Integrand size = 51, antiderivative size = 21 \[ \int \frac {e^{16/625}+e^{\frac {2 \left (-3-x+e^{16/625} (9+x)\right )}{e^{16/625}}} \left (-2 x+e^{16/625} (1+2 x)\right )}{e^{16/625}} \, dx=x+e^{18+2 x-\frac {2 (3+x)}{e^{16/625}}} x \]
[Out]
Leaf count is larger than twice the leaf count of optimal. \(112\) vs. \(2(21)=42\).
Time = 0.09 (sec) , antiderivative size = 112, normalized size of antiderivative = 5.33, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.078, Rules used = {12, 2218, 2207, 2225} \[ \int \frac {e^{16/625}+e^{\frac {2 \left (-3-x+e^{16/625} (9+x)\right )}{e^{16/625}}} \left (-2 x+e^{16/625} (1+2 x)\right )}{e^{16/625}} \, dx=\frac {\exp \left (2 \left (\left (1-\frac {1}{e^{16/625}}\right ) x+3 \left (3-\frac {1}{e^{16/625}}\right )\right )+\frac {16}{625}\right )}{2 \left (1-e^{16/625}\right )}+\frac {\left (e^{16/625}-2 \left (1-e^{16/625}\right ) x\right ) \exp \left (2 \left (\left (1-\frac {1}{e^{16/625}}\right ) x+3 \left (3-\frac {1}{e^{16/625}}\right )\right )-\frac {16}{625}\right )}{2 \left (1-\frac {1}{e^{16/625}}\right )}+x \]
[In]
[Out]
Rule 12
Rule 2207
Rule 2218
Rule 2225
Rubi steps \begin{align*} \text {integral}& = \frac {\int \left (e^{16/625}+e^{\frac {2 \left (-3-x+e^{16/625} (9+x)\right )}{e^{16/625}}} \left (-2 x+e^{16/625} (1+2 x)\right )\right ) \, dx}{e^{16/625}} \\ & = x+\frac {\int e^{\frac {2 \left (-3-x+e^{16/625} (9+x)\right )}{e^{16/625}}} \left (-2 x+e^{16/625} (1+2 x)\right ) \, dx}{e^{16/625}} \\ & = x+\frac {\int \exp \left (2 \left (3 \left (3-\frac {1}{e^{16/625}}\right )+\left (1-\frac {1}{e^{16/625}}\right ) x\right )\right ) \left (e^{16/625}-2 \left (1-e^{16/625}\right ) x\right ) \, dx}{e^{16/625}} \\ & = x+\frac {\exp \left (-\frac {16}{625}+2 \left (3 \left (3-\frac {1}{e^{16/625}}\right )+\left (1-\frac {1}{e^{16/625}}\right ) x\right )\right ) \left (e^{16/625}-2 \left (1-e^{16/625}\right ) x\right )}{2 \left (1-\frac {1}{e^{16/625}}\right )}-\int \exp \left (2 \left (3 \left (3-\frac {1}{e^{16/625}}\right )+\left (1-\frac {1}{e^{16/625}}\right ) x\right )\right ) \, dx \\ & = -\frac {\exp \left (2 \left (3 \left (3-\frac {1}{e^{16/625}}\right )+\left (1-\frac {1}{e^{16/625}}\right ) x\right )\right )}{2 \left (1-\frac {1}{e^{16/625}}\right )}+x+\frac {\exp \left (-\frac {16}{625}+2 \left (3 \left (3-\frac {1}{e^{16/625}}\right )+\left (1-\frac {1}{e^{16/625}}\right ) x\right )\right ) \left (e^{16/625}-2 \left (1-e^{16/625}\right ) x\right )}{2 \left (1-\frac {1}{e^{16/625}}\right )} \\ \end{align*}
Time = 0.72 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {e^{16/625}+e^{\frac {2 \left (-3-x+e^{16/625} (9+x)\right )}{e^{16/625}}} \left (-2 x+e^{16/625} (1+2 x)\right )}{e^{16/625}} \, dx=x+e^{2 \left (9+x-\frac {3+x}{e^{16/625}}\right )} x \]
[In]
[Out]
Time = 1.77 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10
method | result | size |
risch | \(x +x \,{\mathrm e}^{2 \left ({\mathrm e}^{\frac {16}{625}} x +9 \,{\mathrm e}^{\frac {16}{625}}-x -3\right ) {\mathrm e}^{-\frac {16}{625}}}\) | \(23\) |
norman | \(x +x \,{\mathrm e}^{2 \left (\left (x +9\right ) {\mathrm e}^{\frac {16}{625}}-3-x \right ) {\mathrm e}^{-\frac {16}{625}}}\) | \(24\) |
parallelrisch | \({\mathrm e}^{-\frac {16}{625}} \left ({\mathrm e}^{\frac {16}{625}} x \,{\mathrm e}^{2 \left (\left (x +9\right ) {\mathrm e}^{\frac {16}{625}}-3-x \right ) {\mathrm e}^{-\frac {16}{625}}}+{\mathrm e}^{\frac {16}{625}} x \right )\) | \(34\) |
