Optimal. Leaf size=21 \[ -\frac {e^c \sqrt {\pi }}{2 b \text {Erf}(b x)} \]
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Rubi [A]
time = 0.02, 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 = {6508, 30}
\begin {gather*} -\frac {\sqrt {\pi } e^c}{2 b \text {Erf}(b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 6508
Rubi steps
\begin {align*} \int \frac {e^{c-b^2 x^2}}{\text {erf}(b x)^2} \, dx &=\frac {\left (e^c \sqrt {\pi }\right ) \text {Subst}\left (\int \frac {1}{x^2} \, dx,x,\text {erf}(b x)\right )}{2 b}\\ &=-\frac {e^c \sqrt {\pi }}{2 b \text {erf}(b x)}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 21, normalized size = 1.00 \begin {gather*} -\frac {e^c \sqrt {\pi }}{2 b \text {Erf}(b x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 17, normalized size = 0.81
method | result | size |
default | \(-\frac {{\mathrm e}^{c} \sqrt {\pi }}{2 b \erf \left (b x \right )}\) | \(17\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 16, normalized size = 0.76 \begin {gather*} -\frac {\sqrt {\pi } e^{c}}{2 \, b \operatorname {erf}\left (b x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.43, size = 16, normalized size = 0.76 \begin {gather*} -\frac {\sqrt {\pi } e^{c}}{2 \, b \operatorname {erf}\left (b x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.31, size = 17, normalized size = 0.81 \begin {gather*} - \frac {\sqrt {\pi } e^{c}}{2 b \operatorname {erf}{\left (b x \right )}} \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 [B]
time = 0.14, size = 16, normalized size = 0.76 \begin {gather*} -\frac {\sqrt {\pi }\,{\mathrm {e}}^c}{2\,b\,\mathrm {erf}\left (b\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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