Integrand size = 115, antiderivative size = 18 \[ \int \frac {-18-36 x+36 x^3+108 x^4+216 x^5+216 x^6+72 x^7+e^{2 x} \left (18 x^3+54 x^4+54 x^5+18 x^6\right )+e^x \left (-18 x-36 x^2-36 x^3-126 x^4-216 x^5-144 x^6-36 x^7\right )}{x^3+3 x^4+3 x^5+x^6} \, dx=9 \left (e^x-2 x+\frac {1}{x+x^2}\right )^2 \]
[Out]
Leaf count is larger than twice the leaf count of optimal. \(71\) vs. \(2(18)=36\).
Time = 0.84 (sec) , antiderivative size = 71, normalized size of antiderivative = 3.94, number of steps used = 26, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.078, Rules used = {6873, 6874, 2225, 46, 37, 45, 2208, 2209, 2207} \[ \int \frac {-18-36 x+36 x^3+108 x^4+216 x^5+216 x^6+72 x^7+e^{2 x} \left (18 x^3+54 x^4+54 x^5+18 x^6\right )+e^x \left (-18 x-36 x^2-36 x^3-126 x^4-216 x^5-144 x^6-36 x^7\right )}{x^3+3 x^4+3 x^5+x^6} \, dx=\frac {54 x^2}{(x+1)^2}+36 x^2+\frac {9}{x^2}-36 e^x x+9 e^{2 x}-\frac {18 e^x}{x+1}+\frac {90}{x+1}-\frac {45}{(x+1)^2}+\frac {18 e^x}{x}-\frac {18}{x} \]
[In]
[Out]
Rule 37
Rule 45
Rule 46
Rule 2207
Rule 2208
Rule 2209
Rule 2225
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {-18-36 x+36 x^3+108 x^4+216 x^5+216 x^6+72 x^7+e^{2 x} \left (18 x^3+54 x^4+54 x^5+18 x^6\right )+e^x \left (-18 x-36 x^2-36 x^3-126 x^4-216 x^5-144 x^6-36 x^7\right )}{x^3 (1+x)^3} \, dx \\ & = \int \left (18 e^{2 x}+\frac {36}{(1+x)^3}-\frac {18}{x^3 (1+x)^3}-\frac {36}{x^2 (1+x)^3}+\frac {108 x}{(1+x)^3}+\frac {216 x^2}{(1+x)^3}+\frac {216 x^3}{(1+x)^3}+\frac {72 x^4}{(1+x)^3}-\frac {18 e^x \left (1+x+x^2+6 x^3+6 x^4+2 x^5\right )}{x^2 (1+x)^2}\right ) \, dx \\ & = -\frac {18}{(1+x)^2}+18 \int e^{2 x} \, dx-18 \int \frac {1}{x^3 (1+x)^3} \, dx-18 \int \frac {e^x \left (1+x+x^2+6 x^3+6 x^4+2 x^5\right )}{x^2 (1+x)^2} \, dx-36 \int \frac {1}{x^2 (1+x)^3} \, dx+72 \int \frac {x^4}{(1+x)^3} \, dx+108 \int \frac {x}{(1+x)^3} \, dx+216 \int \frac {x^2}{(1+x)^3} \, dx+216 \int \frac {x^3}{(1+x)^3} \, dx \\ & = 9 e^{2 x}-\frac {18}{(1+x)^2}+\frac {54 x^2}{(1+x)^2}-18 \int \left (\frac {1}{x^3}-\frac {3}{x^2}+\frac {6}{x}-\frac {1}{(1+x)^3}-\frac {3}{(1+x)^2}-\frac {6}{1+x}\right ) \, dx-18 \int \left (2 e^x+\frac {e^x}{x^2}-\frac {e^x}{x}+2 e^x x-\frac {e^x}{(1+x)^2}+\frac {e^x}{1+x}\right ) \, dx-36 \int \left (\frac {1}{x^2}-\frac {3}{x}+\frac {1}{(1+x)^3}+\frac {2}{(1+x)^2}+\frac {3}{1+x}\right ) \, dx+72 \int \left (-3+x+\frac {1}{(1+x)^3}-\frac {4}{(1+x)^2}+\frac {6}{1+x}\right ) \, dx+216 \int \left (1-\frac {1}{(1+x)^3}+\frac {3}{(1+x)^2}-\frac {3}{1+x}\right ) \, dx+216 \int \left (\frac {1}{(1+x)^3}-\frac {2}{(1+x)^2}+\frac {1}{1+x}\right ) \, dx \\ & = 9 e^{2 x}+\frac {9}{x^2}-\frac {18}{x}+36 x^2-\frac {45}{(1+x)^2}+\frac {54 x^2}{(1+x)^2}+\frac {90}{1+x}-18 \int \frac {e^x}{x^2} \, dx+18 \int \frac {e^x}{x} \, dx+18 \int \frac {e^x}{(1+x)^2} \, dx-18 \int \frac {e^x}{1+x} \, dx-36 \int e^x \, dx-36 \int e^x x \, dx \\ & = -36 e^x+9 e^{2 x}+\frac {9}{x^2}-\frac {18}{x}+\frac {18 e^x}{x}-36 e^x x+36 x^2-\frac {45}{(1+x)^2}+\frac {54 x^2}{(1+x)^2}+\frac {90}{1+x}-\frac {18 e^x}{1+x}+18 \text {Ei}(x)-\frac {18 \text {Ei}(1+x)}{e}-18 \int \frac {e^x}{x} \, dx+18 \int \frac {e^x}{1+x} \, dx+36 \int e^x \, dx \\ & = 9 e^{2 x}+\frac {9}{x^2}-\frac {18}{x}+\frac {18 e^x}{x}-36 e^x x+36 x^2-\frac {45}{(1+x)^2}+\frac {54 x^2}{(1+x)^2}+\frac {90}{1+x}-\frac {18 e^x}{1+x} \\ \end{align*}
Time = 5.51 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.78 \[ \int \frac {-18-36 x+36 x^3+108 x^4+216 x^5+216 x^6+72 x^7+e^{2 x} \left (18 x^3+54 x^4+54 x^5+18 x^6\right )+e^x \left (-18 x-36 x^2-36 x^3-126 x^4-216 x^5-144 x^6-36 x^7\right )}{x^3+3 x^4+3 x^5+x^6} \, dx=\frac {9 \left (1-2 x^2-2 x^3+e^x x (1+x)\right )^2}{x^2 (1+x)^2} \]
[In]
[Out]
Time = 0.16 (sec) , antiderivative size = 58, normalized size of antiderivative = 3.22
method | result | size |
default | \(\frac {9}{\left (1+x \right )^{2}}-\frac {18}{1+x}+\frac {9}{x^{2}}-\frac {18}{x}+36 x^{2}-\frac {18 \,{\mathrm e}^{x}}{1+x}+9 \,{\mathrm e}^{2 x}-36 \,{\mathrm e}^{x} x +\frac {18 \,{\mathrm e}^{x}}{x}\) | \(58\) |
parts | \(\frac {9}{\left (1+x \right )^{2}}-\frac {18}{1+x}+\frac {9}{x^{2}}-\frac {18}{x}+36 x^{2}-\frac {18 \,{\mathrm e}^{x}}{1+x}+9 \,{\mathrm e}^{2 x}-36 \,{\mathrm e}^{x} x +\frac {18 \,{\mathrm e}^{x}}{x}\) | \(58\) |
risch | \(36 x^{2}+\frac {-36 x^{3}-36 x^{2}+9}{x^{2} \left (x^{2}+2 x +1\right )}+9 \,{\mathrm e}^{2 x}-\frac {18 \left (2 x^{3}+2 x^{2}-1\right ) {\mathrm e}^{x}}{\left (1+x \right ) x}\) | \(63\) |
norman | \(\frac {9-108 x^{3}-72 x^{2}+72 x^{5}+36 x^{6}-36 x^{5} {\mathrm e}^{x}+18 \,{\mathrm e}^{x} x +18 \,{\mathrm e}^{x} x^{2}-36 \,{\mathrm e}^{x} x^{3}-72 \,{\mathrm e}^{x} x^{4}+9 \,{\mathrm e}^{2 x} x^{2}+18 \,{\mathrm e}^{2 x} x^{3}+9 \,{\mathrm e}^{2 x} x^{4}}{x^{2} \left (1+x \right )^{2}}\) | \(92\) |
parallelrisch | \(\frac {9-108 x^{3}-72 x^{2}+72 x^{5}+36 x^{6}-36 x^{5} {\mathrm e}^{x}+18 \,{\mathrm e}^{x} x +18 \,{\mathrm e}^{x} x^{2}-36 \,{\mathrm e}^{x} x^{3}-72 \,{\mathrm e}^{x} x^{4}+9 \,{\mathrm e}^{2 x} x^{2}+18 \,{\mathrm e}^{2 x} x^{3}+9 \,{\mathrm e}^{2 x} x^{4}}{x^{2} \left (x^{2}+2 x +1\right )}\) | \(97\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (21) = 42\).
Time = 0.28 (sec) , antiderivative size = 88, normalized size of antiderivative = 4.89 \[ \int \frac {-18-36 x+36 x^3+108 x^4+216 x^5+216 x^6+72 x^7+e^{2 x} \left (18 x^3+54 x^4+54 x^5+18 x^6\right )+e^x \left (-18 x-36 x^2-36 x^3-126 x^4-216 x^5-144 x^6-36 x^7\right )}{x^3+3 x^4+3 x^5+x^6} \, dx=\frac {9 \, {\left (4 \, x^{6} + 8 \, x^{5} + 4 \, x^{4} - 4 \, x^{3} - 4 \, x^{2} + {\left (x^{4} + 2 \, x^{3} + x^{2}\right )} e^{\left (2 \, x\right )} - 2 \, {\left (2 \, x^{5} + 4 \, x^{4} + 2 \, x^{3} - x^{2} - x\right )} e^{x} + 1\right )}}{x^{4} + 2 \, x^{3} + x^{2}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (15) = 30\).
