Optimal. Leaf size=34 \[ -x^2+\frac {4}{-3+x^2-\left (e^x-\frac {5}{x}-\frac {x}{3}\right ) (2+x)} \]
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Rubi [F] time = 2.78, 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 {360-1800 x-744 x^2-408 x^3-528 x^4-104 x^5-32 x^6-32 x^7+e^{2 x} \left (-72 x^3-72 x^4-18 x^5\right )+e^x \left (828 x^2+540 x^3+120 x^4+120 x^5+48 x^6\right )}{900+360 x+156 x^2+264 x^3+52 x^4+16 x^5+16 x^6+e^{2 x} \left (36 x^2+36 x^3+9 x^4\right )+e^x \left (-360 x-252 x^2-60 x^3-60 x^4-24 x^5\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 {2 \left (180-900 x+6 \left (-62+69 e^x\right ) x^2-6 \left (34-45 e^x+6 e^{2 x}\right ) x^3-12 \left (22-5 e^x+3 e^{2 x}\right ) x^4-\left (52-60 e^x+9 e^{2 x}\right ) x^5+8 \left (-2+3 e^x\right ) x^6-16 x^7\right )}{\left (30-6 \left (-1+e^x\right ) x+\left (2-3 e^x\right ) x^2+4 x^3\right )^2} \, dx\\ &=2 \int \frac {180-900 x+6 \left (-62+69 e^x\right ) x^2-6 \left (34-45 e^x+6 e^{2 x}\right ) x^3-12 \left (22-5 e^x+3 e^{2 x}\right ) x^4-\left (52-60 e^x+9 e^{2 x}\right ) x^5+8 \left (-2+3 e^x\right ) x^6-16 x^7}{\left (30-6 \left (-1+e^x\right ) x+\left (2-3 e^x\right ) x^2+4 x^3\right )^2} \, dx\\ &=2 \int \left (-x-\frac {6 x (3+x)}{(2+x) \left (30+6 x-6 e^x x+2 x^2-3 e^x x^2+4 x^3\right )}+\frac {12 \left (30+60 x+22 x^2-3 x^3+3 x^4+2 x^5\right )}{(2+x) \left (30+6 x-6 e^x x+2 x^2-3 e^x x^2+4 x^3\right )^2}\right ) \, dx\\ &=-x^2-12 \int \frac {x (3+x)}{(2+x) \left (30+6 x-6 e^x x+2 x^2-3 e^x x^2+4 x^3\right )} \, dx+24 \int \frac {30+60 x+22 x^2-3 x^3+3 x^4+2 x^5}{(2+x) \left (30+6 x-6 e^x x+2 x^2-3 e^x x^2+4 x^3\right )^2} \, dx\\ &=-x^2-12 \int \left (\frac {1}{30+6 x-6 e^x x+2 x^2-3 e^x x^2+4 x^3}+\frac {x}{30+6 x-6 e^x x+2 x^2-3 e^x x^2+4 x^3}-\frac {2}{(2+x) \left (30+6 x-6 e^x x+2 x^2-3 e^x x^2+4 x^3\right )}\right ) \, dx+24 \int \left (\frac {12}{\left (30+6 x-6 e^x x+2 x^2-3 e^x x^2+4 x^3\right )^2}+\frac {24 x}{\left (30+6 x-6 e^x x+2 x^2-3 e^x x^2+4 x^3\right )^2}-\frac {x^2}{\left (30+6 x-6 e^x x+2 x^2-3 e^x x^2+4 x^3\right )^2}-\frac {x^3}{\left (30+6 x-6 e^x x+2 x^2-3 e^x x^2+4 x^3\right )^2}+\frac {2 x^4}{\left (30+6 x-6 e^x x+2 x^2-3 e^x x^2+4 x^3\right )^2}+\frac {6}{(2+x) \left (30+6 x-6 e^x x+2 x^2-3 e^x x^2+4 x^3\right )^2}\right ) \, dx\\ &=-x^2-12 \int \frac {1}{30+6 x-6 e^x x+2 x^2-3 e^x x^2+4 x^3} \, dx-12 \int \frac {x}{30+6 x-6 e^x x+2 x^2-3 e^x x^2+4 x^3} \, dx-24 \int \frac {x^2}{\left (30+6 x-6 e^x x+2 x^2-3 e^x x^2+4 x^3\right )^2} \, dx-24 \int \frac {x^3}{\left (30+6 x-6 e^x x+2 x^2-3 e^x x^2+4 x^3\right )^2} \, dx+24 \int \frac {1}{(2+x) \left (30+6 x-6 