Optimal. Leaf size=21 \[ x-\log \left (x-e^{e^2+x} (e+x)^2\right ) \]
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Rubi [F]
time = 0.98, 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 {1+e^{e^2+x} (-2 e-2 x)-x}{-x+e^{e^2+x} \left (e^2+2 e x+x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-1+x+2 e^{e^2+x} (e+x)}{x-e^{e^2+x} (e+x)^2} \, dx\\ &=\int \left (-\frac {2}{e+x}+\frac {e-(1+e) x-x^2}{(e+x) \left (e^{2+e^2+x}-x+2 e^{1+e^2+x} x+e^{e^2+x} x^2\right )}\right ) \, dx\\ &=-2 \log (e+x)+\int \frac {e-(1+e) x-x^2}{(e+x) \left (e^{2+e^2+x}-x+2 e^{1+e^2+x} x+e^{e^2+x} x^2\right )} \, dx\\ &=-2 \log (e+x)+\int \left (-\frac {1}{e^{2+e^2+x}-x+2 e^{1+e^2+x} x+e^{e^2+x} x^2}-\frac {x}{e^{2+e^2+x}-x+2 e^{1+e^2+x} x+e^{e^2+x} x^2}+\frac {2 e}{(e+x) \left (e^{2+e^2+x}-x+2 e^{1+e^2+x} x+e^{e^2+x} x^2\right )}\right ) \, dx\\ &=-2 \log (e+x)+(2 e) \int \frac {1}{(e+x) \left (e^{2+e^2+x}-x+2 e^{1+e^2+x} x+e^{e^2+x} x^2\right )} \, dx-\int \frac {1}{e^{2+e^2+x}-x+2 e^{1+e^2+x} x+e^{e^2+x} x^2} \, dx-\int \frac {x}{e^{2+e^2+x}-x+2 e^{1+e^2+x} x+e^{e^2+x} x^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 4.92, size = 39, normalized size = 1.86 \begin {gather*} x-\log \left (e^{2+e^2+x}-x+2 e^{1+e^2+x} x+e^{e^2+x} x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 3.14, size = 37, normalized size = 1.76
method | result | size |
risch | \(-2 \ln \left (x +{\mathrm e}\right )+{\mathrm e}^{2}+x -\ln \left ({\mathrm e}^{x +{\mathrm e}^{2}}-\frac {x}{{\mathrm e}^{2}+2 x \,{\mathrm e}+x^{2}}\right )\) | \(37\) |
norman | \(x -\ln \left (2 \,{\mathrm e} \,{\mathrm e}^{x +{\mathrm e}^{2}} x +{\mathrm e}^{x +{\mathrm e}^{2}} x^{2}+{\mathrm e}^{2} {\mathrm e}^{x +{\mathrm e}^{2}}-x \right )\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 64 vs.
\(2 (20) = 40\).
time = 0.32, size = 64, normalized size = 3.05 \begin {gather*} x - 2 \, \log \left (x + e\right ) - \log \left (\frac {{\left (x^{2} e^{\left (e^{2}\right )} + 2 \, x e^{\left (e^{2} + 1\right )} + e^{\left (e^{2} + 2\right )}\right )} e^{x} - x}{x^{2} e^{\left (e^{2}\right )} + 2 \, x e^{\left (e^{2} + 1\right )} + e^{\left (e^{2} + 2\right )}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 47 vs.
\(2 (20) = 40\).
time = 0.44, size = 47, normalized size = 2.24 \begin {gather*} x - 2 \, \log \left (x + e\right ) - \log \left (\frac {{\left (x^{2} + 2 \, x e + e^{2}\right )} e^{\left (x + e^{2}\right )} - x}{x^{2} + 2 \, x e + e^{2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.19, size = 32, normalized size = 1.52 \begin {gather*} x - 2 \log {\left (x + e \right )} - \log {\left (- \frac {x}{x^{2} + 2 e x + e^{2}} + e^{x + e^{2}} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 36, normalized size = 1.71 \begin {gather*} x - \log \left (x^{2} e^{\left (x + e^{2} + 1\right )} - x e + 2 \, x e^{\left (x + e^{2} + 2\right )} + e^{\left (x + e^{2} + 3\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.31, size = 33, normalized size = 1.57 \begin {gather*} x-\ln \left ({\mathrm {e}}^{x+{\mathrm {e}}^2+2}-x+2\,x\,{\mathrm {e}}^{x+{\mathrm {e}}^2+1}+x^2\,{\mathrm {e}}^{x+{\mathrm {e}}^2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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