Optimal. Leaf size=31 \[ e^{4 x^2}+\frac {e^{x^2} \left (-1+5 e^{-x}-e^x\right )}{x} \]
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Rubi [B] time = 3.11, antiderivative size = 72, normalized size of antiderivative = 2.32, number of steps used = 8, number of rules used = 3, integrand size = 67, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {6742, 2209, 2288} \begin {gather*} \frac {5 e^{x^2-x} \left (x-2 x^2\right )}{(1-2 x) x^2}+e^{4 x^2}-\frac {e^{x^2+x} \left (2 x^2+x\right )}{x^2 (2 x+1)}-\frac {e^{x^2}}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 2209
Rule 2288
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (8 e^{4 x^2} x-\frac {e^{-x+x^2} \left (5-e^x-e^{2 x}+5 x+e^{2 x} x-10 x^2+2 e^x x^2+2 e^{2 x} x^2\right )}{x^2}\right ) \, dx\\ &=8 \int e^{4 x^2} x \, dx-\int \frac {e^{-x+x^2} \left (5-e^x-e^{2 x}+5 x+e^{2 x} x-10 x^2+2 e^x x^2+2 e^{2 x} x^2\right )}{x^2} \, dx\\ &=e^{4 x^2}-\int \left (\frac {e^{x^2} \left (-1+2 x^2\right )}{x^2}-\frac {5 e^{-x+x^2} \left (-1-x+2 x^2\right )}{x^2}+\frac {e^{x+x^2} \left (-1+x+2 x^2\right )}{x^2}\right ) \, dx\\ &=e^{4 x^2}+5 \int \frac {e^{-x+x^2} \left (-1-x+2 x^2\right )}{x^2} \, dx-\int \frac {e^{x^2} \left (-1+2 x^2\right )}{x^2} \, dx-\int \frac {e^{x+x^2} \left (-1+x+2 x^2\right )}{x^2} \, dx\\ &=e^{4 x^2}-\frac {e^{x^2}}{x}+\frac {5 e^{-x+x^2} \left (x-2 x^2\right )}{(1-2 x) x^2}-\frac {e^{x+x^2} \left (x+2 x^2\right )}{x^2 (1+2 x)}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 3.39, size = 34, normalized size = 1.10 \begin {gather*} -\frac {e^{(-1+x) x} \left (-5+e^x+e^{2 x}-e^{x+3 x^2} x\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.81, size = 61, normalized size = 1.97 \begin {gather*} \frac {{\left ({\left (x e^{\left (7 \, x^{2}\right )} - e^{\left (4 \, x^{2}\right )}\right )} e^{\left (4 \, x^{2} + x\right )} + 5 \, e^{\left (8 \, x^{2}\right )} - e^{\left (8 \, x^{2} + 2 \, x\right )}\right )} e^{\left (-7 \, x^{2} - x\right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 37, normalized size = 1.19 \begin {gather*} \frac {x e^{\left (4 \, x^{2}\right )} - e^{\left (x^{2} + x\right )} + 5 \, e^{\left (x^{2} - x\right )} - e^{\left (x^{2}\right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 27, normalized size = 0.87
method | result | size |
risch | \({\mathrm e}^{4 x^{2}}-\frac {\left ({\mathrm e}^{2 x}+{\mathrm e}^{x}-5\right ) {\mathrm e}^{x \left (x -1\right )}}{x}\) | \(27\) |
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*} -5 i \, \sqrt {\pi } \operatorname {erf}\left (i \, x - \frac {1}{2} i\right ) e^{\left (-\frac {1}{4}\right )} + i \, \sqrt {\pi } \operatorname {erf}\left (i \, x\right ) - \frac {\sqrt {-x^{2}} \Gamma \left (-\frac {1}{2}, -x^{2}\right )}{2 \, x} - \frac {e^{\left (x^{2} + x\right )}}{x} + e^{\left (4 \, x^{2}\right )} - \int \frac {5 \, {\left (x + 1\right )} e^{\left (x^{2} - x\right )}}{x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 28, normalized size = 0.90 \begin {gather*} {\mathrm {e}}^{4\,x^2}-\frac {{\mathrm {e}}^{x^2-x}\,\left ({\mathrm {e}}^{2\,x}+{\mathrm {e}}^x-5\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.26, size = 31, normalized size = 1.00 \begin {gather*} \frac {\left (x e^{x} e^{4 x^{2}} + \left (- e^{2 x} - e^{x} + 5\right ) e^{x^{2}}\right ) e^{- x}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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