Optimal. Leaf size=32 \[ 1-\frac {1}{(-4+x) \left (x^2+\frac {1}{2} x \left (\frac {e^2}{4 x}+x\right )\right )} \]
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Rubi [A] time = 0.11, antiderivative size = 40, normalized size of antiderivative = 1.25, number of steps used = 5, number of rules used = 3, integrand size = 65, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.046, Rules used = {2074, 639, 203} \begin {gather*} \frac {96 (x+4)}{\left (192+e^2\right ) \left (12 x^2+e^2\right )}+\frac {8}{\left (192+e^2\right ) (4-x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 639
Rule 2074
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {8}{\left (192+e^2\right ) (-4+x)^2}+\frac {192 \left (e^2-48 x\right )}{\left (192+e^2\right ) \left (e^2+12 x^2\right )^2}-\frac {96}{\left (192+e^2\right ) \left (e^2+12 x^2\right )}\right ) \, dx\\ &=\frac {8}{\left (192+e^2\right ) (4-x)}-\frac {96 \int \frac {1}{e^2+12 x^2} \, dx}{192+e^2}+\frac {192 \int \frac {e^2-48 x}{\left (e^2+12 x^2\right )^2} \, dx}{192+e^2}\\ &=\frac {8}{\left (192+e^2\right ) (4-x)}+\frac {96 (4+x)}{\left (192+e^2\right ) \left (e^2+12 x^2\right )}-\frac {16 \sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt {3} x}{e}\right )}{e \left (192+e^2\right )}+\frac {96 \int \frac {1}{e^2+12 x^2} \, dx}{192+e^2}\\ &=\frac {8}{\left (192+e^2\right ) (4-x)}+\frac {96 (4+x)}{\left (192+e^2\right ) \left (e^2+12 x^2\right )}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.03, size = 18, normalized size = 0.56 \begin {gather*} -\frac {8}{(-4+x) \left (e^2+12 x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 21, normalized size = 0.66 \begin {gather*} -\frac {8}{12 \, x^{3} - 48 \, x^{2} + {\left (x - 4\right )} e^{2}} \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 = 0.14, size = 18, normalized size = 0.56
method | result | size |
norman | \(-\frac {8}{\left (x -4\right ) \left (12 x^{2}+{\mathrm e}^{2}\right )}\) | \(18\) |
gosper | \(-\frac {8}{12 x^{3}+{\mathrm e}^{2} x -48 x^{2}-4 \,{\mathrm e}^{2}}\) | \(24\) |
risch | \(-\frac {8}{12 x^{3}+{\mathrm e}^{2} x -48 x^{2}-4 \,{\mathrm e}^{2}}\) | \(24\) |
default | \(-\frac {8 \left (576 \,{\mathrm e}^{4}+{\mathrm e}^{6}+110592 \,{\mathrm e}^{2}+7077888\right )}{\left (36864+384 \,{\mathrm e}^{2}+{\mathrm e}^{4}\right )^{2} \left (x -4\right )}+\frac {-\frac {96 \left (-\frac {{\mathrm e}^{-2} \left (221184 \,{\mathrm e}^{4}+{\mathrm e}^{2} {\mathrm e}^{6}+576 \,{\mathrm e}^{2} {\mathrm e}^{4}+14155776 \,{\mathrm e}^{2}+576 \,{\mathrm e}^{6}+{\mathrm e}^{8}\right ) x}{2}-442368 \,{\mathrm e}^{2}-4 \,{\mathrm e}^{6}-2304 \,{\mathrm e}^{4}-28311552\right )}{12 x^{2}+{\mathrm e}^{2}}-8 \left ({\mathrm e}^{2} {\mathrm e}^{6}+576 \,{\mathrm e}^{2} {\mathrm e}^{4}-576 \,{\mathrm e}^{6}-{\mathrm e}^{8}\right ) {\mathrm e}^{-2} \sqrt {3}\, {\mathrm e}^{-1} \arctan \left (2 x \sqrt {3}\, {\mathrm e}^{-1}\right )}{\left (36864+384 \,{\mathrm e}^{2}+{\mathrm e}^{4}\right )^{2}}\) | \(163\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.39, size = 23, normalized size = 0.72 \begin {gather*} -\frac {8}{12 \, x^{3} - 48 \, x^{2} + x e^{2} - 4 \, e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.94, size = 17, normalized size = 0.53 \begin {gather*} -\frac {8}{\left (x-4\right )\,\left (12\,x^2+{\mathrm {e}}^2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.61, size = 22, normalized size = 0.69 \begin {gather*} - \frac {8}{12 x^{3} - 48 x^{2} + x e^{2} - 4 e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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