Time = 0.13 (sec) , antiderivative size = 61, normalized size of antiderivative = 3.39 \[ \int \frac {-18-36 x+36 x^3+108 x^4+216 x^5+216 x^6+72 x^7+e^{2 x} \left (18 x^3+54 x^4+54 x^5+18 x^6\right )+e^x \left (-18 x-36 x^2-36 x^3-126 x^4-216 x^5-144 x^6-36 x^7\right )}{x^3+3 x^4+3 x^5+x^6} \, dx=36 x^{2} + \frac {- 36 x^{3} - 36 x^{2} + 9}{x^{4} + 2 x^{3} + x^{2}} + \frac {\left (9 x^{2} + 9 x\right ) e^{2 x} + \left (- 36 x^{3} - 36 x^{2} + 18\right ) e^{x}}{x^{2} + x} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 177 vs. \(2 (21) = 42\).
Time = 0.25 (sec) , antiderivative size = 177, normalized size of antiderivative = 9.83 \[ \int \frac {-18-36 x+36 x^3+108 x^4+216 x^5+216 x^6+72 x^7+e^{2 x} \left (18 x^3+54 x^4+54 x^5+18 x^6\right )+e^x \left (-18 x-36 x^2-36 x^3-126 x^4-216 x^5-144 x^6-36 x^7\right )}{x^3+3 x^4+3 x^5+x^6} \, dx=36 \, x^{2} - \frac {9 \, {\left (12 \, x^{3} + 18 \, x^{2} + 4 \, x - 1\right )}}{x^{4} + 2 \, x^{3} + x^{2}} + \frac {18 \, {\left (6 \, x^{2} + 9 \, x + 2\right )}}{x^{3} + 2 \, x^{2} + x} + \frac {9 \, {\left ({\left (x^{2} + x\right )} e^{\left (2 \, x\right )} - 2 \, {\left (2 \, x^{3} + 2 \, x^{2} - 1\right )} e^{x}\right )}}{x^{2} + x} + \frac {36 \, {\left (8 \, x + 7\right )}}{x^{2} + 2 \, x + 1} - \frac {108 \, {\left (6 \, x + 5\right )}}{x^{2} + 2 \, x + 1} + \frac {108 \, {\left (4 \, x + 3\right )}}{x^{2} + 2 \, x + 1} - \frac {54 \, {\left (2 \, x + 1\right )}}{x^{2} + 2 \, x + 1} - \frac {18}{x^{2} + 2 \, x + 1} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 101 vs. \(2 (21) = 42\).
Time = 0.28 (sec) , antiderivative size = 101, normalized size of antiderivative = 5.61 \[ \int \frac {-18-36 x+36 x^3+108 x^4+216 x^5+216 x^6+72 x^7+e^{2 x} \left (18 x^3+54 x^4+54 x^5+18 x^6\right )+e^x \left (-18 x-36 x^2-36 x^3-126 x^4-216 x^5-144 x^6-36 x^7\right )}{x^3+3 x^4+3 x^5+x^6} \, dx=\frac {9 \, {\left (4 \, x^{6} - 4 \, x^{5} e^{x} + 8 \, x^{5} + x^{4} e^{\left (2 \, x\right )} - 8 \, x^{4} e^{x} + 4 \, x^{4} + 2 \, x^{3} e^{\left (2 \, x\right )} - 4 \, x^{3} e^{x} - 4 \, x^{3} + x^{2} e^{\left (2 \, x\right )} + 2 \, x^{2} e^{x} - 4 \, x^{2} + 2 \, x e^{x} + 1\right )}}{x^{4} + 2 \, x^{3} + x^{2}} \]
[In]
[Out]
Time = 0.23 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.67 \[ \int \frac {-18-36 x+36 x^3+108 x^4+216 x^5+216 x^6+72 x^7+e^{2 x} \left (18 x^3+54 x^4+54 x^5+18 x^6\right )+e^x \left (-18 x-36 x^2-36 x^3-126 x^4-216 x^5-144 x^6-36 x^7\right )}{x^3+3 x^4+3 x^5+x^6} \, dx=9\,{\mathrm {e}}^{2\,x}-36\,x\,{\mathrm {e}}^x+36\,x^2+\frac {18\,x\,{\mathrm {e}}^x+x^2\,\left (18\,{\mathrm {e}}^x-36\right )-36\,x^3+9}{x^2\,{\left (x+1\right )}^2} \]
[In]
[Out]