{3}+6 x^{2}+9 x}}-x^{5}}{x \left (3+{\mathrm e}^{2}+x \right )^{2}}\)	\(274\)

[In]

int(((15*exp(2)^3+(45*x+135)*exp(2)^2+(45*x^2+270*x+395)*exp(2)+15*x^3+135*x^2+375*x+375)*exp((-15*exp(2)^2+(-
30*x-90)*exp(2)-15*x^2-90*x-125)/(x*exp(2)^2+(2*x^2+6*x)*exp(2)+x^3+6*x^2+9*x))+(-2*x^3+x^2)*exp(2)^3+(-6*x^4-
15*x^3+9*x^2)*exp(2)^2+(-6*x^5-33*x^4-36*x^3+27*x^2)*exp(2)-2*x^6-17*x^5-45*x^4-27*x^3+27*x^2)/(x^2*exp(2)^3+(
3*x^3+9*x^2)*exp(2)^2+(3*x^4+18*x^3+27*x^2)*exp(2)+x^5+9*x^4+27*x^3+27*x^2),x,method=_RETURNVERBOSE)

[Out]

-x^2+x+exp(-5*(6*exp(2)*x+3*x^2+18*exp(2)+3*exp(4)+18*x+25)/x/(2*exp(2)*x+x^2+6*exp(2)+exp(4)+6*x+9))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (25) = 50\).

Time = 0.26 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.22 \[ \int \frac {27 x^2-27 x^3-45 x^4-17 x^5-2 x^6+e^6 \left (x^2-2 x^3\right )+e^4 \left (9 x^2-15 x^3-6 x^4\right )+e^2 \left (27 x^2-36 x^3-33 x^4-6 x^5\right )+e^{\frac {-125-15 e^4+e^2 (-90-30 x)-90 x-15 x^2}{9 x+e^4 x+6 x^2+x^3+e^2 \left (6 x+2 x^2\right )}} \left (375+15 e^6+375 x+135 x^2+15 x^3+e^4 (135+45 x)+e^2 \left (395+270 x+45 x^2\right )\right )}{27 x^2+e^6 x^2+27 x^3+9 x^4+x^5+e^4 \left (9 x^2+3 x^3\right )+e^2 \left (27 x^2+18 x^3+3 x^4\right )} \, dx=-x^{2} + x + e^{\left (-\frac {5 \, {\left (3 \, x^{2} + 6 \, {\left (x + 3\right )} e^{2} + 18 \, x + 3 \, e^{4} + 25\right )}}{x^{3} + 6 \, x^{2} + x e^{4} + 2 \, {\left (x^{2} + 3 \, x\right )} e^{2} + 9 \, x}\right )} \]

[In]

integrate(((15*exp(2)^3+(45*x+135)*exp(2)^2+(45*x^2+270*x+395)*exp(2)+15*x^3+135*x^2+375*x+375)*exp((-15*exp(2
)^2+(-30*x-90)*exp(2)-15*x^2-90*x-125)/(x*exp(2)^2+(2*x^2+6*x)*exp(2)+x^3+6*x^2+9*x))+(-2*x^3+x^2)*exp(2)^3+(-
6*x^4-15*x^3+9*x^2)*exp(2)^2+(-6*x^5-33*x^4-36*x^3+27*x^2)*exp(2)-2*x^6-17*x^5-45*x^4-27*x^3+27*x^2)/(x^2*exp(
2)^3+(3*x^3+9*x^2)*exp(2)^2+(3*x^4+18*x^3+27*x^2)*exp(2)+x^5+9*x^4+27*x^3+27*x^2),x, algorithm="fricas")

[Out]

-x^2 + x + e^(-5*(3*x^2 + 6*(x + 3)*e^2 + 18*x + 3*e^4 + 25)/(x^3 + 6*x^2 + x*e^4 + 2*(x^2 + 3*x)*e^2 + 9*x))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (20) = 40\).

