Optimal. Leaf size=25 \[ -2+\frac {2 (-3+x) x}{3-\sqrt {e}-x+x^2} \]
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Rubi [B] time = 0.12, antiderivative size = 61, normalized size of antiderivative = 2.44, number of steps used = 4, number of rules used = 4, integrand size = 59, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.068, Rules used = {1680, 12, 1814, 8} \begin {gather*} -\frac {8 \left (2 \left (11-4 \sqrt {e}\right ) \left (x-\frac {1}{2}\right )+4 e-27 \sqrt {e}+44\right )}{\left (11-4 \sqrt {e}\right ) \left (4 \left (x-\frac {1}{2}\right )^2-4 \sqrt {e}+11\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 12
Rule 1680
Rule 1814
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\operatorname {Subst}\left (\int \frac {16 \left (-11+4 \sqrt {e}+4 \left (4-\sqrt {e}\right ) x+4 x^2\right )}{\left (11-4 \sqrt {e}+4 x^2\right )^2} \, dx,x,-\frac {1}{2}+x\right )\\ &=16 \operatorname {Subst}\left (\int \frac {-11+4 \sqrt {e}+4 \left (4-\sqrt {e}\right ) x+4 x^2}{\left (11-4 \sqrt {e}+4 x^2\right )^2} \, dx,x,-\frac {1}{2}+x\right )\\ &=-\frac {8 \left (44-27 \sqrt {e}+4 e-\left (11-4 \sqrt {e}\right ) (1-2 x)\right )}{\left (11-4 \sqrt {e}\right ) \left (11-4 \sqrt {e}+(-1+2 x)^2\right )}-\frac {8 \operatorname {Subst}\left (\int 0 \, dx,x,-\frac {1}{2}+x\right )}{11-4 \sqrt {e}}\\ &=-\frac {8 \left (44-27 \sqrt {e}+4 e-\left (11-4 \sqrt {e}\right ) (1-2 x)\right )}{\left (11-4 \sqrt {e}\right ) \left (11-4 \sqrt {e}+(-1+2 x)^2\right )}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.02, size = 29, normalized size = 1.16 \begin {gather*} \frac {2 \left (-3+\sqrt {e}-2 x\right )}{3-\sqrt {e}-x+x^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 25, normalized size = 1.00 \begin {gather*} -\frac {2 \, {\left (2 \, x - e^{\frac {1}{2}} + 3\right )}}{x^{2} - x - e^{\frac {1}{2}} + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, {\left (2 \, x^{2} - {\left (2 \, x - 3\right )} e^{\frac {1}{2}} + 6 \, x - 9\right )}}{x^{4} - 2 \, x^{3} + 7 \, x^{2} - 2 \, {\left (x^{2} - x + 3\right )} e^{\frac {1}{2}} - 6 \, x + e + 9}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 22, normalized size = 0.88
method | result | size |
gosper | \(-\frac {2 \left (-3-2 x +{\mathrm e}^{\frac {1}{2}}\right )}{-x^{2}+{\mathrm e}^{\frac {1}{2}}+x -3}\) | \(22\) |
norman | \(\frac {4 x -2 \,{\mathrm e}^{\frac {1}{2}}+6}{-x^{2}+{\mathrm e}^{\frac {1}{2}}+x -3}\) | \(23\) |
risch | \(\frac {4 x -2 \,{\mathrm e}^{\frac {1}{2}}+6}{-x^{2}+{\mathrm e}^{\frac {1}{2}}+x -3}\) | \(23\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} 2 \, \int \frac {2 \, x^{2} - {\left (2 \, x - 3\right )} e^{\frac {1}{2}} + 6 \, x - 9}{x^{4} - 2 \, x^{3} + 7 \, x^{2} - 2 \, {\left (x^{2} - x + 3\right )} e^{\frac {1}{2}} - 6 \, x + e + 9}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 22, normalized size = 0.88 \begin {gather*} \frac {4\,x-2\,\sqrt {\mathrm {e}}+6}{-x^2+x+\sqrt {\mathrm {e}}-3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.60, size = 87, normalized size = 3.48 \begin {gather*} \frac {x \left (-484 - 64 e + 352 e^{\frac {1}{2}}\right ) - 272 e - 726 + 32 e^{\frac {3}{2}} + 770 e^{\frac {1}{2}}}{x^{2} \left (- 88 e^{\frac {1}{2}} + 16 e + 121\right ) + x \left (-121 - 16 e + 88 e^{\frac {1}{2}}\right ) - 385 e^{\frac {1}{2}} - 16 e^{\frac {3}{2}} + 363 + 136 e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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