Optimal. Leaf size=24 \[ e^{x^2} x+\frac {e^{x^2}}{2 \left (x^2+1\right )} \]
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Rubi [A] time = 0.36, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6742, 2204, 2212, 6715, 2177, 2178} \[ e^{x^2} x+\frac {e^{x^2}}{2 \left (x^2+1\right )} \]
Antiderivative was successfully verified.
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Rule 2177
Rule 2178
Rule 2204
Rule 2212
Rule 6715
Rule 6742
Rubi steps
\begin {align*} \int \frac {e^{x^2} \left (1+4 x^2+x^3+5 x^4+2 x^6\right )}{\left (1+x^2\right )^2} \, dx &=\int \left (e^{x^2}+2 e^{x^2} x^2-\frac {e^{x^2} x}{\left (1+x^2\right )^2}+\frac {e^{x^2} x}{1+x^2}\right ) \, dx\\ &=2 \int e^{x^2} x^2 \, dx+\int e^{x^2} \, dx-\int \frac {e^{x^2} x}{\left (1+x^2\right )^2} \, dx+\int \frac {e^{x^2} x}{1+x^2} \, dx\\ &=e^{x^2} x+\frac {1}{2} \sqrt {\pi } \text {erfi}(x)-\frac {1}{2} \operatorname {Subst}\left (\int \frac {e^x}{(1+x)^2} \, dx,x,x^2\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {e^x}{1+x} \, dx,x,x^2\right )-\int e^{x^2} \, dx\\ &=e^{x^2} x+\frac {e^{x^2}}{2 \left (1+x^2\right )}+\frac {\text {Ei}\left (1+x^2\right )}{2 e}-\frac {1}{2} \operatorname {Subst}\left (\int \frac {e^x}{1+x} \, dx,x,x^2\right )\\ &=e^{x^2} x+\frac {e^{x^2}}{2 \left (1+x^2\right )}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 20, normalized size = 0.83 \[ \frac {1}{2} e^{x^2} \left (\frac {1}{x^2+1}+2 x\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 23, normalized size = 0.96 \[ \frac {{\left (2 \, x^{3} + 2 \, x + 1\right )} e^{\left (x^{2}\right )}}{2 \, {\left (x^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.06, size = 30, normalized size = 1.25 \[ \frac {2 \, x^{3} e^{\left (x^{2}\right )} + 2 \, x e^{\left (x^{2}\right )} + e^{\left (x^{2}\right )}}{2 \, {\left (x^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 24, normalized size = 1.00 \[ \frac {\left (2 x^{3}+2 x +1\right ) {\mathrm e}^{x^{2}}}{2 x^{2}+2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.80, size = 23, normalized size = 0.96 \[ \frac {{\left (2 \, x^{3} + 2 \, x + 1\right )} e^{\left (x^{2}\right )}}{2 \, {\left (x^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.22, size = 24, normalized size = 1.00 \[ \frac {{\mathrm {e}}^{x^2}\,\left (2\,x^3+2\,x+1\right )}{2\,\left (x^2+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 20, normalized size = 0.83 \[ \frac {\left (2 x^{3} + 2 x + 1\right ) e^{x^{2}}}{2 x^{2} + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
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