Optimal. Leaf size=19 \[ \frac {1}{4} \left (-1+x+\log \left (\frac {237}{4}+e^x+x+x^2\right )\right ) \]
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Rubi [F] time = 0.36, 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 {241+8 e^x+12 x+4 x^2}{948+16 e^x+16 x+16 x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {241+8 e^x+12 x+4 x^2}{4 \left (237+4 e^x+4 x+4 x^2\right )} \, dx\\ &=\frac {1}{4} \int \frac {241+8 e^x+12 x+4 x^2}{237+4 e^x+4 x+4 x^2} \, dx\\ &=\frac {1}{4} \int \left (2-\frac {233-4 x+4 x^2}{237+4 e^x+4 x+4 x^2}\right ) \, dx\\ &=\frac {x}{2}-\frac {1}{4} \int \frac {233-4 x+4 x^2}{237+4 e^x+4 x+4 x^2} \, dx\\ &=\frac {x}{2}-\frac {1}{4} \int \left (\frac {233}{237+4 e^x+4 x+4 x^2}-\frac {4 x}{237+4 e^x+4 x+4 x^2}+\frac {4 x^2}{237+4 e^x+4 x+4 x^2}\right ) \, dx\\ &=\frac {x}{2}-\frac {233}{4} \int \frac {1}{237+4 e^x+4 x+4 x^2} \, dx+\int \frac {x}{237+4 e^x+4 x+4 x^2} \, dx-\int \frac {x^2}{237+4 e^x+4 x+4 x^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.13, size = 26, normalized size = 1.37 \begin {gather*} \frac {x}{4}+\frac {1}{4} \log \left (237+4 e^x+4 x+4 x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 21, normalized size = 1.11 \begin {gather*} \frac {1}{4} \, x + \frac {1}{4} \, \log \left (4 \, x^{2} + 4 \, x + 4 \, e^{x} + 237\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 21, normalized size = 1.11 \begin {gather*} \frac {1}{4} \, x + \frac {1}{4} \, \log \left (4 \, x^{2} + 4 \, x + 4 \, e^{x} + 237\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 16, normalized size = 0.84
method | result | size |
risch | \(\frac {x}{4}+\frac {\ln \left ({\mathrm e}^{x}+x +\frac {237}{4}+x^{2}\right )}{4}\) | \(16\) |
norman | \(\frac {x}{4}+\frac {\ln \left (16 \,{\mathrm e}^{x}+16 x^{2}+16 x +948\right )}{4}\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 15, normalized size = 0.79 \begin {gather*} \frac {1}{4} \, x + \frac {1}{4} \, \log \left (x^{2} + x + e^{x} + \frac {237}{4}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 15, normalized size = 0.79 \begin {gather*} \frac {x}{4}+\frac {\ln \left (x+{\mathrm {e}}^x+x^2+\frac {237}{4}\right )}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 17, normalized size = 0.89 \begin {gather*} \frac {x}{4} + \frac {\log {\left (x^{2} + x + e^{x} + \frac {237}{4} \right )}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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