Optimal. Leaf size=21 \[ 3-4 e^x x+\left (-e^{5 x^2}+x\right )^2 \]
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Rubi [A]
time = 0.04, antiderivative size = 34, normalized size of antiderivative = 1.62, number of steps
used = 9, number of rules used = 6, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2207, 2225,
2240, 2258, 2235, 2243} \begin {gather*} x^2-2 e^{5 x^2} x+e^{10 x^2}+4 e^x-4 e^x (x+1) \end {gather*}
Antiderivative was successfully verified.
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Rule 2207
Rule 2225
Rule 2235
Rule 2240
Rule 2243
Rule 2258
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=x^2+20 \int e^{10 x^2} x \, dx+\int e^x (-4-4 x) \, dx+\int e^{5 x^2} \left (-2-20 x^2\right ) \, dx\\ &=e^{10 x^2}+x^2-4 e^x (1+x)+4 \int e^x \, dx+\int \left (-2 e^{5 x^2}-20 e^{5 x^2} x^2\right ) \, dx\\ &=4 e^x+e^{10 x^2}+x^2-4 e^x (1+x)-2 \int e^{5 x^2} \, dx-20 \int e^{5 x^2} x^2 \, dx\\ &=4 e^x+e^{10 x^2}-2 e^{5 x^2} x+x^2-4 e^x (1+x)-\sqrt {\frac {\pi }{5}} \text {erfi}\left (\sqrt {5} x\right )+2 \int e^{5 x^2} \, dx\\ &=4 e^x+e^{10 x^2}-2 e^{5 x^2} x+x^2-4 e^x (1+x)\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.06, size = 27, normalized size = 1.29 \begin {gather*} e^{10 x^2}-4 e^x x-2 e^{5 x^2} x+x^2 \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 27, normalized size = 1.29
method | result | size |
risch | \(x^{2}+{\mathrm e}^{10 x^{2}}-2 x \,{\mathrm e}^{5 x^{2}}-4 \,{\mathrm e}^{x} x\) | \(25\) |
default | \(x^{2}+{\mathrm e}^{10 x^{2}}-2 x \,{\mathrm e}^{5 x^{2}}-4 \,{\mathrm e}^{x} x\) | \(27\) |
norman | \(x^{2}+{\mathrm e}^{10 x^{2}}-2 x \,{\mathrm e}^{5 x^{2}}-4 \,{\mathrm e}^{x} x\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 30, normalized size = 1.43 \begin {gather*} x^{2} - 2 \, x e^{\left (5 \, x^{2}\right )} - 4 \, {\left (x - 1\right )} e^{x} + e^{\left (10 \, x^{2}\right )} - 4 \, e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 24, normalized size = 1.14 \begin {gather*} x^{2} - 2 \, x e^{\left (5 \, x^{2}\right )} - 4 \, x e^{x} + e^{\left (10 \, x^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 26, normalized size = 1.24 \begin {gather*} x^{2} - 4 x e^{x} - 2 x e^{5 x^{2}} + e^{10 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 24, normalized size = 1.14 \begin {gather*} x^{2} - 2 \, x e^{\left (5 \, x^{2}\right )} - 4 \, x e^{x} + e^{\left (10 \, x^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.09, size = 24, normalized size = 1.14 \begin {gather*} {\mathrm {e}}^{10\,x^2}-2\,x\,{\mathrm {e}}^{5\,x^2}-4\,x\,{\mathrm {e}}^x+x^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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