e^x x+2 x^2-3 e^x x^2+4 x^3\right )} \, dx+48 \int \frac {x^4}{\left (30+6 x-6 e^x x+2 x^2-3 e^x x^2+4 x^3\right )^2} \, dx+144 \int \frac {1}{(2+x) \left (30+6 x-6 e^x x+2 x^2-3 e^x x^2+4 x^3\right )^2} \, dx+288 \int \frac {1}{\left (30+6 x-6 e^x x+2 x^2-3 e^x x^2+4 x^3\right )^2} \, dx+576 \int \frac {x}{\left (30+6 x-6 e^x x+2 x^2-3 e^x x^2+4 x^3\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.08, size = 35, normalized size = 1.03 \begin {gather*} -x \left (x-\frac {12}{30-6 \left (-1+e^x\right ) x+\left (2-3 e^x\right ) x^2+4 x^3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.04, size = 67, normalized size = 1.97 \begin {gather*} -\frac {4 \, x^{5} + 2 \, x^{4} + 6 \, x^{3} + 30 \, x^{2} - 3 \, {\left (x^{4} + 2 \, x^{3}\right )} e^{x} - 12 \, x}{4 \, x^{3} + 2 \, x^{2} - 3 \, {\left (x^{2} + 2 \, x\right )} e^{x} + 6 \, x + 30} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 69, normalized size = 2.03 \begin {gather*} -\frac {4 \, x^{5} - 3 \, x^{4} e^{x} + 2 \, x^{4} - 6 \, x^{3} e^{x} + 6 \, x^{3} + 30 \, x^{2} - 12 \, x}{4 \, x^{3} - 3 \, x^{2} e^{x} + 2 \, x^{2} - 6 \, x e^{x} + 6 \, x + 30} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 39, normalized size = 1.15
method | result | size |
risch | \(-x^{2}+\frac {12 x}{4 x^{3}-3 \,{\mathrm e}^{x} x^{2}+2 x^{2}-6 \,{\mathrm e}^{x} x +6 x +30}\) | \(39\) |
norman | \(\frac {-30 x^{2}-6 x^{3}+12 x -2 x^{4}-4 x^{5}+6 \,{\mathrm e}^{x} x^{3}+3 \,{\mathrm e}^{x} x^{4}}{4 x^{3}-3 \,{\mathrm e}^{x} x^{2}+2 x^{2}-6 \,{\mathrm e}^{x} x +6 x +30}\) | \(69\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.42, size = 67, normalized size = 1.97 \begin {gather*} -\frac {4 \, x^{5} + 2 \, x^{4} + 6 \, x^{3} + 30 \, x^{2} - 3 \, {\left (x^{4} + 2 \, x^{3}\right )} e^{x} - 12 \, x}{4 \, x^{3} + 2 \, x^{2} - 3 \, {\left (x^{2} + 2 \, x\right )} e^{x} + 6 \, x + 30} \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 {1800\,x+{\mathrm {e}}^{2\,x}\,\left (18\,x^5+72\,x^4+72\,x^3\right )-{\mathrm {e}}^x\,\left (48\,x^6+120\,x^5+120\,x^4+540\,x^3+828\,x^2\right )+744\,x^2+408\,x^3+528\,x^4+104\,x^5+32\,x^6+32\,x^7-360}{360\,x-{\mathrm {e}}^x\,\left (24\,x^5+60\,x^4+60\,x^3+252\,x^2+360\,x\right )+{\mathrm {e}}^{2\,x}\,\left (9\,x^4+36\,x^3+36\,x^2\right )+156\,x^2+264\,x^3+52\,x^4+16\,x^5+16\,x^6+900} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.34, size = 34, normalized size = 1.00 \begin {gather*} - x^{2} - \frac {12 x}{- 4 x^{3} - 2 x^{2} - 6 x + \left (3 x^{2} + 6 x\right ) e^{x} - 30} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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