Optimal. Leaf size=27 \[ 3+\frac {3}{2 \left (2+\frac {1}{x}\right ) \left (-\frac {e}{3}+\frac {x}{3}\right )}+x \]
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Rubi [A] time = 0.14, antiderivative size = 23, normalized size of antiderivative = 0.85, number of steps used = 4, number of rules used = 4, integrand size = 96, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {1680, 1814, 21, 8} \begin {gather*} x-\frac {9 x}{2 \left (-2 x^2+(2 e-1) x+e\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 21
Rule 1680
Rule 1814
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\operatorname {Subst}\left (\int \frac {-(1+2 e)^2 \left (35-4 e-4 e^2\right )+288 (1-2 e) x-32 \left (19+4 e+4 e^2\right ) x^2+256 x^4}{\left (1+4 e+4 e^2-16 x^2\right )^2} \, dx,x,\frac {1}{32} (8-16 e)+x\right )\\ &=-\frac {9 x}{2 \left (e+(-1+2 e) x-2 x^2\right )}-\frac {\operatorname {Subst}\left (\int \frac {-2 (1+2 e)^4+32 (1+2 e)^2 x^2}{1+4 e+4 e^2-16 x^2} \, dx,x,\frac {1}{32} (8-16 e)+x\right )}{2 (1+2 e)^2}\\ &=-\frac {9 x}{2 \left (e+(-1+2 e) x-2 x^2\right )}+\operatorname {Subst}\left (\int 1 \, dx,x,\frac {1}{32} (8-16 e)+x\right )\\ &=x-\frac {9 x}{2 \left (e+(-1+2 e) x-2 x^2\right )}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.04, size = 52, normalized size = 1.93 \begin {gather*} \frac {1}{2} \left (-\frac {9}{(1+2 e) (-1-2 e+2 (e-x))}-\frac {9 e}{(1+2 e) (e-x)}-2 (e-x)\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.53, size = 45, normalized size = 1.67 \begin {gather*} \frac {4 \, x^{3} + 2 \, x^{2} - 2 \, {\left (2 \, x^{2} + x\right )} e + 9 \, x}{2 \, {\left (2 \, x^{2} - {\left (2 \, x + 1\right )} e + x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 25, normalized size = 0.93
method | result | size |
risch | \(x -\frac {9 x}{4 \left (x \,{\mathrm e}-x^{2}+\frac {{\mathrm e}}{2}-\frac {x}{2}\right )}\) | \(25\) |
norman | \(\frac {\left (2 \,{\mathrm e}^{2}-{\mathrm e}-4\right ) x -2 x^{3}+{\mathrm e}^{2}-\frac {{\mathrm e}}{2}}{\left (2 x +1\right ) \left ({\mathrm e}-x \right )}\) | \(45\) |
gosper | \(\frac {4 \,{\mathrm e}^{2} x -4 x^{3}+2 \,{\mathrm e}^{2}-2 x \,{\mathrm e}-{\mathrm e}-8 x}{4 x \,{\mathrm e}-4 x^{2}+2 \,{\mathrm e}-2 x}\) | \(52\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 26, normalized size = 0.96 \begin {gather*} x + \frac {9 \, x}{2 \, {\left (2 \, x^{2} - x {\left (2 \, e - 1\right )} - e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.86, size = 20, normalized size = 0.74 \begin {gather*} x+\frac {9\,x}{2\,\left (2\,x+1\right )\,\left (x-\mathrm {e}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.46, size = 22, normalized size = 0.81 \begin {gather*} x + \frac {9 x}{4 x^{2} + x \left (2 - 4 e\right ) - 2 e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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