Optimal. Leaf size=21 \[ -\frac {\sqrt {\pi } e^c}{4 b \text {erf}(b x)^2} \]
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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 = {6373, 30} \[ -\frac {\sqrt {\pi } e^c}{4 b \text {Erf}(b x)^2} \]
Antiderivative was successfully verified.
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Rule 30
Rule 6373
Rubi steps
\begin {align*} \int \frac {e^{c-b^2 x^2}}{\text {erf}(b x)^3} \, dx &=\frac {\left (e^c \sqrt {\pi }\right ) \operatorname {Subst}\left (\int \frac {1}{x^3} \, dx,x,\text {erf}(b x)\right )}{2 b}\\ &=-\frac {e^c \sqrt {\pi }}{4 b \text {erf}(b x)^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 21, normalized size = 1.00 \[ -\frac {\sqrt {\pi } e^c}{4 b \text {erf}(b x)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 16, normalized size = 0.76 \[ -\frac {\sqrt {\pi } e^{c}}{4 \, b \operatorname {erf}\left (b x\right )^{2}} \]
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 \[ \int \frac {e^{\left (-b^{2} x^{2} + c\right )}}{\operatorname {erf}\left (b x\right )^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 17, normalized size = 0.81 \[ -\frac {{\mathrm e}^{c} \sqrt {\pi }}{4 b \erf \left (b x \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 16, normalized size = 0.76 \[ -\frac {\sqrt {\pi } e^{c}}{4 \, b \operatorname {erf}\left (b x\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 16, normalized size = 0.76 \[ -\frac {\sqrt {\pi }\,{\mathrm {e}}^c}{4\,b\,{\mathrm {erf}\left (b\,x\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.30, size = 19, normalized size = 0.90 \[ - \frac {\sqrt {\pi } e^{c}}{4 b \operatorname {erf}^{2}{\left (b x \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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