parts | \(x +\frac {\frac {{\mathrm e}^{\frac {16}{625}} {\mathrm e}^{2 \left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +2 \left (9 \,{\mathrm e}^{\frac {16}{625}}-3\right ) {\mathrm e}^{-\frac {16}{625}}}}{2}-\frac {3 \,{\mathrm e}^{2 \left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +2 \left (9 \,{\mathrm e}^{\frac {16}{625}}-3\right ) {\mathrm e}^{-\frac {16}{625}}}}{{\mathrm e}^{\frac {16}{625}}-1}+\frac {12 \,{\mathrm e}^{2 \left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +2 \left (9 \,{\mathrm e}^{\frac {16}{625}}-3\right ) {\mathrm e}^{-\frac {16}{625}}} {\mathrm e}^{\frac {16}{625}}}{{\mathrm e}^{\frac {16}{625}}-1}-\frac {9 \,{\mathrm e}^{2 \left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +2 \left (9 \,{\mathrm e}^{\frac {16}{625}}-3\right ) {\mathrm e}^{-\frac {16}{625}}} {\mathrm e}^{\frac {32}{625}}}{{\mathrm e}^{\frac {16}{625}}-1}-\frac {2 \,{\mathrm e}^{\frac {16}{625}} \left (\frac {{\mathrm e}^{2 \left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +2 \left (9 \,{\mathrm e}^{\frac {16}{625}}-3\right ) {\mathrm e}^{-\frac {16}{625}}} \left (\left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +\left (9 \,{\mathrm e}^{\frac {16}{625}}-3\right ) {\mathrm e}^{-\frac {16}{625}}\right )}{2}-\frac {{\mathrm e}^{2 \left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +2 \left (9 \,{\mathrm e}^{\frac {16}{625}}-3\right ) {\mathrm e}^{-\frac {16}{625}}}}{4}\right )}{{\mathrm e}^{\frac {16}{625}}-1}+\frac {2 \,{\mathrm e}^{\frac {32}{625}} \left (\frac {{\mathrm e}^{2 \left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +2 \left (9 \,{\mathrm e}^{\frac {16}{625}}-3\right ) {\mathrm e}^{-\frac {16}{625}}} \left (\left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +\left (9 \,{\mathrm e}^{\frac {16}{625}}-3\right ) {\mathrm e}^{-\frac {16}{625}}\right )}{2}-\frac {{\mathrm e}^{2 \left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +2 \left (9 \,{\mathrm e}^{\frac {16}{625}}-3\right ) {\mathrm e}^{-\frac {16}{625}}}}{4}\right )}{{\mathrm e}^{\frac {16}{625}}-1}}{{\mathrm e}^{\frac {16}{625}}-1}\) | \(321\) |
default | \({\mathrm e}^{-\frac {16}{625}} \left (\frac {{\mathrm e}^{\frac {16}{625}} \left (\frac {{\mathrm e}^{\frac {16}{625}} {\mathrm e}^{2 \left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +2 \left (9 \,{\mathrm e}^{\frac {16}{625}}-3\right ) {\mathrm e}^{-\frac {16}{625}}}}{2}-\frac {3 \,{\mathrm e}^{2 \left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +2 \left (9 \,{\mathrm e}^{\frac {16}{625}}-3\right ) {\mathrm e}^{-\frac {16}{625}}}}{{\mathrm e}^{\frac {16}{625}}-1}+\frac {12 \,{\mathrm e}^{2 \left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +2 \left (9 \,{\mathrm e}^{\frac {16}{625}}-3\right ) {\mathrm e}^{-\frac {16}{625}}} {\mathrm e}^{\frac {16}{625}}}{{\mathrm e}^{\frac {16}{625}}-1}-\frac {9 \,{\mathrm e}^{2 \left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +2 \left (9 \,{\mathrm e}^{\frac {16}{625}}-3\right ) {\mathrm e}^{-\frac {16}{625}}} {\mathrm e}^{\frac {32}{625}}}{{\mathrm e}^{\frac {16}{625}}-1}-\frac {2 \,{\mathrm e}^{\frac {16}{625}} \left (\frac {{\mathrm e}^{2 \left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +2 \left (9 \,{\mathrm e}^{\frac {16}{625}}-3\right ) {\mathrm e}^{-\frac {16}{625}}} \left (\left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +\left (9 \,{\mathrm e}^{\frac {16}{625}}-3\right ) {\mathrm e}^{-\frac {16}{625}}\right )}{2}-\frac {{\mathrm e}^{2 \left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +2 \left (9 \,{\mathrm e}^{\frac {16}{625}}-3\right ) {\mathrm e}^{-\frac {16}{625}}}}{4}\right )}{{\mathrm e}^{\frac {16}{625}}-1}+\frac {2 \,{\mathrm e}^{\frac {32}{625}} \left (\frac {{\mathrm e}^{2 \left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +2 \left (9 \,{\mathrm e}^{\frac {16}{625}}-3\right ) {\mathrm e}^{-\frac {16}{625}}} \left (\left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +\left (9 \,{\mathrm e}^{\frac {16}{625}}-3\right ) {\mathrm e}^{-\frac {16}{625}}\right )}{2}-\frac {{\mathrm e}^{2 \left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +2 \left (9 \,{\mathrm e}^{\frac {16}{625}}-3\right ) {\mathrm e}^{-\frac {16}{625}}}}{4}\right )}{{\mathrm e}^{\frac {16}{625}}-1}\right )}{{\mathrm e}^{\frac {16}{625}}-1}+{\mathrm e}^{\frac {16}{625}} x \right )\) | \(331\) |
derivativedivides | \(\frac {\frac {{\mathrm e}^{\frac {16}{625}} {\mathrm e}^{2 \left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +2 \left (9 \,{\mathrm e}^{\frac {16}{625}}-3\right ) {\mathrm e}^{-\frac {16}{625}}}}{2}-\frac {3 \,{\mathrm e}^{2 \left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +2 \left (9 \,{\mathrm e}^{\frac {16}{625}}-3\right ) {\mathrm e}^{-\frac {16}{625}}}}{{\mathrm e}^{\frac {16}{625}}-1}+\frac {12 \,{\mathrm e}^{2 \left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +2 \left (9 \,{\mathrm e}^{\frac {16}{625}}-3\right ) {\mathrm e}^{-\frac {16}{625}}} {\mathrm e}^{\frac {16}{625}}}{{\mathrm e}^{\frac {16}{625}}-1}-\frac {9 \,{\mathrm e}^{2 \left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +2 \left (9 \,{\mathrm e}^{\frac {16}{625}}-3\right ) {\mathrm e}^{-\frac {16}{625}}} {\mathrm e}^{\frac {32}{625}}}{{\mathrm e}^{\frac {16}{625}}-1}-\frac {2 \,{\mathrm e}^{\frac {16}{625}} \left (\frac {{\mathrm e}^{2 \left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +2 \left (9 \,{\mathrm e}^{\frac {16}{625}}-3\right ) {\mathrm e}^{-\frac {16}{625}}} \left (\left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +\left (9 \,{\mathrm e}^{\frac {16}{625}}-3\right ) {\mathrm e}^{-\frac {16}{625}}\right )}{2}-\frac {{\mathrm e}^{2 \left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +2 \left (9 \,{\mathrm e}^{\frac {16}{625}}-3\right ) {\mathrm e}^{-\frac {16}{625}}}}{4}\right )}{{\mathrm e}^{\frac {16}{625}}-1}+\frac {2 \,{\mathrm e}^{\frac {32}{625}} \left (\frac {{\mathrm e}^{2 \left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +2 \left (9 \,{\mathrm e}^{\frac {16}{625}}-3\right ) {\mathrm e}^{-\frac {16}{625}}} \left (\left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +\left (9 \,{\mathrm e}^{\frac {16}{625}}-3\right ) {\mathrm e}^{-\frac {16}{625}}\right )}{2}-\frac {{\mathrm e}^{2 \left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +2 \left (9 \,{\mathrm e}^{\frac {16}{625}}-3\right ) {\mathrm e}^{-\frac {16}{625}}}}{4}\right )}{{\mathrm e}^{\frac {16}{625}}-1}+\left (\left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +\left (9 \,{\mathrm e}^{\frac {16}{625}}-3\right ) {\mathrm e}^{-\frac {16}{625}}\right ) {\mathrm e}^{\frac {16}{625}}}{{\mathrm e}^{\frac {16}{625}}-1}\) | \(344\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {e^{16/625}+e^{\frac {2 \left (-3-x+e^{16/625} (9+x)\right )}{e^{16/625}}} \left (-2 x+e^{16/625} (1+2 x)\right )}{e^{16/625}} \, dx=x e^{\left (2 \, {\left ({\left (x + 9\right )} e^{\frac {16}{625}} - x - 3\right )} e^{\left (-\frac {16}{625}\right )}\right )} + x \]
[In]
[Out]
Time = 0.07 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14 \[ \int \frac {e^{16/625}+e^{\frac {2 \left (-3-x+e^{16/625} (9+x)\right )}{e^{16/625}}} \left (-2 x+e^{16/625} (1+2 x)\right )}{e^{16/625}} \, dx=x e^{\frac {2 \left (- x + \left (x + 9\right ) e^{\frac {16}{625}} - 3\right )}{e^{\frac {16}{625}}}} + x \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (17) = 34\).