Time = 0.78 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.22 \[ \int \frac {27 x^2-27 x^3-45 x^4-17 x^5-2 x^6+e^6 \left (x^2-2 x^3\right )+e^4 \left (9 x^2-15 x^3-6 x^4\right )+e^2 \left (27 x^2-36 x^3-33 x^4-6 x^5\right )+e^{\frac {-125-15 e^4+e^2 (-90-30 x)-90 x-15 x^2}{9 x+e^4 x+6 x^2+x^3+e^2 \left (6 x+2 x^2\right )}} \left (375+15 e^6+375 x+135 x^2+15 x^3+e^4 (135+45 x)+e^2 \left (395+270 x+45 x^2\right )\right )}{27 x^2+e^6 x^2+27 x^3+9 x^4+x^5+e^4 \left (9 x^2+3 x^3\right )+e^2 \left (27 x^2+18 x^3+3 x^4\right )} \, dx=- x^{2} + x + e^{\frac {- 15 x^{2} - 90 x + \left (- 30 x - 90\right ) e^{2} - 15 e^{4} - 125}{x^{3} + 6 x^{2} + 9 x + x e^{4} + \left (2 x^{2} + 6 x\right ) e^{2}}} \]

[In]

integrate(((15*exp(2)**3+(45*x+135)*exp(2)**2+(45*x**2+270*x+395)*exp(2)+15*x**3+135*x**2+375*x+375)*exp((-15*
exp(2)**2+(-30*x-90)*exp(2)-15*x**2-90*x-125)/(x*exp(2)**2+(2*x**2+6*x)*exp(2)+x**3+6*x**2+9*x))+(-2*x**3+x**2
)*exp(2)**3+(-6*x**4-15*x**3+9*x**2)*exp(2)**2+(-6*x**5-33*x**4-36*x**3+27*x**2)*exp(2)-2*x**6-17*x**5-45*x**4
-27*x**3+27*x**2)/(x**2*exp(2)**3+(3*x**3+9*x**2)*exp(2)**2+(3*x**4+18*x**3+27*x**2)*exp(2)+x**5+9*x**4+27*x**
3+27*x**2),x)

[Out]

-x**2 + x + exp((-15*x**2 - 90*x + (-30*x - 90)*exp(2) - 15*exp(4) - 125)/(x**3 + 6*x**2 + 9*x + x*exp(4) + (2
*x**2 + 6*x)*exp(2)))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 873 vs. \(2 (25) = 50\).

Time = 0.95 (sec) , antiderivative size = 873, normalized size of antiderivative = 32.33 \[ \int \frac {27 x^2-27 x^3-45 x^4-17 x^5-2 x^6+e^6 \left (x^2-2 x^3\right )+e^4 \left (9 x^2-15 x^3-6 x^4\right )+e^2 \left (27 x^2-36 x^3-33 x^4-6 x^5\right )+e^{\frac {-125-15 e^4+e^2 (-90-30 x)-90 x-15 x^2}{9 x+e^4 x+6 x^2+x^3+e^2 \left (6 x+2 x^2\right )}} \left (375+15 e^6+375 x+135 x^2+15 x^3+e^4 (135+45 x)+e^2 \left (395+270 x+45 x^2\right )\right )}{27 x^2+e^6 x^2+27 x^3+9 x^4+x^5+e^4 \left (9 x^2+3 x^3\right )+e^2 \left (27 x^2+18 x^3+3 x^4\right )} \, dx=\text {Too large to display} \]

[In]

integrate(((15*exp(2)^3+(45*x+135)*exp(2)^2+(45*x^2+270*x+395)*exp(2)+15*x^3+135*x^2+375*x+375)*exp((-15*exp(2
)^2+(-30*x-90)*exp(2)-15*x^2-90*x-125)/(x*exp(2)^2+(2*x^2+6*x)*exp(2)+x^3+6*x^2+9*x))+(-2*x^3+x^2)*exp(2)^3+(-
6*x^4-15*x^3+9*x^2)*exp(2)^2+(-6*x^5-33*x^4-36*x^3+27*x^2)*exp(2)-2*x^6-17*x^5-45*x^4-27*x^3+27*x^2)/(x^2*exp(
2)^3+(3*x^3+9*x^2)*exp(2)^2+(3*x^4+18*x^3+27*x^2)*exp(2)+x^5+9*x^4+27*x^3+27*x^2),x, algorithm="maxima")

[Out]

