Optimal. Leaf size=28 \[ -5+3 e^{\frac {5}{4 x^2}}-\left (-1-e^x+x+x^2\right )^2 \]
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Rubi [A] time = 0.13, antiderivative size = 54, normalized size of antiderivative = 1.93, number of steps used = 15, number of rules used = 6, integrand size = 64, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {12, 14, 2194, 2196, 2176, 2209} \begin {gather*} -x^4-2 x^3+2 e^x x^2+x^2+3 e^{\frac {5}{4 x^2}}+2 e^x x+2 x-2 e^x-e^{2 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 2176
Rule 2194
Rule 2196
Rule 2209
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int \frac {-15 e^{\frac {5}{4 x^2}}+4 x^3-4 e^{2 x} x^3+4 x^4-12 x^5-8 x^6+e^x \left (12 x^4+4 x^5\right )}{x^3} \, dx\\ &=\frac {1}{2} \int \left (-4 e^{2 x}+4 e^x x (3+x)-\frac {15 e^{\frac {5}{4 x^2}}-4 x^3-4 x^4+12 x^5+8 x^6}{x^3}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {15 e^{\frac {5}{4 x^2}}-4 x^3-4 x^4+12 x^5+8 x^6}{x^3} \, dx\right )-2 \int e^{2 x} \, dx+2 \int e^x x (3+x) \, dx\\ &=-e^{2 x}-\frac {1}{2} \int \left (\frac {15 e^{\frac {5}{4 x^2}}}{x^3}+4 \left (-1-x+3 x^2+2 x^3\right )\right ) \, dx+2 \int \left (3 e^x x+e^x x^2\right ) \, dx\\ &=-e^{2 x}+2 \int e^x x^2 \, dx-2 \int \left (-1-x+3 x^2+2 x^3\right ) \, dx+6 \int e^x x \, dx-\frac {15}{2} \int \frac {e^{\frac {5}{4 x^2}}}{x^3} \, dx\\ &=3 e^{\frac {5}{4 x^2}}-e^{2 x}+2 x+6 e^x x+x^2+2 e^x x^2-2 x^3-x^4-4 \int e^x x \, dx-6 \int e^x \, dx\\ &=3 e^{\frac {5}{4 x^2}}-6 e^x-e^{2 x}+2 x+2 e^x x+x^2+2 e^x x^2-2 x^3-x^4+4 \int e^x \, dx\\ &=3 e^{\frac {5}{4 x^2}}-2 e^x-e^{2 x}+2 x+2 e^x x+x^2+2 e^x x^2-2 x^3-x^4\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.05, size = 46, normalized size = 1.64 \begin {gather*} 3 e^{\frac {5}{4 x^2}}-e^{2 x}+2 x+x^2-2 x^3-x^4+2 e^x \left (-1+x+x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 41, normalized size = 1.46 \begin {gather*} -x^{4} - 2 \, x^{3} + x^{2} + 2 \, {\left (x^{2} + x - 1\right )} e^{x} + 2 \, x - e^{\left (2 \, x\right )} + 3 \, e^{\left (\frac {5}{4 \, x^{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 47, normalized size = 1.68 \begin {gather*} -x^{4} - 2 \, x^{3} + 2 \, x^{2} e^{x} + x^{2} + 2 \, x e^{x} + 2 \, x - e^{\left (2 \, x\right )} - 2 \, e^{x} + 3 \, e^{\left (\frac {5}{4 \, x^{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 46, normalized size = 1.64
method | result | size |
risch | \(-x^{4}-2 x^{3}+x^{2}-{\mathrm e}^{2 x}+2 x +\frac {\left (4 x^{2}+4 x -4\right ) {\mathrm e}^{x}}{2}+3 \,{\mathrm e}^{\frac {5}{4 x^{2}}}\) | \(46\) |
default | \(-x^{4}-2 x^{3}+x^{2}+2 x -{\mathrm e}^{2 x}+3 \,{\mathrm e}^{\frac {5}{4 x^{2}}}+2 \,{\mathrm e}^{x} x -2 \,{\mathrm e}^{x}+2 \,{\mathrm e}^{x} x^{2}\) | \(48\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.86, size = 50, normalized size = 1.79 \begin {gather*} -x^{4} - 2 \, x^{3} + x^{2} + 2 \, {\left (x^{2} - 2 \, x + 2\right )} e^{x} + 6 \, {\left (x - 1\right )} e^{x} + 2 \, x - e^{\left (2 \, x\right )} + 3 \, e^{\left (\frac {5}{4 \, x^{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.39, size = 41, normalized size = 1.46 \begin {gather*} 2\,x-{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^{\frac {5}{4\,x^2}}+x^2-2\,x^3-x^4+2\,{\mathrm {e}}^x\,\left (x^2+x-1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.38, size = 42, normalized size = 1.50 \begin {gather*} - x^{4} - 2 x^{3} + x^{2} + 2 x + \left (2 x^{2} + 2 x - 2\right ) e^{x} + 3 e^{\frac {5}{4 x^{2}}} - e^{2 x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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