Optimal. Leaf size=17 \[ \frac {1}{3} \left (e^x+\frac {1}{16 x}\right )+x \]
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Rubi [A] time = 0.01, antiderivative size = 16, normalized size of antiderivative = 0.94, number of steps used = 6, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {12, 14, 2194} \begin {gather*} x+\frac {e^x}{3}+\frac {1}{48 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 2194
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{48} \int \frac {-1+48 x^2+16 e^x x^2}{x^2} \, dx\\ &=\frac {1}{48} \int \left (16 e^x+\frac {-1+48 x^2}{x^2}\right ) \, dx\\ &=\frac {1}{48} \int \frac {-1+48 x^2}{x^2} \, dx+\frac {\int e^x \, dx}{3}\\ &=\frac {e^x}{3}+\frac {1}{48} \int \left (48-\frac {1}{x^2}\right ) \, dx\\ &=\frac {e^x}{3}+\frac {1}{48 x}+x\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.01, size = 16, normalized size = 0.94 \begin {gather*} \frac {e^x}{3}+\frac {1}{48 x}+x \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 17, normalized size = 1.00 \begin {gather*} \frac {48 \, x^{2} + 16 \, x e^{x} + 1}{48 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.93, size = 17, normalized size = 1.00 \begin {gather*} \frac {48 \, x^{2} + 16 \, x e^{x} + 1}{48 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 12, normalized size = 0.71
method | result | size |
default | \(\frac {1}{48 x}+\frac {{\mathrm e}^{x}}{3}+x\) | \(12\) |
risch | \(\frac {1}{48 x}+\frac {{\mathrm e}^{x}}{3}+x\) | \(12\) |
norman | \(\frac {\frac {1}{48}+x^{2}+\frac {{\mathrm e}^{x} x}{3}}{x}\) | \(15\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 11, normalized size = 0.65 \begin {gather*} x + \frac {1}{48 \, x} + \frac {1}{3} \, e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.12, size = 11, normalized size = 0.65 \begin {gather*} x+\frac {{\mathrm {e}}^x}{3}+\frac {1}{48\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 10, normalized size = 0.59 \begin {gather*} x + \frac {e^{x}}{3} + \frac {1}{48 x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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