Optimal. Leaf size=21 \[ \frac {\sqrt {\pi } e^c}{4 b \text {erfc}(b x)^2} \]
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Rubi [A] time = 0.03, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {6374, 30} \[ \frac {\sqrt {\pi } e^c}{4 b \text {Erfc}(b x)^2} \]
Antiderivative was successfully verified.
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Rule 30
Rule 6374
Rubi steps
\begin {align*} \int \frac {e^{c-b^2 x^2}}{\text {erfc}(b x)^3} \, dx &=-\frac {\left (e^c \sqrt {\pi }\right ) \operatorname {Subst}\left (\int \frac {1}{x^3} \, dx,x,\text {erfc}(b x)\right )}{2 b}\\ &=\frac {e^c \sqrt {\pi }}{4 b \text {erfc}(b x)^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 21, normalized size = 1.00 \[ \frac {\sqrt {\pi } e^c}{4 b \text {erfc}(b x)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.41, size = 26, normalized size = 1.24 \[ \frac {\sqrt {\pi } e^{c}}{4 \, {\left (b \operatorname {erf}\left (b x\right )^{2} - 2 \, b \operatorname {erf}\left (b x\right ) + b\right )}} \]
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 \[ \int \frac {e^{\left (-b^{2} x^{2} + c\right )}}{\operatorname {erfc}\left (b x\right )^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.36, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{-b^{2} x^{2}+c}}{\mathrm {erfc}\left (b x \right )^{3}}\, dx \]
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 \[ \int \frac {e^{\left (-b^{2} x^{2} + c\right )}}{\operatorname {erfc}\left (b x\right )^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 16, normalized size = 0.76 \[ \frac {\sqrt {\pi }\,{\mathrm {e}}^c}{4\,b\,{\mathrm {erfc}\left (b\,x\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.06, size = 22, normalized size = 1.05 \[ \begin {cases} \frac {\sqrt {\pi } e^{c}}{4 b \operatorname {erfc}^{2}{\left (b x \right )}} & \text {for}\: b \neq 0 \\x e^{c} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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