Optimal. Leaf size=32 \[ \frac {x}{b \sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {Erfi}(b x)}{2 b^2} \]
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Rubi [A]
time = 0.02, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6519, 8}
\begin {gather*} \frac {x}{\sqrt {\pi } b}-\frac {e^{-b^2 x^2} \text {Erfi}(b x)}{2 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 6519
Rubi steps
\begin {align*} \int e^{-b^2 x^2} x \text {erfi}(b x) \, dx &=-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{2 b^2}+\frac {\int 1 \, dx}{b \sqrt {\pi }}\\ &=\frac {x}{b \sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{2 b^2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 32, normalized size = 1.00 \begin {gather*} \frac {x}{b \sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {Erfi}(b x)}{2 b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 41, normalized size = 1.28
method | result | size |
default | \(\frac {\left (2 \,{\mathrm e}^{b^{2} x^{2}} b x -\erfi \left (b x \right ) \sqrt {\pi }\right ) {\mathrm e}^{-b^{2} x^{2}}}{2 \sqrt {\pi }\, b^{2}}\) | \(41\) |
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 \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 = 0.37, size = 40, normalized size = 1.25 \begin {gather*} \frac {{\left (2 \, \sqrt {\pi } b x e^{\left (b^{2} x^{2}\right )} - \pi \operatorname {erfi}\left (b x\right )\right )} e^{\left (-b^{2} x^{2}\right )}}{2 \, \pi b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 3.15, size = 27, normalized size = 0.84 \begin {gather*} \begin {cases} \frac {x}{\sqrt {\pi } b} - \frac {e^{- b^{2} x^{2}} \operatorname {erfi}{\left (b x \right )}}{2 b^{2}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \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 = 0.05, size = 27, normalized size = 0.84 \begin {gather*} \frac {x}{b\,\sqrt {\pi }}-\frac {{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erfi}\left (b\,x\right )}{2\,b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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