Optimal. Leaf size=21 \[ x+e^{18+2 x-\frac {2 (3+x)}{e^{16/625}}} x \]
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Rubi [B] time = 0.16, antiderivative size = 112, normalized size of antiderivative = 5.33, number of steps used = 5, number of rules used = 4, integrand size = 51, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.078, Rules used = {12, 2187, 2176, 2194} \begin {gather*} \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 \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2176
Rule 2187
Rule 2194
Rubi steps
\begin {gather*} \begin {aligned} \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 {aligned} \end {gather*}
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Mathematica [A] time = 0.57, size = 21, normalized size = 1.00 \begin {gather*} x+e^{2 \left (9+x-\frac {3+x}{e^{16/625}}\right )} x \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 20, normalized size = 0.95 \begin {gather*} x e^{\left (2 \, {\left ({\left (x + 9\right )} e^{\frac {16}{625}} - x - 3\right )} e^{\left (-\frac {16}{625}\right )}\right )} + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.34, size = 88, normalized size = 4.19 \begin {gather*} \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 23, normalized size = 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 | \(x +x \,{\mathrm e}^{2 \left (\left (x +9\right ) {\mathrm e}^{\frac {16}{625}}-3-x \right ) {\mathrm e}^{-\frac {16}{625}}}\) | \(24\) |
default | \({\mathrm e}^{-\frac {16}{625}} \left (\frac {{\mathrm e}^{\frac {16}{625}} \left (\frac {{\mathrm e}^{2 \left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +6 \left (3 \,{\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}}} {\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 +6 \left (3 \,{\mathrm e}^{\frac {16}{625}}-1\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 +6 \left (3 \,{\mathrm e}^{\frac {16}{625}}-1\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 +6 \left (3 \,{\mathrm e}^{\frac {16}{625}}-1\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 +6 \left (3 \,{\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}}} \left (\left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +3 \left (3 \,{\mathrm e}^{\frac {16}{625}}-1\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 +6 \left (3 \,{\mathrm e}^{\frac {16}{625}}-1\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 +6 \left (3 \,{\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}}} \left (\left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +3 \left (3 \,{\mathrm e}^{\frac {16}{625}}-1\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 +6 \left (3 \,{\mathrm e}^{\frac {16}{625}}-1\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 )\) | \(341\) |
derivativedivides | \(\frac {\frac {{\mathrm e}^{2 \left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +6 \left (3 \,{\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}}} {\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 +6 \left (3 \,{\mathrm e}^{\frac {16}{625}}-1\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 +6 \left (3 \,{\mathrm e}^{\frac {16}{625}}-1\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 +6 \left (3 \,{\mathrm e}^{\frac {16}{625}}-1\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 +6 \left (3 \,{\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}}} \left (\left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +3 \left (3 \,{\mathrm e}^{\frac {16}{625}}-1\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 +6 \left (3 \,{\mathrm e}^{\frac {16}{625}}-1\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 +6 \left (3 \,{\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}}} \left (\left ({\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}} x +3 \left (3 \,{\mathrm e}^{\frac {16}{625}}-1\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 +6 \left (3 \,{\mathrm e}^{\frac {16}{625}}-1\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 +3 \left (3 \,{\mathrm e}^{\frac {16}{625}}-1\right ) {\mathrm e}^{-\frac {16}{625}}\right ) {\mathrm e}^{\frac {16}{625}}}{{\mathrm e}^{\frac {16}{625}}-1}\) | \(355\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.48, size = 134, normalized size = 6.38 \begin {gather*} \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 22, normalized size = 1.05 \begin {gather*} 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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 24, normalized size = 1.14 \begin {gather*} x e^{\frac {2 \left (- x + \left (x + 9\right ) e^{\frac {16}{625}} - 3\right )}{e^{\frac {16}{625}}}} + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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