-x^2 + 6*x*(e^2 + 3) - 3*((4*x*(e^2 + 3) + 3*e^4 + 18*e^2 + 27)/(x^2 + 2*x*(e^2 + 3) + e^4 + 6*e^2 + 9) + 2*lo
g(x + e^2 + 3))*e^4 + 3*(6*(e^2 + 3)*log(x + e^2 + 3) - 2*x + (6*x*(e^4 + 6*e^2 + 9) + 5*e^6 + 45*e^4 + 135*e^
2 + 135)/(x^2 + 2*x*(e^2 + 3) + e^4 + 6*e^2 + 9))*e^2 - 33/2*((4*x*(e^2 + 3) + 3*e^4 + 18*e^2 + 27)/(x^2 + 2*x
*(e^2 + 3) + e^4 + 6*e^2 + 9) + 2*log(x + e^2 + 3))*e^2 - 12*(e^4 + 6*e^2 + 9)*log(x + e^2 + 3) + 51*(e^2 + 3)
*log(x + e^2 + 3) - 17*x + (2*x + e^2 + 3)*e^6/(x^2 + 2*x*(e^2 + 3) + e^4 + 6*e^2 + 9) + 15/2*(2*x + e^2 + 3)*
e^4/(x^2 + 2*x*(e^2 + 3) + e^4 + 6*e^2 + 9) + 18*(2*x + e^2 + 3)*e^2/(x^2 + 2*x*(e^2 + 3) + e^4 + 6*e^2 + 9) -
(8*x*(e^6 + 9*e^4 + 27*e^2 + 27) + 7*e^8 + 84*e^6 + 378*e^4 + 756*e^2 + 567)/(x^2 + 2*x*(e^2 + 3) + e^4 + 6*e
^2 + 9) + 17/2*(6*x*(e^4 + 6*e^2 + 9) + 5*e^6 + 45*e^4 + 135*e^2 + 135)/(x^2 + 2*x*(e^2 + 3) + e^4 + 6*e^2 + 9
) - 45/2*(4*x*(e^2 + 3) + 3*e^4 + 18*e^2 + 27)/(x^2 + 2*x*(e^2 + 3) + e^4 + 6*e^2 + 9) + 27/2*(2*x + e^2 + 3)/
(x^2 + 2*x*(e^2 + 3) + e^4 + 6*e^2 + 9) - 1/2*e^6/(x^2 + 2*x*(e^2 + 3) + e^4 + 6*e^2 + 9) - 9/2*e^4/(x^2 + 2*x
*(e^2 + 3) + e^4 + 6*e^2 + 9) - 27/2*e^2/(x^2 + 2*x*(e^2 + 3) + e^4 + 6*e^2 + 9) - 27/2/(x^2 + 2*x*(e^2 + 3) +
e^4 + 6*e^2 + 9) + e^(15*e^4/(x^2*(e^2 + 3) + 2*x*(e^4 + 6*e^2 + 9) + e^6 + 9*e^4 + 27*e^2 + 27) + 15*e^4/(x*
(e^4 + 6*e^2 + 9) + e^6 + 9*e^4 + 27*e^2 + 27) + 90*e^2/(x^2*(e^2 + 3) + 2*x*(e^4 + 6*e^2 + 9) + e^6 + 9*e^4 +
27*e^2 + 27) - 15*e^2/(x^2 + 2*x*(e^2 + 3) + e^4 + 6*e^2 + 9) + 90*e^2/(x*(e^4 + 6*e^2 + 9) + e^6 + 9*e^4 + 2
7*e^2 + 27) + 125/(x^2*(e^2 + 3) + 2*x*(e^4 + 6*e^2 + 9) + e^6 + 9*e^4 + 27*e^2 + 27) - 45/(x^2 + 2*x*(e^2 + 3
) + e^4 + 6*e^2 + 9) + 125/(x*(e^4 + 6*e^2 + 9) + e^6 + 9*e^4 + 27*e^2 + 27) - 15/(x + e^2 + 3) - 15*e^4/(x*(e
^4 + 6*e^2 + 9)) - 90*e^2/(x*(e^4 + 6*e^2 + 9)) - 125/(x*(e^4 + 6*e^2 + 9))) - 45*log(x + e^2 + 3)

