Optimal. Leaf size=22 \[ -e^{-4+\frac {2}{x^2}}+\frac {1}{e x}+2 x \]
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Rubi [A]
time = 0.04, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 3, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {12, 14, 2240}
\begin {gather*} -e^{\frac {2}{x^2}-4}+2 x+\frac {1}{e x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 2240
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {-x^3+e x^2 \left (4 e^{\frac {2-4 x^2}{x^2}}+2 x^3\right )}{x^5} \, dx}{e}\\ &=\frac {\int \left (\frac {4 e^{-3+\frac {2}{x^2}}}{x^3}+\frac {-1+2 e x^2}{x^2}\right ) \, dx}{e}\\ &=\frac {\int \frac {-1+2 e x^2}{x^2} \, dx}{e}+\frac {4 \int \frac {e^{-3+\frac {2}{x^2}}}{x^3} \, dx}{e}\\ &=-e^{-4+\frac {2}{x^2}}+\frac {\int \left (2 e-\frac {1}{x^2}\right ) \, dx}{e}\\ &=-e^{-4+\frac {2}{x^2}}+\frac {1}{e x}+2 x\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.01, size = 22, normalized size = 1.00 \begin {gather*} -e^{-4+\frac {2}{x^2}}+\frac {1}{e x}+2 x \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 32, normalized size = 1.45
method | result | size |
default | \(2 x +\frac {{\mathrm e}^{-1-\ln \left (x^{2}\right )+2 \ln \left (x \right )}}{x}-{\mathrm e}^{\frac {2}{x^{2}}-4}\) | \(32\) |
norman | \(\frac {{\mathrm e}^{-1} x^{3}+2 x^{5}-x^{4} {\mathrm e}^{\frac {-4 x^{2}+2}{x^{2}}}}{x^{4}}\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 22, normalized size = 1.00 \begin {gather*} {\left (2 \, x e + \frac {1}{x} - e^{\left (\frac {2}{x^{2}} - 3\right )}\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 33, normalized size = 1.50 \begin {gather*} \frac {{\left (2 \, x^{2} e - x e^{\left (-\frac {2 \, {\left (2 \, x^{2} - 1\right )}}{x^{2}} + 1\right )} + 1\right )} e^{\left (-1\right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 24, normalized size = 1.09 \begin {gather*} \frac {2 e x + \frac {1}{x}}{e} - e^{\frac {2 - 4 x^{2}}{x^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 26, normalized size = 1.18 \begin {gather*} \frac {{\left (2 \, x^{2} e + 1\right )} e^{\left (-1\right )}}{x} - e^{\left (\frac {2}{x^{2}} - 4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.27, size = 20, normalized size = 0.91 \begin {gather*} 2\,x-{\mathrm {e}}^{-4}\,{\mathrm {e}}^{\frac {2}{x^2}}+\frac {{\mathrm {e}}^{-1}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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