Optimal. Leaf size=11 \[ \frac {1}{2} \sqrt {\pi } \text {erfi}(x) \]
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Rubi [A]
time = 0.00, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2235}
\begin {gather*} \frac {1}{2} \sqrt {\pi } \text {erfi}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 2235
Rubi steps
\begin {align*} \int e^{x^2} \, dx &=\frac {1}{2} \sqrt {\pi } \text {erfi}(x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 11, normalized size = 1.00 \begin {gather*} \frac {1}{2} \sqrt {\pi } \text {erfi}(x) \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 4 in
optimal.
time = 1.88, size = 7, normalized size = 0.64 \begin {gather*} \frac {\sqrt {\text {Pi}} \text {erfi}\left [x\right ]}{2} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.02, size = 8, normalized size = 0.73
method | result | size |
default | \(\frac {\erfi \left (x \right ) \sqrt {\pi }}{2}\) | \(8\) |
meijerg | \(\frac {\erfi \left (x \right ) \sqrt {\pi }}{2}\) | \(8\) |
risch | \(\frac {\erfi \left (x \right ) \sqrt {\pi }}{2}\) | \(8\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.27, size = 9, normalized size = 0.82 \begin {gather*} -\frac {1}{2} i \, \sqrt {\pi } \operatorname {erf}\left (i \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.31, size = 7, normalized size = 0.64 \begin {gather*} \frac {1}{2} \, \sqrt {\pi } \operatorname {erfi}\left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 8, normalized size = 0.73 \begin {gather*} \frac {\sqrt {\pi } \operatorname {erfi}{\left (x \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 0.00, size = 11, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {\pi } \mathrm {erf}\left (-\mathrm {i} x\right )}{\mathrm {i}\cdot 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.02, size = 7, normalized size = 0.64 \begin {gather*} \frac {\sqrt {\pi }\,\mathrm {erfi}\left (x\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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