Giac [F]

\[ \int \frac {27 x^2-27 x^3-45 x^4-17 x^5-2 x^6+e^6 \left (x^2-2 x^3\right )+e^4 \left (9 x^2-15 x^3-6 x^4\right )+e^2 \left (27 x^2-36 x^3-33 x^4-6 x^5\right )+e^{\frac {-125-15 e^4+e^2 (-90-30 x)-90 x-15 x^2}{9 x+e^4 x+6 x^2+x^3+e^2 \left (6 x+2 x^2\right )}} \left (375+15 e^6+375 x+135 x^2+15 x^3+e^4 (135+45 x)+e^2 \left (395+270 x+45 x^2\right )\right )}{27 x^2+e^6 x^2+27 x^3+9 x^4+x^5+e^4 \left (9 x^2+3 x^3\right )+e^2 \left (27 x^2+18 x^3+3 x^4\right )} \, dx=\int { -\frac {2 \, x^{6} + 17 \, x^{5} + 45 \, x^{4} + 27 \, x^{3} - 27 \, x^{2} + {\left (2 \, x^{3} - x^{2}\right )} e^{6} + 3 \, {\left (2 \, x^{4} + 5 \, x^{3} - 3 \, x^{2}\right )} e^{4} + 3 \, {\left (2 \, x^{5} + 11 \, x^{4} + 12 \, x^{3} - 9 \, x^{2}\right )} e^{2} - 5 \, {\left (3 \, x^{3} + 27 \, x^{2} + 9 \, {\left (x + 3\right )} e^{4} + {\left (9 \, x^{2} + 54 \, x + 79\right )} e^{2} + 75 \, x + 3 \, e^{6} + 75\right )} e^{\left (-\frac {5 \, {\left (3 \, x^{2} + 6 \, {\left (x + 3\right )} e^{2} + 18 \, x + 3 \, e^{4} + 25\right )}}{x^{3} + 6 \, x^{2} + x e^{4} + 2 \, {\left (x^{2} + 3 \, x\right )} e^{2} + 9 \, x}\right )}}{x^{5} + 9 \, x^{4} + 27 \, x^{3} + x^{2} e^{6} + 27 \, x^{2} + 3 \, {\left (x^{3} + 3 \, x^{2}\right )} e^{4} + 3 \, {\left (x^{4} + 6 \, x^{3} + 9 \, x^{2}\right )} e^{2}} \,d x } \]

[In]

integrate(((15*exp(2)^3+(45*x+135)*exp(2)^2+(45*x^2+270*x+395)*exp(2)+15*x^3+135*x^2+375*x+375)*exp((-15*exp(2
)^2+(-30*x-90)*exp(2)-15*x^2-90*x-125)/(x*exp(2)^2+(2*x^2+6*x)*exp(2)+x^3+6*x^2+9*x))+(-2*x^3+x^2)*exp(2)^3+(-
6*x^4-15*x^3+9*x^2)*exp(2)^2+(-6*x^5-33*x^4-36*x^3+27*x^2)*exp(2)-2*x^6-17*x^5-45*x^4-27*x^3+27*x^2)/(x^2*exp(
2)^3+(3*x^3+9*x^2)*exp(2)^2+(3*x^4+18*x^3+27*x^2)*exp(2)+x^5+9*x^4+27*x^3+27*x^2),x, algorithm="giac")

[Out]

integrate(-(2*x^6 + 17*x^5 + 45*x^4 + 27*x^3 - 27*x^2 + (2*x^3 - x^2)*e^6 + 3*(2*x^4 + 5*x^3 - 3*x^2)*e^4 + 3*
(2*x^5 + 11*x^4 + 12*x^3 - 9*x^2)*e^2 - 5*(3*x^3 + 27*x^2 + 9*(x + 3)*e^4 + (9*x^2 + 54*x + 79)*e^2 + 75*x + 3
*e^6 + 75)*e^(-5*(3*x^2 + 6*(x + 3)*e^2 + 18*x + 3*e^4 + 25)/(x^3 + 6*x^2 + x*e^4 + 2*(x^2 + 3*x)*e^2 + 9*x)))
/(x^5 + 9*x^4 + 27*x^3 + x^2*e^6 + 27*x^2 + 3*(x^3 + 3*x^2)*e^4 + 3*(x^4 + 6*x^3 + 9*x^2)*e^2), x)

Mupad [B] (verification not implemented)

