Optimal. Leaf size=32 \[ 4-x+\frac {3 x (-x+4 (1+x))}{e^{2+2 x}-x^2} \]
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Rubi [F] time = 0.88, 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 {-e^{4+4 x}+12 x^2-x^4+e^{2+2 x} \left (12-6 x-16 x^2\right )}{e^{4+4 x}-2 e^{2+2 x} x^2+x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-e^{4+4 x}+12 x^2-x^4+e^{2+2 x} \left (12-6 x-16 x^2\right )}{\left (e^{2+2 x}-x^2\right )^2} \, dx\\ &=\int \left (-1-\frac {3 (1+3 x)}{2 \left (e^{1+x}-x\right )}+\frac {3 (1+3 x)}{2 \left (e^{1+x}+x\right )}-\frac {3 \left (-4+x+3 x^2\right )}{2 \left (e^{1+x}-x\right )^2}-\frac {3 \left (-4+x+3 x^2\right )}{2 \left (e^{1+x}+x\right )^2}\right ) \, dx\\ &=-x-\frac {3}{2} \int \frac {1+3 x}{e^{1+x}-x} \, dx+\frac {3}{2} \int \frac {1+3 x}{e^{1+x}+x} \, dx-\frac {3}{2} \int \frac {-4+x+3 x^2}{\left (e^{1+x}-x\right )^2} \, dx-\frac {3}{2} \int \frac {-4+x+3 x^2}{\left (e^{1+x}+x\right )^2} \, dx\\ &=-x-\frac {3}{2} \int \left (\frac {1}{e^{1+x}-x}+\frac {3 x}{e^{1+x}-x}\right ) \, dx-\frac {3}{2} \int \left (-\frac {4}{\left (e^{1+x}-x\right )^2}+\frac {x}{\left (e^{1+x}-x\right )^2}+\frac {3 x^2}{\left (e^{1+x}-x\right )^2}\right ) \, dx-\frac {3}{2} \int \left (-\frac {4}{\left (e^{1+x}+x\right )^2}+\frac {x}{\left (e^{1+x}+x\right )^2}+\frac {3 x^2}{\left (e^{1+x}+x\right )^2}\right ) \, dx+\frac {3}{2} \int \left (\frac {1}{e^{1+x}+x}+\frac {3 x}{e^{1+x}+x}\right ) \, dx\\ &=-x-\frac {3}{2} \int \frac {1}{e^{1+x}-x} \, dx-\frac {3}{2} \int \frac {x}{\left (e^{1+x}-x\right )^2} \, dx-\frac {3}{2} \int \frac {x}{\left (e^{1+x}+x\right )^2} \, dx+\frac {3}{2} \int \frac {1}{e^{1+x}+x} \, dx-\frac {9}{2} \int \frac {x}{e^{1+x}-x} \, dx-\frac {9}{2} \int \frac {x^2}{\left (e^{1+x}-x\right )^2} \, dx-\frac {9}{2} \int \frac {x^2}{\left (e^{1+x}+x\right )^2} \, dx+\frac {9}{2} \int \frac {x}{e^{1+x}+x} \, dx+6 \int \frac {1}{\left (e^{1+x}-x\right )^2} \, dx+6 \int \frac {1}{\left (e^{1+x}+x\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.28, size = 27, normalized size = 0.84 \begin {gather*} -x-\frac {3 x (4+3 x)}{-e^{2+2 x}+x^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 37, normalized size = 1.16 \begin {gather*} -\frac {x^{3} + 9 \, x^{2} - x e^{\left (2 \, x + 2\right )} + 12 \, x}{x^{2} - e^{\left (2 \, x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 37, normalized size = 1.16 \begin {gather*} -\frac {x^{3} + 9 \, x^{2} - x e^{\left (2 \, x + 2\right )} + 12 \, x}{x^{2} - e^{\left (2 \, x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.28, size = 27, normalized size = 0.84
method | result | size |
risch | \(-x -\frac {3 \left (4+3 x \right ) x}{x^{2}-{\mathrm e}^{2 x +2}}\) | \(27\) |
norman | \(\frac {x \,{\mathrm e}^{2 x +2}-9 \,{\mathrm e}^{2 x +2}-12 x -x^{3}}{x^{2}-{\mathrm e}^{2 x +2}}\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 37, normalized size = 1.16 \begin {gather*} -\frac {x^{3} + 9 \, x^{2} - x e^{\left (2 \, x + 2\right )} + 12 \, x}{x^{2} - e^{\left (2 \, x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.42, size = 28, normalized size = 0.88 \begin {gather*} \frac {9\,x^2+12\,x}{{\mathrm {e}}^{2\,x+2}-x^2}-x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 19, normalized size = 0.59 \begin {gather*} - x + \frac {9 x^{2} + 12 x}{- x^{2} + e^{2 x + 2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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