Optimal. Leaf size=21 \[ \frac {\sqrt {\pi } e^c \text {erf}(b x)^3}{6 b} \]
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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 = {6373, 30} \[ \frac {\sqrt {\pi } e^c \text {Erf}(b x)^3}{6 b} \]
Antiderivative was successfully verified.
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Rule 30
Rule 6373
Rubi steps
\begin {align*} \int e^{c-b^2 x^2} \text {erf}(b x)^2 \, dx &=\frac {\left (e^c \sqrt {\pi }\right ) \operatorname {Subst}\left (\int x^2 \, dx,x,\text {erf}(b x)\right )}{2 b}\\ &=\frac {e^c \sqrt {\pi } \text {erf}(b x)^3}{6 b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 21, normalized size = 1.00 \[ \frac {\sqrt {\pi } e^c \text {erf}(b x)^3}{6 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 16, normalized size = 0.76 \[ \frac {\sqrt {\pi } \operatorname {erf}\left (b x\right )^{3} e^{c}}{6 \, b} \]
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 \[ \int \operatorname {erf}\left (b x\right )^{2} e^{\left (-b^{2} x^{2} + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 17, normalized size = 0.81 \[ \frac {{\mathrm e}^{c} \erf \left (b x \right )^{3} \sqrt {\pi }}{6 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 16, normalized size = 0.76 \[ \frac {\sqrt {\pi } \operatorname {erf}\left (b x\right )^{3} e^{c}}{6 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 16, normalized size = 0.76 \[ \frac {\sqrt {\pi }\,{\mathrm {e}}^c\,{\mathrm {erf}\left (b\,x\right )}^3}{6\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.74, size = 19, normalized size = 0.90 \[ \begin {cases} \frac {\sqrt {\pi } e^{c} \operatorname {erf}^{3}{\left (b x \right )}}{6 b} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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