Optimal. Leaf size=23 \[ -2+x+\frac {1}{3} \left (\frac {13-e+e^{2 x}}{x}+x\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 30, normalized size of antiderivative = 1.30, number of steps used = 6, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {12, 14, 2197} \begin {gather*} \frac {4 x}{3}+\frac {e^{2 x}}{3 x}+\frac {13-e}{3 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 2197
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int \frac {-13+e+4 x^2+e^{2 x} (-1+2 x)}{x^2} \, dx\\ &=\frac {1}{3} \int \left (\frac {e^{2 x} (-1+2 x)}{x^2}+\frac {-13+e+4 x^2}{x^2}\right ) \, dx\\ &=\frac {1}{3} \int \frac {e^{2 x} (-1+2 x)}{x^2} \, dx+\frac {1}{3} \int \frac {-13+e+4 x^2}{x^2} \, dx\\ &=\frac {e^{2 x}}{3 x}+\frac {1}{3} \int \left (4+\frac {-13+e}{x^2}\right ) \, dx\\ &=\frac {13-e}{3 x}+\frac {e^{2 x}}{3 x}+\frac {4 x}{3}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.02, size = 22, normalized size = 0.96 \begin {gather*} \frac {13-e+e^{2 x}+4 x^2}{3 x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 20, normalized size = 0.87 \begin {gather*} \frac {4 \, x^{2} - e + e^{\left (2 \, x\right )} + 13}{3 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 20, normalized size = 0.87 \begin {gather*} \frac {4 \, x^{2} - e + e^{\left (2 \, x\right )} + 13}{3 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 22, normalized size = 0.96
method | result | size |
norman | \(\frac {\frac {4 x^{2}}{3}+\frac {{\mathrm e}^{2 x}}{3}-\frac {{\mathrm e}}{3}+\frac {13}{3}}{x}\) | \(22\) |
default | \(\frac {4 x}{3}-\frac {{\mathrm e}}{3 x}+\frac {13}{3 x}+\frac {{\mathrm e}^{2 x}}{3 x}\) | \(26\) |
risch | \(\frac {4 x}{3}-\frac {{\mathrm e}}{3 x}+\frac {13}{3 x}+\frac {{\mathrm e}^{2 x}}{3 x}\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.38, size = 29, normalized size = 1.26 \begin {gather*} \frac {4}{3} \, x - \frac {e}{3 \, x} + \frac {13}{3 \, x} + \frac {2}{3} \, {\rm Ei}\left (2 \, x\right ) - \frac {2}{3} \, \Gamma \left (-1, -2 \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 20, normalized size = 0.87 \begin {gather*} \frac {4\,x}{3}+\frac {\frac {{\mathrm {e}}^{2\,x}}{3}-\frac {\mathrm {e}}{3}+\frac {13}{3}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 20, normalized size = 0.87 \begin {gather*} \frac {4 x}{3} + \frac {e^{2 x}}{3 x} + \frac {13 - e}{3 x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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