Time = 9.91 (sec) , antiderivative size = 217, normalized size of antiderivative = 8.04 \[ \int \frac {27 x^2-27 x^3-45 x^4-17 x^5-2 x^6+e^6 \left (x^2-2 x^3\right )+e^4 \left (9 x^2-15 x^3-6 x^4\right )+e^2 \left (27 x^2-36 x^3-33 x^4-6 x^5\right )+e^{\frac {-125-15 e^4+e^2 (-90-30 x)-90 x-15 x^2}{9 x+e^4 x+6 x^2+x^3+e^2 \left (6 x+2 x^2\right )}} \left (375+15 e^6+375 x+135 x^2+15 x^3+e^4 (135+45 x)+e^2 \left (395+270 x+45 x^2\right )\right )}{27 x^2+e^6 x^2+27 x^3+9 x^4+x^5+e^4 \left (9 x^2+3 x^3\right )+e^2 \left (27 x^2+18 x^3+3 x^4\right )} \, dx=x-x^2+{\mathrm {e}}^{-\frac {15\,x^2}{9\,x+6\,x\,{\mathrm {e}}^2+x\,{\mathrm {e}}^4+2\,x^2\,{\mathrm {e}}^2+6\,x^2+x^3}}\,{\mathrm {e}}^{-\frac {30\,x\,{\mathrm {e}}^2}{9\,x+6\,x\,{\mathrm {e}}^2+x\,{\mathrm {e}}^4+2\,x^2\,{\mathrm {e}}^2+6\,x^2+x^3}}\,{\mathrm {e}}^{-\frac {125}{9\,x+6\,x\,{\mathrm {e}}^2+x\,{\mathrm {e}}^4+2\,x^2\,{\mathrm {e}}^2+6\,x^2+x^3}}\,{\mathrm {e}}^{-\frac {15\,{\mathrm {e}}^4}{9\,x+6\,x\,{\mathrm {e}}^2+x\,{\mathrm {e}}^4+2\,x^2\,{\mathrm {e}}^2+6\,x^2+x^3}}\,{\mathrm {e}}^{-\frac {90\,{\mathrm {e}}^2}{9\,x+6\,x\,{\mathrm {e}}^2+x\,{\mathrm {e}}^4+2\,x^2\,{\mathrm {e}}^2+6\,x^2+x^3}}\,{\mathrm {e}}^{-\frac {90\,x}{9\,x+6\,x\,{\mathrm {e}}^2+x\,{\mathrm {e}}^4+2\,x^2\,{\mathrm {e}}^2+6\,x^2+x^3}} \]

[In]

int(-(exp(4)*(15*x^3 - 9*x^2 + 6*x^4) - exp(6)*(x^2 - 2*x^3) - 27*x^2 + 27*x^3 + 45*x^4 + 17*x^5 + 2*x^6 - exp
(-(90*x + 15*exp(4) + 15*x^2 + exp(2)*(30*x + 90) + 125)/(9*x + exp(2)*(6*x + 2*x^2) + x*exp(4) + 6*x^2 + x^3)
)*(375*x + 15*exp(6) + exp(2)*(270*x + 45*x^2 + 395) + 135*x^2 + 15*x^3 + exp(4)*(45*x + 135) + 375) + exp(2)*
(36*x^3 - 27*x^2 + 33*x^4 + 6*x^5))/(exp(4)*(9*x^2 + 3*x^3) + x^2*exp(6) + exp(2)*(27*x^2 + 18*x^3 + 3*x^4) +
27*x^2 + 27*x^3 + 9*x^4 + x^5),x)

[Out]

x - x^2 + exp(-(15*x^2)/(9*x + 6*x*exp(2) + x*exp(4) + 2*x^2*exp(2) + 6*x^2 + x^3))*exp(-(30*x*exp(2))/(9*x +
6*x*exp(2) + x*exp(4) + 2*x^2*exp(2) + 6*x^2 + x^3))*exp(-125/(9*x + 6*x*exp(2) + x*exp(4) + 2*x^2*exp(2) + 6*
x^2 + x^3))*exp(-(15*exp(4))/(9*x + 6*x*exp(2) + x*exp(4) + 2*x^2*exp(2) + 6*x^2 + x^3))*exp(-(90*exp(2))/(9*x
+ 6*x*exp(2) + x*exp(4) + 2*x^2*exp(2) + 6*x^2 + x^3))*exp(-(90*x)/(9*x + 6*x*exp(2) + x*exp(4) + 2*x^2*exp(2
) + 6*x^2 + x^3))