Time = 0.21 (sec) , antiderivative size = 134, normalized size of antiderivative = 6.38 \[ \int \frac {e^{16/625}+e^{\frac {2 \left (-3-x+e^{16/625} (9+x)\right )}{e^{16/625}}} \left (-2 x+e^{16/625} (1+2 x)\right )}{e^{16/625}} \, dx=\frac {1}{2} \, {\left (2 \, x e^{\frac {16}{625}} + \frac {{\left (2 \, x {\left (e^{\frac {11298}{625}} - e^{\frac {11282}{625}}\right )} - e^{\frac {11298}{625}}\right )} e^{\left (-2 \, x e^{\left (-\frac {16}{625}\right )} + 2 \, x\right )}}{e^{\left (6 \, e^{\left (-\frac {16}{625}\right )}\right )} + e^{\left (6 \, e^{\left (-\frac {16}{625}\right )} + \frac {32}{625}\right )} - 2 \, e^{\left (6 \, e^{\left (-\frac {16}{625}\right )} + \frac {16}{625}\right )}} - \frac {{\left (2 \, x {\left (e^{\frac {11282}{625}} - e^{\frac {11266}{625}}\right )} - e^{\frac {11282}{625}}\right )} e^{\left (-2 \, x e^{\left (-\frac {16}{625}\right )} + 2 \, x\right )}}{e^{\left (6 \, e^{\left (-\frac {16}{625}\right )}\right )} + e^{\left (6 \, e^{\left (-\frac {16}{625}\right )} + \frac {32}{625}\right )} - 2 \, e^{\left (6 \, e^{\left (-\frac {16}{625}\right )} + \frac {16}{625}\right )}} - \frac {e^{\left (-2 \, x e^{\left (-\frac {16}{625}\right )} + 2 \, x - 6 \, e^{\left (-\frac {16}{625}\right )} + \frac {11266}{625}\right )}}{e^{\left (-\frac {16}{625}\right )} - 1}\right )} e^{\left (-\frac {16}{625}\right )} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (17) = 34\).
Time = 0.27 (sec) , antiderivative size = 88, normalized size of antiderivative = 4.19 \[ \int \frac {e^{16/625}+e^{\frac {2 \left (-3-x+e^{16/625} (9+x)\right )}{e^{16/625}}} \left (-2 x+e^{16/625} (1+2 x)\right )}{e^{16/625}} \, dx=\frac {1}{2} \, {\left (2 \, x e^{\frac {16}{625}} + \frac {{\left (2 \, x e^{\left (-\frac {16}{625}\right )} - 2 \, x + e^{\left (-\frac {16}{625}\right )}\right )} e^{\left (-2 \, x e^{\left (-\frac {16}{625}\right )} + 2 \, x - 6 \, e^{\left (-\frac {16}{625}\right )} + \frac {11266}{625}\right )}}{2 \, e^{\left (-\frac {16}{625}\right )} - e^{\left (-\frac {32}{625}\right )} - 1} - \frac {{\left (2 \, x e^{\left (-\frac {16}{625}\right )} - 2 \, x + 1\right )} e^{\left (-2 \, x e^{\left (-\frac {16}{625}\right )} + 2 \, x - 6 \, e^{\left (-\frac {16}{625}\right )} + 18\right )}}{2 \, e^{\left (-\frac {16}{625}\right )} - e^{\left (-\frac {32}{625}\right )} - 1}\right )} e^{\left (-\frac {16}{625}\right )} \]
[In]
[Out]
Time = 0.15 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.05 \[ \int \frac {e^{16/625}+e^{\frac {2 \left (-3-x+e^{16/625} (9+x)\right )}{e^{16/625}}} \left (-2 x+e^{16/625} (1+2 x)\right )}{e^{16/625}} \, dx=x\,\left ({\mathrm {e}}^{-6\,{\mathrm {e}}^{-\frac {16}{625}}}\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{18}\,{\mathrm {e}}^{-2\,x\,{\mathrm {e}}^{-\frac {16}{625}}}+1\right ) \]
[In]
[Out]