Optimal. Leaf size=17 \[ \frac {e^{\frac {x}{2 e^x+x}}}{x} \]
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Rubi [F]
time = 1.50, 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 {e^{\frac {x}{2 e^x+x}} \left (-4 e^{2 x}-x^2+e^x \left (-2 x-2 x^2\right )\right )}{4 e^{2 x} x^2+4 e^x x^3+x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{\frac {x}{2 e^x+x}} \left (-4 e^{2 x}-x^2-2 e^x x (1+x)\right )}{x^2 \left (2 e^x+x\right )^2} \, dx\\ &=\int \left (-\frac {e^{\frac {x}{2 e^x+x}}}{x^2}+\frac {e^{\frac {x}{2 e^x+x}} (-1+x)}{\left (2 e^x+x\right )^2}-\frac {e^{\frac {x}{2 e^x+x}} (-1+x)}{x \left (2 e^x+x\right )}\right ) \, dx\\ &=-\int \frac {e^{\frac {x}{2 e^x+x}}}{x^2} \, dx+\int \frac {e^{\frac {x}{2 e^x+x}} (-1+x)}{\left (2 e^x+x\right )^2} \, dx-\int \frac {e^{\frac {x}{2 e^x+x}} (-1+x)}{x \left (2 e^x+x\right )} \, dx\\ &=-\int \frac {e^{\frac {x}{2 e^x+x}}}{x^2} \, dx+\int \left (-\frac {e^{\frac {x}{2 e^x+x}}}{\left (2 e^x+x\right )^2}+\frac {e^{\frac {x}{2 e^x+x}} x}{\left (2 e^x+x\right )^2}\right ) \, dx-\int \left (\frac {e^{\frac {x}{2 e^x+x}}}{2 e^x+x}-\frac {e^{\frac {x}{2 e^x+x}}}{x \left (2 e^x+x\right )}\right ) \, dx\\ &=-\int \frac {e^{\frac {x}{2 e^x+x}}}{x^2} \, dx-\int \frac {e^{\frac {x}{2 e^x+x}}}{\left (2 e^x+x\right )^2} \, dx+\int \frac {e^{\frac {x}{2 e^x+x}} x}{\left (2 e^x+x\right )^2} \, dx-\int \frac {e^{\frac {x}{2 e^x+x}}}{2 e^x+x} \, dx+\int \frac {e^{\frac {x}{2 e^x+x}}}{x \left (2 e^x+x\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.14, size = 17, normalized size = 1.00 \begin {gather*} \frac {e^{\frac {x}{2 e^x+x}}}{x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 16, normalized size = 0.94
method | result | size |
risch | \(\frac {{\mathrm e}^{\frac {x}{2 \,{\mathrm e}^{x}+x}}}{x}\) | \(16\) |
norman | \(\frac {x \,{\mathrm e}^{\frac {x}{2 \,{\mathrm e}^{x}+x}}+2 \,{\mathrm e}^{x} {\mathrm e}^{\frac {x}{2 \,{\mathrm e}^{x}+x}}}{x \left (2 \,{\mathrm e}^{x}+x \right )}\) | \(42\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.30, size = 15, normalized size = 0.88 \begin {gather*} \frac {e^{\left (\frac {x}{x + 2 \, e^{x}}\right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.11, size = 10, normalized size = 0.59 \begin {gather*} \frac {e^{\frac {x}{x + 2 e^{x}}}}{x} \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 = 0.88 \begin {gather*} \frac {e^{\left (\frac {x}{x + 2 \, e^{x}}\right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.77, size = 15, normalized size = 0.88 \begin {gather*} \frac {{\mathrm {e}}^{\frac {x}{x+2\,{\mathrm {e}}^x}}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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