Optimal. Leaf size=35 \[ -\frac {e^{(a+b x)^2}}{b \sqrt {\pi }}+\frac {(a+b x) \text {Erfi}(a+b x)}{b} \]
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Rubi [A]
time = 0.00, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6486}
\begin {gather*} \frac {(a+b x) \text {Erfi}(a+b x)}{b}-\frac {e^{(a+b x)^2}}{\sqrt {\pi } b} \end {gather*}
Antiderivative was successfully verified.
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Rule 6486
Rubi steps
\begin {align*} \int \text {erfi}(a+b x) \, dx &=-\frac {e^{(a+b x)^2}}{b \sqrt {\pi }}+\frac {(a+b x) \text {erfi}(a+b x)}{b}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 33, normalized size = 0.94 \begin {gather*} \frac {-\frac {e^{(a+b x)^2}}{\sqrt {\pi }}+(a+b x) \text {Erfi}(a+b x)}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.23, size = 31, normalized size = 0.89
method | result | size |
derivativedivides | \(\frac {\left (b x +a \right ) \erfi \left (b x +a \right )-\frac {{\mathrm e}^{\left (b x +a \right )^{2}}}{\sqrt {\pi }}}{b}\) | \(31\) |
default | \(\frac {\left (b x +a \right ) \erfi \left (b x +a \right )-\frac {{\mathrm e}^{\left (b x +a \right )^{2}}}{\sqrt {\pi }}}{b}\) | \(31\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 30, normalized size = 0.86 \begin {gather*} \frac {{\left (b x + a\right )} \operatorname {erfi}\left (b x + a\right ) - \frac {e^{\left ({\left (b x + a\right )}^{2}\right )}}{\sqrt {\pi }}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 45, normalized size = 1.29 \begin {gather*} \frac {{\left (\pi b x + \pi a\right )} \operatorname {erfi}\left (b x + a\right ) - \sqrt {\pi } e^{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}}{\pi b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.14, size = 51, normalized size = 1.46 \begin {gather*} \begin {cases} \frac {a \operatorname {erfi}{\left (a + b x \right )}}{b} + x \operatorname {erfi}{\left (a + b x \right )} - \frac {e^{a^{2}} e^{b^{2} x^{2}} e^{2 a b x}}{\sqrt {\pi } b} & \text {for}\: b \neq 0 \\x \operatorname {erfi}{\left (a \right )} & \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 = 0.14, size = 46, normalized size = 1.31 \begin {gather*} x\,\mathrm {erfi}\left (a+b\,x\right )+\frac {a\,\mathrm {erfi}\left (a+b\,x\right )}{b}-\frac {{\mathrm {e}}^{a^2}\,{\mathrm {e}}^{b^2\,x^2}\,{\mathrm {e}}^{2\,a\,b\,x}}{b\,\sqrt {\pi }} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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