Optimal. Leaf size=95 \[ \frac {e^c x^2}{2 b \sqrt {\pi }}+\frac {e^{c+b^2 x^2} x \text {Erfc}(b x)}{2 b^2}-\frac {e^c \sqrt {\pi } \text {Erfi}(b x)}{4 b^3}+\frac {e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{2 b \sqrt {\pi }} \]
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Rubi [A]
time = 0.06, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {6521, 6512,
2235, 6511, 12, 30} \begin {gather*} \frac {e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{2 \sqrt {\pi } b}-\frac {\sqrt {\pi } e^c \text {Erfi}(b x)}{4 b^3}+\frac {x e^{b^2 x^2+c} \text {Erfc}(b x)}{2 b^2}+\frac {e^c x^2}{2 \sqrt {\pi } b} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 2235
Rule 6511
Rule 6512
Rule 6521
Rubi steps
\begin {align*} \int e^{c+b^2 x^2} x^2 \text {erfc}(b x) \, dx &=\frac {e^{c+b^2 x^2} x \text {erfc}(b x)}{2 b^2}-\frac {\int e^{c+b^2 x^2} \text {erfc}(b x) \, dx}{2 b^2}+\frac {\int e^c x \, dx}{b \sqrt {\pi }}\\ &=\frac {e^{c+b^2 x^2} x \text {erfc}(b x)}{2 b^2}-\frac {\int e^{c+b^2 x^2} \, dx}{2 b^2}+\frac {\int e^{c+b^2 x^2} \text {erf}(b x) \, dx}{2 b^2}+\frac {e^c \int x \, dx}{b \sqrt {\pi }}\\ &=\frac {e^c x^2}{2 b \sqrt {\pi }}+\frac {e^{c+b^2 x^2} x \text {erfc}(b x)}{2 b^2}-\frac {e^c \sqrt {\pi } \text {erfi}(b x)}{4 b^3}+\frac {e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{2 b \sqrt {\pi }}\\ \end {align*}
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Mathematica [A]
time = 0.22, size = 104, normalized size = 1.09 \begin {gather*} -\frac {e^c \left (-2 b e^{b^2 x^2} \sqrt {\pi } x-2 b^2 x^2+\pi \text {Erfi}(b x)+\text {Erf}(b x) \left (2 b e^{b^2 x^2} \sqrt {\pi } x-\pi \text {Erfi}(b x)\right )+2 b^2 x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )\right )}{4 b^3 \sqrt {\pi }} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.12, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{b^{2} x^{2}+c} x^{2} \mathrm {erfc}\left (b x \right )\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 32.90, size = 65, normalized size = 0.68 \begin {gather*} - \frac {b x^{4} e^{c} {{}_{2}F_{2}\left (\begin {matrix} 1, 2 \\ \frac {3}{2}, 3 \end {matrix}\middle | {b^{2} x^{2}} \right )}}{2 \sqrt {\pi }} + \frac {x e^{c} e^{b^{2} x^{2}}}{2 b^{2}} + \frac {i \sqrt {\pi } e^{c} \operatorname {erf}{\left (i b x \right )}}{4 b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^2\,{\mathrm {e}}^{b^2\,x^2+c}\,\mathrm {erfc}\left (b\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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