Optimal. Leaf size=186 \[ \frac {3 b e^{c-\left (b^2-d\right ) x^2}}{4 \left (b^2-d\right ) d^2 \sqrt {\pi }}-\frac {b e^{c-\left (b^2-d\right ) x^2}}{2 \left (b^2-d\right )^2 d \sqrt {\pi }}-\frac {b e^{c-\left (b^2-d\right ) x^2} x^2}{2 \left (b^2-d\right ) d \sqrt {\pi }}-\frac {3 e^{c+d x^2} x \text {Erfc}(b x)}{4 d^2}+\frac {e^{c+d x^2} x^3 \text {Erfc}(b x)}{2 d}+\frac {3 \text {Int}\left (e^{c+d x^2} \text {Erfc}(b x),x\right )}{4 d^2} \]
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Rubi [A]
time = 0.16, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int e^{c+d x^2} x^4 \text {Erfc}(b x) \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int e^{c+d x^2} x^4 \text {erfc}(b x) \, dx &=\frac {e^{c+d x^2} x^3 \text {erfc}(b x)}{2 d}-\frac {3 \int e^{c+d x^2} x^2 \text {erfc}(b x) \, dx}{2 d}+\frac {b \int e^{c-\left (b^2-d\right ) x^2} x^3 \, dx}{d \sqrt {\pi }}\\ &=-\frac {b e^{c-\left (b^2-d\right ) x^2} x^2}{2 \left (b^2-d\right ) d \sqrt {\pi }}-\frac {3 e^{c+d x^2} x \text {erfc}(b x)}{4 d^2}+\frac {e^{c+d x^2} x^3 \text {erfc}(b x)}{2 d}+\frac {3 \int e^{c+d x^2} \text {erfc}(b x) \, dx}{4 d^2}-\frac {(3 b) \int e^{c-\left (b^2-d\right ) x^2} x \, dx}{2 d^2 \sqrt {\pi }}+\frac {b \int e^{c+\left (-b^2+d\right ) x^2} x \, dx}{\left (b^2-d\right ) d \sqrt {\pi }}\\ &=\frac {3 b e^{c-\left (b^2-d\right ) x^2}}{4 \left (b^2-d\right ) d^2 \sqrt {\pi }}-\frac {b e^{c-\left (b^2-d\right ) x^2}}{2 \left (b^2-d\right )^2 d \sqrt {\pi }}-\frac {b e^{c-\left (b^2-d\right ) x^2} x^2}{2 \left (b^2-d\right ) d \sqrt {\pi }}-\frac {3 e^{c+d x^2} x \text {erfc}(b x)}{4 d^2}+\frac {e^{c+d x^2} x^3 \text {erfc}(b x)}{2 d}+\frac {3 \int e^{c+d x^2} \text {erfc}(b x) \, dx}{4 d^2}\\ \end {align*}
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Mathematica [A]
time = 0.58, size = 0, normalized size = 0.00 \begin {gather*} \int e^{c+d x^2} x^4 \text {Erfc}(b x) \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 0.07, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{d \,x^{2}+c} x^{4} \mathrm {erfc}\left (b x \right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
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 [A]
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 [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} e^{c} \int x^{4} e^{d x^{2}} \operatorname {erfc}{\left (b x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
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 [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^4\,{\mathrm {e}}^{d\,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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