Optimal. Leaf size=38 \[ \frac {4}{-e^3+3 x-\frac {5-x}{-e^{5-4 x}-x+x^2}} \]
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Rubi [F] time = 20.62, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {20-12 e^{10-8 x}-40 x-8 x^2+24 x^3-12 x^4+e^{5-4 x} \left (-76-8 x+24 x^2\right )}{25-10 x+31 x^2-36 x^3+15 x^4-18 x^5+9 x^6+e^{10-8 x} \left (e^6-6 e^3 x+9 x^2\right )+e^6 \left (x^2-2 x^3+x^4\right )+e^3 \left (-10 x+12 x^2-8 x^3+12 x^4-6 x^5\right )+e^{5-4 x} \left (30 x-6 x^2+18 x^3-18 x^4+e^6 \left (2 x-2 x^2\right )+e^3 \left (-10+2 x-12 x^2+12 x^3\right )\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 \left (-3 e^{10}-e^{5+4 x} \left (19+2 x-6 x^2\right )-e^{8 x} \left (-5+10 x+2 x^2-6 x^3+3 x^4\right )\right )}{\left (e^8-3 e^5 x-e^{3+4 x} (-1+x) x+e^{4 x} \left (-5+x-3 x^2+3 x^3\right )\right )^2} \, dx\\ &=4 \int \frac {-3 e^{10}-e^{5+4 x} \left (19+2 x-6 x^2\right )-e^{8 x} \left (-5+10 x+2 x^2-6 x^3+3 x^4\right )}{\left (e^8-3 e^5 x-e^{3+4 x} (-1+x) x+e^{4 x} \left (-5+x-3 x^2+3 x^3\right )\right )^2} \, dx\\ &=4 \int \left (\frac {5-10 x-2 x^2+6 x^3-3 x^4}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right )^2}+\frac {e^5 \left (5 \left (19-2 e^3\right )+\left (21+e^3\right ) x-7 \left (5-3 e^3\right ) x^2-\left (57+4 e^3\right ) x^3+12 x^4\right )}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right )^2 \left (e^8-5 e^{4 x}-3 e^5 x+e^{4 x} \left (1+e^3\right ) x-3 e^{4 x} \left (1+\frac {e^3}{3}\right ) x^2+3 e^{4 x} x^3\right )}+\frac {e^{10} \left (-75 \left (1-\frac {1}{15} e^3 \left (-19+e^3\right )\right )+315 \left (1+\frac {1}{35} e^3 \left (1+e^3\right )\right ) x-75 \left (1+\frac {2}{75} e^3 \left (29+11 e^3\right )\right ) x^2+93 \left (1+\frac {4}{93} e^3 \left (33+e^3\right )\right ) x^3-198 \left (1+\frac {4 e^3}{33}\right ) x^4+36 x^5\right )}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right )^2 \left (e^8-5 e^{4 x}-3 e^5 x+e^{4 x} \left (1+e^3\right ) x-3 e^{4 x} \left (1+\frac {e^3}{3}\right ) x^2+3 e^{4 x} x^3\right )^2}\right ) \, dx\\ &=4 \int \frac {5-10 x-2 x^2+6 x^3-3 x^4}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right )^2} \, dx+\left (4 e^5\right ) \int \frac {5 \left (19-2 e^3\right )+\left (21+e^3\right ) x-7 \left (5-3 e^3\right ) x^2-\left (57+4 e^3\right ) x^3+12 x^4}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right )^2 \left (e^8-5 e^{4 x}-3 e^5 x+e^{4 x} \left (1+e^3\right ) x-3 e^{4 x} \left (1+\frac {e^3}{3}\right ) x^2+3 e^{4 x} x^3\right )} \, dx+\left (4 e^{10}\right ) \int \frac {-75 \left (1-\frac {1}{15} e^3 \left (-19+e^3\right )\right )+315 \left (1+\frac {1}{35} e^3 \left (1+e^3\right )\right ) x-75 \left (1+\frac {2}{75} e^3 \left (29+11 e^3\right )\right ) x^2+93 \left (1+\frac {4}{93} e^3 \left (33+e^3\right )\right ) x^3-198 \left (1+\frac {4 e^3}{33}\right ) x^4+36 x^5}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right )^2 \left (e^8-5 e^{4 x}-3 e^5 x+e^{4 x} \left (1+e^3\right ) x-3 e^{4 x} \left (1+\frac {e^3}{3}\right ) x^2+3 e^{4 x} x^3\right )^2} \, dx\\ &=-\frac {4 x^2}{5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3}+\frac {4}{3} \int \frac {15-3 \left (3+e^3\right ) x^2+18 x^3}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right )^2} \, dx+\left (4 e^5\right ) \int \frac {5 \left (19-2 e^3\right )+\left (21+e^3\right ) x-7 \left (5-3 e^3\right ) x^2-\left (57+4 e^3\right ) x^3+12 x^4}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right )^2 \left (e^8-3 e^5 x-e^{3+4 x} (-1+x) x+e^{4 x} \left (-5+x-3 x^2+3 x^3\right )\right )} \, dx+\left (4 e^{10}\right ) \int \frac {-75+315 x-75 x^2+93 x^3-198 x^4+36 x^5+e^6 \left (5+9 x-22 x^2+4 x^3\right )+e^3 \left (-95+9 x-58 x^2+132 x^3-24 x^4\right )}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right )^2 \left (e^8-3 e^5 x-e^{3+4 x} (-1+x) x+e^{4 x} \left (-5+x-3 x^2+3 x^3\right )\right )^2} \, dx\\ &=\frac {4 x}{5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3}-\frac {4 x^2}{5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3}+\left (4 e^5\right ) \int \left (\frac {2 \left (5 \left (2-e^3\right )+4 \left (7+2 e^3\right ) x-\left (42-e^3\right ) x^2\right )}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right )^2 \left (e^8-5 e^{4 x}-3 e^5 x+e^{4 x} \left (1+e^3\right ) x-3 e^{4 x} \left (1+\frac {e^3}{3}\right ) x^2+3 e^{4 x} x^3\right )}+\frac {15-4 x}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right ) \left (e^8-5 e^{4 x}-3 e^5 x+e^{4 x} \left (1+e^3\right ) x-3 e^{4 x} \left (1+\frac {e^3}{3}\right ) x^2+3 e^{4 x} x^3\right )}\right ) \, dx+\left (4 e^{10}\right ) \int \left (\frac {-5 \left (42+e^3-e^6\right )+\left (72-2 e^3-9 e^6\right ) x-3 \left (14-9 e^3\right ) x^2}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right )^2 \left (e^8-5 e^{4 x}-3 e^5 x+e^{4 x} \left (1+e^3\right ) x-3 e^{4 x} \left (1+\frac {e^3}{3}\right ) x^2+3 e^{4 x} x^3\right )^2}+\frac {9 \left (3-2 e^3\right )+2 \left (27+2 e^3\right ) x-12 x^2}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right ) \left (e^8-5 e^{4 x}-3 e^5 x+e^{4 x} \left (1+e^3\right ) x-3 e^{4 x} \left (1+\frac {e^3}{3}\right ) x^2+3 e^{4 x} x^3\right )^2}\right ) \, dx\\ &=\frac {4 x}{5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3}-\frac {4 x^2}{5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3}+\left (4 e^5\right ) \int \frac {15-4 x}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right ) \left (e^8-5 e^{4 x}-3 e^5 x+e^{4 x} \left (1+e^3\right ) x-3 e^{4 x} \left (1+\frac {e^3}{3}\right ) x^2+3 e^{4 x} x^3\right )} \, dx+\left (8 e^5\right ) \int \frac {5 \left (2-e^3\right )+4 \left (7+2 e^3\right ) x-\left (42-e^3\right ) x^2}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right )^2 \left (e^8-5 e^{4 x}-3 e^5 x+e^{4 x} \left (1+e^3\right ) x-3 e^{4 x} \left (1+\frac {e^3}{3}\right ) x^2+3 e^{4 x} x^3\right )} \, dx+\left (4 e^{10}\right ) \int \frac {-5 \left (42+e^3-e^6\right )+\left (72-2 e^3-9 e^6\right ) x-3 \left (14-9 e^3\right ) x^2}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right )^2 \left (e^8-5 e^{4 x}-3 e^5 x+e^{4 x} \left (1+e^3\right ) x-3 e^{4 x} \left (1+\frac {e^3}{3}\right ) x^2+3 e^{4 x} x^3\right )^2} \, dx+\left (4 e^{10}\right ) \int \frac {9 \left (3-2 e^3\right )+2 \left (27+2 e^3\right ) x-12 x^2}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right ) \left (e^8-5 e^{4 x}-3 e^5 x+e^{4 x} \left (1+e^3\right ) x-3 e^{4 x} \left (1+\frac {e^3}{3}\right ) x^2+3 e^{4 x} x^3\right )^2} \, dx\\ &=\frac {4 x}{5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3}-\frac {4 x^2}{5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3}+\left (4 e^5\right ) \int \frac {15-4 x}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right ) \left (e^8-3 e^5 x-e^{3+4 x} (-1+x) x+e^{4 x} \left (-5+x-3 x^2+3 x^3\right )\right )} \, dx+\left (8 e^5\right ) \int \frac {5 \left (2-e^3\right )+4 \left (7+2 e^3\right ) x-\left (42-e^3\right ) x^2}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right )^2 \left (e^8-3 e^5 x-e^{3+4 x} (-1+x) x+e^{4 x} \left (-5+x-3 x^2+3 x^3\right )\right )} \, dx+\left (4 e^{10}\right ) \int \frac {-5 \left (42+e^3-e^6\right )+\left (72-2 e^3-9 e^6\right ) x-3 \left (14-9 e^3\right ) x^2}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right )^2 \left (e^8-3 e^5 x-e^{3+4 x} (-1+x) x+e^{4 x} \left (-5+x-3 x^2+3 x^3\right )\right )^2} \, dx+\left (4 e^{10}\right ) \int \frac {9 \left (3-2 e^3\right )+2 \left (27+2 e^3\right ) x-12 x^2}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right ) \left (e^8-3 e^5 x-e^{3+4 x} (-1+x) x+e^{4 x} \left (-5+x-3 x^2+3 x^3\right )\right )^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [A] time = 0.12, size = 61, normalized size = 1.61 \begin {gather*} -\frac {4 \left (e^5-e^{4 x} (-1+x) x\right )}{e^8-3 e^5 x-e^{3+4 x} (-1+x) x+e^{4 x} \left (-5+x-3 x^2+3 x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 59, normalized size = 1.55 \begin {gather*} \frac {4 \, {\left (x^{2} - x - e^{\left (-4 \, x + 5\right )}\right )}}{3 \, x^{3} - 3 \, x^{2} - {\left (x^{2} - x\right )} e^{3} - {\left (3 \, x - e^{3}\right )} e^{\left (-4 \, x + 5\right )} + x - 5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.84, size = 66, normalized size = 1.74
method | result | size |
norman | \(\frac {-4 x^{2}+4 x +4 \,{\mathrm e}^{-4 x +5}}{x^{2} {\mathrm e}^{3}-3 x^{3}-x \,{\mathrm e}^{3}-{\mathrm e}^{3} {\mathrm e}^{-4 x +5}+3 x^{2}+3 \,{\mathrm e}^{-4 x +5} x -x +5}\) | \(66\) |
risch | \(-\frac {4}{{\mathrm e}^{3}-3 x}-\frac {4 \left (x -5\right )}{\left ({\mathrm e}^{3}-3 x \right ) \left (x^{2} {\mathrm e}^{3}-3 x^{3}-x \,{\mathrm e}^{3}-{\mathrm e}^{-4 x +8}+3 x^{2}+3 \,{\mathrm e}^{-4 x +5} x -x +5\right )}\) | \(70\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 59, normalized size = 1.55 \begin {gather*} -\frac {4 \, {\left ({\left (x^{2} - x\right )} e^{\left (4 \, x\right )} - e^{5}\right )}}{3 \, x e^{5} - {\left (3 \, x^{3} - x^{2} {\left (e^{3} + 3\right )} + x {\left (e^{3} + 1\right )} - 5\right )} e^{\left (4 \, x\right )} - e^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {40\,x+12\,{\mathrm {e}}^{10-8\,x}+{\mathrm {e}}^{5-4\,x}\,\left (-24\,x^2+8\,x+76\right )+8\,x^2-24\,x^3+12\,x^4-20}{{\mathrm {e}}^6\,\left (x^4-2\,x^3+x^2\right )-10\,x+{\mathrm {e}}^{10-8\,x}\,\left (9\,x^2-6\,{\mathrm {e}}^3\,x+{\mathrm {e}}^6\right )+{\mathrm {e}}^{5-4\,x}\,\left (30\,x+{\mathrm {e}}^6\,\left (2\,x-2\,x^2\right )+{\mathrm {e}}^3\,\left (12\,x^3-12\,x^2+2\,x-10\right )-6\,x^2+18\,x^3-18\,x^4\right )-{\mathrm {e}}^3\,\left (6\,x^5-12\,x^4+8\,x^3-12\,x^2+10\,x\right )+31\,x^2-36\,x^3+15\,x^4-18\,x^5+9\,x^6+25} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.68, size = 94, normalized size = 2.47 \begin {gather*} \frac {4 x - 20}{- 9 x^{4} + 9 x^{3} + 6 x^{3} e^{3} - x^{2} e^{6} - 6 x^{2} e^{3} - 3 x^{2} + 15 x + x e^{3} + x e^{6} + \left (9 x^{2} - 6 x e^{3} + e^{6}\right ) e^{5 - 4 x} - 5 e^{3}} + \frac {12}{9 x - 3 e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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