Optimal. Leaf size=21 \[ 5+e-\frac {e^3}{-1+x-\frac {4+x}{x}} \]
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Rubi [A]
time = 0.05, antiderivative size = 18, normalized size of antiderivative = 0.86, number of steps
used = 4, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {12, 1694, 1828,
8} \begin {gather*} \frac {e^3 x}{5-(1-x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 12
Rule 1694
Rule 1828
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=e^3 \int \frac {4+x^2}{16+16 x-4 x^2-4 x^3+x^4} \, dx\\ &=e^3 \text {Subst}\left (\int \frac {5+2 x+x^2}{\left (5-x^2\right )^2} \, dx,x,-1+x\right )\\ &=\frac {e^3 x}{5-(1-x)^2}-\frac {1}{10} e^3 \text {Subst}(\int 0 \, dx,x,-1+x)\\ &=\frac {e^3 x}{5-(1-x)^2}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.01, size = 16, normalized size = 0.76 \begin {gather*} -\frac {e^3 x}{-4-2 x+x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 16, normalized size = 0.76
method | result | size |
gosper | \(-\frac {x \,{\mathrm e}^{3}}{x^{2}-2 x -4}\) | \(16\) |
default | \(-\frac {x \,{\mathrm e}^{3}}{x^{2}-2 x -4}\) | \(16\) |
norman | \(-\frac {x \,{\mathrm e}^{3}}{x^{2}-2 x -4}\) | \(16\) |
risch | \(-\frac {x \,{\mathrm e}^{3}}{x^{2}-2 x -4}\) | \(16\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 15, normalized size = 0.71 \begin {gather*} -\frac {x e^{3}}{x^{2} - 2 \, x - 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 15, normalized size = 0.71 \begin {gather*} -\frac {x e^{3}}{x^{2} - 2 \, x - 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 14, normalized size = 0.67 \begin {gather*} - \frac {x e^{3}}{x^{2} - 2 x - 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 15, normalized size = 0.71 \begin {gather*} -\frac {x e^{3}}{x^{2} - 2 \, x - 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.98, size = 16, normalized size = 0.76 \begin {gather*} \frac {x\,{\mathrm {e}}^3}{-x^2+2\,x+4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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